THE EFFECTIVE VISCOSITY OF A SUSPENSION OF POROUS PARTICLES BASED ON THE DARCY-STOKES COUPLING MODEL

Particle suspensions exist widely in nature and engineering applications, and their viscous characteristics have an important influence on their flow behavior. Based on the Darcy-Stokes coupling model, the analytical formulas of effective viscosity of dilute suspensions containing porous particles are derived in this paper. Firstly, an auxiliary problem is solved, that is, the disturbance caused by porous media spheres in the flow field with linear distribution under the condition of low Reynolds number. The fluid flows in the free-flow domain and porous medium are governed by the Stokes equation and Darcy’ law, respectively. The mass conservation law, the balance of normal forces, and the Beavers-Joseph (-Saffman) interface condition are used at the fluid–porous interface. An analytical solution for the present coupled free-flow and porous-medium system is derived by using the undetermined coefficient method. Then the additional heat dissipation rate caused by the porous media particle is calculated. Intrinsic viscosity of the porous media suspension under the condition of low concentration is determined as a function of the Darcy number and the Beavers-Joseph coefficient, which is based on the additional heat dissipation rate under the condition of low concentration. It is found that the intrinsic viscosity increases with increasing the Beavers-Joseph coefficient, and the larger the Beavers-Joseph coefficient is, the slower the increase of the intrinsic viscosity. When Darcy number is in the range of 10^ - 6 to 10^ - 4, the intrinsic viscosity is close to 2.5, which was consistent with the classical Einstein viscosity formula. When Darcy number is in the range of 10^ - 4 to 10^ - 1, the intrinsic viscosity decreases rapidly, so the effective viscosity coefficient of porous media suspension is closer to the viscosity of the based fluid. At last, the present effective viscosity formula is compared with that obtained by the Darcy-Brinkman equation coupling with the shear stress jump condition. It can be found that the effective viscosities obtained two different models agree well with each other in the low Darcy number regime when the sum of the Beavers-Joseph coefficient and the shear stress jump coefficient is unity.