Shear Flow Against a Wall (Fluid Mechanics)

[Isobel2]

Homework Statement

Assume a shear flow against a wall, given by U= Uo (2y/ax - y^2/((ax)^2) where a is a constant. Derive the velocity component V (x; y) assuming incompressibility.

Homework Equations

Haven't been able to find any in my course notes.

The Attempt at a Solution

Some googling has taught me that shear flow is the flow induced by a shear stress force gradient. But I really need some sort of equation to solve this I think.

 

[MaxManus]

I'm sorry but I am not able to help, but an incomrepssible fluid have divergence = 0 and the navier stoke given in this link:
http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node19.html
I'm not sure if the equations help, but they are all I remember from my course, and the bernoulli Equation.

 

[Chestermiller]

Hi Isobel2. Welcome to Physics Forums.

You need to make use of MaxManus' suggestion, and set the divergence of the velocity vector equal to zero:

$$\frac{\partial V}{\partial y}=-\frac{\partial U}{\partial x}$$

But, before you start trying to do this by brute force, first define the following parameter:

$$\eta=\frac{y}{ax}$$

so that $$U = U_0(2\eta - \eta ^2)$$

Also note that $$\frac{\partial U}{\partial x}=\frac{\partial U}{\partial \eta}\frac{\partial \eta}{\partial x}=-\frac{\partial U}{\partial \eta}\frac{\eta}{x}$$
$$\frac{\partial V}{\partial y}=\frac{\partial V}{\partial \eta}\frac{\partial \eta}{\partial y}=\frac{\partial V}{\partial \eta}\frac{1}{ax}$$

Working with the parameter η in this way will make the "arithmetic" much simpler and less prone to error.