Two-Dimensional Uniform Flow

Consider a steady two-dimensional flow pattern that is uniform: i.e., such that the fluid velocity is the same everywhere in the $x$-$y$ plane. For instance, suppose that the common fluid velocity is

$${\bf v} = V_0\,\cos\theta_0\,{\bf e}_x + V_0\,\sin\theta_0\,{\bf e}_y,$$ (483)

which corresponds to flow at the uniform speed $V_0$ in a fixed direction that subtends a (counter-clockwise) angle $\theta_0$ with the $x$-axis. It follows, from Equations (472) and (473), that the stream function for steady uniform flow takes the form

$$\psi(x,y) = V_0\left(\sin\theta_0\,x-\cos\theta_0\,y\right).$$ (484)

When written in terms of cylindrical coordinates, this becomes

$$\psi(r,\theta)=-V_0\,r\,\sin(\theta-\theta_0).$$ (485)

Note, from (484), that $\partial^2\psi/\partial x^2=\partial^2\psi/\partial y^2=0$. Thus, it follows from Equation (477) that uniform flow is irrotational. Hence, according to Section 5.7, such flow can also be derived from a velocity potential. In fact, it is easily demonstrated that

$$\phi(r,\theta) = -V_0\,r\,\cos(\theta-\theta_0).$$ (486)