# American Institute of Mathematical Sciences

December  2012, 5(6): 1133-1145. doi: 10.3934/dcdss.2012.5.1133

## The thermo-mechanics of rate-type fluids

 1 Department of Mechanical Engineering, Texas A&M University, College Station, TX-77843, United States

Received  April 2012 Revised  June 2012 Published  August 2012

In this short paper a brief review is provided concerning the modeling of the thermo-mechanical response of rate type fluid models. Recently, two different approaches have been used to develop thermodynamically compatible rate type fluid models, one that assumes the Helmholtz potential and the other the Gibbs potential for the fluids. These two perspectives are complimentary, not all models that can be modeled within the first procedure can be obtained from the second one, and vice-versa. The two approaches greatly enlarge the arsenal of a modeler and most models that are used can be derived within the purview of these two approaches. More importantly, the two methodologies lead to interesting and useful new models which can be used to describe the behavior of materials that have hitherto defied proper description.
Citation: K. R. Rajagopal. The thermo-mechanics of rate-type fluids. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1133-1145. doi: 10.3934/dcdss.2012.5.1133
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##### References:
 [1] Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161 [2] Manuel Falconi, E. A. Lacomba, C. Vidal. The flow of classical mechanical cubic potential systems. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 827-842. doi: 10.3934/dcds.2004.11.827 [3] Robert S. Strichartz. A fractal quantum mechanical model with Coulomb potential. Communications on Pure & Applied Analysis, 2009, 8 (2) : 743-755. doi: 10.3934/cpaa.2009.8.743 [4] Horst Heck, Gunther Uhlmann, Jenn-Nan Wang. Reconstruction of obstacles immersed in an incompressible fluid. Inverse Problems & Imaging, 2007, 1 (1) : 63-76. doi: 10.3934/ipi.2007.1.63 [5] Youcef Amirat, Kamel Hamdache. On a heated incompressible magnetic fluid model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 675-696. doi: 10.3934/cpaa.2012.11.675 [6] Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911 [7] Scott W. Hansen, Andrei A. Lyashenko. Exact controllability of a beam in an incompressible inviscid fluid. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 59-78. doi: 10.3934/dcds.1997.3.59 [8] I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759 [9] D. L. Denny. Existence of solutions to equations for the flow of an incompressible fluid with capillary effects. Communications on Pure & Applied Analysis, 2004, 3 (2) : 197-216. doi: 10.3934/cpaa.2004.3.197 [10] Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138 [11] Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067 [12] Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215 [13] Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079 [14] Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 [15] Huanyao Wen, Changjiang Zhu. Remarks on global weak solutions to a two-fluid type model. Communications on Pure & Applied Analysis, 2021, 20 (7-8) : 2839-2856. doi: 10.3934/cpaa.2021072 [16] Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068 [17] Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024 [18] Bernard Ducomet, Šárka Nečasová. On the motion of rigid bodies in an incompressible or compressible viscous fluid under the action of gravitational forces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1193-1213. doi: 10.3934/dcdss.2013.6.1193 [19] Guowei Liu, Rui Xue. Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2193-2216. doi: 10.3934/dcdsb.2018231 [20] Mohamed Alahyane, Abdelilah Hakim, Amine Laghrib, Said Raghay. Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems & Imaging, 2018, 12 (5) : 1055-1081. doi: 10.3934/ipi.2018044

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