Smoluchowski Theory of Microstructure for Concentrated Sheared Colloidal Suspensions

This work seeks to develop a theory for predicting microstructure of colloidal suspensions as a function of dimensionless quantities: Pe =6¹ηγ^.a³/k_bT and particle volume fraction, ϕ, where γ^. is the shear rate, a is the particle radius, η is the viscosity of the fluid matrix and k_bT is the thermal energy. This problem was pursued by conditionally averaging probability distribution function conservation equation (Smoluchowski equation) for two particles. Many-body effects in the averaged hydrodynamic and thermodynamic coefficients of the conservation equation were then formulated self-consistently through probabilistic third-particle integrals.  The resulting differential-integral conservation equation was solved using an iterative method. Comparison between the theory predictions and simulation results showed that the theory was able to predict the well-known near equilibrium (Pe << 1) and dilute suspensions (ϕ <<1) results. In addition this theory was capable of predicting the details of microstructure at Pe >1 and concentrated regime, which differentiates it from the previous theoretical works in the field. The important rheological quantities such as shear stress, τ, first and second normal stress differences, N₁ and N₂ were calculated based on the obtained microstructure and were compared with simulation results. The results of this work suggest that the proposed self-consistent approach is able to capture the many-body hydrodynamic interactions in colloidal suspensions.