Terminal Velocity for Intermediate Value of Re_p

Solving for the transient velocity profile for intermediate values of Re, equa- tions proposed by Putnam, Schiller-Naumann and similar have to be incor- porated and solved accordingly and incorporating them yields 

$\rho_pV_p\frac{du}{dt}=V_pg(\rho_p-\rho_f)-/frac{1}{2}\pi r^2\rho_f(C_1u+C2u^k)$

which by introducing the constants a,b and c yields

$a=\frac{V_pg(\rho_p-\rho_f)}{\rho_pV_p}$

$b=\frac{\frac{1}{2}\pi r^2\rho_fC_1}{\rho_pV_p}$

$c=\frac{\frac{1}{2}\pi r^2 \rho_f C_2}{\rho_pV_p}$

$\frac{du}{dt}=a-bu-cu^k$

to solve with respect to a varying velocity , a solution for the integral is required. Hence a method for generally approximating the integral with respect to u is proposed.

$\int\frac{du}{a-bu-cu^k}$

This integral is to be solved in the open interval for which u varies , where u=z>0 . 

Solving & approximation of the integral.Having proved that the approximation is reasoable, and processding with calculating the integral by susbituing the approximation into the integral, yields the following

$\int \frac{du}{a-bu-cu^k}$

This integral is trivial to solve and when computed yields

$\frac{A}{2(u_{top}-z}ln(\frac{u-2u_{top}+z}{w-z})+Bu+C$

​where C is a constant of integration.