Internally and externally driven flows of complex fluids: viscoelastic active matter, flows in porous media and contact line dynamics

Abstract

We consider three varied soft matter topics from a continuum fluid mechanics perspective, namely: viscoelastic active matter, viscoelastic flows in porous media, and contact line dynamics.

Active matter. For the purposes of this thesis, the term active matter describes a collection of active particles which absorb energy from their local environment or from an internal fuel tank and dissipate it to the surrounding fluid. We explore the stability and dynamics of active matter in a biological context in the presence of a polymeric background fluid. Using a novel coarse-grained model, we generalise earlier linear stability analyses (without polymer) and demonstrate that the bulk orientationally ordered phase remains intrinsically unstable to spontaneous flow instabilities. This instability remains even as one takes an ’elastomeric limit’ in which the polymer relaxation time τC → ∞. The 1D nonlinear dynamics in this limit are oscillatory on a timescale set by the rate of active forcing.
Then, by considering the rheological response of our model under shear, we explore the mechanism behind the above generic flow instability, which we show exists not only for orientationally ordered phases but also for disordered states deep in the isotropic phase. Our linear stability analysis in 1D for sheared suspensions predicts that initially homogeneous states represented by negatively sloping regions of the constitutive curve are unstable to shear-banding flow instabilities. In some cases, the shear-bands themselves are unstable which leads to a secondary instability that produces rheochaotic flow states. Consistent with recent experiments on active cellular extracts (without applied shear) which show apparently chaotic flow states, we find that the dynamics of active matter are significantly more complex in 2D. Focusing on the turbulent phase that occurs when the activity ζ (or energy input) is large, we show that the characteristic lengthscale of structure in the fluid l∗ scales as l∗ ∝ 1/ √ζ. While this lengthscale decreases with ζ, it also increases with the polymer relaxation time. This can produce a novel ‘drag reduction’ effect in confined geometries where the system forms more coherent flow states, characterised by net material transport. In the elastomeric limit spontaneous flows may still occur, though these appear to be transient in nature. Examples of exotic states that arise when the polymer is strongly coupled to the active particles are also given.

Flows in porous media. The second topic treats viscoelastic flows in porous media, which we approximate numerically using geometries consisting of periodic arrays of cylinders. Experimentally, the normalised drag χ (i.e., the ratio of the pressure drop to the flow rate) is observed to undergo a large increase as the Weissenberg number We (which describes the ratio of the polymer relaxation time to the characteristic velocity-gradient timescale) is increased. An analysis of steady flow in the Newtonian limit identifies regions dominated by shear and extension; these are mapped to the rheological behaviour of several popular models for polymer viscoelasticity in simple viscometric protocols, allowing us to study and influence the upturn in the drag. We also attempt to reproduce a recent study in the literature which reported fluctuations for cylinders confined to a channel at high We. At low numerical resolution, we observe fluctuations which increase in magnitude with the same scaling observed in that study. However, these disappear at very high resolutions, suggesting that numerical convergence was not properly obtained by the earlier authors.

Contact line dynamics. We finish by investigating the dynamics of the contact line, i.e., the point at which a fluid-fluid interface meets a solid surface, under an externally applied shear flow. The contact line moves relative to the wall, apparently contradicting the conventional no-slip boundary conditions employed in continuum fluid dynamics. A mechanism where material is transported within a ‘slip region’ via diffusive processes resolves this paradox, though the question of how the size of this region (i.e., slip length ξ) scales with fluid properties such as the viscosity η and the width of the interface between phases l, remains disputed within the literature. We reconcile two apparently contradictory scalings, which are shown to describe different limits: (a) a diffuse interface limit where ξ/l is small and (b) a sharp interface limit for large ξ/l. We demonstrate that the physics of the latter (which more closely resembles real fluids in macroscopic experimental geometries) can be captured using simulations in the former regime (which are numerically more accessible).