Radial pressure variations in spinning—M. Example 6.8 assumed that the variation of pressure across the filament was negligible. Investigate the validity of this assumption by starting with the...

Radial pressure variations in spinning—M. Example 6.8 assumed that the variation of pressure across the filament was negligible. Investigate the validity of this assumption by starting with the suitably simplified momentum balance: ∂p ∂r = μ ∂ ∂r 1 r ∂(rvr) ∂r + ∂2vr ∂z2 . If vz is the local axial velocity, prove that the corresponding increase of pressure from just inside the free surface (pR) to the centerline (p0) is: p0 − pR = μvz(ln β)3R2 4L3 , where β = vzL/vz0. Obtain an expression for the ratio ξ = (pR −p0)/(μdvz/dz), in which the denominator is the pressure decrease due to viscosity when crossing the interface from the air into the filament, and which was accounted for in Example 6.6. Estimate ξ at the beginning of the filament for the situation in which μ = 104 P, L = 1 m, RL (exit radius) = 0.0002 m, vz0 = 0.02 m/s, and vzL = 2 m/s. Comment briefly on your findings.