Session MT: Non-Newtonian Flows II

A particle-based model for yield-stress fluids

For soft materials that exhibit yielding, by far the most popular constitutive model is the Bingham model. Deceptively simple, such yield-stress models introduce a mathematical singularity into the problem, which creates great difficulties for flow simulations. The actual physics of yielding is specific to the microstructure of the material, and differs for gels and colloids, say. But it certainly contains no singularity. Inspired by this observation, we propose a mesoscopic model that represents the microstructures by particles. They interact via a potential force as well as a frictional force. We model the former by simple elastic springs that fail at a maximum forcing, and the latter using ideas from granular materials. The motions of the particles are computed using a procedure similar to smoothed particle hydrodynamics. This model predicts yielding in simple flows that agree with the picture based on a macroscopic yield stress. When applied to complex flows, the yield surface is captured naturally without having to deal with the artificial singularity in Bingham-like models.   

Inverse Modeling of Magnetorheological Dampers: Bingham and Hershel-Bulkley Models

Magnetorheological fluids (MR) are increasingly used in damper design when a given response is critical for desired performance. MR fluids flow through narrow passages in dampers subject to a magnetic field applied across the passages. The inverse problem of the determination of the required constitutive properties of the MR fluid together with the corresponding flow pattern for the efficient damping of a given load is solved when the required performance is specified a priori. The fluid is modeled either as a Bingham plastic or is assumed to obey Herschel-Buckley constitutive structure both with time-varying yield-stress. Flow is governed by the continuously adjustable constitutive parameters of the MR fluid which are determined to generate variable resistance to flow to dampen the selected load efficiently. The method developed leads to the determination of the magnetic field variation required to achieve a predetermined displacement of the piston in the damper. The governing equations are solved for any time history of the dimensionless yield stress of the fluid. Relationships that correlate damping load and magnetic field time variations are obtained.   

Matched high-De expansions of the Hookean dumbbell configuration tensor downstream of neutral stagnation points

In steady flow at high Deborah number, one may specify a regular asymptotic expansion of the Hookean dumbbell configuration tensor provided a high-De solution exists upstream. However, downstream of neutral stagnation points, where the strain rate is either zero or negated by vorticity, this expansion is singular despite the existence of a solution at the stagnation point. Following the work of M. Renardy (JNNFM, 90, 13-23), we present a method for obtaining an inner expansion in the boundary layer downstream of neutral stagnation points where affine deformation is checked by dumbbell recoil at leading order. A generic outer expansion may then be matched to this inner expansion to obtain a composite solution. We consider two such matched expansions in the limit of infinite dilution where the flow is Newtonian: 1.) on the centerline downstream of an immobile rigid sphere in axisymmetric flow and 2.) on the separating streamline around a freely-mobile circle in shear flow.   

The viscoelastic range in wall turbulence of dilute polymers

A small amount of long chain polymers dissolved in an otherwise Newtonian flow is known to reduce dramatically drag in wall bounded flows. This corresponds to an alteration in the mean velocity profile, where the slope the log-law of the wall passes from $2.5$ to $11.7$ One of the relevant parameters in such flows is the ratio of two times scales, namely the Deborah (or Weissemberg) number $De_* = \tau_{\rm p}/\tau_*$. Here $\tau_* = \nu/u_*^2$ is the friction time scale and $\tau_{\rm p}$ is the principal relaxation time of the polymeric chain, the estimated time needed to recover equilibrium after the external strain is removed. This opens the way for an hyper-simplified description of the polymer, as a system with a single internal degree of freedom, the resulting model is called Oldroid-B. A set of numerical simulations at constant friction Reynolds number $Re_* = u_* h/\nu$, where $h$ is the channel half-width, have been performed varying the value of Deborah, $De_*$. From these data, the viscoelastic ranges at different distances from the wall will be discussed by means of a scale by scale kinetic energy budget.   

Hi Fidelity Multiscale Flow Simulation of Sedimentation of a Sphere in Dilute Polymeric Solutions

Modeling flow of dilute polymeric solutions in complex kinematics flows using closed form constitutive equations or single segment elastic dumbbell models has attracted considerable attention in the past decade. However, to date most of these simulations have not been able to \textbf{\textit{quantitatively}} describe the experimentally observed flow dynamics\textbf{.} This failure can be attributed to the fact that these models can at best qualitatively describe the polymer dynamics and rheological properties of dilute polymer solutions. However, multi-segment bead-rod and bead spring descriptions of dilute polymeric solutions have been shown to describe both single molecule dynamics as well as the solution rheological properties. Our recent success in quantitatively describing a contraction/expansion flow behavior has motivated extension of this approach to other complex kinematics flows, namely, sedimentation of a sphere in a tube. In particular, we have carried out extensive multiscale flow simulations in tubes of various diameters. Comparison of computational and experimental results clearly demonstrate that multiscale simulations with micromechanical models that accurately describe the internal degrees of freedom of the macromolecules as well as their polydispersity are capable of providing accurate prediction of the drag coefficient of the sphere over a broad range of De and tube diameters.   

Rodless Weissenberg effect

The climbing effect of a viscoelastic fluid when stirred by a spinning rod is well documented. It is often called the Weissenberg effect. This phenomenon is related to the elasticity of the fluid. We have observed that the fluid can also climb when stirred without a rod. In this work, a comparison of the flow around a spinning disk immersed in a shallow fluid layer between a Newtonian and a non-Newtonian liquid is presented. The flow is visualized with dye and bubbles as fluid path tracers. For Newtonian fluids, a classical parabolic surface profile is observed; for the non- Newtonian case, the fluid rises near the center. The height of the fluid hump increases with rotational speed and the fluid elasticity. The flow visualization indicates that a source-like flow pattern appears on top of the rotating disk for the non- Newtonian fluid. The toroidal vortices which appear in the flow, spin in opposite directions for Newtonian and non- Newtonian cases.   

Study of the properties of bubbly non Newtonian liquids

It has been reported that the rise velocity of a swarm of bubbles in shear-thinning fluids is mainly influenced by the reduction in local viscosity and the population of bubbles, i.e., gas hold up. On the other hand, while many authors have studied the elastic effects in the abrupt change of velocity and shape in single bubbles, little has been done in bubble columns. By using a high-speed camera and digital image analysis we measured the rise velocity of bubbles in inelastic and elastic shear-thinning fluids in a bubble column. Great care was taken to produce nearly mono-dispersed bubble swarms. Preliminary results show that the convex-to-concave shape can also be observed in bubble swarms. This change of shape, which is associated with the so-called single bubble velocity discontinuity, has a significant influence in the general behavior of the bubbly flow (cluster formation, pseudo-turbulence, etc).   

Numerical and experimental study of Newtonian and non-Newtonian flow in a spiral viscous pump

The need to transport small volumes of viscous media is a vital part of microfluidic applications in biotechnology, chemistry and electronics. A novel Archimedean viscous micro-pump was developed in an attempt to achieve the precise and accurate delivery of fluid in a robust and industrially viable package. The pump consists of a two-disc system, where one is patterned with a spiral rectangular channel and the other is smooth and has a rate of rotation \( \Omega \) in order to pump the fluid. The width of the channel is variable along its length in order to achieve a constant local Reynolds number and avoid recirculation zones along the spiral, which is described $r = a + b \theta^{c}$, where \( r \) is the radius at the spiral centerline and \( \theta \) is the azimuthal angle. Numerical and analytical studies of the proposed model exhibiting a linear relationship between the flow \( Q \) and \( \Omega \) will be presented, as well as results from experiments with a simplified prototype supporting the analytical and numerical studies.