Determine the stream function for a forced vortex. A forced vortex is obtained by rotating the fluid as a whole. The flow is characterised by the equation v t = – ω r and v r = 0 (– sign for clockwise...

Determine the stream function for a forced vortex.

A forced vortex is obtained by rotating the fluid as a whole. The flow is characterised by the equation v_t
= – ω_r
and v_r
= 0 (– sign for clockwise vortex)

taking ψ = 0 for the stream line at r = 0, C will vanish and so ψ = ωr2/2

An important aspect of the flow is that the flow is rotational

This can be shown by considering an element in the flow as shown in Fig. P.5.3 and calculating the circulation and then vorticity

d Γ = – ω (r + dr) (r + dr) d θ + 0 + ω r rd θ + 0 = – 2 ω rd rd θ

as vorticity = d Γ/area and as area = r dθ dr

Vorticity = – 2 ω . Hence flow is rotational. The vorticity is directly related to the angular velocity of the mass.