Uniform Flows

A uniform flow is characterized by constant velocity in a single direction, with velocity components $u=U,v=0,w=0$.

The stream function for uniform flow is:

$$u=U=\frac{\partial \psi }{\partial y}$$

Integrating with respect to $y$:

$$\psi =Uy+C$$

This means that the streamlines are parallel and spaced linearly with $y$. We can also write $\psi =Ur\sin \theta +C$ in cylindrical coordinates.

The velocity potential for uniform flow is:

$$u=U=\frac{\partial \varphi }{\partial x}$$

Integrating with respect to $x$:

$$\varphi =Ux+C$$

This means the equipotential lines are vertical and spaced linearly with $x$. We can also write $\varphi =Ur\cos \theta +C$ in cylindrical coordinates.