A More Complete Thermodynamic Framework For Fluent Continua

Polar decomposition of the changing velocity gradient tensor in a deforming fluent continua into pure stretch rates and rates of rotations shows that a location and its neighboring locations can experience different rates of rotations during evolution. Alternatively, we can also consider decomposition of the velocity gradient tensor into symmetric and skew symmetric tensors. The skew symmetric tensor is also a measure of pure rates of rotations whereas the symmetric tensor is a measure of strain rates. The measures of the internal rates of rotations due to deformation in the two approaches describe the same physics but in different forms. Polar decomposition gives the rate of rotation matrix and not the rates of rotation angles whereas the skew symmetric part of the velocity gradient tensor yields rates of rotation angles that are explicitly defined in terms of velocity gradients. These varying rates of rotations at neighboring locations arise due to varying deformation of the continua, hence are internal to the volume of matter and are explicitly defined by deformation. If the internal varying rates of rotations are resisted by the continua, then there must exist internal moments corresponding to these. The internal rates of rotations and the corresponding moments can result in additional rate of energy storage or rate of dissipation. This physics is all internal to the deforming continua and exists in all deforming isotropic, homogeneous fluent continua but is completely neglected in the presently used thermodynamic framework for fluent continua. In this paper we present derivation of a more complete thermodynamic framework in which the derivation of the conservation and balance laws consider additional physics due to varying rates of rotations. The currently used thermodynamic framework for fluent continua is a subset of the thermodynamic framework presented in this paper. The continuum theory presented here considers internal varying rates of rotations and the associated conjugate moments in the derivation of conservation and balance laws, thus the theory presented in this paper can be called “a polar continuum theory” but is different than micropolar continuum theories published currently in which material points have six external degrees of freedom i.e. the rotation rates are additional external degrees of freedom. In the remainder of the paper we refer to this new thermodynamic framework as ‘a polar continuum theory’. The continuum theory presented here only accounts for internal rotation rates and associated moments that exist as a consequence of deformation but are neglected in the present theories hence this theory results in a more complete thermodynamic framework. The polar continuum theory and the resulting thermodynamic framework presented in this paper is suitable for compressible as well as incompressible thermoviscous fluent continua such as Newtonian, Power law, Carreau-Yasuda fluids etc. and thermoviscoelastic fluent continua such as Maxwell, Oldroyd-B, Giesekus etc. The thermodynamic framework presented here is applicable to all isotropic, homogeneous fluent continua. Obviously the constitutive theories will vary depending upon the choice of physics. These are considered in subsequent papers

A MORE COMPLETE THERMODYNAMIC FRAMEWORK FOR SOLID CONTINUA

The Jacobian of deformation at a material point can be decomposed into the stretch tensor and the rotation tensor. Thus, varying Jacobians of deformation at the neighboring material points in the deforming volume of solid continua would yield varying stretch and rotation tensors at the material points. Measures of strain, such as Green’s strain, at a material point are purely a function of the stretch tensor, i.e. the rotation tensor plays no role in these measures. Alternatively, we could also consider decomposition of displacement gradient tensor into symmetric and skew symmetric tensors. Skew symmetric tensor is also a measure of pure rotations whereas symmetric tensor is a measure of strains, i.e. stretches. The measures of rotations in these two approaches describe the same physics but are in different forms. Polar decomposition gives the rotation matrix and not the rotation angles whereas the skew symmetric part of the displacement gradient tensor yields rotation angles that are explicitly and conveniently defined in terms of the displacement gradients. The varying rotations and rotation rates arise in all deforming solid continua due to varying deformation of the continua at neighboring material points, hence are internal to the volume of solid continua and are explicitly defined by the deformation, therefore do not require additional degrees of freedom to define them. If the internal varying rotations and their rates are resisted by the continua, then there must exist internal moments corresponding to these. The internal rotations and their rates and the corresponding moments can result in additional energy storage and dissipation. This physics is all internal to the deforming continua (hence does not require consideration of additional external degrees of freedom and associated external moments) and is neglected in the presently used continuum theories for isotropic, homogeneous solid continua. The continuum theory presented in this paper considers internal varying rotations and associated conjugate moments in the derivation of the conservation and balance laws, thus the theory presented in this paper is “a polar theory for solid continua” but is different than the micropolar theories published currently in which material points have six external degrees of freedom i.e. rotations are additional external degrees of freedom. This polar continuum theory only accounts for internal rotations and associated moments that exist as a consequence of deformation but are neglected in the present theories. We call this theory “a polar continuum theory” as it considers rotations and moments as conjugate pairs in a deforming solid continua though these are internal, hence are purely related to the deformation of the solid. It is shown that the polar continuum theory presented in this paper is not the same as the strain gradient theories reported in the literature. The differences are obviously in terms of the physics described by them and the mathematical details associated with conservation and balance laws. In this paper, we only consider polar continuum theory for small deformation and small strain. This polar continuum theory presented here is a more complete thermodynamic framework as it accounts for additional physics of internally varying rotations that is neglected in the currently used thermodynamic framework. This thermodynamic framework is suitable for isotropic, homogeneous solid matter such as thermoelastic and thermoviscoelastic solid continua with and without memory when the deformation is small. The paper also presents preliminary material helpful in consideration of the constitutive theories for polar continua

Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs

This paper presents derivation of a priori error estimates and convergence rates of finite element processes for boundary value problems (BVPs) described by self adjoint, non-self adjoint, and nonlinear differential operators. A posteriori error estimates are discussed in context with local approximations in higher order scalar product spaces. A posteriori error computational framework (without the knowledge of theoretical solution) is presented for all BVPs regardless of the method of approximation employed in constructing the integral form. This enables computations of local errors as well as the global errors in the computed finite element solutions. The two most significant and essential aspects of the research presented in this paper that enable all of the features described above are: 1) ensuring variational consistency of the integral form(s) resulting from the methods of approximation for self adjoint, non-self adjoint, and nonlinear differential operators and 2) choosing local approximations for the elements of a discretization in a subspace of a higher order scalar product space that is minimally conforming, hence ensuring desired global differentiability of the approximations over the discretizations. It is shown that when the theoretical solution of a BVP is analytic, the a priori error estimate (in the asymptotic range, discussed in a later section of the paper) is independent of the method of approximation or the nature of the differential operator provided the resulting integral form is variationally consistent. Thus, the finite element processes utilizing integral forms based on different methods of approximation but resulting in VC integral forms result in the same a priori error estimate and convergence rate. It is shown that a variationally consistent (VC) integral form has best approximation property in some norm, conversely an integral form with best approximation property in some norm is variationally consistent. That is best approximation property of the integral form and the VC of the integral form is equivalent, one cannot exist without the other, hence can be used interchangeably. Dimensional model problems consisting of diffusion equation, convection-diffusion equation, and Burgers equation described by self adjoint, non-self adjoint, and nonlinear differential operators are considered to present extensive numerical studies using Galerkin method with weak form (GM/WF) and least squares process (LSP) to determine computed convergence rates of various error norms and present comparisons with the theoretical convergence rates

Restrictions on the Material Coefficients in the Constitutive Theories for Non-Classical Viscous Fluent Continua

This paper considers conservation and balance laws and the constitutive theo-ries for non-classical viscous fluent continua without memory, in which in-ternal rotation rates due to the velocity gradient tensor are incorporated in the thermodynamic framework. The constitutive theories for the deviatoric part of the symmetric Cauchy stress tensor and the Cauchy moment tensor are de-rived based on integrity. The constitutive theories for the Cauchy moment tensor are considered when the balance of moments of moments 1) is not a balance law and 2) is a balance law. The constitutive theory for heat vector based on integrity is also considered. Restrictions on the material coefficients in the constitutive theories for the stress tensor, moment tensor, and heat vector are established using the conditions resulting from the entropy inequa-lity, keeping in mind that the constitutive theories derived here based on inte-grity are in fact nonlinear constitutive theories. It is shown that in the case of the simplest linear constitutive theory for stress tensor used predominantly for compressible viscous fluids, Stokes’ hypothesis or Stokes’ assumption has no thermodynamic basis, hence may be viewed incorrect. Thermodynamically consistent derivations of the restrictions on various material coefficients are presented for non-classical as well as classical theories that are applicable to nonlinear constitutive theories, which are inevitable if the constitutive theo-ries are derived based on integrity

Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory

In recent papers, Surana et al. presented internal polar non-classical Continuum theory in which velocity gradient tensor in its entirety was incorporated in the conservation and balance laws. Thus, this theory incorporated symmetric part of the velocity gradient tensor (as done in classical theories) as well as skew symmetric part representing varying internal rotation rates between material points which when resisted by deforming continua result in dissipation (and/or storage) of mechanical work. This physics referred as internal polar physics is neglected in classical continuum theories but can be quite significant for some materials. In another recent paper Surana et al. presented ordered rate constitutive theories for internal polar non-classical fluent continua without memory derived using deviatoric Cauchy stress tensor and conjugate strain rate tensors of up to orders n and Cauchy moment tensor and its conjugate symmetric part of the first convected derivative of the rotation gradient tensor. In this constitutive theory higher order convected derivatives of the symmetric part of the rotation gradient tensor are assumed not to contribute to dissipation. Secondly, the skew symmetric part of the velocity gradient tensor is used as rotation rates to determine rate of rotation gradient tensor. This is an approximation to true convected time derivatives of the rotation gradient tensor. The resulting constitutive theory: (1) is incomplete as it neglects the second and higher order convected time derivatives of the symmetric part of the rotation gradient tensor; (2) first convected derivative of the symmetric part of the rotation gradient tensor as used by Surana et al. is only approximate; (3) has inconsistent treatment of dissipation due to Cauchy moment tensor when compared with the dissipation mechanism due to deviatoric part of symmetric Cauchy stress tensor in which convected time derivatives of up to order n are considered in the theory. The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain tensor, moment tensor and heat vector for thermofluids without memory in which convected time derivatives of strain tensors up to order n are conjugate with the Cauchy stress tensor and the convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n are conjugate with the moment tensor. Conservation and balance laws are used to determine the choice of dependent variables in the constitutive theories: Helmholtz free energy density Φ, entropy density η, Cauchy stress tensor, moment tensor and heat vector. Stress tensor is decomposed into symmetric and skew symmetric parts and the symmetric part of the stress tensor and the moment tensor are further decomposed into equilibrium and deviatoric tensors. It is established through conjugate pairs in entropy inequality that the constitutive theories only need to be derived for symmetric stress tensor, moment tensor and heat vector. Density in the current configuration, convected time derivatives of the strain tensor up to order n, convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n, temperature gradient tensor and temperature are considered as argument tensors of all dependent variables in the constitutive theories based on entropy inequality and principle of equipresence. The constitutive theories are derived in contravariant and covariant bases as well as using Jaumann rates. The nth and 1nth order rate constitutive theories for internal polar non-classical thermofluids without memory are specialized for n = 1 and 1n = 1 to demonstrate fundamental differences in the constitutive theories presented here and those used presently for classical thermofluids without memory and those published by Surana et al. for internal polar non-classical incompressible thermofluids

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

This is the published version. Copyright © 2011 Scientific Research PublishingThe present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method (GM ), Galerkin method with weak form (/GMWF ), Petrov-Galerkin method (PGM), weighted residual method (WRY ), and least squares method or process (LSM or LSP ) to construct finite element approximations in time. A correspondence is estab- lished between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) com- putational processes for which types of operators and, 2) to establish which integral forms do not yield un- conditionally stable computations (variationally inconsistent integral forms, VIC). It is shown that varia- tionally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approxima- tions as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson’s method as well as Newmark method to demonstrate highly meritorious features of the pro- posed methodology

Mathematical Models and Numerical Solutions of Liquid-Solid and Solid-Liquid Phase Change

This paper presents numerical simulations of liquid-solid and solid-liquid phase change processes using mathematical models in Lagrangian and Eulerian descriptions. The mathematical models are derived by assuming a smooth interface or transition region between the solid and liquid phases in which the specific heat, density, thermal conductivity, and latent heat of fusion are continuous and differentiable functions of temperature. In the derivations of the mathematical models we assume the matter to be homogeneous, isotropic, and incompressible in all phases. The change in volume due to change in density during phase transition is neglected in all mathematical models considered in this paper. This paper describes various approaches of deriving mathematical models that incorporate phase transition physics in various ways, hence results in different mathematical models. In the present work we only consider the following two types of mathematical models: (i)We assume the velocity field to be zero i.e. no flow assumption, and free boundaries i.e. zero stress field in all phases. Under these assumptions the mathematical models reduce to first law of thermodynamics i.e. the energy equation, a nonlinear diffusion equation in temperature if we assume Fourier heat conduction law relating temperature gradient to the heat vector. These mathematical models are invariant of the type of description i.e. Lagrangian or Eulerian due to absence of velocities and stress field. (ii) The second class of mathematical models are derived with the assumption that stress field and velocity field are nonzero in the fluid region but in the solid region stress field is assumed constant and the velocity field is assumed zero. In the transition region the stress field and the velocity field transition in a continuous and differentiable manner from nonzero at the liquid state to constant and zero in the solid state based on temperature in the transition zone. Both of these models are consistent with the principles of continuum mechanics, hence provide correct interaction between the regions and are shown to work well in the numerical simulations of phase transition applications with flow. Details of other mathematical models, problems associated with them, and their limitations are also discussed in this paper. Numerical solutions of phase transition model problems in R1and R2 are presented using these two types of mathematical models. Numerical solutions are obtained using h; p;k space-time finite element processes based on residual functional for an increment of time with time marching in which variationally consistent space-time integral forms ensure unconditionally stable computations during the entire evolution

A Nonlinear Constitutive Theory for Deviatoric Cauchy Stress Tensor for Incrompressible Viscous Fluids

Newton’s law of visocosity is a commonly used constitutive theory for deviatoric Cauchy stress tensor. In this constitutive theory originally constructed based on experimental observation, the deviatoric Cauchy stress is proportional to the symmetric part of the velocity gradient tensor. The constant of proportionality is the viscosity of the fluid. For all continuous media if the deforming matter is in thermodynamic equilibrium then all constitutive theories including those considered here must satisfy conservation and balance laws. It is well known that only the second law of thermodynamics provides possible conditions or mechanisms for deriving constitutive theories. The constitutive theory for deviatoric stress tensor used here can be shown to be a simplified form of the constitutive theory derived using conditions resulting from the entropy inequality in conjunction with the theory of generators and invariants that contains up to fifth degree terms in the components of the symmetric part of the velocity gradient tensor. In general the constitutive theory for deviatoric stress tensor is basis (covariant, contravariant, or Jaumann) dependent as it uses convected time derivatives of the Green and Almansi strain tensors of orders higher than one. However, the first convected time derivative of the Green and Almansi strain tensors are in fact symmetric part of the velocity gradient tensor which is basis independent. Thus, if the constitutive theory for deviatoric Cauchy stress tensor is only dependent on the symmetric part of the velocity gradient tensor, then it is basis independent. This is the case for the theory presented in this paper. In this paper we limit the constitutive theory for deviatoric Cauchy stress tensor to contain only up to quadratic terms in the components of the symmetric part of the velocity gradient tensor. The objective is to study the resulting flow physics due to the constitutive theory for deviatoric Cauchy stress tensor that contains up to quadratic terms in the velocity gradient tensor. Model problems consisting of fully developed flow between parallel plates, square lid-driven cavity, and asymmetric sudden expansion are used to present numerical solutions. Numerical solutions of the model problems are calculated using least squares finite element formulation based on residual functional in which the local approximations are considered in higher order scalar product spaces that permit higher order global differentiability solutions. Nonlinear system of algebraic equations are solved using Newton’s linear method with line search

A Nonlinear Constitutive Theory for Heat Conduction in Lagrangian Description Based on Integrity

If the deforming matter is to be in thermodynamic equilibrium, then all constitutive theories, including those for heat vector, must satisfy conservation and balance laws. It is well known that only the second law of thermodynamics provides possible conditions or mechanisms for deriving constitutive theories, but the constitutive theories so derived also must not violate other conservation and balance laws. In the work presented here constitutive theories for heat vector in Lagrangian description are derived (i) strictly using the conditions resulting from the entropy inequality and (ii) using theory of generators and invariants in conjunction with the conditions resulting from the entropy inequality. Both theories are used in the energy equation to construct a mathematical model in R1 that is utilized to present numerical studies using p-version least squares finite element method based on residual functional in which the local approximations are considered in higher order scalar product spaces that permit higher order global differentiability approximations. The constitutive theory for heat vector resulting from the theory of generators and invariants contains up to cubic powers of temperature gradients and is based on integrity, hence complete. The constitutive theory in approach (i) is linear in temperature gradient, standard Fourier heat conduction law, and shown to be subset of the constitutive theory for heat vector resulting from the theory of generators and invariants

Pharmacognostic and Phytochemical Investigation of Ficus carica Linn.

Ficus carica Linn. (Syn: Ficus sycomorous; family: Moraceae) is grows in tropical and subtropical regions of India, used for varity of purpose in traditional medicine. The usefulness of this plant is scientifically evidenced, and different biologically active phytoconstituents were isolated form plant. But no reports are available on morphoanatomy, and phytochemical studies, hence present attempt was undertaken to investigate the microscopical and preliminary phytochemical studies. The study revels the midrib is biconvex and lamina is dorsiventral, shows presence of nonglandular trichome, anomocytic stomata, prismatic calcium oxalate crystals. It shows presence of steroids, triterpenoids, cumarines, flavanoids and glycoside