Session G27: Non-Newtonian Flows: Theory and Analysis

A viscoelastic model with thixotropic yield stress behavior for filament stretching

The transient behavior of filament stretching is studied for a viscoelastic constitutive model that combines a Partially Extending strand Convection model with a Newtonian solvent. The vertical filament is fixed at the bottom and the top is pulled up and held. Gravity and surface tension are also included in the model though they are not the primary mechanisms in this study. An axisymmetric circular slender jet approximation is applied. An asymptotic analysis for the initial stages of evolution is performed for large relaxation time, so that an interplay of fast and slow time scales emerges, and gives a criterion for whether the fluid yields immediately or whether slow dynamics ensues, depending on elastic stresses, gravity and capillary stress.   

Advection of nematic liquid crystals by chaotic flow

Consideration is given to the effects of inhomogeneous shear flow (both regular and chaotic) on nematic liquid crystals in a planar two-dimensional geometry. The Landau-de Gennes equation coupled to an externally-prescribed flow field is the basis for the study: this is solved numerically in a periodic spatial domain. The focus is on a limiting case where the advection is passive, such that variations in the liquid-crystal properties do not feed back into the equation of motion for the fluid velocity. The numerical simulations demonstrate that the coarsening of the liquid-crystal domains is arrested by the flow. The nature of the arrest is different depending on whether the flow is regular or chaotic. For the specific case where tumbling is important, the flow has a strong effect on the the liquid-crystal morphology: this provides a mechanism for controlling the shape of the liquid-crystal domains.   

Inertial migration of elastic particles in a pressure-driven power-law fluid

Using three-dimensional computer simulations, we study the cross-stream migration of deformable particles in a channel filled with a non-Newtonian fluid driven by a pressure gradient. Our numerical approach integrates lattice Boltzmann method and lattice spring method in order to model fluid structural interactions of the elastic particle and the surrounding power fluid in the channel. The particles are modeled as elastic shells filled with a viscous fluid that are initially spherical. We focus on the regimes where the inertial effects cannot be neglected and cause cross-stream drift of particles. We probe the flow with different power law indexes including both the shear thickening and thinning fluids. We also examine migration of particles of with different elasticity and relative size. To isolate the non-Newtonian effects on particle migration, we compare the results with the inertial migration results found in the case where the channel is filled with a simple Newtonian fluid. The results can be useful for applications requiring high throughput separation, sorting, and focusing of both synthetic particles and biological cells in microfluidic devices.   

Viscous-elastic dynamics of power-law fluids within an elastic cylinder

We study the fluid-structure interaction dynamics of non-Newtonian flow through a slender linearly elastic cylinder at the creeping flow regime. Specifically, considering power-law fluids and applying the thin shell approximation for the elastic cylinder, we obtain a non-homogeneous p-Laplacian equation governing the viscous-elastic dynamics. We obtain exact solutions for the pressure and deformation fields for various initial and boundary conditions, for both shear thinning and shear thickening fluids. In particular, impulse or a step in inlet pressure yield self-similar solutions, which exhibit a compactly supported propagation front solely for shear thinning fluids. Applying asymptotic expansions, we provide approximations for weakly non-Newtonian behavior showing good agreement with the exact solutions sufficiently far from the front.   

Viscoplastic boundary layers

Viscoplastic fluids are characterized by a yield stress, below which they do not deform. If the yield stress is large, viscoplastic flows can can develop narrow boundary layers that provide surfaces of failure between rigid or almost rigid regions, or between such regions and rigid boundaries. Oldroyd (1947) presented a theoretical discussion of these viscoplastic boundary layers, but they have been largely ignored since then, in part because of the complexity of the nonlinear boundary-layer equations. We revisit Oldroyd’s analysis, and consider various examples of flow, including a jet-like intrusion, flow past a thin plate, and flow down channels with topography. By comparison with detailed numerical solutions, we verify Oldroyd’s original theory, and also reveal its shortcomings. Where these exist, we present an alternative theory more akin to classical lubrication solutions. We also relate these viscoplastic flow solutions to slipline constructions in classical plasticity theory.   

Dynamics of two disks settling in a two-dimensional narrow channel: From periodic motion to vertical chain in Oldroyd-B fluid

In this talk we present a numerical study of the dynamics of two disks settling in a narrow vertical channel filled with an Oldroyd-B fluid. Two kinds of particle dynamics are obtained: (i) periodic interaction between two disks and (ii) the formation of the chain of two disks. For the periodic interaction of two disks, two different motions are obtained: (a) two disks stay far apart and interact is periodically, which is similar to one of the motions of two disks settling in a narrow channel filled with a Newtonian fluid discussed by Aidun & Ding (Phys. Fluids, 15 (2003), 1612) and (b) two disks draft, kiss and break away periodically and the chain is not formed due to not strong enough elastic force. For the formation of two disk chain occurred at higher values of the elasticity number, it is either a tilted chain or a vertical chain. The tilted chain can be obtained for either that the elasticity number is less than the critical value for having the vertical chain or that the Mach number is greater than the critical value for a long body to fall broadside-on, which is consistent with the results for the elliptic particles settling in Oldroyd-B fluids.   

Effect of the Convected Terms in the Transient Viscoelastic Flow

Influence of fluid elasticity is examined for the plane Couette flow (PCF) of an improved Johnson Segalman (J.S) fluid through introduction of coefficients in the convected terms.The flow field is obtained from the conservation and constitutive equations using the Galerkin projection method. Effect of several values of governing parameters such as introduced coefficients, Reynolds number and Weissenberg number on velocity and normal and shear stresses profiles are explored. The results show that the oscillating behavior of velocity profile tends to grow as the coefficients increase. For higher Wiessenberg, the oscillations are more intensive, whereas the amplitude of oscillation tends to reduce. This reveals that, the deviation decreases by increasing the coefficients. The amplitude of normal stress differences tend to grow as the coefficients of the convected terms grow, revealing more elastic behavior in the fluid. On the other hand; the effect of the convected terms on the steady behavior of normal stress difference is strongly dependent on the value of Weissenberg number. The shear stress behavior is also dependent on the coefficients of the convected terms and the flow properties, that is, for higher Reynolds the shear stress reaches a maximum and then decreases to minimum. For lower Reynolds, the opposite occurs.