lectures in fluid mechanics · 2021. 1. 6. · preface this document consists of a series of...

Lectures in Fluid Mechanics

Dancing Jellyfish edition (Spring 2021)

David S. Ancalle, P.E.

Author website: www.ancalle.net

About this title: www.learnfluidmechanics.com

David S. Ancalle

LECTURES IN FLUID MECHANICS

Dancing Jellyfish edition (Spring 2021)

Last Updated January 5, 2021

© 2021, David S. Ancalle. ALL RIGHTS RESERVED

Cover photo obtained from www.pexels.com. Shot by Dean Ha.

Other figures in the text by David S. Ancalle unless otherwise

stated.

Figures and data from external sources are presented in this work

under “Fair Use” for educational and nonprofit purposes. Figures

obtained from external sources are referenced within the text.

This work is distributed freely to students and is not for commercial

use. This work may not be reproduced without the author’s

permission. This work is continuously being updated. Ensure you

have the most recent version at www.learnfluidmechanics.com.

Please send notices of typos & errors, and requests for modifications

or additions to [email protected].

2

Contributors

David S. Ancalle, P.E., CA Engineering and Kennesaw State University

Marguerite Matherne, Georgia Institute of Technology (co-author, Ch. 8)

Blake J. Landry, Ph.D., U.S. Naval Research Laboratory (Ch. 11-12)

Zack Anderson, Vanderlande (Ch. 1-2)

Bill T. Ngo, Kennesaw State University (Ch. 3-4)

3

Preface

This document consists of a series of lectures and notes I use for my undergraduate fluid mechanics course,

in addition to supplementary information that I normally don't have time to cover in a single semester. It

is not meant to replace traditional classroom instruction or any of the good textbooks on the topic. I

encourage diversifying study resources, which can include solving problems from various textbooks or

considering alternative means of instruction like online videos. Thus, while I have tried my best to make

these lectures as comprehensive as can be, I still recommend students to consider any other resource

available to them, as long as their academic integrity is not compromised. I also strongly recommend that

students use these lectures in tandem with a textbook.

To make the most out of this book, I recommend you have a copy of your calculus, physics, and statics

textbooks at hand. This work is continuously being updated. Ensure you have the most recent version at

www.learnfluidmechanics.com. Please send notices of typos & errors, and requests for modifications or

additions to [email protected].

4

Table of Contents

The page number corresponds to the number on the PDF file. There are several sections that are still in

development and not included in this draft.

Ch. 1 Introduction to Fluids ......................................................................................................................... 7

1 The Field of Fluid Mechanics

2 Approaching Fluid Mechanics

3 Fluid Density

4 Viscosity

5 Compressibility

6 Surface Tension

7 Vapor Pressure

8 Ideal Gas Law

9 First Law of Thermodynamics

10 Other Thermodynamic Properties

11 Other Properties

Ch. 2 Fluid Statics ....................................................................................................................................... 39

1 Approaching Pressure

2 Hydrostatic Pressure

3 Measuring Pressure

4 Buoyancy

5 Stability

6 Forces on Submerged Areas

7 Pressure on Linearly Accelerating Containers

8 Pressure on Rotating Containers

Ch. 3 Introduction to Fluid Dynamics ......................................................................................................... 69

1 Description of Flows

2 Classification of Flows

3 Kinematics

4 Coordinate Systems

5 Ideal Fluids

Ch. 4 Integral Analysis ................................................................................................................................. 89

1 Control Volume Approach

2 Conservation of Mass

3 Conservation of Momentum

4 Conservation of Energy

Ch. 5 Differential Analysis ......................................................................................................................... 117

1 Review of Differential Equations

2 Introduction to Differential Analysis

5

3 Conservation of Mass

4 Conservation of Momentum

5 Vorticity

6 Conservation of Energy

Ch. 6 Dimensional Analysis and Similitude ............................................................................................... 135

1 Dimensional Analysis

2 Similitude

3 Normalized Differential Equations

Ch. 7 Internal Flows ................................................................................................................................... 147

1 Entrance Region

2 Laminar Flow in Pipes

3 Laminar Flow between Parallel Plates

4 Laminar Flow Between Rotating Cylinders

5 Navier-Stokes Solutions for Laminar Flow

6 Turbulent Flow in Pipes

Ch. 8 External Flows.................................................................................................................................. 167

1 Introduction to External Flows

2 Boundary Layer

3 Forces on Immersed Bodies

Ch. 9 Potential Flow ................................................................................................................... not included

Ch. 10 Compressible Flow ........................................................................................................... not included

Ch. 11 Pipe Systems ................................................................................................................................... 177

1 Head Losses

2 Simple Pipe Systems

3 Analysis of Pipe Networks

Ch. 12 Open Channel Flow ........................................................................................................................ 189

1 Introduction to Open Channel Flow

2 Steady Uniform Flow

3 Energy in Open Channels

4 Momentum in Open Channels

Ch. 13 Turbomachinery.............................................................................................................................. 207

1 Pumps in Piping Systems

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updated 1/1/2021 Ch. 1

Introduction to Fluids David S. Ancalle, P.E.

1 The Field of Fluid Mechanics

1.1 Defining Fluids

Fluid: a substance that deforms continuously under any shear stress.

examples: liquids, gases, plasma

properties: ability to flow (to deform continuously under any shear stress)

1.2 Outline of Fluid Mechanics

Fluid mechanics is a broad, exciting, and complex area of study whose applications span many fields. Its

concepts allow us to study cell movement, the circulatory system, bird flight, cars, pumps, ships, planes,

windmills, hurricanes, rockets, and even stars and nebulae in space. You may be wondering how these topics

could be related to one another, or why anyone would even want to study such a complex field. My hope is

that after completing this book, you have the answers to both questions, and may even be motivated to

continue studying this field.

Fluid Mechanics: the study of fluids at rest and in motion.

1.3 Applications of Fluid Mechanics:

• Civil Engineering – dams, sewers, rivers, coastal

• Mechanical Engineering – HVAC, turbomachinery, oil & gas

• Aerospace Engineering – propulsion, aerodynamics

• Biomedical Engineering – blood flow monitoring, cardiovascular pathology

1.4 History of Fluid Mechanics

• 200 B.C. – 400 Archimedes (buoyancy); Roman aqueduct

• 1400 – 1500 Leonardo Da Vinci

• 1500 – 1700 d’Alembert paradox

fluid

statics

fluid

dynamics

7

Fluid Mechanics Lecture Notes


• 1700 – 1800 drainage and flood control

• 1800 – 1900 Navier-Stokes equations

• 1900 – 2000 aerospace, aeronautics, computational fluid dynamics

• 2000 – present climate change, magnetohydrodynamics, bioengineering, sustainability

2 Approaching Fluid Mechanics

2.1 Philosophy of Fluid Mechanics

Everybody has a worldview, a way to observe, interpret and understand the events that happen around us.

While most scientifically minded students will seek a pure study of any STEM field, unadulterated by

emotional, philosophical, moral, or religious convictions, the reality is that these convictions, which both

shape and reflect our worldview, stand at the center of how we approach any field. The best way to reconcile

our desire for scientific study with our worldview is not to try to separate the two, but to accept that the

two cannot be separated; and instead to be fully aware of our initial biases and convictions and how they

may or may not impact the conclusions reached. An understanding of the philosophy of our field is then

necessary to engage in a critical and analytical study of it.

Science is inherently observational. While we can certainly complement our science with analytical theory,

which relies heavily on mathematics; we cannot deny the empirical nature of scientific study. Because of

this, scientific study is somewhat limited by our ability to perform experiments, that is, our ability to

measure, observe, and interpret. The nature of science sometimes causes confusion, because mathematics,

statistics and other fields (usually called formal sciences) study abstract concepts that cannot be observed

empirically. These fields function by outlining axioms (truths) and deducing theorems from them. While

these formal sciences are related and helpful to science (sometimes intrinsically so), they do not constitute

science, per se. Yet, we cannot have a full understanding of science without these formal treatments. What

this shows is that the scientific method alone is not sufficient to study any field in its entirety.

This brings us to the study of fluids. A thorough study requires us to know the what, the why, and the how

when it comes to studying fluids. While my philosophy may not be identical to yours, knowing how I’d

answer these questions will help you understand my worldview as it relates to fluids; and in turn, will help

you apply what you learn in this course to fit in with your own worldview, interests, and goals. In this

course, the focus of the study of fluids will be in its mechanics, where we will seek to understand how fluids

respond to their environment, specifically to the forces and stresses acting on them. While this is the main

focus of this course, we will also touch on other aspects of fluids, albeit briefly. We will look at the

thermodynamics of fluids, the biology of fluids, and to a much lesser extent, the chemistry of fluids. A

thorough study of fluids would require us to master all of the fields above, which is a hard if not impossible

task.

Next, we look at why study fluids. This will be the answer with the most variation among those in this

course. The answers can range from wanting to pass the course in order to obtain a degree, to wanting to

learn the skills necessary to engage in engineering work related to fluids in the future. For me, the answer

is best articulated in a Scientific American article:

8



Excerpt from “Tackling turbulence with supercomputers” published in Scientific American. Used without permission.

In short, fluids are a part of us and the world we live in. I study fluids to understand myself, and the world

around me. “The simple believes everything, but the prudent gives thought to his steps.” (Proverbs 14:15)

While I gladly accept that I live thanks to the flow of three important fluids (blood, water, and air), I still

see it prudent to want to understand the mechanisms behind these flows. Your approach to fluids will play

a role in determining how much or how little you take advantage of this course, so it is best to think about

it early on.

Finally, we look at how to study fluids. The pursuit of knowledge, be it scientific or otherwise, is a search

for truth. Contrary to most science communicators (and even some scientists), this search is not exclusive

to the scientific method. The search for truth (ἀξίωμα) has been part of humanity since the beginning of

recorded history. This search led to the development of religion, philosophy, and logic, all of which preceded

and strongly influenced the scientific method. Unless you are merely completing this course for a grade, you

have probably participated in that search as well. Unfortunately, we cannot know the complete truth. As

Richard Feynman puts it, “there is an expanding frontier of ignorance.” With that in mind, the teacher faces

a choice between teaching the complete truth (within the scope of our current knowledge), or teaching an

approximate truth (within what is known by the student, at the time). It may seem dishonest, but the

reality is what most of what is taught in introductory courses is at best an approximate truth, or at worst

a comforting lie. The reality is that knowledge is acquired and built upon by the individual. So, if the

purpose of this course is to teach the undergraduate student how to study fluids, then we must begin with

basic assumptions of what the student knows before joining the course. I cannot assume that all my students

have a strong grasp of relativity or quantum physics, because most of my students have only learned classical

mechanics. Because of this, we will have to make assumptions that we know are wrong. These include the

notion that gravity is a force, or that it is constant, or that solids and fluids are continuous substances. All

of these statements are incorrect, but we will treat them as correct in order to begin to understand fluids.

Once we do understand how fluids operate within the scope of these assumptions, we can begin to

deconstruct these statements and add to our knowledge of fluids.

Earlier, I said that the scientific method alone isn’t sufficient, and this applies to our search for truth. Our

study of fluids will be strongly complemented by analytical theory, and will strongly depend on starting

assumptions (for example, that the universe and everything in it follows a set of laws that can be studied

9



and understood with our current knowledge and technology). In most cases, empirical observations will

match analytical theory as well as our starting assumptions, but in other cases, they won’t, and it is up to

you: the engineer or scientist, to choose which approach to follow. This will be governed by your own

worldview, but I cannot make that decision for you.

2.2 Dimensions and Units

A physical quantity is any physical property that can be quantified numerically, that is, a property that

can be measured. We can quantify physical quantities through the use of dimensions, and each physical

quantity is regarded as having its own dimension or dimensions.

In physics, nine fundamental dimensions are traditionally recognized. These are: length, mass, time,

temperature, amount of substance, electric current, luminous intensity, plane angle, and solid angle. The

first seven are known as base quantities, as they represent real physical quantities that are not defined in

terms of other quantities. The last two are dimensionless quantities, which do not have dimensions associated

with them, but are still identified as part of these quantities as they are measurable, in contrast to pure

numbers. In our initial study of fluid mechanics, we will focus on five of these fundamental dimensions:

1. length (�) 2. mass (�) 3. time (�) 4. temperature (Θ) 5. plane angle

Since plane angle is a dimensionless quantity, it does not have a variable traditionally associated with it,

but you will note that in these lectures we will use various Greek letters to identify angles.

Mathematically, any other physical quantity can be expressed as a product of one or more of the nine

fundamental dimensions raised to a rational power. For example, the dimensions of density, which we

colloquially define as "mass over unit volume" can be expressed as:

�� = �� = ��

Notice that we use square brackets � � when we want to show the dimensions of a physical quantity. The term �� stands for the dimensions (brackets) of volume (physical quantity). Another way of showing the dimensions of density is:

�� = ��Θ�

Here, we included all fundamental dimensions used in the course. This method of writing down dimensions

will become useful in the study of dimensional analysis.

In summary, physical quantities:

10



are used to quantify the world around us

can be quantified using dimensions

Fundamental Dimensions: length (L), mass (M), time (T), temperature (Θ), plane angle

Units are standardized measurements of dimensions. Groups of units that are either related to each other

or simply belong to the same standardized system are called unit systems. These lectures will place

emphasis on two unit systems: the International System of Units (SI) which uses metric units, and the

United States Customary System (US) which uses English units.

Quantity S.I. unit U.S. unit

length meter, foot, �� mass kilogram, �� slug, �� time second, � second, �

Temperature Kelvin, � Degree Rankine, °� plane angle radian, �� radian, ��

We see that second is in both the SI system as well as in the US system. In fact, some units may be used

in multiple unit systems. It should also be noted that, while we may be accustomed to using "SI units" and

"metric units" interchangeably, these terms refer to two different things: the SI unit system is a system

that uses metric units. Other systems of measurements that use metric units include the cgs system, whereas

other systems of measurements that use English units include the English Engineering system. We see that

physical quantities can be expressed numerically as the product of the measurement of its dimension and

the unit in which it is being measured. For example, to express the distance between two points in space,

we write: � = 2 �� where � is the variable that represents the physical quantity, 2 is the numerical measurement of that quantity, and �� is the unit in which the dimension is being measured.

Physical measurements are limited to the precision of the measuring instrument. For example, if you are

measuring a length using a ruler that only has centimeter marks, your measurement cannot be more precise

than centimeters. That means that you cannot obtain a measurement in millimeters from that ruler. Now,

say you measure a length that is somewhere between the 2 � mark and the 3 � mark of your ruler. What value would you give that measurement? You can try to estimate a value in between 2 and 3 centimeters,

but your estimation will be at best an educated guess and at worst an erroneous one. Thus, when a

measurement is declared, we must consider the potential error associated with that measurement.

11



We know that the result of a calculation can be no more precise than the least precise measurement. This

is because any precision beyond that of the least precise measurement will introduce an error in the

calculation. We can avoid errors in precision by the use of significant figures, or sig. figs. The significant

figures of a calculation refer to the precision of the least precise measurement in the calculations.

Example. If we want to find the average velocity of a remote-controlled car, we can do so by measuring

the time it takes for the car to travel a known distance. Say the distance is measured using a measuring

tape that measures up to centimeters, and the time is measured with a stopwatch that measures up to

milliseconds. What is the average velocity of the car? Your measurements are as follows:

� = 3.05 � = 2.104 �

Solving this problem should be easy enough for any student. However, we will focus our attention on

expressing the answer correctly, using the correct number of significant figures. In This example, the least

precise measurement is that of distance, which is precise up to 0.01 , that is, three significant figures. Therefore, our answer cannot be more precise than three significant figures. Performing the calculation

yields:

v&'( = �� = 3.05 2.104 � = 1.44961977 … � ≅ 1.45 �

Note that, by presenting our answer up to three significant figures, we do not consider any precision beyond

the second decimal value. Because the third decimal value was a number greater than five, we have rounded

the second decimal value up. Rounding can be straightforward for numbers smaller than or greater than

five, but it becomes a mystery when we have a five involved. There are different rules used by different

authors. I typically round up when I see a five. If you are using a calculator for your work, take advantage

of its memory and avoid rounding until your final answer.

In our use of dimensions and units, two methods will be helpful in presenting our data. The first is scientific

notation, which consists of expressing large or small numbers as products of a coefficient by a power of 10.

For example, a measurement of two million Newtons can be expressed in scientific notation as:

2,000,000 / = 2 × 101 /

Note that in these lectures we will use the comma as a delimiter for every thousandth order of magnitude

(for clarity only) and we will use the point to denote the delimiter between decimal and whole values.

Scientific notation allows us to work directly with the order of magnitude of our quantities, and can lead to

clearer and simpler calculations when dealing with very large or very small numbers. While not a rule in its

own right, we will try to keep our orders of magnitude in products of three (i.e., 10�, 101, 102, …) Now, what if you want to express a quantity of two thousand eight hundred and fifty-nine kilograms? In scientific

notation, you may write 2,859 �� = 2.859 × 10� ��, but if we want to use this measurement in tandem with other measurements, we will need to consider their number significant figures.

12



Another helpful tool is the use of SI Prefixes. These prefixes are appended to the beginning of a metric

unit to represent powers of 10. SI prefixes, as the name implies, are almost exclusively applied to metric

units; with very few exceptions, like the kip (kilopound, 10� �4).

Common SI Prefixes:

mega- M × 101 kilo- k × 10� centi- c × 105 (only used w/ meters or Poises) milli- m × 10�

2.3 Dimensional Homogeneity

in a dimensionally homogeneous equation, units must be consistent among terms

Consider the equation � + 4 = �. If a and b are lengths, then c must also be a length. This is called dimensional homogeneity, and it will be a very useful and important tool to use in the field of fluid

mechanics. Most equations that describe physical processes are (or, at least, should be) dimensionally

homogeneous. When we find equations that aren’t (usually those that are obtained from empirical data),

there is usually a unit conversion coefficient added that makes the equation dimensionally homogeneous for

use among multiple unit systems. For the purposes of an introductory fluids course, it’s safe to assume that

all equations encountered are dimensionally homogeneous unless otherwise stated.

2.4 Density and States of Matter

All substances are made of atoms. These atoms naturally exist in groups that we call molecules, and these

molecules attract each other at a certain distance, but repel each other when squeezed too closely. The

atoms and molecules vibrate and move continuously, but remain attracted to each other such that they do

not “break off” easily. This group of atoms and molecules form a substance.

A substance has mass, and this mass is determined by the number of atoms and molecules in it. If we

define a three-dimensional region in space bounded by a two-dimensional surface (what we call a volume)

and count the number of atoms and molecules in it, or measure the amount of mass in it, we can determine

the substance’s density. Since atoms and molecules are continuously moving and vibrating, it’s possible

that the mass in a volume may change with time, such that the density of a substance also changes

depending on when you measure it. This problem will be addressed in a future section.

The vibrations of the atoms and molecules is what we call heat, the measure of which we call temperature.

We can identify three states for a substance, defined by the relative motion of the atoms and molecules in

it. First, we consider a substance with moving and vibrating molecules. The motion of these molecules is

seemingly random such that we cannot identify any pattern within them. However, they are close enough

such that they all remain within the “boundary” of the substance as a whole. This is what we call a liquid.

When the temperature of a substance increases, the vibrations and movement of these atoms increases,

which also increases the distance between the atoms and molecules. At some point, it’s possible for an

increase in temperature to be such that the distance between molecules exceeds the zone of influence for

13



their forces of attraction to operate. In this case, molecules can break free from their group. This group of

freely moving molecules is called a gas. On the other hand, if the temperature of a substance decreases, the

motion of its molecules decreases and the distance between them shrinks. As these molecules move closer

and closer to each other, they begin to “accommodate” themselves in noticeable patterns, to a point where

all molecules have an “assigned space” within the substance and their movement is limited to small vibrations,

but their arrangement does not change. This is called a solid.

This distinction between the three states of matter depends on the energy (heat) of its molecules. This

course takes a classical mechanics approach to substances, and therefore, we will distinguish between two

phases of matter: solid and fluid, which depends on how a substance reacts when exerted to a stress. We’ve

already defined fluids as substances that deform continuously when subjected to any shear stress. This is

the behavior of liquids and gases, because their molecules do not have a fixed position to adhere to. However,

in the case of solids, molecules do have fixed positions and will try to maintain those positions, such that

when a shear stress is exerted, the solid may resist this stress if the magnitude is mall, or may attempt to

return to its original position for larger magnitudes, or maybe rearrange the order of its molecules to adapt

to even larger stresses, or may “fracture”, that is, break off into various substances in order to keep the

molecules’ relative positions.

2.5 Gravitational Acceleration

Up to this point I’ve tried to be as rigorous as this course allows regarding some of the basic topics. From

this point, we will begin dealing with “approximate truths” that will allow us to study fluids at the

undergraduate level.

As far as we know, gravity represents the effect of mass moving along spacetime. The path through which

matter moves is curved, due to uneven distribution of mass. However, in local space, the path appears to

be straight. This is the description of gravity in general relativity, or Einstein’s Law of Gravity, and as

far as we know, it is the description closest to reality.

Before general relativity, gravity was defined as a force of attraction between two substances, proportional

to their mass and inversely proportional to the square of the distance between their center of mass. This

description, known as Newton’s Law of Gravity, is the basis of Newtonian physics.

We know that the Newtonian approach to gravity is not correct, however, it accurately describes how

substances interact with each other at several scales, including all the scales that we will study in this course.

It is sometimes said that Newtonian physics accurately describes 99% of the universe and what occurs in it,

whereas relativity and quantum physics describes the other 1%. Because of this, we will adopt a Newtonian

view of gravity in this course. In other words, we will treat gravity as a force that fluids experience.

From this description, we define gravity as 78 = 9 :;: = � where � is the gravitational acceleration.

acceleration resulting from forces of attraction

varies with location (elevation)

14



we will use a standard value of � = 9.81 /�5 = 32.2 ��/�5 can be used to determine weight: 7@ = �

2.6 Absolute and Relative Scales

Pressure is:

stress that acts normal to an area

acts into the substance it is being measured at

A = 7/B (A = limFG→� FIFG = JIJG) acts over an entire area

A = 7B A&K:

A(

A&LM = A&K: + A(

We can specify that a pressure is absolute in metric units using �A�&LM and in English units using N�O�.

Absolute Temperature

measure of thermal energy / kinetic activity / specific heat / vibration of molecules

units: �, °�

Relative Temperature

measured relative to the freezing and boiling points of water

units: ℃, ℉

system

absolute

temperature

units

relative

temperature

units

freezing

point of

water

boiling

point of

water

SI � ℃ 0 ℃ 100 ℃ US °� ℉ 32 ℉ 212 ℉

(remember!)

pressure from the atmosphere

varies with elevation

any pressure above and below the atmospheric

can be measured with a gage

15



For most engineering problems, if no pressure or temperature data are available for the environment, we

will use the old Standard Temperature and Pressure (STP) values:

• Standard engineering pressure: A&K: = 1 �� = 101.3 �A� = 2116 N�� (mean sea level pressure) • Standard engineering temperature: � = 15 ℃ = 59 ℉

These values, combined with nominal Earth values (like gravitational acceleration of 9.81 /�5 ) are sometimes also known as standard engineering conditions, a term we will use in this course.

2.7 Continuum Approach

We have already defined a volume as a region in 3D space bounded by a 2D surface. We can further expand

this to define a control volume as a region of interest delineated by a boundary (the surface).

A substance formed out of molecules can be visualized as a continuous substance.

We have discussed that substances are formed of many molecules vibrating and/or moving around freely.

The study of these individual molecules is called the statistical approach and is the basis for statistical

mechanics, a topic that we will not go over in this course. Instead, we can now introduce another of these

approximate truths, which is the continuum approach, a study that assumes substances to be continuous.

Statistical approach: studies fluid molecules individually; incomplete theory for dense gases and liquids

Continuum approach: assumes fluid to be a continuous substance (i.e., no distance between molecules); the

control volume has to be much larger than the free path between two fluid molecules.

The continuum approach allows us to easily describe properties such as density and pressure, because we

no longer look at substances as collections of molecules, but as continuous amounts of matter, with a clearly

defined mass.

A way to numerically assess the validity of the continuum approach is by using the Knudsen number.

16



Kn = ℓ�

where ℓ is the mean free path between fluid molecules, and � is a macroscopic length scale of the system. When the Knudsen number is very small, the continuum approach can be employed.

2.8 Laws of Conservation

We define a system as a fixed quantity of matter. Three laws of conservation for systems are applied to

this course:

Conservation of Mass: Matter cannot be created nor destroyed. This means that the amount of mass in

a substance can only change if there is an exchange of matter in that substance. By definition, the mass in

a system is fixed and does not change, so:

U�� VMWM = 0

Conservation of Momentum: Momentum is a conserved quantity. Newton’s second law of motion states

that the change of motion of a body is proportional to the net force acting on it, and acts along the direction

of that force. If a system is isolated, there will be no external forces acting on it, and momentum does not

change. In this course, most systems we will study are not isolated, so our study of conservation of

momentum will lead us to use the more general form of Newton’s second law. Furthermore, because a

system is a fixed quantity of matter, we can treat mass as a constant and express the law as:

∑7⃑MWM − [�\v]⃑ ^�� _MWM = 0

∑7⃑MWM = ` �v]⃑��aMWM ∑7⃑MWM = \�⃑^MWM

Conservation of Energy: Energy is a conserved quantity. The first law of thermodynamics allows us to

express the conservation of energy, by stating that a chance in internal energy of a system is equal to the

energy added to a system due to heat minus the energy lost from a system due to work done by the system:

�bMWM − �cMWM + �dMWM = 0 �bMWM = �cMWM − �dMWM

3 Fluid Density

17



3.1 Density

○1 Density

� = limef→ef∗ hh = ��

We can express mass as a function of density by solving the equation above: = ∭ � �.

Density of water under standard engineering conditions

�j



Do it yourself: Determine the units for density, specific weight, and temperature for the empirical

equations above. Determine the percent error for these equations (by comparing results to tabulated values.)

2.3 Specific Gravity

○3 Specific Gravity

n = ��j



Newton’s Law of Viscosity

z = | ��v

Sample velocity gradient (velocity is the horizontal axis and depth is the vertical axis) from White, Fluid Mechanics.

Used without permission.

• linear velocity profile: if the thickness of the fluid, } is very small, we can assume a linear velocity profile. In this case, the velocity gradient and velocity profile are, respectively:

��v = �:&~}

� = �:&~} v

Linear velocity gradient. (From White, Fluid Mechanics. Used without permission.)

4.2 Non-Newtonian Fluids

Definitions:

• Non-Newtonian fluid: shear stress is not proportional to strain rate (e.g. blood, whipped cream)

• Dilatant (shear thickening): viscosity increases with shear stress (e.g. quicksand, oobleck)

20



• Pseudoplastic (shear thinning): viscosity decreases with shear stress (e.g. ketchup, paint)

• Ideal plastic (Bingham plastic): requires a minimum shear stress to cause motion

• Bingham pseudoplastic: requires a minimum shear stress and behaves like a pseudoplastic (e.g.

toothpaste)

• Kinematic viscosity (momentum diffusivity): = |/�

The relationship between shear stress and strain rate in non-Newtonian fluids takes one of the following

forms:

z = � U��vV = [� U��vV

�_ ��v = |& ��v

…where � is the consistency index, and is the power-law index (or flow behavior index). This relationship can be expressed in terms of Newton’s law of viscosity by reducing the variable viscosity to a value called

apparent viscosity, |& = � JyJW�.

Diagram of shear stress and strain rate for various fluids. Source: Wikimedia Commons.

4.3 Viscometers

Viscometer: an instrument used to measure viscosity.

Consider two concentric cylinders with a fluid between them. The thickness of the fluid is }, and the inner cylinder rotates at an angular velocity :

21



Diagram of a viscometer.

In order to rotate the inner cylinder, a torque � is needed. Recall that � = 7�, 7 = zB, and B = 2�ℎ, therefore � = 2�5ℎ| JyJW or � = 5= .

4.4 Variation of Viscosity with Temperature

• Viscosity varies with temperature, not pressure.

• Viscosity of gases increases with temperature.

• Viscosity of liquids decreases with temperature

The viscosity of liquids can be approximated by Andrade’s Equation

| = /

where and are empirical constants and � is the absolute temperature.

The viscosity of gases can be approximated by the Sutherland equation

| = ��/5� +

where � and are empirical constants. For air under standard engineering conditions, � = 1.458 ×101 ��/\ ⋅ � ⋅ ��.^ and = 110.4 �. If the equation is applied at a reference condition, it becomes ||� = U ��V

�/5 �� + � +

As an alternative, a power law relationship is sometimes assumed:

22



||� = U ��V

where the exponent is commonly taken as 0.7.

4.5 To Slip or not to Slip

We have explained that for most fluids, we can assume that the fluid particles in contact with a solid

boundary do not slip. For a fluid flowing in the u-direction along a wall on the uw-plane, this is expressed mathematically as:

v]⃑ |W>& = v]⃑ >&

That is, the velocity of the fluid at the wall is equal to the velocity of the wall. If the wall isn’t moving,

then the velocity of the fluid particles at the wall is zero. For inviscid fluids (fluids with no viscosity), we

have to assume slip to allow for mathematical consistency, and so, we don’t apply the no-slip condition.

However, if the wall is not permeable, then we can say that the fluid does not penetrate the wall. This no-

penetration boundary condition is expressed for the same fluid above as:

o|W>& = 0

In rarefied gases, where there aren’t enough molecules to obtain momentum equilibrium with the wall, then

there is a realistic wall slop. The slip velocity can be predicted with James Clerk Maxwell’s equation:

h�|W>& ≈ ℓ �vW>&

where ℓ is the mean free path of the gas. If ℓ is very small, then the slip velocity approaches zero.

5 Compressibility

5.1 Bulk Modulus of Elasticity

Definitions

• compressibility: a fluid’s resistance to deformation from normal stress.

• Bulk Modulus of Elasticity: the reciprocal of compressibility

Volumetric Bulk Modulus of Elasticity

bf = − A ≅ − A/

Unit-mass Bulk Modulus of Elasticity

23



bf = � A� ≅ A�/�

We can use the bulk modulus to determine the speed of sound! (we will learn more about this in a future

lesson)

� = A� = bf�

Some fluids exhibit very little change with respect to pressure, so we treat them as if they had no

compressibility. If temperature doesn’t change either, then we can take the fluid’s density to be constant

with respect to pressure; we call such fluids incompressible fluids.

5.2 Empirical Compressibility of Liquids

If we neglect changes in temperature, the density of a liquid can be related to pressure empirically by:

AA& ≈ \ + 1^ U ��&V −

where �& is the density of the fluid under standard engineering conditions, A& is the atmospheric pressure, and and are empirical values.

For water, ≈ 3000 and ≈ 7.

6 Surface Tension

6.1 Forces at the Surface

Definitions

• surface tension: a measure of force per unit length in a liquid surface

• surface: the portion of a liquid in contact with a gas

• capillary rise/drop: liquid column that results from forces of attraction between a fluid and a solid

• static rise/drop: liquid column that results from pressure forces

• contact angle: the angle of the direction in which a force of attraction acts. It depends on the liquid,

solid, and gas in contact.

= 7� ⟹ 7 = �

6.2 Contact Angle

24



When a liquid droplet contacts a solid surface, the angle formed between the liquid and solid surface is

called the contact angle. When ¡ > 90°, the liquid is a wetting liquid, otherwise, it is a nonwetting liquid.

Droplet on a solid surface. From Vennard, Elementary Fluid Mechanics. Used without permission.

6.3 Capillary Rise and Drop

Capillary rise on a glass tube. From White, Fluid Mechanics. Used without permission.

Consider a tube inserted in water:

The forces of attraction 7 between the water and the tube will cause it to “rise”. However, the weight of the water in the tube will exert a force in the downward direction. The force of attraction is a function of surface

tension and contact angle £. 7 = � cos £

The system will reach equilibrium when 7 = 7@ � cos £ = m

§ cos £ = m 4 §5 ℎ

Note that depending on the fluids and solids in contact, the cosine of the contact angle may be negative,

which means we have a “drop” instead of a rise. This happens with mercury.

25



6.4 Liquid Droplets

Consider a small liquid droplet with diameter §. In order for this droplet to maintain its shape, a difference in pressure must exist between the external pressure and the internal pressure. When the external pressure

is atmospheric, then this difference is called gage pressure.

A − Ä ≠ 0

Analyzing the forces acting on half of the droplet results in a force from the surface tension, acting along

the circumference, and a force from gage pressure acting on the circular area.

Diagram of the forces on a liquid droplet. From Vennard, Elementary Fluid Mechanics. Used without permission.

ª 7 = 7« − 7¬ = 0

§ = 4 §5A

A = 4§

This shows that the gage pressure inside a liquid droplet is inversely proportional to the diameter of the

droplet.

We can also apply this analysis to droplets with double curvature. Consider a small element of dimensions �u �v on a surface of a droplet with curvature radii �� and �5. The sum of forces normal to the element becomes:

ª 7 = 7« − 7¬ = 0 A �u �v = 2�v sin + 2�u sin £

…where and £ are small angles corresponding to the radii and half of the element dimensions, so that

sin = �u2�� = �u2��

26



sin £ = �v2�5 = �v2�5

Substituting these values in the equation above yields:

A = U 1�� + 1�5V

6.5 Bubbles in Gases

For bubbles in gases (i.e. air), there is a thin layer of liquid between the inside and outside of the bubble,

which creates two surfaces in which surface tension acts. Keeping in mind that A� − A5 = A Analyzing the forces acting on half of a bubble of diameter D yields:

Diagram of the forces of a bubble in a gas. From Chin, Fluid Mechanics for Engineers. Used without permission.

ª 7 = 7« − 7¬ = 0

§ + § + 4 §5A5 − 4 §5A� = 0

2§ = 4 §5A

A = 8§

From this equation, we see that the pressure inside a bubble is twice as large as the pressure inside a droplet

of the same diameter.

6.6 Bubbles in Liquids

The difference in pressure for bubbles in liquids yields the same value as that of a liquid droplet. However,

the external pressure will be higher than the atmospheric pressure (as a hydrostatic pressure is also added

to it), and so the equation takes the following notation

27



ΔA = A − Ä = 4§

A bubble in a liquid must have an inside pressure higher than that on the outside. Therefore, in order to

generate a bubble, a pressure of A = Ä + ΔA = Ä + ¯«° is necessary where Ä is the pressure in the liquid at the location the bubble is to be created.

7 Vapor Pressure

7.1 Defining Vapor Pressure

• vapor pressure: the pressure of the gaseous phase of the fluid that is in contact with the liquid phase

of the fluid and in equilibrium, Af • Vapor pressure increases as temperature increases.

• When the absolute pressure in a liquid is less than its vapor pressure (A&LM < Af), it evaporates. • When the absolute pressure in a gas is greater than its vapor pressure (A&LM > Af), it condenses. • The vapor pressure of water at 100 ℃ is 101.3 �A�, which means that, at sea level, water will

evaporate at 100 ℃.

7.2 Cavitation

Definitions:

• Cavitation: formation of low-pressure pockets (vapor cavities) when the pressure within a liquid is

equal or lower than the vapor pressure.

• Boiling: the spontaneous formation of vapor cavities within a liquid

• Boiling occurs when the temperature of a liquid is raised or when the pressure of a liquid is lowered.

Note: Vapor pressure increases with temperature, so at higher fluid temperatures there is greater danger of

cavitation.

Cavitation can cause damage to conduits and propellers, but it can also have positive uses, such as in

ultrasonic cleaning and supercavitating torpedoes, and shock wave lithotripsy.

The potential for cavitation in a flowing liquid is measured by the cavitation number:

Ca = A − Af12 �o5

7.3 Humidity

• saturation pressure: the pressure at which a pure substance changes phase, AM&K • In pure substances, the vapor pressure and saturation pressure are the same.

28



In mixtures, the partial pressure of each component must be equal to their respective vapor pressures in

order to achieve phase equilibrium. When this occurs, the system is said to be saturated. Thus, the rate of

evaporation from open water bodies (e.g. lakes) is controlled by the difference between the vapor pressure

and the partial pressure.

In hydrologic and agricultural applications, the relative humidity is used to relate the vapor pressure and

partial pressures of a substance:

�´ = AAf × 100

Based on this equation, evaporation can only occur when the relative humidity is less than 100%. Some sources use the variables and M to denote partial pressure and vapor pressure, respectively.

8 Ideal Gas Law

An ideal gas (or perfect gas) has the following relationship between pressure, temperature, and density:

A = �� A = ��

where pressure and temperature are in their absolute scales, and � is the gas constant. The gas constant is determined by

� = �y�

where �y is the universal gas constant, 8.314 �¶/\�·� ⋅ �^ = 49,710 �� ⋅ �4/\��·� ⋅ °�^, and � is the molar mass. In prior decades, �y was expressed in terms of pounds instead of slugs, so the ideal gas law would be written in terms of specific weight instead of density. We will not worry about this in class but

students should be aware of this when reading older literature on the subject. The ideal gas law is also

known as the equation of state (or state relation) for gases.

When a mixture contains various gases (such as in air), Dalton’s Law of Partial Pressures states that each

gas exerts its own pressure as if the others were not present.

In engineering applications, most gases can be treated as ideal gases. However, vapors cannot be treated as

ideal gases.

9 First Law of Thermodynamics

29



• First Law of Thermodynamics: when a system changes from one state to another, its energy content

changes by energy exchange with its surroundings.

• Energy exchange: occurs in the form of heat transfer of work

In this course, we will define heat transfer to the system as positive and work done by the system as negative,

therefore:

�c − �d = �b

where �c is the amount of heat transfer to the system, �d is the work done by the system, and �b is the change in internal energy in the system. In this course, we consider three types of energy: internal, kinetic,

and potential energy:

b = �̧ + o52 + �w = `�̧ + o5

2 + �wa

where �̧ is the internal energy (and by default, �̧ is the internal energy per unit mass), :'



10 Other Thermodynamic Properties

10.1 Enthalpy

• Enthalpy: the sum of a system’s internal energy and the product of its pressure and volume

´ = �̧ + A

Its corresponding intensive property is:

ℎ = ́ = �̧ + A�

10.2 Specific Heat

• Constant-pressure specific heat: a property used to calculate the change in enthalpy in an ideal gas

Δℎ = ¹ �¬ �� ℎ = �¬ ��

• Constant-volume specific heat: a property used to calculate the change in internal energy in an

ideal gas

Δ�̧ = ¹ �f �� ̧ = �f ��

The relationship between the two specific heats can be found through applying the ideal gas law to the

enthalpy equation and differentiating:

ℎ = �̧ + A� = �̧ + �� ℎ = ��̧ + �� ¬�� = �f�� + �� ¬ = �f + �

The ratio of specific heats (or adiabatic exponent) can be expressed as:

� = �¬�f

31



In many situations, particularly with liquids, the specific heats can be assumed to be constant. For air and

other diatomic gases, � = 1.4.

10.3 Isothermal and Isentropic Processes

• Quasi-equilibrium process (quasi-static process): a process in which pressure, temperature, and other

properties are constant at any instant throughout the system (e.g. compression and expansion in

the cylinder of an internal combustion engine)

• Isothermal process: a process in which temperature is constant

• Isobaric process: a process in which pressure is constant

• Adiabatic Process: a quasi-equilibrium process in which there is no heat transfer.

• Isentropic Process: a frictionless and reversible adiabatic process

For perfect gases, the relationship between pressure and density can be given by:

A5A� = U�5��V

which also gives the following relationships:

��5 = UA�A5V\�^/ = U��5V

�

where is a nonnegative value from zero to infinity that is determined by the process to which the gas is subjected.

An isothermal process occurs when the temperature does not change from one state to another, so �� = �5 and

;< = 1. From the equation above, this also means that º;º



An isobaric process occurs when ¬;¬< = 1 and = 0. An expansion with friction occurs when < � . A

compression with friction occurs when > �.

10.4 Compressibility of Perfect Gases

In thermodynamic applications, the volumetric bulk modulus can be expressed as a function of specific

volume:

bf = − A = − oo A

The pressure/density relationship for perfect gases can also be expressed as a function of specific volume:

A5A� = U�5��V = Uo�o5V

A�o� = A5o5 = �·�� ∴ Ao = �·��

Differentiating this equation gives:

Ao��o + o�A = 0

�A = − Ao�o �o = − Ao �o

Combining with the volumetric bulk modulus gives:

bf = − o�o U− Ao �oV = A

So, for an isothermal process: bf = A and for an isentropic process, bf = �A.

The inverse of the bulk modulus is called the isothermal compressibility and is denoted by:

= 1bf = − 1o oA

10.5 Speed of Sound

If a fluid is inelastic (bf → ∞), pressure disturbances are transmitted instantaneously. In elastic fluids, small pressure changes travel at a finite velocity, known as celerity. The celerity is often called the sonic velocity,

acoustic velocity, or speed of sound.

• Sound: a pressure wave that travels through a medium that can be perceived audibly

33



For small pressure waves travelling in a perfect gas at low frequencies (e.g. a sound wave), the disturbance

is so small and rapid that heat exchange may be neglected, and the wave speed is given by an isentropic

process so that:

� = A� = �A� = √��

where � is the ratio of specific heats. For high frequencies, entropy is not constant, and the equation becomes:

� = A� = A� = √��

The ratio of an object’s speed to the speed of sound is called the Mach number, Ma:

Ma = v�

If Ma < 1, then an object is at subsonic speed, and if Ma > 1, an object is at supersonic speed. For an object at constant speed, the Mach number decreases as the temperature increases. Fluids with Ma ≤ 0.3 can be assumed to be incompressible.

10.6 Heat Transfer in an Isothermal Expansion

The ideal gas law can be expressed in terms of the number of moles of the gas, and the universal gas

constant as follows:

A = �� = �y� � = �y�

where is the number of moles of the gas and is determined by = /�.

For an isothermal expansion of an ideal gas, the heat transfer can be determined by combining the 1st Law

of Thermodynamics with the Ideal Gas Law (expressed in terms of number of moles):

c�5 − d�5 = b5 − b�

Where d�5 = Á A �f



account changes in temperature (while keeping pressure constant) is the coefficient of thermal expansion

and is denoted as:

£ = 1o Uo�V ≅ Δo/oΔ�

£ = − 1� U��V ≅ − Δ�/�Δ�

The thermal expansion coefficient for an ideal gas is equivalent to the inverse of the temperature:

£ = 1�

In natural convection currents, the temperature and density of the fluid body that surrounds a finite hot or

cold region is given a subscript of ∞, and the thermal expansion coefficient is approximated by:

£ ≈ − \�Â − �^/�ÃÂ − �

The combined effects of pressure and temperature changes on the volume change of a fluid is determined

by taking the specific volume to be a function of pressure and temperature, which can then be related to

the isothermal compressibility and the thermal expansion coefficient:

�o = o� �� + oA �A = £o�� − o�A = o\£�� − �A^

The fractional change in volume can be approximated as:

�oo = £�� − �A Δoo = £Δ� − ΔA

Similarly, for density: �� = �A − £�� Δ�� = ΔA − £Δ�

10.8 Latent Heat

• Latent heat: energy per unit mass that is absorbed or released by a fluid upon a change in phase

at a constant temperature and pressure.

35



• Latent heat of vaporization (enthalpy of vaporization), �f: amount of heat required to convert a unit mass of a fluid from the liquid to a vapor phase at a given temperature.

• Latent heat of fusion (enthalpy of fusion), �Ä: amount of heat required to convert a unit mass of a solid to a liquid at the melting point.

10.9 State Equations for Liquids

We have discussed the equation of state for gases (or ideal gas law), and have looked at other thermodynamic

properties which apply to liquids and gases. For most liquids, we assume the following state relations

(covered in detail in the previous sections). Assuming no temperature changes:

� = const �¬ = �f = const �ℎ = �¬�� AA& ≈ \ + 1^ U ��&V

−

In this course, when dealing with liquids, if no temperature change is specified, we will assume that the

equations above hold.

11 Other Properties

11.1 Standard Atmosphere

Atmospheric properties vary with time and latitude, therefore, engineering calculations are based on a

standard atmosphere, which is at 40° latitude. The standard atmosphere is a set of standard values (pressure, temperature, etc.) that were adopted in the 1920’s to standardize aircraft instruments and aircraft

performance. These values have been extended and improved, with the latest accepted iterations being the

International Civil Aviation Organization (ICAO) standard atmosphere adopted in 1964, the International

Standards Organization (ISO) standard atmosphere adopted in 1973, and the United States (US) standard

atmosphere adopted in 1976.

The atmosphere is divided into four layers: the troposphere, stratosphere, mesosphere, and the ionosphere

(which is itself composed of the thermosphere, exosphere, and part of the mesosphere). In the troposphere,

at altitudes of 0 – 11.02 km (36,200 ft), temperature decreases linearly at a lapse rate . The temperature can be expressed as a function of elevation with the equation

�\w^ = �� − w

where �� = 288 � = 518 °� and = −6.489 ℃/� = −3.560 ℉/\1000 ��^. In the stratosphere, at altitudes of 11.02 – 20.06 km, the temperature remains constant at −56.5 ℃ (−69.7 ℉). In the mesosphere, at

36



altitudes of 20.06 – 31.16 km, the temperature increases linearly to −44.5 ℃, then from 31.16 – 47.35 km, it increases linearly to −2.5 ℃ (27.5 ℉). From 47.35 – 51.41 km, the temperature remains constant. From 51.41 – 71.80 km, the temperature decreases linearly to −58.5 ℃. From 71.80 – 86.00 km, the temperature decreases linearly to −86.28 ℃ (−123.30 ℉).

The standard pressure decreases rapidly to almost zero at an altitude of 30 km (98,000 ft). The pressure

profile can be determined using fluid statics calculations, which are covered in future lessons.

Temperature and pressure in the atmosphere. From White, Fluid Mechanics. Used without permission.

11.2 Salinity

When salt is added to water, its density is increased and its freezing point is decreased. This is why salt is

added to roads to prevent ice formation. The salt content in a body of water is measured by its salinity,

which is the ratio of the weight of the salt to the total weight of the mixture:

n = 7@M&K7@:~Ky=Æ

The average salinity of seawater is 0.035, commonly expressed as 35‰ (parts per thousand).

References

1. Chin, D. A., Fluid Mechanics for Engineers, Pearson Higher Education, 2017.

2. Çengel, Y. A., Cimbala, J. M., Fluid Mechanics: Fundamentals and Applications, 4th ed.,

McGraw-Hill Education, 2018.

37



3. Elger, D. F., Williams, B. C., Crowe, C. T., Roberson, J. A., Engineering Fluid Mechanics, 10th

ed., John Wiley & Sons, 2013.

4. Finnemore, E. J., Franzini, J. B., Fluid Mechanics with Engineering Applications, 10th ed.,

McGraw-Hill, 2002.

5. Feynman, R. P., Leighton, R. B., Sands, M., The Feynman Lectures on Physics, New Millennium

ed., California Institute of Technology, 2010.

6. Fox, R. W., McDonald, A. T., Pritchard, P. J., Introduction to Fluid Mechanics, 6th ed., John

Wiley & Sons, 2004.

7. Gerhart, P. M., Gerhart, A. L., Hochstein, J. I., Munson, Young, and Okiishi’s Fundamentals of

Fluid Mechanics, 8th ed., 2016.

8. Giancoli, D. C., Physics: Principles with Applications, 7th ed., Pearson Education, 2014.

9. Hibbeler, R. C., Fluid Mechanics, 2nd ed., Pearson Higher Education, 2018.

10. Moin, P., Kim, J., “Tackling Turbulence with Supercomputers.” Scientific American, vol. 276, no.

1, 1997, pp. 62-68.

11. Panton, R. L., Incompressible Flow, 4th ed., Wiley, 2013.

12. Potter, M. C., Wiggert, D. C., Schaum’s Outline of Fluid Mechanics, McGraw-Hill, 2008.

13. Potter, M. C., Wiggert, D. C., Ramadan, B. H., Mechanics of Fluids, 5th ed., Cengage Learning,

2015.

14. Tokaty, G. A., A History & Philosophy of Fluid Mechanics, G. T. Foulis & Co, 1971.

15. Vennard, J. K., Elementary Fluid Mechanics, 4th ed., John Wiley & Sons, 1961.

16. White, F. M., Xue, H., Fluid Mechanics, 9th ed., McGraw-Hill, 2021.

38


Fluid Statics David S. Ancalle, P.E.

1 Approaching Pressure

1.1 Pressure at a Point

Pressure at a point can be defined as an infinitesimal normal compressive force divided by an infinitesimal

area over which it acts.

Pressure at a point does not vary with direction. Consider a wedge-shaped element with a uniform width

��, sides Δ� and Δ�, and hypotenuse Δ�. A force, �� results from a pressure � on the hypotenuse, and forces � and �� act on the other sides. If we draw a free body diagram and take forces in the x and z directions, we get:

Diagram of pressure acting on a wedge-shaped element.

Then, taking a sum of forces in the x-axis:

� � = �� ⟹ � − �� sin � = ��

Δ�Δ� − �Δ�Δ� sin � = 12 �Δ�Δ�Δ��

Δ�Δ� − �Δ�Δ� = 12 �Δ�Δ�Δ��

− � = 12 �� Δ�

39



As the element shrinks to a point, Δ� → 0 and the limit becomes:

− � = 0 ⟹ = �

Repeating this process for the z-axis:

� �� = �� ⟹ �� − �� cos � − �� = ��

�Δ�Δ� − �Δ�Δ� cos � − �� Δ�Δ�Δ�2 =12 �Δ�Δ�Δ��

�Δ�Δ� − �ΔxΔ� − �� Δ�Δ�Δ�2 =12 �Δ�Δ�Δ��

� − � − �� Δ�2 =12 �Δ��

� − � = 12 �!�� + �#Δ�

As the element shrinks to a point, Δ� → 0 and the limit becomes:

� − � = 0 ⟹ � = �

So, we see that, at a point, the pressure in all directions does not vary. This analysis holds true for other

shapes.

1.2 Pressure Variation

Consider an infinitesimally small element with dimensions $�, $�, $�, that undergoes a pressure !�, �, �#, where & is the pressure at the center of the element. We can determine the pressure at the sides of the element using the chain rule:

$ = ''� $� +''� $� +

''� $�

So, if we move from the center of the element to an arbitrary side � (where � can be any side, �, �, or �), then the pressure at that side is:

= & + ''�$�2

Note that the distance moved is (�) because we are moving from the center of the element to the side, so we

only move half of the entire length of the element in that direction. We can use this equation to find the

pressure on all of the sides of the element, and we can use those pressures to determine the forces acting on

the element:

40



Infinitesimal cube-shaped element.

On the top face: = & + ''�$�2 � = − *& +

''�

$�2 + $� $�

On the bottom face: = & − ''�$�2 � = *& −

''�

$�2 + $� $�

On the front face: = & + ''�$�2 � = − *& +

''�

$�2 + $� $�

On the back face: = & − ''�$�2 � = *& −

''�

$�2 + $� $�

On the left face: = & − ''�$�2 � = *& −

''�

$�2 + $� $�

On the right face: = & + ''�$�2 � = − *& +

''�

$�2 + $� $�

Newton’s second law can now be applied in each direction, taking � = �, = � $� $� $� and �� = ��,:

� � = *& − ''�$�2 + $� $� − *& +

''�

$�2 + $� $� = � $� $� $� �

− ''� $� $� $� = � $� $� $� �

''� = −��

� �- = *& − ''�$�2 + $� $� − *& +

''�

$�2 + $� $� = � $� $� $� �-

''� = −��-

� �� = *& − ''�$�2 + $� $� − *& +

''�

$�2 + $� $� − �� $� $� $� = � $� $� $� ��

− ''� $� $� $� − �� $� $� $� = � $� $� $� ��

41



''� = −�� − �� = −�!�� + �#

We can now apply these expressions to find the pressure differential in any direction:

$ = ''� $� +''� $� +

''� $� = −��$� − ��-$� − �!�� + �#$�

1.3 Pressure in Fluids at Rest

An object at rest does not undergo acceleration in any direction. Therefore, for a fluid at rest, the pressure

differential becomes:

$ = −��$� − ��-$� − �!�� + �#$� = −��$�

which we can express as:

$$� = −�� = −.

This tells us the following: there is no pressure variation in the x and y directions, pressure only varies in

the vertical direction; pressure decreases as we move up and increases as we move down; and pressure

variation for a fluid at rest depends on the specific weight of the fluid.

If the specific weight of the fluid is constant (e.g. an incompressible liquid with no variations in temperature

or composition), then we can integrate the pressure differential:

/ $ = / −.$� = −. / $�

Δ = −.Δ�

2 Hydrostatic Pressure

2.1 Pascal’s Law

Fluid statics: the study of fluids at rest.

When a system is at rest: ∑ �⃑ = 0, ∑ 233⃑ = 0

We define pressure as = $�/$5. Following this definition, we can express the average pressure over an area as 678 = �/5.

42



In 1647-1648, Blaise Pascal established that a change in pressure in an enclosed fluid at rest is equal at all

points in the fluid. This is known as Pascal’s Law.

Consider an enclosed container full of a fluid:

Diagram of a closed container filled with a fluid.

If we apply a force �9 on the left area 59, then the pressure applied on the fluid is 9 = �9/59. Pascal’s Law states that the change in pressure will be equal at all points in the fluid, which means that on the right area

5), the pressure will be increased by ) = 9. We can then determine the resulting force on the right area: �) = )5). Mathematically:

9 = �959 = ) =�)5) ∴ �) = �9

5)59

Notice that if 5) is larger than 59, then the force applied on the right side will be increased. This makes hydraulic lifts possible.

Ex. In the figure above, �9 = 1 ;



Diagram of a contained with an incompressible fluid, open to the atmosphere, and an infinitesimal cube of that fluid.

We can determine the pressure acting on the top face of the fluid as follows:

= ��5 =.,

$� $� =. $� $� ℎ

$� $� = .ℎ

where:

• �� is the weight of the column of water above the cube • 5 is the area of the top face of the cube • . is the specific weight of the column of water above the cube • , is the volume of the column of water above the cube

Notice that the pressure exerted by the fluid on the top of the cube only depends on the depth of the cube,

and not its geometry. We also did not include atmospheric pressure (which can be conceptually defined as

the pressure exerted on the water by the column of air above it).

This equation agrees with the equation for pressure in fluids at rest that we derived in the previous section

Δ = −.Δ�, where Δ� = −ℎ (since ℎ is measured from top to bottom).

2.3 Pressure Head

For incompressible fluids, the pressure can be expressed in dimensions of length by solving for ℎ:

= .ℎ ⟹ ℎ = .

This quantity is called the pressure head, and is used as a unit in pressure measurements. Conceptually,

this head represents the height of a column of liquid with specific gravity . that produces a gage pressure . Pressure is commonly measured as the length of a column of water (? = 1) or a column of mercury (? =13.6). The unit conversions are as follows:

1 ��CDE = 9.81 �

44



1 ��C8 = 133 �

From the conversions, it can be deduced that mercury is used to measure higher pressures than water.

Pressure head can also be measured in English units.

2.4 Pressure in Compressible Fluids

On compressible fluids, density and specific weight vary with pressure. Therefore, for a compressible fluid

(such as an ideal gas), we will have to express density as a function of pressure. Applying the ideal gas law

to the pressure differential yields:

$$� = −�� = −

HI �

If we take the temperature of the fluid to be constant, we can integrate the above expression to get:

/ 1 $J

JK= / − �HI $�

��K

ln & = −�

HI !� − �

= & MN 8OP!�N�K#

where & is a reference pressure at an elevation �&.

2.5 Pressure in the Atmosphere

We have learned that the temperature in the troposphere decreases linearly by I = I& − Q�. We can now determine the pressure variation in the troposphere. Applying the ideal gas law to the pressure differential

yields

$$� = −�� = −

HI �

Rearranging, expressing I as a function of �, integrating between an elevation of 0 and z, and solving for pressure:

/ 1 $J

JRST= / − �H!I& − Q�# $�

�&

ln 6UV =�

QH lnI& − Q�I&

45



= 6UV *I& − Q�I& +8WO

Solving for values between 0 < � ≤ 1000 shows that the pressure decrease is very small, and changes in static air pressure can be neglected unless the elevation difference is relatively large.

The temperature in the stratosphere, I�, is constant. So, we determine the pressure by integrating from ��, the lowest elevation in the stratosphere to an elevation �.

= � MN 8OPZ!�N�Z#

3 Measuring Pressure

3.1 U-tube Manometers

Consider pressurized flow in a pipe. If we punch open a hole in the top wall of the pipe and insert a tube,

water will flow upward until the pressure of the water is equal to atmospheric pressure. The fluid in the

tube will be static, and so we can use hydrostatic equations to find the pressure at different points in the

tube and in the pipe.

Profile view of a pressurized pipe with a tube inserted on its top side.

Manometers are instruments that use this principle to measure pressures. In the figure below, a cross-

sectional view of the pipe is shown, and a manometer is connected to the side of the pipe. The manometer

connects the pipe to a point with a known pressure (i.e. ○2 , where the pressure is atmospheric). By measuring the pressure difference between points ○1 and ○2 , we can determine the pressure at the pipe. If we want to measure gage pressures, then ) = 6UV = 0.

46



U-Tube Manometer. From Potter, Wiggert, Ramadan, Mechanics of Fluids. Used without permission.

) + Δ = 9

0 + .ℎ = 9

9 = .ℎ

Remember that pressure increases with depth. Therefore, since we move “downward” from ) to 9, the pressure difference is positive. If we would have started measuring from the pipe to the tube, our equation

would have been:

9 − .ℎ = ) = 0

9 = .ℎ

Ex. In the U-tube manometer above, ℎ = 2[\ and the fluid is water. What is the pressure in the pipe?

Ans:

. = 62.4 ]^/[\_

= .ℎ = !62.4#!2# = 124.8 ]^/[\)

These types of manometers are called U-tube manometers. Manometers that use a single fluid are usually

only used to measure very small pressures. To measure larger pressures, a heavier fluid can be inserted in

the manometer. Measuring the pressure differences from ○1 to ○3 yields:

47



U-tube manometer with two different fluids. From Potter, Wiggert, Ramadan, Mechanics of Fluids. Used without

permission.

9 + .9ℎ − .)` = _ = 0

9 = .)` − .9ℎ

Notice that the pressure at ○2 and ○2’ is the same, since there is no difference in elevation.

3.2 Differential Manometers

A differential manometer is used to measure pressure differences between two pipes or between two points

in a conduit. The pressure difference is computed in the same way as the U-tube manometer, but the end

of the manometer will not be open to the atmosphere.

3.3 Piezometer

A piezometer is a simple type of manometer that consists of a vertical tube that is inserted in a vessel with

a fluid. Piezometers are useful for measuring small pressures. The static pressure of a point in the liquid at

a depth $ is:

Piezometer. From Hibbeler, Fluid Mechanics. Used without permission.

= .!ℎ + $#

3.4 Micromanometers

Another type of manometer is the micromanometer, which is used to measure very small pressure changes.

Applying the manometer equations, we get:

48



Micromanometer. From Potter, Wiggert, Ramadan, Mechanics of Fluids. Used without permission.

9 + .9!�9 − �)# + .)!�) − �_# − ._!�a − �_# − .)!�b − �a# = b

9 + .9!�9 − �)# + .)!�) − �_ + �a − �b# − ._!�a − �_# = b

Note that b = 6UV = 0, ` = �a − �_, and ℎ = �b − �).

9 + .9!�9 − �)# + .)!` − ℎ# − ._` = 0

9 = .9!�) − �9# + .)ℎ + !._ − .)#`

A small change in pressure Δ will result in a change in the elevation �) of Δ�, as well as a change in ℎ of 2Δ�, and a change in ` of 2Δ�c)/$). Therefore, we can evaluate a pressure change by:

9 = .9!−Δ�# + .)Δℎ + !._ − .)#Δ`

Δ9 = .9!−Δ�# + .)!2Δ�# + !._ − .)#2Δ�c)

$)

And so, the rate of change of ` with respect to is:

Δ`Δ9 =

2Δ�c)/$).9!−Δ�# + .)!2Δ�# + !._ − .)#2Δ�c)$)

= 2c)/$)−.9 + 2.) + 2 !._ − .)#c)$)

3.5 Barometers

A barometer is an instrument used to measure atmospheric pressure. It was invented by Evangelista

Torricelli.

49



Barometer. From Hibbeler, Fluid Mechanics. Used without permission.

A barometer consists of a glass tube filled with mercury. The tube is inserted in a reservoir filled with

mercury and turned upside down. The weight of the mercury in the tube causes it to move downward and

creates a vacuum in the top of the tube where ≈ 0. The atmospheric pressure can be measured by measuring the height of the mercury column inside the tube, so that:

6UV = .C8ℎ

Note: Under standard engineering conditions, the atmospheric pressure is 760 ��C8.

3.6 Pressure Gages

In cases where static pressures are too high to measure with manometers, a pressure gage can be used.

Different types of pressure gages are:

Bourdon gage: uses an elastic, coiled metal (Bourdon tube) to determine pressures.

Bourdon gage. From Hibbeler, Fluid Mechanics. Used without permission.

Pressure Transducer: uses an electrical strain gage to determine deformation in its diaphragm and converts

the electrical current measure into a pressure measure.

50



Pressure transducer. From Hibbeler, Fluid Mechanics. Used without permission.

Fused Quartz Force-Balance Bourdon tube: uses the same concept as the Bourdon gage, but determines

pressures using a magnetic field that returns the Bourdon tube to its original position.

Piezoelectric gages: devices that change their electric potential when subjected to small pressure changes.

4 Buoyancy

4.1 Buoyant Force

Consider an object with specific weight .f, submerged in a fluid with specific weight .g:

Diagram of a contained with an incompressible fluid, open to the atmosphere, and an infinitesimal cube of that fluid.

The force from the surrounding liquid acting on top of the object can be computed by multiplying the

pressure on the top face times the area of the top face:

�Ufh = Ufh5Ufh = .gℎ $� $�

The force acting on the bottom face is:

�ifU = ifU5ifU = .g!ℎ + $�#$� $�

51



We don’t consider the forces on the sides since they will cancel out. The net force acting on the object is:

�jkU = �ifU − �Ufh = .g!ℎ + $�#$� $� − .gℎ $� $� = .g!ℎ + $� − ℎ#$� $� = .g $� $� $�

Notice that the net force is then the specific weight of the fluid by the volume of the submerged object (i.e.

the displaced volume). We call this force the buoyant force, �l and define it as:

�l = .g,

Where , is the displaced volume. In order for an object to float, the buoyant force has to be larger than its weight, applying static equilibrium, we see that a submerged or partially submerged object is static when

its buoyant force is equal to its weight:

� � = �l − �� = 0 ⟹ �l = ��

4.2 Hydrometers

A hydrometer is an instrument used to measure the specific gravity of liquids that uses the principle of

buoyancy. It consists of a stem with a constant area 5. When placed in water, the hydrometer will submerge until:

�� = .CDE,&

where ,& is the initial submerged volume. When submerged in a different fluid with specific weight ., the force balance is:

�� = .!,& − 5Δℎ#

where Δℎ is the change in submerged portion of the stem. Combining both equations gives:

.CDE,& = .!,& − 5Δℎ# ⟹ S = V&,& − 5Δℎ =1

1 − 5Δℎ,&

5 Stability

Consider a submerged object:

• Center of gravity: the centroid of an object, o • Center of buoyancy: the centroid of the displaced volume, pl

Gravitational force acts on the center of gravity, and buoyant force acts on the center of buoyancy. We

have already shown that a floating object is in vertical equilibrium if �l = ��. However, we must also

52



consider if an object has rotational equilibrium, which results when no moment is formed by the two forces.

We will consider three types of rotational equilibrium:

Stable Equilibrium

Occurs when o is below pl . If the object is slightly rotated, a coupled moment between �� and �l will restore the object to its original position.

Unstable Equilibrium

Occurs when o is above pl . If the object is slightly rotated, a coupled moment between �� and �l will continue to rotate the object.

From Potter, Wiggert, and Ramadan, Mechanics

of Fluids. Used without permission.

Neutral Equilibrium

Occurs when o and pl coincide. No coupled moment forms from a rotation.

53



For objects that are partially submerged, a rotation may move the center of buoyancy. This allows for

stable equilibrium even in instances where o is above pl.

Consider a partially submerged object

(like a ship) in unstable equilibrium. We

will define the point about which the

object rotates as the origin, q , which lies at the water surface on the vertical

line that crosses pl to o. We will call this line the line of action.

When a rotation is applied to the object,

the line of action rotates along with the

object, and the pl can shift. If we trace a vertical line from the new position of

the center of buoyancy, it will intersect

the line of action at the metacenter, 2.

If 2 is above o in the line of action, then the object is in stable equilibrium,

since a counteracting moment acting

between 2 and o will return the object to its original position.

From Hibbeler, Fluid Mechanics. Used without permission.

If, however, 2 is below o in the line of action, then the object is in unstable

equilibrium, as the resulting moment

will continue the rotation and cause the

object to overturn.

While the theory for stability is relatively simple to grasp at a conceptual level, determining the stability

of objects becomes complicated when it is time to determine the location of 2. In this class, we will consider the method taught by Potter et. al.

Let’s define the metacentric height, o2rrrrr as the distance from o to 2. A positive o2rrrrr value indicates that the metacenter is above the center of gravity. A negative o2rrrrr value indicates that 2 is below o. Consider a floating body with a uniform cross section, having rotated as shown below:

54



Diagram of a tilted, partially submerged body. From Potter, Wiggert, and Ramadan, Mechanics of Fluids. Used

without permission.

The x-coordinate of the center of buoyancy, �̅ can be found by considering the volume of the original submerged volume (,&), plus the volume from added wedge cqt, ,9 minus the volume from the subtracted wedge 5qu, ,):

�̅ = �̅&,& + �̅9,9 − �̅),),

�̅, = �̅&,& + �̅9,9 − �̅),)

Taking the x-coordinate of the original center of buoyancy to be zero, we get �̅& = 0:

�̅, = �̅9,9 − �̅),)

�̅, = / � $,vw

− / � $,vD

Now, to find an expression for the infinitesimal volume $,, let’s take a step back and take a look into the basics of integrating. Suppose you want to find the area of the triangle formed by the lines � = ��, � = 0, and � = �9. You could easily do this by integrating � = �� from 0 to �9. Your integral would look like:

/ �� $�w&

55



Triangular area.

We know that in a linear equation, the constant represents the slope of the line. If we define Q as the angle of the line measured from the x-axis, then the slope is � = x-y = tan Q. Giving us an integral of | � tan Q $�w& . Now, thi

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