Effect of Confinement-Geometry On the Suspension-Rheology and the Pressure-Drop Inside a Conduit

Hydrodynamic interactions between a confining boundary and a suspended or deposited particle affect the dynamics of a number of colloidal systems. For example, a fixed particle deposited on the wall of a cylindrical conduit enhances the flow-resistance in the channel by inducing additional pressure-drop. Similarly, required pressure-gradient to drive a flow through a duct increases in presence of suspended particles leading to an increase in effective viscosity which is greater than the nominal viscosity of the solvent. In this talk, we present our results which provide a unified description of these phenomena.

In our analysis, we consider a sphere in a cylinder under creeping motion assumption. The key step of the mathematical formulation is a semianalytical approach where the Stokesian field is expanded in terms of two sets of basis functions corresponding to the particle and the conduit geometry, respectively. The obtained flow-solution is used in determination of far-field pressure governing either the pressure-drop across a fixed blockage or the rheology of the suspension of freely moving particles. Furthermore, we use a novel lubrication theory to independently verify these analytical results.

Our results include the following components. First, we consider a spherical obstruction inside a conduit and determine the pressure-drop across it in presence of a parabolic flow quantifying the increase in resistivity of the channel. Then, we account for the pressure generated by an axially translating as well as azimuthally rotating sphere in static fluid inside the cylinder. Finally, we combine these findings to compute the pressure-field across a freely translating and rotating particle in parabolic flow. We relate the last result to the effective viscosity of a dilute suspension, and describe the effect of the confinement on the rheology. Such geometric effect is absent in an unbounded domain where the viscosity of the solution is expressed by Einstein's relation. We have checked that under the limit of very large conduit-diameter our calculation approaches to the Einstein's result.