Given V@ = (gamma/2*pi*r)*[1 - exp(-a^2r)] for a two-dimensional incompressible flow, determine the radial velocity Vr.
Work:
Continuity equation in cylindrical form:
(1/r)*d/dr(r*Vr)+(1/r)*d/d@(V@) = 0, where d/dr and d/d@ are partial derivatives and @ = theta
Subtituting the given V@ into the continuity equation:
(1/r)*d/dr(r*Vr)+(1/r)*d/d@{(gamma/2*pi*r)*[1-exp(-a^2r)]} = 0
Term 1: (1/r)*d/dr(r*Vr) = (r^-1)*Vr
Term 2: (1/r)*d/d@{gamma/2*pi*r)*[1-exp(-a^2r)]} = 0
r^-1*Vr = 0
Integrating both sides:
Vr*r = r
Vr = r/r = 1
Vr = r/r = 1