Computer extended series for a thermally driven gas centrifuge

Linearized, multidimensional, thermally driven flow in a gas centrifuge can be approximately described in regions away from the ends by Onsager's homogeneous pancake equation.¹ Upon reformulation of the general problem, we find a new, simple and rigorous closed form, analytical solution by assuming a special separable solution and replacing the usual Ekman end cap boundary conditions with idealized impermeable, free slip boundary conditions. Then the flow may be described by an ordinary differential equation with solutions in terms of simple, classical functions. By identifying a small parameter, say ?, defining the semi-long bowl approximation, and assuming a power series expansion in ?, a sequence of asymptotic approximations to the master potential is obtained. Not surprisingly, the leading order term involves the well known ‘long bowl’ solution. Using the so-called ‘solving’ property of the 1-D pancake Green's function,² we determine the next higher order solution. This recursive process is carried out on the computer to find all the terms up to O(?⁴). Consequently, the solution of some complex rotating, viscous, heat conducting flow problems that normally require large mainframe computers can be better understood.