Francis J. Rooney

Connections Between Stability, Convexity of Internal Energy, and the Second Law for Compressible Newtonian Fluids

In this note we provide proofs of the following statements for a compressible Newtonian fluid: (i) internal energy being a convex function of entropy and specific volume is equivalent to nonnegativity of both specific heat at constant volume and isothermal bulk modulus; (ii) convexity of internal energy together with the second law of thermodynamics imply linear stability of the rest state; and (iii) linear stability of the rest state together with the second law imply convexity of internal energy.

Thermal expansion models of viscous fluids based on limits of free energy

Many viscous flows are mechanically incompressible, yet thermally expand and shrink. Approximations of the compressible Navier–Stokes equations are routinely utilized to model diverse phenomena that share these properties, with a primary goal to remove rapid timescales associated with sound waves. Most models are derived from thermodynamic assumptions coupled with application-specific, scale separation assumptions in time and space. The Boussinesq model for laboratory-scale, buoyancy-driven thermal convection patterns [Spiegel and Veronis, Astrophys. J. 131, 442 (1960)] and the anelastic model for atmospheric-scale, density-stratification phenomena [Emanuel, Atmospheric Convection (Oxford University Press, New York, 1994)] are two important examples. Some engineering models of thermal expansion are fluid specific, e.g., for molten glasses and polymers, and rest upon thermodynamic assumptions alone. These models postpone specification of flow conditions, since applications range from slow confined flows fo...

Constraints, Constitutive Limits, and Instability in Finite Thermoelasticity

This paper examines all the possible types of thermomechanical constraints in finite-deformational elasticity. By a thermomechanical constraint we mean a functional relationship between a mechanical variable, either the deformation gradient or the stress, and a thermal variable, temperature, entropy or one of the energy potentials; internal energy, Helmholtz free energy, Gibbs free energy or enthalpy. It is shown that for the temperature-deformation, entropy-stress, enthalpy-deformation, and Helmholtz free energy-stress constraints equilibrium states are unstable, in the sense that certain perturbations of the equilibrium state grow exponentially. By considering the constrained materials as constitutive limits of unconstrained materials, it is shown that the instability is associated with the violation of the Legendre–Hadamard condition on the internal energy. The entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints do not exhibit this instability. It is proposed that stability of the rest state (or equivalently convexity of internal energy) is a necessary requirement for a model to be physically valid, and hence entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints are physical, whereas temperature-deformation constraints (including the customary notion of thermal expansion that density is a function of temperature only), entropy-stress constraints, enthalpy-deformation constraints, and Helmholtz free energy-stress constraints are not.

Internal constraint theories for the thermal expansion of viscous fluids

Abstract Viscous fluids in many processes are mechanically incompressible but experience significant thermally induced volume change. We focus here on models for viscous fluids that impose this behavior through a posited internal constraint, following the formalism which introduces constraint responses that produce no entropy. We investigate four different thermal expansion constraints, whereby the independent mechanical variable in the problem formulation (density ρ or pressure p ) is assumed to be specified completely by the independent thermal variable (temperature θ or entropy η ). The internal constraint approach yields simpler constitutive relations, and therefore easier material characterization; the resulting model equations are alternatives to the compressible Navier–Stokes equations for simulation and analysis of thermal expansion phenomena. However, whereas the internal constraint formalism preserves consistency with the thermomechanical balance laws, second law, and invariance by fiat, there is no a priori guarantee that stability of the rest state and nonnegativity of specific heat and bulk modulus are preserved. Here we first derive the four possible internal constraint models for thermal expansion. Next we show that each of these four models is equivalent to a specific constitutive limit of the compressible theory, namely two of no pressure dependence of density and their duals of no density dependence of pressure. From this connection, we deduce that two constraints, namely ρ = ρ ( η ) and p = p ( θ ), offer physically viable candidates for the modeling of thermal expansion, whereas two others (the customary density–temperature constraint ρ = ρ ( θ ), as well as p = p ( η )) are unphysical in the absence of any other conditions on the process, predicting catastrophic instability of the rest state. This analysis reveals that the internal constraint formalism must be further conditioned to preserve fundamental irreversible (second law) and reversible (stability) inequalities.

On the St. Venant Problems for Inhomogeneous Circular Bars

The classical St. Venant problems, i.e., simple tension, pure bending, and flexure by a transverse force, are considered for circular bars with elastic moduli that vary as a function of the radial coordinate. The problems are reduced to second-order ordinary differential equations, which are solved for a particular choice of elastic moduli. The special case of a bar with a constant shear modulus and the Poissons ratio varying is also considered and for this situation the problems are solved completely. The solutions are then used to obtain homogeneous effective moduli for inhomogeneous cylinders. Material inhomogeneities associated with spatially variable distributions of the reinforcing phase in a composite are considered. It is demonstrated that uniform distribution of the reinforcement leads to a minimum of the Youngs modulus in the class of spatial variations in the concentration considered.

Investigation of simplified thermal expansion models for compressible Newtonian fluids applied to nonisothermal plane Couette and Poiseuille flows

In this paper six different theories of a Newtonian viscous fluid are investigated and compared, namely, the theory of a compressible Newtonian fluid, and five constitutive limits of this theory: the incompressible theory, the limit where density changes only due to changes in temperature, the limit where density changes only with changes in entropy, the limit where pressure is a function only of temperature, and the limit of pressure a function only of entropy. The six theories are compared through their ability to model two test problems: (i) steady flow between moving parallel isothermal planes separated by a fixed distance with no pressure gradient in the flow direction, and (ii) steady flow between stationary isothermal parallel planes with a pressure gradient. The incompressible theory admits solutions to these problems of the plane Couette/Poiseuille flow form: a single nonzero velocity component in a direction parallel to the bounding planes, and velocity and temperature varying only in the direct...

Rotation of cylinders of special compressible materials

Abstract In this paper we seek closed form solutions to the boundary value problems associated with the rotation of solid cylinders of finitely-deforming compressible elastic materials. Two exact solutions to the equations of motion will be obtained and some of the qualitative features of these solutions will be studied. The strain energy functions used are quadratic in one stretch invariant and linear in another.

Stability of radially symmetric deformations of spheres of compressible non-linearly elastic materials

The stability of homogeneous, isotropic, compressible pressurized non-linearly elastic spheres is considered. A simplified form of the classical method of adjacent equilibria is employed, where the assumed perturbed solution depends only on the radial co-ordinate of the undeformed configuration. The differential equation describing such perturbations and the associated boundary conditions are obtained for all compressible materials and an explicit instability criterion is then obtained. This criterion is studied for three general isotropic materials. The most general form of the strain-energy function for each of the corresponding three radially symmetric deformations is obtained. It is shown that the instability criterion also corresponds to the turning points of the corresponding pressure-inflation relation.

Exact solutions in finite compressible elasticity via the complementary energy function

The purpose of this paper is to establish a method of obtaining closed-form solutions in isotropic hyperelasticity using the complementary energy, the Legendre transform of the strain energy function. Using the complementary energy, the stress becomes the independent variable and the strain the dependent variable. Some of the implications of this formulation of the equations are explored and illustrative examples of solutions for spherical and cylindrical inflation for several forms of the complementary energy are presented.

Spherical inflation for a class of compressible thermoelastic materials

In this paper we consider spherical inflation for two families of constrained thermoelastic materials. The two families correspond to different ideas of mechanical incompressibility. For the first family the material is assumed incompressible under mechanical loading in isothermal conditions, and thus the density is independent of the deformation gradient and is a function of temperature alone. The second family consists of materials that are incompressible under adiabatic deformations, in other words materials whose density is a function of the specific entropy alone, independent of the deformation gradient. The first family, with density a function of temperature, has been shown to exhibit instability. We introduce a modification of the constitutive equations for the first family which satisfies the stability criterion. We show that the solutions of thermostatic boundary value problems for the modified material are identical to the solutions obtained from the second family, where density is a function of entropy, with the appropriate identifications.