A model for the dissipation rate in linear unsteady flow through porous media

We present a model for the volume-averaged dissipation rate in linear unsteady flow through porous media. The model is derived by blending a new small-time asymptotic expression for the dissipation rate obtained from boundary layer theory with the known large-time asymptotic expression obtained from Darcy's law. The resulting model is a second-order Volterra functional of the volume-averaged acceleration. We validate the model with an analytical solution for transient flow through a porous medium composed of circular tubes and with numerical simulations of transient and oscillatory flow through a cylinder array and through a hexagonal sphere pack.