Comprehensive numerical computations are made of a homogenous spin-up in a cylindrical cavity with a time-dependent rotation rate. Numerical solutions are acquired to the governing axisymmetric cylindrical Navier-Stokes equations. A rotation rate formula is $\Omega_f$=$\Omega_i$+ΔΩ(1-exp(-t/$t_c$)). If &t_c& is large, it implies that a rotation change rate is small. The Ekman number, E, is set to $10^{-4}& and the aspect ratio, R/H, fixed to 1. For a linear spin-up(ε＜＜1), the major contributor to spin-up in the interior is not viscous-diffusion term but inviscid term, especially Coriolis term, though $t_c$ is very large. The viscous-diffusion term only works near sidewall. But for spin-up from rest, when $t_c$ is very large, viscous-diffusion term affects interior area as well as sidewall, initially. So azimuthal velocity of interior for large $t_c$ appears faster than that of inteior for relatively small $t_c$. However, the viscous-diffusion term of interior decreases as time increases. Instead, inviscid term appears in the interior.
