The Apparent Stress‐Deformation Behavior of a Dilute Suspension of Spheres in a Power Model Fluid

Einstein computed the apparent viscosity of a dilute suspension of neutrally buoyant spheres in an incompressible Newtonian fluid. The objective here is to determine the apparent stress‐deformation behavior of a dilute suspension of neutrally buoyant spheres in an incompressible power model fluid. With the assumption that $η=mγn−1$ describes the apparent viscosity of the continuous phase, our analysis predicts $η(sus)=(1+c0x)mγn−1$ as the apparent viscosity of the dilute suspension of spheres. Here γ is the shear rate, m and n are the power model parameters for the continuous phase, and x is the volume fraction of spheres in the suspension. The coefficient $c0$ is defined by the rate of energy dissipation within the neighborhood of a typical sphere in the suspension. In order to evaluate this rate of energy dissipation, the velocity distribution is required. In the limit $n=1,$ corresponding to an incompressible Newtonian fluid, Einstein was able to solve the equations of motion and found $c0=2.5.$ For $n≠1,$ the equations of motion are nonlinear and have not yet been integrated. Instead, we have used previously available principles to calculate upper and lower bounds for the rate of energy dissipation within the neighborhood for a single sphere. This results in upper and lower bounds for $c0$ as a function of n. The average of these bounds for $c0$ is compared with available experimental data.