Consider steady, incompressible, two-dimensional flow due to a line source at the origin (Fig. P976). Fluid is

Question:

Consider steady, incompressible, two-dimensional flow due to a line source at the origin (Fig. P9–76). Fluid is created at the origin and spreads out radially in all directions in the xy-plane. The net volume flow rate of created fluid per unit width is V̇/L (into the page of Fig. P9–76), where L is the width of the line source into the page in Fig. P9–76. Since mass must be conserved everywhere except at the origin (a singular point), the volume flow rate per unit width through a circle of any radius r must also be V̇/L. If we (arbitrarily) specify stream function ψ to be zero along the positive x-axis (u = 0), what is the value of ψ along the positive y-axis (θ = 90°)? What is the value of ψ along the negative x-axis (θ = 180°)?

FIGURE P9–76