Session GN: Non-Newtonian Flows I

Analytical solution for creeping channel flow of non-Newtonian compressible fluid subject to wall slip

Creeping channel flows of compressible viscoplastic fluids subject to wall slip are important in many industries as well as presenting significant academic challenges. Here we present analytical solutions for pressure-driven steady flows of viscoplastic fluids within planar and circular channels. Herschel-Bulkley constitutive equation is employed in conjunction with constant or pressure dependent wall slip coefficients. Simplifications of the Hershel-Bulkley fluid provide other generalized Newtonian fluids including power law and Newtonian fluids. Under the assumption that pressure only changes in the flow direction and its gradient deviates slightly from a constant, explicit solutions are derived for distributions of pressure, velocity, and slip velocity within the channels. The analytical solutions are compared against numerical solutions as well as experimental data collected using rectangular slit dies. The effects of compressibility and wall slip on the flows are elucidated. A distinctive feature of such a flow is that, when the slip coefficient is considered to be inversely proportional to pressure, the slip velocity increases rapidly in the flow direction and the flow can evolve into a pure plug flow at the exit, removing the stress singularity that is presumed to exist in the transition from the channel flow into a free surface flow at the exit.   

Transient growth without inertia

We study transient growth in inertialess plane Couette and Poiseuille flows of viscoelastic fluids. For streamwise-constant 3D fluctuations, we demonstrate analytically the existence of initial conditions that lead to quadratic scaling of both the kinetic energy density and the elastic energy with the Weissenberg number, $We$. This shows that in strongly elastic channel flows, both velocity and polymer stress fluctuations can exhibit significant transient growth even in the absence of inertia. Our analysis identifies the spatial structure of the initial conditions (i.e., components of the polymer stress tensor at $t = 0$) responsible for this large transient growth. Furthermore, we show that the fluctuations in streamwise velocity and the streamwise component of the polymer stress tensor achieve $O(We)$ and $O(We^2)$ growth, respectively, over a time scale $O (We)$ before eventual asymptotic decay. We also demonstrate that the large transient responses originate from the stretching of polymer stress fluctuations by a background shear and draw parallels between streamwise-constant inertial flows of Newtonian fluids and streamwise-constant creeping flows of viscoelastic fluids. One of the main messages of this work is that, at the level of velocity fluctuation dynamics, polymer stretching and the Weissenberg number in elasticity-dominated flows effectively assume the role of vortex tilting and the Reynolds number in inertia-dominated flows of Newtonian fluids.   

Extensional dynamics of viscoplastic filaments

A one-dimensional slender-thread model is used to explore viscoplastic dripping under gravity and the controlled extension of a liquid bridge. We describe dynamics up to pinch-off and consider the possibility of using measurements in the two configurations (eg. drop volume) to infer rheological parameters. The model results are compared with experiments using aqueous solutions of Carbopol and kaolin suspensions.   

Multiscale simulation of polymer melt flows between parallel plates

The behaviors of polymer melt composed of short chains with ten beads in parallel plates are simulated by using a hybrid method of molecular dynamics and computational fluid dynamics. The creep motion under a constant shear stress and recovery after removing the stress, the pressure driven flows and the flows in rapidly oscillating plates are simulated. The flow profiles of polymer melt are quite different from those of the Newtonian fluid due to the elasticity or the shear thinning. The delayed elastic deformation and plug-like velocity profile are reproduced, respectively, in the creep and pressure driven flow. In the rapidly oscillating plates the viscous boundary layer of the melt is much thinner than that of the Newtonian fluid due to the shear thinning of the melt. Three different rheological regimes, i.e., the viscous fluid, viscoelastic liquid, and viscoelastic solid regimes, form over the oscillating plate according to the local Deborah numbers. The melt behaves as a viscous fluid in a region for $\omega\tau^R\la 1$, and the crossover between the liquid-like and solid-like regime takes place around $\omega\tau^{\alpha}\simeq 1$ (where $\omega$ is the angular frequency of the plate and $\tau^R$ and $\tau^{\alpha}$ are Rouse and $\alpha$ relaxation time, respectively).   

Experimental study of the hydrodynamic interaction between a pair of bubbles ascending in a non-Newtonian liquid

We present some experimental results about the interaction of a pair of bubbles ascending in non-Newtonian fluids. A high speed camera was used to follow in-line and off-line rising motion of two bubbles in a Newtonian fluid (a glycerin-water solution), a Boger fluid (aqueous polyacrylamide solution), and a shear-thinning fluid (aqueous xanthan solution). For the case of shear-thinning fluids, the power index, n, affects the tendency of the bubble pair to aggregate. Therefore, in addition to bubble separation, orientation and Reynolds number, the hydrodynamic force depends strongly on the shear-thinning nature of the fluid. Several examples will be shown. For elastic fluids, the Deborah number affects the hydrodynamic interaction. We found that the appearance of the negative wake changes the nature of the interaction substantially. Some examples and comparisons with numerical results will be presented.   

A numerical study of the hydrodynamic interaction of bubble pairs ascending in non-Newtonian liquids

This talk presents computational results on the interaction of a pair of bubbles immersed in non-Newtonian fluids. The Arbitrary Lagrangian-Eulerian (ALE) technique was used to simulate two bubbles rising in tandem or side by side in shear-thinning and Oldroyd-B fluids. In the shear-thinning fluid, the pairwise interaction is affected by the the Eotvos and Reynolds numbers as well as the initial orientation of two bubbles. In particular, two in-line bubbles will rise together and form a doublet as the trailing bubble catches up with the leading one. In a viscoelastic fluid, a negative wake may appear depending on the initial separation between the bubbles. The capillary number, which can be an indicator of the bubble deformability, seems to play a secondary role in the bubble interaction. The numerical simulations complement previous experiments done with bubble swarms by our group.   

Selective withdrawal of non-Newtonian fluids: surface deformation induced by a sink flow

This talk reports experiment and numerical studies of selective withdrawal in a fluid-gas system. Using visual observation and finite element simulations based on an Arbitrary Lagrangian-Eulerian scheme, we have explored the effects of viscoelasticity on the deformation of free surface when the fluid is polymer solution (experiment) or Giesekus fluid (simulation). In the experiments, we find a thin air jet emanating from the tip of the free surface for polymer solutions when the distance between the free surface and the sink is below a critical value. This does not occur for the free surface of Newtonian liquids, and is caused by the additional elongational stress due to the polymer. In the simulations, the effects of elasticity on the surface deformation have been captured. The balance between surface and viscoelastic forces may potential be used for measuring extensional viscosity.   

Active and hibernating turbulence in minimal channel flow of Newtonian and polymeric fluids

The experimental observation of minute amount of polymers reducing turbulent drag has been long established. In this study, we isolate the turbulent self-sustaining process by conducting direct numerical simulations (DNS) in minimal flow units (MFU). These solutions obtained at various polymer parameters recover all key transitions in viscoelastic turbulent flows reported previously in experiments at much higher Re, including the onset of drag reduction, low degree of DR (LDR), high degree of DR (HDR) and maximum drag reduction (MDR). At MDR, the mean velocity profile is insensitive to changing polymer parameters. The LDR-HDR transition is characterized by a sudden increase in the minimal box size of sustaining turbulence, which may correspond to a qualitative change in the self-sustaining mechanism. Dynamics of turbulence show intermittent appearance of ``hibernation'' periods, which are characterized by long-lasting flow structures with low instantaneous wall shear stress and low turbulence intensity. These periods appear both in Newtonian and viscoelastic fluids; however they are observed much more frequently in HDR and MDR stages, which contribute substantially to the relatively high level of DR. Instantaneous velocity profiles during hibernation periods resemble the Virk MDR profile, including the disappearance of the log-law layer and a comparable slope with the Virk MDR asymptote.   

Entangled chain dynamics of polymer knots in extensional flow

We formulate a coarse grained molecular dynamics model of polymer chains in solution that includes hydrodynamic interactions, thermal fluctuations, nonlinear elasticity, and topology-preserving solvent mediated excluded volume interactions. The latter involve a combination of potential forces with explicit geometric detection and tracking of chain entanglements. By solving this model with numerical and computational methods, we study the physics of polymer knots in strong extensional flow (Deborah number, $De=1.6$). We show that knots slow down the stretching of individual polymers by obstructing via entanglements the ``natural", unraveling, flow-induced chain motions. Moreover, the steady state polymer length and polymer-induced stress values are smaller in knotted chains than in topologically trivial chains. We indicate the molecular processes via which the rate of knot tightening affects the rheology of the solution.   

A Finite Volume Solver for Non-Newtonian flow on Unstructured Grid with Application in Blood Flow

In order to simulate blood flow in complex vessel, a finite volume solver for Casson fluid flow based on SIMPLE algorithm of Newtonian fluid on unstructured collocated grid is developed. For the discretization of convective fluxes and source term, it is similar with Newtonian fluid. For the discretization of diffusion fluxes, viscosity will take the value calculated from the flow field of previous iteration in order to avoid the complexity caused by the complicated viscosity expression as a function of shear rate. Then the discretization of momentum equation is similar with that of Newtonian fluid with variable viscosity and SIMPLE algorithm can be used to resolve the pressure-velocity coupling. Casson fluid flow through a symmetric sudden expansion channel is compared with literature and the good agreement between simulated velocity distributions with literature confirms the validation of present algorithm. Afterwards, blood flows in T-type bifurcation are simulated by our proposed algorithm. The simulation result of Casson fluid is more consistent with experiment than that of Newtonian fluid, which indicates that using Casson model to simulate on-Newtonian characteristics of blood is successful and necessary.