Fluid mechanics: simple calculus issue

[K29]

Homework Statement

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 $v_{y}\equiv0$

Homework Equations

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

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?

 

[qbert]

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

 

[K29]

Thank you. Solved