Flow of yield-stress fluids through channels

Abstract

We propose a new way to study the dynamics of glass-forming colloidal suspensions by successfully combining mode-coupling theory (MCT) with a Lattice Boltzmann (LB) scheme. The mode-coupling theory provides a constitutive equation which is fully determined by the microscopic interactions of the system. We use a schematic MCT model, which incorporates the essential features of the full model, to close the Navier-Stokes (NS) equation. The LB method presents an elegant and computational efficient way to find a solution for the NS equation even for complex flow problems. We propose a modified LB algorithm capable of handling tensorial and integral constitutive equations. Combining a mesoscopic flow solver with microscopic dynamics offers some significant advantages. We are now able to study complex flow problems which would be inaccessible when using microscopic approaches such as Molecular Dynamics simulations.To test our method, we have studied the pressure-driven Poiseuille flow through a straight channel. Further approximations to theschematic MCT (sMCT) model yield a shear-thinning constitutive equation similar to Maxwell's well-known model of viscoelastic fluids. We have implemented this non-linear Maxwell (nlM) model in its integral formulation and in its differential form (inlM), which assumes a fully developed flow. The inlM model adjusts instantaneously to changes in the flow and is similar to other shear-thinning constitutive equations often used in LB simulations. Under the reasonable constraint of an incompressible flow, the steady state channel flow profile of the non-linear Maxwell model can be solved analytically.The LB results agree very well with the theoretical predictions. In a channel, the shear-thinning fluid forms a no-shear plug in the center and high-shear regions near the walls. Normal stresses working on the plug are balanced by a pressure gradient. The non-linear Maxwell model shows glass-like dynamics already for moderate large drops in the viscosity. For cessation flows, this makes the inlM fluid show nearly finite stopping times hinting at a yield stress, which exists only in the glass limit.In the nlM model, the stress takes the flow history into account, and viscoelastic effects enter the transient dynamics. The intriguing interplay between the evolution of the stress and velocity of the fluid presents an interesting field of study. The velocity can vary more rapidly than the stress, and overshoots and oscillations appear in the transient profiles. For the stopping flow, this is especially interesting as resident stresses make the stopped flow start moving again, but in opposite direction. Comparing with the linear Maxwell model, we could visualize the shear-thinning setting in. We have identified two different regimes and a characteristic time scale ton we can vary by changing the channel diameter. For narrow channels (small ton ), the starting flow exhibits overshoots in the velocity and stress. In wide channels, the velocity profile develops monotonously again. Increasing the channel diameter restores the scaling with ton on a single master curve, which is the same for the inlM model.Comparing the results of the non-linear Maxwell model with the dynamics of the schematic mode-coupling model, we find the nlM model to capture the dynamics of the velocity and shear stress exceptionally well. The steady state profiles are nearly indistinguishable, and we find the same qualitative change for different channel diameters in the starting flow of the sMCT model. The cessation dynamics are more complex. The correlator becomes non-monotonous due to the oscillating flow and affects the stopping flow after the initial undershoot. Thenormal stress profiles are different from the ones found for the non-linear Maxwell model, but the agreement should improve for simulationscloser to the glass transition.We have shown that the non-linear Maxwell model reproduces the dynamics of the schematic MCT model extremely well. It incorporates most features of the microscopic dynamics known from mode-coupling theory in a relatively simple constitutive equation and presents a good compromise between microscopic details and practicability. We have demonstrated how to use mode-coupling theory, if in a simplified model, to close the Navier-Stokes equation and apply it to flow problems. We have compared our results with theoretical predictions and found them in good agreement. Already in a pressure-driven channel flow, the dynamics are highly non-trivial and promise even more interesting physics for more complex flow geometries.