Diffusion in an elliptical cylinder, and a numerical method for calculating time-varying diffusant sink rates, with special reference to diffusion of oxygen in the frog sartorius muscle (Q761375)

For a substance whose distribution F(x,y,t) within an elliptical cylinder is governed by the diffusion equation, a formal description of the time course of this distribution following a step change at the boundary was given by \textit{N. W. McLachlan} in ''Heat conduction in elliptical cylinder and an analogous electromagnetic problem.'', Philos. Mag., VII. Ser. 36, 600-609 (1945); it is a doubly infinite series of exponentials. Computer programs have been written to evaluate these terms, each of which requires frequent evaluation of the standard and modified Mathieu functions. The exponent of the leading term, together with measurements of F(x,y,t) at a given \((x_ 0,y_ 0,t)\), allows the determination of the diffusion coefficient D. These exponents are tabulated for boundaries spanning the range from a cylinder to a sheet. If the diffusant is produced or consumed at uniformly distributed sites, at the rate Q(t), evaluation of the above solution allows the computation of Q(t) from \(F(x_ 0,y_ 0,t)\). This was investigated in detail for conditions chosen to simulate the diffusion and consumption of oxygen in the frog sartorius muscle. Ten terms of the solution, when used with a convergence-speeding routine, were sufficient for an accurate approximation to Q(t).