New exact solutions for microscale gas flows

Abstract

We present a number of exact solutions to the linearised Grad equations for non-equilibrium rarefied gas flows and heat flows. The solutions include the flow and pressure fields associated to a point force placed in a rarefied gas flow close to a no-slip boundary and the temperature field for a point heat source placed in a heat flow close to a temperature jump boundary. We also derive the solution of the unsteady Grad equations in one dimension with a time-dependent point heat source term and the Grad analogue of the rotlet, a well-known singularity of Stokes flow which corresponds to a point torque.

Neumann and Dirichlet boundary conditions for the one-dimensional wave equation

“…In terms of boundary value problems, the mathematical treatment is slightly involved, but we can set up the Dirichlet boundary value problem where a wave travels along a line and achieves a value of zero at the fixed end point [2]. In fact, if one assumes a kind of 'formal reflection' property of the solution to the problem, the domain is extended to the line which goes beyond the fixed point where the string ends (this is similar to the method of images, where one extends the domain in which the PDE holds beyond the boundary into a 'fictitious region') [4]. The initial displacement splits into a right-moving wave and a left-moving wave, with the wave which travels to the left being reflected back from the fixed point with opposite phase.…”