Two-Dimensional Uniform Flow

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

$${\bf v} = V_0\,\cos\theta_0\,{\bf e}_x + V_0\,\sin\theta_0\,{\bf e}_y,$$ (5.27)

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 (5.5) and (5.6), that the stream function for steady uniform flow takes the form

$$\psi(x,y) = V_0\left(\sin\theta_0\,x-\cos\theta_0\,y\right).$$ (5.28)

When written in terms of cylindrical coordinates, this becomes

$$\psi(r,\theta)=-V_0\,r\,\sin(\theta-\theta_0).$$ (5.29)

Note, from Equation (5.28), that $\partial^{\,2}\psi/\partial x^{\,2}=\partial^{\,2}\psi/\partial y^{\,2}=0$ . Thus, it follows from Equation (5.10) that uniform flow is irrotational. Hence, according to Section 4.15, such flow can also be derived from a velocity potential. In fact, it is easily demonstrated that

$$\phi(r,\theta) = -V_0\,r\,\cos(\theta-\theta_0).$$ (5.30)