A Cartesian grid method for two‐phase gel dynamics on an irregular domain

We develop a numerical method for simulating models of two‐phase gel dynamics in an irregular domain using a regular Cartesian grid. The models consist of transport equations for the volume fractions of the two phases, polymer network and solvent; coupled momentum equations for the two phases; and a volume‐averaged incompressibility constraint. Multigrid with Vanka‐type box relaxation scheme is used as a preconditioner for the Krylov subspace solver (GMRES) to solve the momentum and incompressibility equations. Ghost points are used to enforce no‐slip boundary conditions for the velocity field of each phase, and no‐flux boundary conditions for the volume fractions. The behavior of the new method, including its rate of convergence, is explored through numerical experiments for a problem in which strong phase separation develops from an initially (almost) homogeneous phase distribution. We also use the method to explore situations, motivated by biology, which show that imposed boundary velocities can cause substantial redistribution of network and solvent. Copyright © 2010 John Wiley & Sons, Ltd.     

A high-resolution finite-difference method for simulating two-fluid,viscoelastic gel dynamics

An important class of gels are those composed of a polymer network and fluid solvent. The mechanical and rheological properties of these two-fluid gels can change dramatically in response to temperature, stress, and chemical stimulus. Because of their adaptivity, these gels are important in many biological systems, e.g. gels make up the cytoplasm of cells and the mucus in the respiratory and digestive systems, and they are involved in the formation of blood clots. In this study we consider a mathematical model for gels that treats the network phase as a viscoelastic fluid with spatially and temporally varying material parameters and treats the solvent phase as a viscous Newtonian fluid. The dynamics are governed by a coupled system of time-dependent partial differential equations which consist of transport equations for the two phases, constitutive equations for the viscoelastic stresses, two coupled momentum equations for the velocity fields of the two fluids, and a volume-averaged incompressibility constraint. We present a numerical method based on a staggered grid, second order finite-difference discretization of the momentum equations and a high-resolution unsplit Godunov method for the transport equations. The momentum and incompressibility equations are solved in a coupled manner with the Generalized Minimum Residual (GMRES) method using a multigrid preconditioner based on box-relaxation. We present results on the accuracy and robustness of the method together with an illustration of the interesting behavior of this gel model for the four-roll mill problem.     

Constitutive equations for an electroactive polymer

Ionic electroactive polymers can be used as sensors or actuators. For this purpose, a thin film of polyelectrolyte is saturated with a solvent and sandwiched between two platinum electrodes. The solvent causes a complete dissociation of the polymer and the release of small cations. The application of an electric field across the thickness results in the bending of the strip and vice versa. The material is modeled by a two-phase continuous medium. The solid phase, constituted by the polymer backbone inlaid with anions, is depicted as a deformable porous media. The liquid phase is composed of the free cations and the solvent (usually water). We used a coarse grain model. The conservation laws of this system have been established in a previous work. The entropy balance law and the thermodynamic relations are first written for each phase and then for the complete material using a statistical average technique and the material derivative concept. One deduces the entropy production. Identifying generalized forces and fluxes provides the constitutive equations of the whole system: the stress–strain relations which satisfy a Kelvin–Voigt model, generalized Fourier’s and Darcy’s laws and the Nernst–Planck equation.     

Conservation laws of an electro-active polymer

Ionic electro-active polymer is an active material consisting in a polyelectrolyte (for example Nafion). Such material is usually used as thin film sandwiched between two platinum electrodes. The polymer undergoes large bending motions when an electric field is applied across the thickness. Conversely, a voltage can be detected between both electrodes when the polymer is suddenly bent. The solvent-saturated polymer is fully dissociated, releasing cations of small size. We used a continuous medium approach. The material is modelled by the coexistence of two phases; it can be considered as a porous medium where the deformable solid phase is the polymer backbone with fixed anions; the electrolyte phase is made of a solvent (usually water) with free cations. The microscale conservation laws of mass, linear momentum and energy and the Maxwell’s equations are first written for each phase. The physical quantities linked to the interfaces are deduced. The use of an average technique applied to the two-phase medium finally leads to an Eulerian formulation of the conservation laws of the complete material. Macroscale equations relative to each phase provide exchanges through the interfaces. An analysis of the balance equations of kinetic, potential and internal energy highlights the phenomena responsible of the conversion of one kind of energy into another, especially the dissipative ones : viscous frictions and Joule effect.     

A parallel computational method for simulating two‐phase gel dynamics on a staggered grid

We develop a parallel computational algorithm for simulating models of gel dynamics where the gel is described by two phases, a networked polymer and a fluid solvent. The models consist of transport equations for the two phases, two coupled momentum equations, and a volume‐averaged incompressibility constraint. Multigrid with Vanka‐type box‐relaxation scheme is used as preconditioner for the Krylov subspace solver (GMRES) to solve the momentum and incompressibility equations. Through numerical experiments of a model problem, the efficiency, robustness and scalability of the algorithm are illustrated. Copyright © 2008 John Wiley & Sons, Ltd.     

Statistical continuum theory for inelastic behavior of a two-phase medium

A statistical continuum mechanics formulation is presented to predict the inelastic behavior of a medium consisting of two isotropic phases. The phase distribution and morphology are represented by a two-point probability function. The isotropic behavior of the single phase medium is represented by a power law relationship between the strain rate and the resolved local shear stress. It is assumed that the elastic contribution to deformation is negligible. A Green’s function solution to the equations of stress equilibrium is used to obtain the constitutive law for the heterogeneous medium. This relationship links the local velocity gradient to the macroscopic velocity gradient and local viscoplastic modulus. The statistical continuum theory is introduced into the localization relation to obtain a closed form solution. Using a Taylor series expansion an approximate solution is obtained and compared to the Taylor’s upper-bound for the inelastic effective modulus. The model is applied for the two classical cases of spherical and unidirectional discontinuous fiber-reinforced two-phase media with varying size and orientation.     

Interphase momentum interaction effects in the averaged multifield model: Part I: Void propagation in bubbly flows

The volume-averaged form of the linear momentum conservation equations for twophase flow is examined to clarify momentum interaction effects between phases. The case of an accelerating sphere of varying radius in an accelerating fluid is used to derive the form of the interphase force terms. The analysis is extended to assemblies of noninteracting spheres and an interphase force term related to the spatial gradient of phase volume fraction is seen to arise. For the case of bubbly flow, two real characteristics are obtained for dispersed phase volume fractions less than about 0.25. If the term involving the spatial gradient of the phase volume fraction is neglected, then the characteristics are always complex for velocity differences between the phases. The interphase force model is applied to predict experiments on void propagation in bubbly flows. There are no adjustable constants in the model. The experimental data were obtained in our laboratory using cross correlation of signals from a pair of gamma densitometers. The predictions are in excellent agreement with the data. In addition, the predictions are compared with data from several other laboratories, taken over different sets of flow conditions. The predictions are again in close agreement with the data.     

Stress gradient-induced migration effects in the Taylor–Couette flow of a dilute polymer solution

We present an investigation of the phenomenon of stress-induced polymer migration for dilute polymer solutions in the Taylor–Couette device, consisting of two infinitely long, concentric cylinders rotating at constant angular velocities. The underlying physical model is represented by the dilute limit of a two-fluid Hamiltonian system involving two components: one (the polymer) is viscoelastic and obeys the Oldroyd-B constitutive equation, and the other (the solvent) is viscous Newtonian. The two components are considered to be in thermal, but not mechanical equilibrium, interacting with each other through an isotropic drag coefficient tensor. This allows for stress-induced diffusion of polymer chains. The governing equations consist of the continuity and the momentum equations for the bulk velocity, the constitutive model for the polymer chain conformation tensor and the diffusion equation for the polymer concentration. The diffusion equation contains an extra source term, which is proportional to gradients in the polymer stress, so that polymer concentration gradients can develop even in the absence of externally imposed fluxes in the presence of stress inhomogeneities. The solution to the steady-state purely azimuthal flow is obtained first using a spectral collocation method and an adaptive mesh formulation to track the steep changes of the concentration in the flow domain. The calculations show the development of strong polymer migration towards the inner cylinder with increasing Deborah number (De) in agreement with experimental observations. The migration is enhanced for increasing values of the gap thickness resulting in concentration changes by several orders of magnitude in the area between the inner and outer cylinder walls. The extent of the migration also depends strongly on the ratio of the solvent to the polymer viscosity. In addition to a strongly inhomogeneous polymer concentration, significant deviations from the homogenous flow are also observed in the velocity profile. Next, results are reported from a linear stability analysis around the steady-state solution against axisymmetric disturbances corresponding to various wavenumbers in the axial direction. The calculations show that the steady-state solution remains stable up to moderate values of the Deborah number, explaining why some of the predicted stress-induced migration effects should be experimentally observable. The role of the Peclet number (Pe) on the stability of the system is elucidated.     

A compressible single‐temperature conservative two‐phase model with phase transitions

A model for multidimensional compressible two‐phase flow with pressure and velocity relaxations based on the theory of thermodynamically compatible system is extended to study liquid–gas flows with cavitation. The model assumes for each phase its own pressure and velocity, while a common temperature is considered. The governing equations form a hyperbolic system in conservative form and are derived through the theory of a thermodynamically compatible system. The phase pressure‐equalizing process and the interfacial friction are taken into account in the balance laws for the volume fractions of one phase and for the relative velocity by adding two relaxation source terms, while the phase transition is introduced into the model considering in the balance equation for the mass of one phase the relaxation of the Gibbs free energies of the two phases. A modification of the central finite‐volume Kurganov–Noelle–Petrova method is adopted in this work to solve the homogeneous hyperbolic part, while the relaxation source terms are treated implicitly. In order to investigate the effect of the mass transfer in the solution, a 1D cavitation tube problem is presented. In addition, two 2D numerical simulations regarding cavitation problem are also studied: a cavitating Richtmyer–Meshkov instability and a laser‐induced cavitation problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Kinetics of thermally induced swelling of hydrogels

We present a continuum model for thermally induced volume transitions in stimulus–responsive hydrogels (SRHs). The framework views the transition as proceeding via the motion of a sharp interface separating swollen and collapsed phases of the underlying polymer network. In addition to bulk and interfacial force and energy balances, our model imposes an interfacial normal configurational force balance. To account for the large volume changes exhibited by SRHs during actuation, the governing equations are developed in the setting of finite-strain kinematics. The numerical approximations to the coupled thermomechanical equations are obtained with an extended finite element/level-set method. The solution strategy involves a non-standard operator split and a simplified version of the level-set update. A number of representative problems are considered to investigate the model and compare its predictions to experimental observations. In particular, we consider the thermally induced swelling of spherical and cylindrical specimens. The stability of the interface evolution is also examined.     

Direct numerical simulation of turbulence over resolved and modeled rough walls with irregularly distributed roughness

To simulate turbulent flow over a rough wall without resolving complicated rough geometries, a macroscopic rough wall model is developed based on spatial (plane) averaging theory. The plane-averaged drag force term, which arises through averaging the Navier–Stokes equations in a plane parallel to a rough wall, can be modeled using a plane porosity and a plane hydraulic diameter. To evaluate the developed model, direct and macroscopic model simulations for turbulence over irregularly distributed semi-spheres at Reynolds number of 300 are carried out using the D3Q27 multiple-relaxation time lattice Boltzmann method. The results show that the developed model can be used to predict rough wall skin friction. The results agree quantitatively with standard turbulence statistics such as mean velocity and Reynolds stress profiles with the fully resolved DNS data. Since velocity dispersion occurs inside the rough wall and is found to contribute to turbulence energy dissipation, which the developed model cannot account for, the developed model fails to reproduce dispersion-related turbulence energy dissipation. However, it is found that the plane-averaged drag force term can successfully recover the deficiency of dispersion-related turbulence energy dissipation.     

Saturated Compressible and Incompressible Porous Solids: Macro- and Micromechanical Approaches

In this paper two complementary approaches are used to describe the mechanical behavior of saturated compressible and incompressible porous solids. The macroscopic investigation is based on the mixture theory, restricted by the volume fraction concept. In the micromechanical approach, a hierarchy of conditionally ensemble averaged fluid and solid phase momentum balance equations are derived for a simple model of quasi-static liquid saturated porous media. The ensemble averaged equations for both the phases agree remarkably well with the macroscopic results. A micromechanical basis for Terzhagi's effective stress concept is presented. In addition, an expression for additional partial solid stress modifying the effective stress principle, to account for deformability of solid materials, is also derived.     

Decoupled second-order transient schemes for the flow of viscoelastic fluids without a viscous solvent contribution

The simulation of transient flows is relevant in several applications involving viscoelastic fluids. In the last decades, much effort has been spent on deriving time-marching schemes able to efficiently solve the governing equations at low computational cost. In this direction, decoupling schemes, where the global system is split into smaller subsystems, have been particularly successful. However, most of these techniques only work if inertia and/or a large Newtonian solvent contribution is included in the modeling. This is not the case for polymer melts or concentrated polymer solutions.In this work, we propose two second-order time-integration schemes for discretizing the momentum balance as well as the constitutive equation, based on a Gear and a Crank–Nicolson scheme. The solution of the momentum and continuity equations is decoupled from the constitutive one. The stress tensor term in the momentum balance is replaced by its space-continuous but time-discretized form of the constitutive equation through an Euler scheme implicit in the velocity. This adds velocity unknowns in the momentum equation thus an updating of the velocity field is possible even if inertia and solvent viscosity are not included in the model. To further reduce computational costs, the non-linear relaxation term in the constitutive equation is taken explicitly leading to a linear system of equations for each stress component.Four benchmark problems are considered to test the numerical schemes. The results show that a Crank–Nicolson based discretization for the momentum equation produces oscillations when combined with a Crank–Nicolson based scheme for the constitutive equation whereas, if a Gear based scheme is implemented for the constitutive equation, the stability is found to be dependent on the specific problem. However, the Gear based scheme applied to the momentum balance combined with both second-order methods used for the constitutive equation is stable and accurate and performs much better than a first-order Euler scheme. Finally, a numerical proof of the second-order convergence is also carried out.     

About macroscopic models of instability pattern formation

Various macroscopic models to describe instability pattern formation are discussed in this paper. They are similar to the Ginzburg–Landau envelope equation, but they can remain valid away from the bifurcation and are based on the technique of Fourier series with slowly varying coefficients. We focus on two questions: the need to take phase changes into account and the boundary conditions to be associated with macroscopic models. The analysis is carried out on the basis of numerical simulations for the problem of a compressed beam on a nonlinear foundation that is quite similar to the well known Swift–Hohenberg equation. The first macroscopic model involves a real envelope so that the phase is assumed to be constant. The second model is also macroscopic and it is a sort of Ginzburg–Landau equation with a complex envelope. The third one follows from a multi-scaled approach with a numerical bridging between the full model near the boundary and a macroscopic model in the bulk.     

Finite-Element simulation of the start-up problem for a viscoelastic fluid in an eccentric rotating cylinder geometry using a third-order upwind scheme

We numerically simulate the flow field of a dilute polymeric solution using a finitely extendable nonlinear elastic (FENE) dumbbell model. A third-order accurate finite element upwind scheme is used to discretize the convection term in the FENE dumbbell equations for the configuration tensor. The numerical scheme also avoids unphysical negative values for diagonal components of the configuration tensor. The FENE dumbbell equations are solved along with the momentum and continuity equations at small Reynolds numbers with an accuracy of second order in time. In this work we apply this numerical technique to the motion of a viscoelastic fluid in an eccentric rotating cylinder geometry. We obtain the velocity and the polymer contribution to the stress fields as a function of time, and also examine the steady solutions. A particular focus is the influence of coupling between changes in polymer conformation and changes in the flow that occurs as the polymer concentration is increased to a level where the polymer contribution to the zero-shear viscosity of the solution is equal to that of the solvent.This research was supported under grants from the National Science Foundation and the San Diego Supercomputer Center.     

Microstructure and rheology of polymer melts reinforced by long glass fibres: direct simulations

The two stages of classical modelling (formulation of the governing equations and finding their solutions) become particularly difficult, in fact practically unfeasable, in the case of concentrated suspensions of long fibres in polymer melts. In such situations it may be useful to explore a completely different route. One such route is offered by the idea of direct simulations. On this route the model is formulated by specifying the fibre–solvent and the fibre–fibre interactions. The passage from the single-fibre point of view to the macroscopic view on which rheological properties and the microstructure are observed is made by first calculating trajectories of all fibres in a model-system composed of several hundreds of fibres and making appropriate averages. The preliminary results of such simulations are in a good qualitative agreement with experimental observations for a polypropylene melt reinforced by long glass fibres. For examples, dilution and migration of fibres away from die walls are predicted; it is also observed that in concentrated situations the fibres form bundles which could in the entry flow generate large energy dissipation and oscillations.     

Chemically induced swelling of hydrogels

We consider a continuum model for chemically induced volume transitions in hydrogels. Consistent with experimental observations, the model allows for a sharp interface separating swelled and collapsed phases of the underlying polymer network. The polymer chains are treated as a solute with an associated diffusion potential and their concentration is assumed to be discontinuous across the interface. In addition to the standard bulk and interfacial equations imposing force balance and solute balance, the model involves a supplemental interfacial equation imposing configurational force balance. We present a hybrid eXtended-Finite-Element/Level-Set Method for obtaining approximate solutions to the governing equations of the model. As an application, we consider the swelling of a spherical specimen whose boundary is traction-free and is in contact with a reservoir of uniform chemical potential. Our numerical results exhibit good qualitative comparison with experimental observations and predict characteristic swelling times that are proportional to the square of the specimen radius. Our results also suggest several possible synthetic pathways that might be pursued to engineer hydrogels with optimal response times.