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