fluid mechanics q3 2012 solution -...

of 10

AER210 VECTOR CALCULUS and FLUID MECHANICS

Fluid Mechanics Quiz 3

Duration: 70 minutes

12 November 2012

Closed Book, no aid sheets

Non-programmable calculators allowed

Instructor: Alis Ekmekci

Family Name: __________________________________________

Given Name: __________________________________________

Student #: __________________________________________

TA Name/Tutorial #: ____________________________________

FOR MARKER USE ONLY

Question Marks Earned

1 15

2 6

3 8

4 13

5 10

TOTAL 52 /50

Hints: −∇p−ρg�� = ρ��

VVd

dPEV −=

dy

duµτ =

of 10

1-a) [1 mark] Someone claims that the absolute pressure in a liquid of constant density doubles

when the depth is doubled. Do you agree? Explain.

1-b) [5 marks] Consider two tanks filled with water. The first tank is 8 m high and is stationary,

while the second tank is 2 m high and is moving upward with an acceleration of 5 m/s2. Which

tank will have a higher pressure at the bottom? (Note that gravitational acceleration is 10 m/s2.)

1-c) [3 marks] Explain what is a streamline and a pathline? Are they different or identical in a

steady flow?

of 10

1-d) [3 marks] A shaft 6 cm in diameter and 40 cm long is pulled steadily at a velocity V = 0.4

m/s through a sleeve of 6.02 cm in diameter. The clearance is filled with oil. Estimate the force

required to pull the shaft. The dynamic viscosity of the oil is μ = 2.63 N.s/m2. (Hint: Assume a linear velocity distribution in the clearance)

1-e) [3 marks] The tank, shown below, contains water and immiscinle oil. What is h in cm if the

density of the oil is 898 kg/m3 and the density of the water is 1000 kg/m

3?

2) Circle the true statement in the following:

- [1 mark] If you have a bulk modulus of elasticity that is a very large number, then a small

change in pressure would cause

a. a very large change in volume

b. a very small change in volume

- [1 mark] Consider a balloon filled with helium (case

A) and a balloon filled with air (case B). Which

statement is correct?

a. Buoyant force (case A) > Buoyant force (case B)

b. Buoyant force (case A) < Buoyant force (case B)

c. Buoyant force (case A) = Buoyant force (case B)

- [1 mark] The dynamic viscosity

a. density b. temperature

- [1 mark] The no-slip condition

a. only applies to ideal flow

b. only applies to rough surfaces

c. means velocity of the fluid is equal to zero at the wall

d. means velocity of the fluid at the wall is equal to the velocity of the wall

- [1 mark] Dimensions of the dynamic viscosity

a. L2/T

b. L/(MT2)

c. M/(LT)

d. M/(T2L)

- [1 mark] A tiny neutrally buoyant electronic pressure probe is released into the inlet pipe of a

water pump and transmits 2000 pressure readings every second as it passes through the pump.

This is a

a. Lagrangian measurement

true statement in the following:

If you have a bulk modulus of elasticity that is a very large number, then a small

a very large change in volume

a very small change in volume

Consider a balloon filled with helium (case

A) and a balloon filled with air (case B). Which

Buoyant force (case A) > Buoyant force (case B)

Buoyant force (case A) < Buoyant force (case B)

Buoyant force (case A) = Buoyant force (case B)

The dynamic viscosity μ of a fluid is primarily a function of

temperature c. pressure d. velocity

slip condition

only applies to ideal flow

only applies to rough surfaces

means velocity of the fluid is equal to zero at the wall

means velocity of the fluid at the wall is equal to the velocity of the wall

Dimensions of the dynamic viscosity μ is

A tiny neutrally buoyant electronic pressure probe is released into the inlet pipe of a


measurement b. Eulerian measurement

Page 4 of 10

If you have a bulk modulus of elasticity that is a very large number, then a small

means velocity of the fluid at the wall is equal to the velocity of the wall

A tiny neutrally buoyant electronic pressure probe is released into the inlet pipe of a


of 10

3) The tank, shown below, accelerates to the right with the fluid in rigid-body motion.

(a) [4 marks] The gravitational acceleration can be taken as 10 m/s2. Compute the acceleration

in the direction of motion (ax).

(b) [4 marks] Determine the gage pressure values at points A and B (located at the bottom

corners of the tank) if the fluid has a density of 1000 kg/m3.

of 10

4) A thin layer of particles rests on the bottom of a horizontal tube as shown in the figure below.

When an incompressible fluid flows through the tube, it is observed that at some critical velocity,

the particles will rise and be transported along the tube. The critical velocity Vc is known to

depend on the pipe diameter D, particle diameter d, the fluid density ρ, the fluid viscosity μ, the density of the particles ρp, and the gravitational acceleration g. Thus, Vc = function (D, d, ρ, μ, ρp, g) a) [9 marks] By dimensional analysis, choosing the repeating variables as ρρρρ, D and μμμμ, determine the dimensionless (π) groups for this problem, and re-write the original dimensional relationship in dimensionless terms.

b) [2 marks] A 1/2 scale model is to be used to determine this functional relationship

experimentally. If fluid densities and the gravitational acceleration in the small-scale model test

and the full-scale prototype test are identical, what should be the viscosity ratio of the model

fluid and the prototype fluid (that is, μmodel/μprototype) in order to insure similarity?

c) [2 marks] Assuming all similarity requirements are satisfied between the model and the

prototype tests, what is the ratio of the critical velocity of the model and prototype experiments

(that is, (Vc)model /(Vc)prototype)?

of 10

5) [10 marks] For this gate, ∝ = 45°, y1 = 1 m, and y2 = 3 m. The gate weighs 90 kN and is 1 m wide.

a. [8 marks] Find the moment acting on the gate as a result of the hydrostatic water.

b. [2 marks] Will the gate fall or stay in position under the action of the hydrostatic and gravity

forces?

(Note that the gravitational acceleration is a = 10 m/s2, the water density is ρ = 1000 kg/m3, and

sin(45°) = cos(45°) = √2/2.)

fluid mechanics q3 2012 solution -...

Documents