Non-steady pressure-driven flow of a Bingham fluid through a channel filled with a Darcy–Brinkman medium

An analytical solution for a one-dimensional non-steady pressure-driven flow of a Bingham fluid in a channel filled with a uniform high-porosity porous medium is derived. The porous medium in the channel is described by the Darcy–Brinkman model. The formulation is made dimensionless. Two time scales are introduced, i.e., a viscosity-related time scale and a permeability-related one. Solutions are fully determined by two dimensionless parameters, i.e., a fluid parameter and a permeability parameter. An analytical solution procedure is presented for the general non-steady case as well as for the steady state. Asymptotic steady solutions are found. Start-up flow is considered and evaluated in more detail. The effects of porous medium on the flow are specifically investigated. How velocity decreases with decreasing permeability of the porous medium is specified. The start-up flow evolves in time monotonously into a steady state. The time in which this state is reached is shorter for media of lower permeability. A simple hand-on formula for the total flow rate is derived and compared with the exact solution; good correspondence is found.