-
Previous Article
A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis
- DCDS-S Home
- This Issue
-
Next Article
Stability and interaction of vortices in two-dimensional viscous flows
The thermo-mechanics of rate-type fluids
1. | Department of Mechanical Engineering, Texas A&M University, College Station, TX-77843, United States |
References:
[1] |
G. Barot, I. J. Rao and K. R. Rajagopal, A thermodynamic framework for the modeling of crystallizable shape memory polymers, International Journal of Engineering Science, 46 (2008), 325-351.
doi: 10.1016/j.ijengsci.2007.11.008. |
[2] |
A. N. Beris and S. F. Edwards, "Thermodynamics of Flowing Systems with Internal Microstructure," Oxford Engineering Science Series 36, Oxford University Press, New York, 1994. |
[3] |
D. R. Bland, "The Linear Theory of Viscoelasticity," Pergamon Press, Oxford, 1960. |
[4] |
J. M. Burgers, "Mechanical Considerations-model Systems-Phenomenological Theories of Relaxation and Viscosity," In: First Report on Viscosity and Plasticity, Second ed. Nordemann Publishing Company, Inc., New York, Prepared by the committee of viscosity of the academy of sciences at Amsterdam, 1939. |
[5] |
A. L. Cauchy, Recherches sur lequilibre et le mouvement interieur des corps solides ou fluids, elastiques ou non elastiques, Bull. Soc. Philomath, (1823), 9-13. |
[6] |
J. D. Ferry, "Viscoelastic Properties of Polymers," Wiley, New York, 1980. |
[7] |
J. Finger, Über die allgemeisten Bezeihungen zwischen Deformationen und den zugehoringen Spannungenin aelotropen und isotropen substanzen, Akad. Wiss. Wien Sitzungsber, 103 (1894), 1073-1100. |
[8] |
G. Green, On the laws of reflexion and refraction of light at the common surface of two non-crystallized media, (1837), Trans. Cambr. Phil. Soc. 7 (1839-1842), 1-24. Papers, 245-269 (1839). |
[9] |
A. E. Green and P. M. Naghdi, On thermodynamics and nature of second law, Proc. Roy. Soc. Lond. A, 357 (1977), 253-270.
doi: 10.1098/rspa.1977.0166. |
[10] |
M. Heida and J. Málek, On Korteweg-type compressible fluid-like materials, International Journal of Engineering Science, 48 (2010), 1313-1324.
doi: 10.1016/j.ijengsci.2010.06.031. |
[11] |
M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework, Zeitschrift für Angewandte Mathematik und Physik, 63 (2012), 145-169. |
[12] |
M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynamic framework, Zeitschrift für Angewandte Mathematik und Physik, (in print) (2012). |
[13] |
K. Kannan and K. R. Rajagopal, A thermodynamic framework for chemically reacting systems, Zeitschrift für Angewandte Mathematik und Physik, 62 (2011), 331-362. |
[14] |
J. M. Krishnan and K. R. Rajagopal, On the mechanical behavior of asphalt, Mech. of Materials, 37 (2005), 1085-1100.
doi: 10.1016/j.mechmat.2004.09.005. |
[15] |
J. Málek and K. R. Rajagopal, Incompressible rate type fluids with pressure and shear-rate dependent material moduli, Nonlinear Anal. Real World Appl., 8 (2007), 156-164.
doi: 10.1016/j.nonrwa.2005.06.006. |
[16] |
J. Málek and K. R. Rajagopal, A thermodynamic framework for a mixture of two liquids, 2008 Nonlinear Anal. Real World Appl., 9 (2008), 1649-1660.
doi: 10.1016/j.nonrwa.2007.04.008. |
[17] |
J. C. Maxwell, On the dynamical theory of gases, Philosophical Transactions of the Royal Society, London A, 157 (1866), 26-78. |
[18] |
W. Noll, On the foundations of mechanics of continuous media, Carnegie Institute of Technology, Department of Mathematics Report 17 (1957). |
[19] |
J. G. Oldroyd, On the formulation of the rheological equations of state, Proc. Roy. Soc. Lond. A, 200 (1950), 523-541.
doi: 10.1098/rspa.1950.0035. |
[20] |
S. C. Prasad, K. R. Rajagopal and I. J. Rao, A continuum model for the creep of single crystal nickel-base superalloys, Acta Mater., 53 (2005), 669-679.
doi: 10.1016/j.actamat.2004.10.020. |
[21] |
S. C. Prasad, K. R. Rajagopal and I. J. Rao, A continuum model for the anisotropic creep of single crystal nickel-based superalloys, Acta Mater., 54 (2006), 1487-1500.
doi: 10.1016/j.actamat.2005.11.016. |
[22] |
K. R. Rajagopal and A. R. Srinivasa, On the inelastic behavior of solids - Part 1: Twinning, Int. J. Plast., 11 (1995), 653-678.
doi: 10.1016/S0749-6419(95)00027-5. |
[23] |
K. R. Rajagopal and A. R. Srinivasa, Inelastic behavior of materials - part II: Energetics associated with discontinuous twinning, Int. J. Plast., 13 (1997), 1-35.
doi: 10.1016/S0749-6419(96)00049-6. |
[24] |
K. R. Rajagopal and A. R. Srinivasa, Mechanics of the inelastic behavior of materials - part I: Theoretical underpinnings, Int. J. Plast., 14) (1998), 945-967. |
[25] |
K. R. Rajagopal and A. R. Srinivasa, Mechanics of the inelastic behavior of materials - part II: Inelastic response, Int. J. Plast., 14) (1998), 969-995. |
[26] |
K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of shape memory wires, Zeitschrift für Angewandte Mathematik und Physik, 50 (1999), 459-496. |
[27] |
K. R. Rajagopal and A. R. Srinivasa, Thermodynamics of Rate type fluid model, Journal of Non-Newtonian Fluid Mechanics, 88 (2000), 207-227.
doi: 10.1016/S0377-0257(99)00023-3. |
[28] |
K. R. Rajagopal and A. R. Srinivasa, Modeling anisotropic fluids within the framework of bodies with multiple natural configurations, Journal of Non-Newtonian Fluid Mechanics, 99 (2001), 109-124.
doi: 10.1016/S0377-0257(01)00116-1. |
[29] |
K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of materials that have multiple natural configurations - part I: Viscoelasticity and classical plasticity, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004), 861-893. |
[30] |
K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of materials that have multiple natural configurations - part II: Twinning and solid to solid phase transformation, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004), 1074-1093. |
[31] |
K. R. Rajagopal and A. R. Srinivasa, On the response of non-dissipative solids, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 463 (2007), 357-367. |
[32] |
K. R. Rajagopal and A. R. Srinivasa, On a class of non-dissipative materials that are not hyperelastic, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 465 (2009), 495-500. |
[33] |
K. R. Rajagopal and A. R. Srinivasa, A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 467 (2011), 39-58. |
[34] |
K. R. Rajagopal and A. R. Srinivasa, "Restrictions Placed on Constitutive Relations by Angular Momentum Balance and Galilean Invariance," In Press, Zeitschrift für Angewandte Mathematik und Mechanik, 2012. |
[35] |
K. R. Rajagopal and A. S. Wineman, "Mechanical Response of Polymers," Cambridge University Press, Cambridge, 2001. |
[36] |
I. J. Rao and K. R. Rajagopal, A thermodynamic framework for the study of crystallization in polymers, Zeitschrift für Angewandte Mathematik und Physik, 53 (2002), 365-406. |
[37] |
C. Truesdell, Mechanical Foundations of Elasticity and Fluid Dynamics, Mechanics I, Gordon and Breach, New York (1966) (Reprinted from Journal of Rational Mechanics, 1, 125-300 (1952) as corrected in 2, 595-616 (1953), and 3, 801 (1954)). |
[38] |
C. Truesdell and W. Noll, "Non-Linear Field Theories of Mechanics," 2nd edition, Springer, Berlin, 1992. |
show all references
References:
[1] |
G. Barot, I. J. Rao and K. R. Rajagopal, A thermodynamic framework for the modeling of crystallizable shape memory polymers, International Journal of Engineering Science, 46 (2008), 325-351.
doi: 10.1016/j.ijengsci.2007.11.008. |
[2] |
A. N. Beris and S. F. Edwards, "Thermodynamics of Flowing Systems with Internal Microstructure," Oxford Engineering Science Series 36, Oxford University Press, New York, 1994. |
[3] |
D. R. Bland, "The Linear Theory of Viscoelasticity," Pergamon Press, Oxford, 1960. |
[4] |
J. M. Burgers, "Mechanical Considerations-model Systems-Phenomenological Theories of Relaxation and Viscosity," In: First Report on Viscosity and Plasticity, Second ed. Nordemann Publishing Company, Inc., New York, Prepared by the committee of viscosity of the academy of sciences at Amsterdam, 1939. |
[5] |
A. L. Cauchy, Recherches sur lequilibre et le mouvement interieur des corps solides ou fluids, elastiques ou non elastiques, Bull. Soc. Philomath, (1823), 9-13. |
[6] |
J. D. Ferry, "Viscoelastic Properties of Polymers," Wiley, New York, 1980. |
[7] |
J. Finger, Über die allgemeisten Bezeihungen zwischen Deformationen und den zugehoringen Spannungenin aelotropen und isotropen substanzen, Akad. Wiss. Wien Sitzungsber, 103 (1894), 1073-1100. |
[8] |
G. Green, On the laws of reflexion and refraction of light at the common surface of two non-crystallized media, (1837), Trans. Cambr. Phil. Soc. 7 (1839-1842), 1-24. Papers, 245-269 (1839). |
[9] |
A. E. Green and P. M. Naghdi, On thermodynamics and nature of second law, Proc. Roy. Soc. Lond. A, 357 (1977), 253-270.
doi: 10.1098/rspa.1977.0166. |
[10] |
M. Heida and J. Málek, On Korteweg-type compressible fluid-like materials, International Journal of Engineering Science, 48 (2010), 1313-1324.
doi: 10.1016/j.ijengsci.2010.06.031. |
[11] |
M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework, Zeitschrift für Angewandte Mathematik und Physik, 63 (2012), 145-169. |
[12] |
M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynamic framework, Zeitschrift für Angewandte Mathematik und Physik, (in print) (2012). |
[13] |
K. Kannan and K. R. Rajagopal, A thermodynamic framework for chemically reacting systems, Zeitschrift für Angewandte Mathematik und Physik, 62 (2011), 331-362. |
[14] |
J. M. Krishnan and K. R. Rajagopal, On the mechanical behavior of asphalt, Mech. of Materials, 37 (2005), 1085-1100.
doi: 10.1016/j.mechmat.2004.09.005. |
[15] |
J. Málek and K. R. Rajagopal, Incompressible rate type fluids with pressure and shear-rate dependent material moduli, Nonlinear Anal. Real World Appl., 8 (2007), 156-164.
doi: 10.1016/j.nonrwa.2005.06.006. |
[16] |
J. Málek and K. R. Rajagopal, A thermodynamic framework for a mixture of two liquids, 2008 Nonlinear Anal. Real World Appl., 9 (2008), 1649-1660.
doi: 10.1016/j.nonrwa.2007.04.008. |
[17] |
J. C. Maxwell, On the dynamical theory of gases, Philosophical Transactions of the Royal Society, London A, 157 (1866), 26-78. |
[18] |
W. Noll, On the foundations of mechanics of continuous media, Carnegie Institute of Technology, Department of Mathematics Report 17 (1957). |
[19] |
J. G. Oldroyd, On the formulation of the rheological equations of state, Proc. Roy. Soc. Lond. A, 200 (1950), 523-541.
doi: 10.1098/rspa.1950.0035. |
[20] |
S. C. Prasad, K. R. Rajagopal and I. J. Rao, A continuum model for the creep of single crystal nickel-base superalloys, Acta Mater., 53 (2005), 669-679.
doi: 10.1016/j.actamat.2004.10.020. |
[21] |
S. C. Prasad, K. R. Rajagopal and I. J. Rao, A continuum model for the anisotropic creep of single crystal nickel-based superalloys, Acta Mater., 54 (2006), 1487-1500.
doi: 10.1016/j.actamat.2005.11.016. |
[22] |
K. R. Rajagopal and A. R. Srinivasa, On the inelastic behavior of solids - Part 1: Twinning, Int. J. Plast., 11 (1995), 653-678.
doi: 10.1016/S0749-6419(95)00027-5. |
[23] |
K. R. Rajagopal and A. R. Srinivasa, Inelastic behavior of materials - part II: Energetics associated with discontinuous twinning, Int. J. Plast., 13 (1997), 1-35.
doi: 10.1016/S0749-6419(96)00049-6. |
[24] |
K. R. Rajagopal and A. R. Srinivasa, Mechanics of the inelastic behavior of materials - part I: Theoretical underpinnings, Int. J. Plast., 14) (1998), 945-967. |
[25] |
K. R. Rajagopal and A. R. Srinivasa, Mechanics of the inelastic behavior of materials - part II: Inelastic response, Int. J. Plast., 14) (1998), 969-995. |
[26] |
K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of shape memory wires, Zeitschrift für Angewandte Mathematik und Physik, 50 (1999), 459-496. |
[27] |
K. R. Rajagopal and A. R. Srinivasa, Thermodynamics of Rate type fluid model, Journal of Non-Newtonian Fluid Mechanics, 88 (2000), 207-227.
doi: 10.1016/S0377-0257(99)00023-3. |
[28] |
K. R. Rajagopal and A. R. Srinivasa, Modeling anisotropic fluids within the framework of bodies with multiple natural configurations, Journal of Non-Newtonian Fluid Mechanics, 99 (2001), 109-124.
doi: 10.1016/S0377-0257(01)00116-1. |
[29] |
K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of materials that have multiple natural configurations - part I: Viscoelasticity and classical plasticity, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004), 861-893. |
[30] |
K. R. Rajagopal and A. R. Srinivasa, On the thermomechanics of materials that have multiple natural configurations - part II: Twinning and solid to solid phase transformation, Zeitschrift für Angewandte Mathematik und Physik, 55 (2004), 1074-1093. |
[31] |
K. R. Rajagopal and A. R. Srinivasa, On the response of non-dissipative solids, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 463 (2007), 357-367. |
[32] |
K. R. Rajagopal and A. R. Srinivasa, On a class of non-dissipative materials that are not hyperelastic, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 465 (2009), 495-500. |
[33] |
K. R. Rajagopal and A. R. Srinivasa, A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 467 (2011), 39-58. |
[34] |
K. R. Rajagopal and A. R. Srinivasa, "Restrictions Placed on Constitutive Relations by Angular Momentum Balance and Galilean Invariance," In Press, Zeitschrift für Angewandte Mathematik und Mechanik, 2012. |
[35] |
K. R. Rajagopal and A. S. Wineman, "Mechanical Response of Polymers," Cambridge University Press, Cambridge, 2001. |
[36] |
I. J. Rao and K. R. Rajagopal, A thermodynamic framework for the study of crystallization in polymers, Zeitschrift für Angewandte Mathematik und Physik, 53 (2002), 365-406. |
[37] |
C. Truesdell, Mechanical Foundations of Elasticity and Fluid Dynamics, Mechanics I, Gordon and Breach, New York (1966) (Reprinted from Journal of Rational Mechanics, 1, 125-300 (1952) as corrected in 2, 595-616 (1953), and 3, 801 (1954)). |
[38] |
C. Truesdell and W. Noll, "Non-Linear Field Theories of Mechanics," 2nd edition, Springer, Berlin, 1992. |
[1] |
Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete and 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 and 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 and 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 and 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 and Applied Analysis, 2012, 11 (2) : 675-696. doi: 10.3934/cpaa.2012.11.675 |
[6] |
Scott W. Hansen, Andrei A. Lyashenko. Exact controllability of a beam in an incompressible inviscid fluid. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 59-78. doi: 10.3934/dcds.1997.3.59 |
[7] |
I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759 |
[8] |
Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911 |
[9] |
D. L. Denny. Existence of solutions to equations for the flow of an incompressible fluid with capillary effects. Communications on Pure and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Imaging, 2018, 12 (5) : 1055-1081. doi: 10.3934/ipi.2018044 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]