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UNSTEADY VISCOUS FLOW. Viscous effects confined to within some finite area near the boundary → boundary layer. In unsteady viscous flows at low Re the boundary layer thickness δ grows with time; but in periodic flows, it remains constant.

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slide1

UNSTEADY VISCOUS FLOW

Viscous effects confined to within some finite area near the boundary → boundary layer

In unsteady viscous flows at low Re the boundary layer thickness δgrows with time; but in periodic flows, it remains constant

If the pressure gradient is zero, Navier-Stokes equation (in x) reduces to:

slide2

Heat Equation– parabolic partial differential equation - linear

Requires one initial condition and two boundary conditions

U

Impulsively started plate –

Stokes first problem

y

Total of three conditions

slide3

Heat Equation– parabolic partial differential equation

Can be solved by “Separation of Variables”

Suppose we have a solution:

Substituting in the diff eq:

May also be written as:

Moving variables to same side:

The two sides have to be equal for any choice of x and t ,

The minus sign in front of k is for convenience

slide6

Alternative solution to“Separation of Variables” – “Similarity Solution”

New independent variable:

from:

Substituting into heat equation:

η is used to transform heat equation:

slide7

2 BC turn into 1

To transform second order into first order:

Or in terms of the error function:

Integrating to obtain u:

With solution:

For η > 2 the error function is nearly 1, so that u → 0

slide8

For η > 2 the error function is nearly 1, so that u → 0

Then, viscous effects are confined to the region η < 2

This is the boundary layer δ

increasing time

δgrows as the squared root of time

slide10

UNSTEADY VISCOUS FLOW

Oscillating Plate

Look for a solution of the form:

Ucos(ωt)

y

Euler’s formula

slide11

B.C. in Y

Substitution into:

Fourier’s transform in the time domain:

slide14

UNSTEADY VISCOUS FLOW

Oscillating Plate

Look for a solution of the form:

Ucos(ωt)

y

W

Euler’s formula

slide15

B.C. in Y

Substitution into:

Fourier’s transform in the time domain:

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