On the Mathematical Modelling of a Compressible Viscoelastic Fluid
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Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.
KeywordsPoisson Bracket Bulk Viscosity Cauchy Stress Maxwell Model Augmented Pressure
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- 2.Beris A.N., Edwards B.J.: Thermodynamics of Flowing Systems. Oxford University Press, New York (1994)Google Scholar
- 7.Grmela M., Öttinger H.-C.: Dynamics and thermodynamics of complex fluids I. Development of the GENERIC formalism. Phys. Rev. E 56, 620–6632 (1997)Google Scholar
- 10.Marsden J.E., Hughes T.J.R.: Mathematical Foundations of Elasticity. Dover Publications, New York (1994)Google Scholar
- 14.Öttinger H.-C., Grmela M.: Dynamics and thermodynamics of complex fluids II. Illustrations of the GENERIC formalism. Phys. Rev. E 56, 6633–6655 (1997)Google Scholar
- 17.Temam R., Miranville A.: Mathematical Modelling in Continuum Mechanics. Cambridge University Press, Cambridge (2000)Google Scholar
- 18.Truesdell C.: Continuum Mechanics I: The Mechanical Foundations of Elasticity and Fluid Systems. International Science Review Series. Gordon and Breach, Inc, New York (1969)Google Scholar
- 21.Wilmanski K.: Continuum Thermodynamics. Series on Advances in Mathematics for Applied Sciences, vol 77. World Scientific (2009)Google Scholar
- 22.Woods, L.C.: The Thermodynamics of Fluid Systems. Oxford Engineering, Science Series 2. Oxford University Press, 1986Google Scholar