On the Mathematical Modelling of a Compressible Viscoelastic Fluid

Abstract

Article:Bollada, PC and Phillips, TN (2012) ReuseUnless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. On the Mathematical Modelling of a Compressible Viscoelastic FluidThe fundamental principles associated with the development of mathematical models for compressible viscoelastic fluids are described using methodology and examples, that are sometimes conflicting, from the literature. The modelling of compressibility introduces two additional issues that need to be addressed over those that are required for incompressible fluids. The first issue is concerned with the role of variable density in the derivation of viscoelastic constitutive equations. The second, and perhaps more controversial issue, is concerned with the definition of pressure in a compressible setting, where it is seen to be dependent on bulk viscosity. A heuristic derivation of a compressible version of the Upper Convected Maxwell (UCM) model, starting from first principles in elasticity continuum mechanics, is presented, which suggests a dependence between dynamic viscosity and bulk viscosity, thereby addressing both issues.

Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

“…Fortunately, the pressure field in an incompressible flow is not a thermodynamic state variable (as opposed to the case of compressible flows); therefore this inconsistency does not affect the dynamics of the flow [66]. Here, the pressure variable rather acts mathematically as a Lagrange multiplier that ensures that the continuity constraint (3) is locally fulfilled [67].…”

Hyperbolic model for the classical Navier‐Stokes equations

“…To remedy this flaw, one should use one of the standard codeformational time derivatives that is consistent with the material frame indifference principle defined in the classic paper by Oldroyd . For compressible relative tensors of weight W , the compressible codeformational time derivative is as follows: where W = 1 for the extra stress tensor . Note that this is the only codeformational time derivative that accounts for volumetric (dilatational) changes.…”

“…where W ¼ 1 for the extra stress tensor. [19] Note that this is the only codeformational time derivative that accounts for volumetric (dilatational) changes. Using the codeformational time derivatives (8) instead of the partial or total time derivatives yields the final form of the modified constitutive relation as follows:…”

Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model

“…In a related context, Jiang, Jiang & Wang [31] have studied the existence of global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions. For a survey of macroscopic models of compressible viscoelastic flow, the reader is referred to the paper by Bollada & Phillips [10]. As was noted there, even for isothermal viscoelastic models, the transition from the incompressible to the compressible case is nontrivial; in fact, the precise form of temperature-dependence in compressible viscoelastic models is not yet properly understood, the development of complete, thermodynamically consistent, models being the subject of ongoing research.…”