How can I generate an analytical flow field for poiseuille flow including a time dependent flow solution?

I was looking for some help in Generating an analytical flow field such as Couette, Poiseuille flow which includes a Time dependent flow solution.

Thanks in Advance.

[Torsten]

What are the equations to be solved ?

[Abhisheik Chanda]

Thankyou for the reply,

I am trying it for the Unsteady Poiseuille flow between Parallel plates.

Answer 1

[William Rose]

@Abhisheik Chanda,

@Torsten did not follow up, because he asked for the equations, and you did not provide actual equations. Would you, please? Then we can do it.

[William Rose]

@Abhisheik Chanda,

Suppose that the flow field for Couette flow is given by

where is the kinematic viscosity. The equation above isn't true, but it is simple, and serves as an example. Then we can compute the flow field at different times. Suppose and and h=1 cm and Vh=10 cm/s.

%constants
Vh=10;              %plate velocity (cm/s)
h=1;                %channel width (cm)
mu=0.04;        %viscosity (Poise=g/(cm-s))
rho=1;              %denisty (g/cm^3)
%derived quantities
nu=mu/rho;          %dynamic viscosity (cm^2/s)
tau=h^2/nu;         %time constant (s)
%Compute the flow at different times
t=tau*(0:.25:4);    %time
y=h*(0:.1:1);       %position
U=(y/h)'*(1-exp(-t/tau)); %x-component of vel.
V=zeros(size(U));   %y-component of velocity=0

Plot it with quiver().

quiver(U,V);

Try it.

[William Rose]

@Abhisheik Chanda, This code ocmputes the actual analytical epxression for Couette flow as it develops from time 0. It displays the data in two ways: as a quiver plot and as a set of line plots.