Stochastic approach to the Cauchy problem of linearized Couette flow
In the present paper it is shown that, assuming the stochastic process of particle diffusion in plane Couette flow and furthermore making use of the modified Chapman-Kolmogorov equation, the source function is readily evaluated by solving the Cauchy system of a nonlinear integrodifferential equation, whose form is similar to the Kolmogorov-Feller equation. The initial value is expressed in terms of the scattering function, whose numerical value can be computed by solving a Riccati-type nonlinear integrodifferential equation. Once the source function has been determined, we can compute successively the one-dimensional perturbation velocity distribution function, mass density, flow velocity, temperature, and other quantities. It is thus shown how such significant physical quantities in the linearized plane Couette flow can be readily evaluated by computing the lower-order moments of the distribution functions.