  We perform numerical investigation of nearly self-similar blowup of
generalized axisymmetric Navier-Stokes equations and Boussinesq system with a
time-dependent fractional dimension. The dynamic change of the space dimension
is proportional to the ratio R(t)/Z(t), where (R(t),Z(t)) is the position at
which the maximum vorticity achieves its global maximum. This choice of space
dimension is to ensure that the advection along the r-direction has the same
scaling as that along the z-direction, thus preventing formation of two-scale
solution structure. For the generalized axisymmetric Navier-Stokes equations
with solution dependent viscosity, we show that the solution develops a
self-similar blowup with dimension equal to 3.188 and the self-similar profile
satisfies the axisymmetric Navier-Stokes equations with constant viscosity. We
also study the nearly self-similar blowup of the axisymmetric Boussinesq system
with constant viscosity. The generalized axisymmetric Boussinesq system
preserves almost all the known properties of the 3D Navier-Stokes equations
except for the conservation of angular momentum. We present convincing
numerical evidence that the generalized axisymmetric Boussinesq system develops
a stable nearly self-similar blowup solution with maximum vorticity increased
by O(10^{30}).
