We examine the local dynamics of nonisothermal viscous flows in the neighborhood of the constant equilibria using the thermomechanically consistent constrained theory for materials with prescribed temperature-dependent density developed by Cao et al.. We discover that the linearized growth rate of small length scale, infinitesimal disturbances near the equilibria is proportional to the reciprocal of their wave length, a classical phenomenon known as the Hadamard instability, indicating the local ill-posedness of the constrained theory. Therefore, the use of the theory to model transient flow phenomenon is not advised. [S0021-8936(00)01901-2]
Issue Section:
Technical Papers
1.
Cao
, D.
, Bechtel
, S. E.
, and Forest
, M. G.
, 1996
, “Thermomechanical Equations Governing a Material with Prescribed Temperature-Dependent Density with Application to Nonisothermal Plane Poiseuille Flow
,” ASME J. Appl. Mech.
, 63
, pp. 1011
–1018
.2.
Winter
, H. H.
, 1977
, “Viscous Dissipation in Shear Flows of Molten Polymers
,” Adv. Heat Transfer
, 13
, pp. 205
–267
.3.
Kase
, S.
and Matsuo
, T.
, 1965
, “Studies on Melt Spinning, I. Fundamental Equations on the Dynamics of Melt Spinning
,” J. Polym. Sci., Part A: Gen. Pap.
, 3
, pp. 2541
–2554
.4.
Sabhapathy
, P.
and Cheng
, K. C.
, 1986
, “The Effects of Temperature-Dependent Viscosity and Coefficient of Thermal Expansion on the Stability of Laminar, Natural Convective Flow Along an Isothermal, Vertical Surface
,” Int. J. Heat Mass Transf.
, 29
, No. 10
, pp. 1521
–1529
.5.
Hayashi
, S.
, Tani
, K.
, Ishihara
, H.
, and Yasuda
, H.
, 1992
, “Fundamental Analysis of Hot Drawing in Spinline and Introduction of Basic Equations for Simulation
,” Sen-I Gakkaishi
, 48
, No. 10
, pp. 541
–549
.6.
Bird, R. B., Armstrong, R. C., and Hassager, O., 1987, Dynamics of Polymer Liquids, Vol. V1, John Wiley and Sons, New York.
7.
Joseph, D. D., 1990, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York.
8.
Drazin, P. G., and Reid, W. H., 1981, Hydrodynamic Stability, Cambridge University Press, Cambridge, U.K.
9.
Wang
, Q.
, Bechtel
, S. E.
, and Forest
, M. G.
, 1995
, “Modeling and Computational of the Onset of Failure in Polymeric Liquid Filaments
,” J. Non-Newtonian Fluid Mech.
, 58
, pp. 97
–129
.10.
Rooney, F. J., Wang, Q., Bechtel, S. E., and Forest, M. G., 1999, “On the Stability of the Rest State of Thermomechanically Constrained Newtonian Fluids,” submitted for publication.
11.
Gibbs, J. W., 1873, “A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surface,” Trans. Connecticut Acad. II, 1873, Collected Works, pp. 33–54.
12.
Gibbs, J. W., 1878, “On the Equilibrium of Heterogeneous Substances,” Trans. Connecticut Acad. III, 1875–1878, Collected Works, pp. 55–354.
13.
Green
, A. E.
, Naghdi
, P. M.
, and Trapp
, J. A.
, 1970
, “Thermodynamics of a Continuum with Internal Constraints
,” Int. J. Eng. Sci.
, 8
, pp. 891
–908
.14.
Trapp
, J. A.
, 1971
, “Reinforced Materials With Thermo-mechanical Constraints
,” Int. J. Eng. Sci.
, 9
, pp. 757
–773
.15.
Scott
, N. H.
, 1992
, “Linear Dynamical Stability in Constrained Thermoelasticity, II. Deformation-Entropy Constraints
,” Q. J. Mech. Appl. Math.
, 45
, pp. 651
–662
.16.
Scott
, N. H.
, 1996
, “Connections Between Deformation-Temperature and Deformation-Entropy Constraints and Near Constraints in Thermoelasticity
,” Int. J. Eng. Sci.
, 34
, pp. 1689
–1704
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