# Fluid mechanics: simple calculus issue

1. Aug 12, 2012

### K29

1. The problem statement, all variables and given/known data
Consider the steady flow of an incompressible viscous fluid down an inclined plane under the action of gravity. The plane makes an angle $\alpha$ with the horizontal. The fluid is infinite in the z-direction (x is down the plane and y is normal to the plane). Look for a solution of the form:
$v_{x}=v_{x}(y), v_{y}=v_{y}(y), v_{z}=0, p=p(y)$

By considering the incompressibility condition show that $v_{y}\equiv0$

2. Relevant equations
Incompressibility condition:
$\bar{\nabla} . \bar{v}=0$

3. The attempt at a solution
$\bar{\nabla} . \bar{v}= \frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{\partial v_{z}}{\partial z}=0$

Since $v_{x}$ is a function of y only and $v_{z} = 0$:
$\frac{\partial v_{y}}{\partial y}=0$

Surely $v_{y}$ can be any constant, and is not necessarily 0???? Or does this have something to do with the use of $\equiv$ instead of $=$ in the question?

2. Aug 12, 2012

### qbert

you need to use a boundary condition to fix the value.
what do you know about vy(y=0) ?

3. Aug 13, 2012

### K29

Thank you. Solved