Maxwell Constitutive Relations Research Articles

Axisymmetric convection flow of fractional Maxwell fluid past a vertical cylinder with velocity slip and temperature jump

A novel finite volume method is developed to investigate the axisymmetric convection flow and heat transfer of fractional viscoelastic fluid past a vertical cylinder. Fractional cylindrical governing equations are formulated by fractional Maxwell model and generalized Fourier's law. The velocity slip and temperature jump boundary conditions are considered across the fluid-solid interface. Numerical results are validated by exact solutions of special case with source terms. The effects of fractional derivative parameter and boundary condition parameters on flow and heat transfer characteristics are discussed. The viscoelastic fluid performs evident shear thickening property in the fractional Maxwell constitutive relation. Moreover, the boundary condition parameters have remarkable influence on velocity and temperature distributions.

Learning the constitutive relation of polymeric flows with memory

We develop a learning strategy to infer the constitutive relation for the stress of polymeric flows with memory. We make no assumptions regarding the functional form of the constitutive relations, except that they should be expressible in differential form as a function of the local stress- and strain-rate tensors. In particular, we use a Gaussian Process regression to infer the constitutive relations from stress trajectories generated from small-scale (fixed strain-rate) microscopic polymer simulations. For simplicity, a Hookean dumbbell representation is used as a microscopic model, but the method itself can be generalized to incorporate more realistic descriptions. The learned constitutive relation is then used to perform macroscopic flow simulations, allowing us to update the stress distribution in the fluid in a manner that accounts for the microscopic polymer dynamics. The results using the learned constitutive relation are in excellent agreement with full Multi-Scale Simulations, which directly couple micro/macro degrees of freedom, as well as the exact analytical solution given by the Maxwell constitutive relation. We are able to fully capture the history dependence of the flow, as well as the elastic effects in the fluid. We expect the proposed learning/simulation approach to be used not only to study the dynamics of entangled polymer flows, but also for the complex dynamics of other Soft Matter systems, which possess a similar hierarchy of length- and time-scales.

Lattice Boltzmann model for three-phase viscoelastic fluid flow.

A lattice Boltzmann (LB) framework is developed for simulation of three-phase viscoelastic fluid flows in complex geometries. This model is based on a Rothman-Keller type model for immiscible multiphase flows which ensures mass conservation of each component in porous media even for a high density ratio. To account for the viscoelastic effects, the Maxwell constitutive relation is correctly introduced into the momentum equation, which leads to a modified lattice Boltzmann evolution equation for Maxwell fluids by removing the normal but excess viscous term. Our simulation tests indicate that this excess viscous term may induce significant errors. After three benchmark cases, the displacement processes of oil by dispersed polymer are studied as a typical example of three-phase viscoelastic fluid flow. The results show that increasing either the polymer intrinsic viscosity or the elastic modulus will enhance the oil recovery.

Newtonian Limit of Maxwell Fluid Flows

In this paper, we revise Maxwell's constitutive relation and formulate a system of first-order partial differential equations with two parameters for compressible viscoelastic fluid flows. The system is shown to possess a nice conservation-dissipation (relaxation) structure and therefore is symmetrizable hyperbolic. Moreover, for smooth flows we rigorously verify that the revised Maxwell's constitutive relations are compatible with Newton's law of viscosity.

Stamping coal cake simulation with Duncan–Kelvin–Maxwell constitutive relations

In stamp charging coke making, the lack of stamping process analysis and design information such as coal box pressure causes debasement of the design quality of related devices. To achieve a radical solution, a simulation is thus recommended, in which the essence is the appropriate description of the behaviour and reaction of stamped coal blend. To accomplish the mission, finite element method was applied in the current study to conduct a simulation on the stamping coal cake process, and a constitutive model was developed to meet the specific requirements of stamping coal. The constitutive model was established by combining Duncan–Chang from soil mechanics and Kelvin–Maxwell model from viscoelastic mechanics, and then extended to three-dimensional, and its numerical solution was also given. The simulation result are consisted with the field test data, which shows that the method is a feasible tool for studying stamping coking coal, and the proposed constitutive model is able to describe to some extent the inherent behaviour of the coal cake.

Large time step numerical modelling of the flow of Maxwell materials

SUMMARY Maxwell viscoelastic materials are commonly simulated numerically in order to model the stresses and deformations associated with large-scale earth processes, such as mantle convection or crustal deformation. Both implicit and explicit time-marching methods require that the time steps used be small compared with the Maxwell relaxation time if accurate solutions are to be obtained. For crustal tectonic modelling, where Maxwell times in a ductile lower crust may be of order of a decade or less, the large number of time steps required to model processes lasting many millions of years imposes a huge computational burden. This burden is avoidable. In this paper I show that, with the appropriate formulation of the problem, time steps may be taken which are much larger than the Maxwell time without loss of accuracy, as long as they are not large compared with the times over which strain rates vary significantly (‘tectonic’ timescales) in the model. The method relies on explicit analytic integration of the Maxwell constitutive relation for the stress over time intervals, which may be longer than the relaxation time as long as they are short compared with the timescale over which crustal stresses and geometries change. The validity of the formulation is also demonstrated numerically by comparison with the analytic solutions for three simple plane-strain models: extension of a uniform block, shear of a composite layer and development of a Rayleigh‐Taylor instability.

Post-seismic relaxation theory on a laterally heterogeneous viscoelastic model

Investigation was carried out into the problem of relaxation of a laterally heterogeneous viscoelastic Earth following an impulsive moment release event. The formal solution utilizes a semi-analytic solution for post-seismic deformation on a laterally homogeneous Earth constructed from viscoelastic normal modes, followed by application of mode coupling theory to derive the response on the aspherical Earth. The solution is constructed in the Laplace transform domain using the correspondence principle and is valid for any linear constitutive relationship between stress and strain. The specific implementation described in this paper is a semi-analytic discretization method which assumes isotropic elastic structure and a Maxwell constitutive relation. It accounts for viscoelastic–gravitational coupling under lateral variations in elastic parameters and viscosity. For a given viscoelastic structure and minimum wavelength scale, the computational effort involved with the numerical algorithm is proportional to the volume of the laterally heterogeneous region. Examples are presented of the calculation of post-seismic relaxation with a shallow, laterally heterogeneous volume following synthetic impulsive seismic events, and they illustrate the potentially large effect of regional 3-D heterogeneities on regional deformation patterns.

Asymptotic behaviour in linear thermoelastic materials

In this paper we consider the linear theory of thermoelastic solids characterized by Cattaneo–Maxwell's constitutive relation for the heat flux in the form proposed by Pao and Banerjee (Pao, Y. H. and Banerjee, D. K., Thermal pulses in dielectric crystals. Lett. Appl. Eng. Sci., 1973, 1, 35–47). Thermodynamic restrictions on the constitutive equations and the domain of dependence are first derived. Then, in the unidimensional case, existence, uniqueness and asymptotic stability properties of the solution are investigated. Finally, we also prove some dissipation properties of the system.

Theory of melt fracture instabilities in the capillary flow of polymer melts

We present a model for the flow of a polymer melt through a capillary with nonlinear slip boundary conditions at the wall of the capillary. The model consists of the linearized Navier-Stokes equations coupled to a Maxwell constitutive relation for the viscoelasticity and a phase-field model for a first-order transition between stick and slip flow at the boundary. Specializing to the case of a two-dimensional capillary, we perform a linear stability analysis about the steady-state solutions and predict in which parameter regimes the steady-state becomes unstable. A numerical study of the model shows regions of steady flow, as well as regimes with periodic oscillations, spatially uniform but temporally chaotic oscillations, and more complicated spatiotemporal behavior. We show that the oscillations can account for the sharkskin texturing and defect structures seen in the extrusion of polymer melts.

Effects of Combined Shearing and Stretching in Viscoelastic Lubrication

Abstract It is argued theoretically that in normal bearing and squeeze film geometries, when using a nonlinear Maxwell constitutive relation, that elongation and normal stress effects are of minor importance.