An Immersed Boundary Method for Calculating the Relative Viscosity of a Suspension of Rigid Particles

In this paper, we provide a numerical framework to calculate the relative viscosity of a suspension of rigid particles. A high-resolution background grid is used to solve the flow around the particles. In order to generate infinite number of particles in the suspension, a particle is placed in the center of a cubic cell and periodic boundary conditions are imposed in two directions. The flow around the particle is solved using the second-order accurate curvilinear immersed boundary (CURVIB) method [1]. The particle is discretized with triangular elements, and is treated as a sharp interface immersed boundary by reconstructing the velocities on the fluid nodes adjacent to interface using a quadratic interpolation method. Hydrodynamic torque on the particle has been calculated, to solve the equation of motion for the particle and obtain its angular velocity. Finally, relative viscosity of the suspension has been calculated based on two different methods: (1) the rate of the energy consumption and (2) bulk stress-bulk strain method. The framework has been validated by simulating a suspension of spheres, and comparing the numerical results with the corresponding analytical ones. Very good agreement has been observed between the analytical and the calculated relative viscosities using both methods. This framework is then used to model a suspension with arbitrary complex particles, which demonstrates the effect of shape on the effective viscosity.