kar : The Von Karman Swirling Flows.

The steady axi-symmetric flow of a viscous incompressible fluid above an infinite rotating disk is modeled by the following ODE boundary value problem (Equation (11) in LeKe:80 LeKe:80 :

$$\begin{array}{cl} u_1' &= T u_2, \\ u_2' &= T u_3, \\ u_3' &=... ...gl[ 2 \gamma u_2 + 2 u_2 u_4 - 2 u_1 u_5 \bigr], \\ \end{array}$$ (15.7)

with left boundary conditions

$$u_1(0)=0, \qquad u_2(0)=0, \qquad u_4(0)=1-\gamma,$$

and (asymptotic) right boundary conditions

$$\begin{array}{cl} & \bigl[ f_\infty + a(f_\infty,\gamma) \big... ...bigr] ~u_4(1) + u_5(1) = 0, \\ & u_1(1) = f_\infty, \end{array}$$ (15.8)

where

$$\begin{array}{cl} & a(f_\infty,\gamma) = \frac{1 }{ \sqrt{2} ... ...4 + 4 \gamma^2)^{1/2} - f_\infty^2 \bigr]^{1/2}. \\ \end{array}$$ (15.9)

Note that there are five differential equations and six boundary conditions. Correspondingly, there are two free parameters in the computation of a solution family, namely $\gamma$ and $f_\infty$. The ``period'' $T$ is fixed; $T=500$. The starting solution is $u_i=0$, $i=1,\cdots,5$, at $\gamma=1$, $f_\infty=0$.

Table 15.6: Commands for running demo kar.

AUTO -COMMAND	ACTION
! mkdir kar	create an empty work directory
cd kar	change directory
demo('kar')	copy the demo files to the work directory
run(c='kar.1')	computation of the solution family
sv('kar')	save output-files as b.kar, s.kar, d.kar