Diffusion in random networks

The ensemble averaging technique is applied to model mass transport by diffusion in random networks. The system consists of an ensemble of random networks, where each network is made of pockets connected by tortuous channels. Inside a channel, fluid transport is assumed to be governed by the one-dimensional diffusion equation. Mass balance leads to an integro-differential equation for the pocket mass density. The so-called dual-porosity model is found to be equivalent to the leading order approximation of the integration kernel when the diffusion time scale inside the channels is small compared to the macroscopic time scale. As a test problem, we consider the one-dimensional mass diffusion in a semi-infinite domain. Because of the required time to establish the linear concentration profile inside a channel, for early times the similarity variable is xt−1/4 rather than xt−1/2 as in the traditional theory. This early time similarity can be explained by random walk theory through the network

Data for Efficient simulation of rarefied gas flow past a particle: A boundary element method for the linearized G13 equations

We develop a novel boundary integral formulation for the steady linearized form of Grad's 13- moment (G13) equations applied to uniform flow of rarefied gas past solid objects at low Mach numbers. Changing variables leads to a system of boundary integral equations that combines integral equations from Stokes ow and potential theory. The strong coupling between the stress deviator and heat flux featured by the G13 equations demands adding a boundary integral equation for the pressure. We specialize the integral equations for axisymmetric flow with no swirl and derive the axisymmetric fundamental solutions for the pressure equation, seemingly absent in the Stokes- flow literature. Using the boundary element method to achieve a numerical solution, we apply this formulation to streaming ow of rarefied gas past prolate or oblate spheroids with their axis of symmetry parallel to the free stream, considering various aspect ratios and Knudsen numbers- the ratio of the molecules' mean free path to the macroscopic length scale. After validating the method, we obtain the surface profiles of the deviations from the unperturbed state of the traction, heat flux, pressure, temperature, and slip velocity, as well as the drag on the spheroid, observing convergence with the number of elements. Rarefaction phenomena such as temperature jump and polarization, Knudsen effects in the drag, and velocity slippage are predicted. This method opens a new path for investigating other gas non-equilibrium phenomena that can be modelled by the same set of equations, such as thermophoresis, and has application in nano- and microfluidics