fluids and related notes

8/11/2019 Fluids and Related Notes

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Fluid Mechanics and Transport Phenomena:

1.

Basic Definitions:

2.

Navier-Stokes Equations:

3.

Types of Flow:

i.

Planar Poiseuille Flowii.

Impulsively started single plate

flow that is semi-infinite

4.

dd

Basic Definitons:

Incompressible: Density, , is constant.

Homogeneous: Density, , is the same for all

fluid elements considered.

Navier-Stokes Equations:

Continuity:

Momentum:

o

: Temporal and inertial/acceleration

term. Equals 0 if steady.

o

: Temporal changes.

o

: Inertial/acceleration term

o : Pressure term (can sometimes

account for gravity,

o : Viscous effects. (If inviscid this term

is zero. Basis for Newton's Equation)

o : Body forces (external body forces - can

include gravity)

Where the material derivative is

Stress Tensor:

Rate of Strain Tensor, :

Types of Flow:

Planar Poiseuille Flow:

Steady

Infinite channel Assume unidirectional flow

Initial Conditions:

o

No Slip:

o

Symmetric: Centre flow has no

shear, where is the

direction of flow

Notes:

In reality typically unstable for

sufficiently high Re.

Introduction of perturbance (knocks,

wall imperfections), the perturbance

grows resulting in unsteady

multidirectional flow.

Navier-Stokes solutions are not unique.

Example: Planar Poiseuille Flow where x is the

direction of flow and y is the height of the pipe

Steps:



1.

Continuity

2.

Navier-Stokes Equation applying

assumptions

3.

Get expression for u. If considering

viscous terms, this will be a second

order pde and hence two b/c or i/c will

be required.

4.

Express for maximum velocity

5.

Determine average velocity

6.

Determine mass flux

Continuity:

Momentum:

Assumptions:

Steady:

Continuity:

Unidirectional:

No body forces cf gravity:

Hence the Navier-Stokes equation reduces to:

Unidirectional:

Assume 2D problem:

Hence take the second integral of

.

To solve, use boundary/initial conditions:

(symmetry and no slip)

Determining constants of integration:

*Note: CHE3167 would express this as

, where

is half the pipe height

and is the length of pipe being considered for

which there is a .

Mass Flux:



Impulsively Started Single Plate for a Semi-

Infinite Fluid:

Plate moves with speed

at

Initially,

Pressure: Far from the plate,

and so it must be

constant everywhere

Assume unidirectional flow

Initial Conditions:

o

No Slip: Fluid moves at the

same speed as plate (it 'sticks'to it)

o

Far Away Fluid:

Example:

Look for a similarity solution such that

Assumptions:

No slip:

Far fluid:

Constant pressure: (p far away is

constant so it must be constant

everywhere)

Unidirectional flow

No body forces

Continuity:

No other components to consider as it is

unidirectional

Momentum:

Similarity Solution/Self-Similar Solution: Results

when a space and time variable are tied

together such that . Solution is of form

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