On the heat flux vector for flowing granular materials—part II: derivation and special cases

2006 ◽  
Vol 29 (13) ◽  
pp. 1599-1613 ◽  
Author(s):  
Mehrdad Massoudi
Keyword(s):  
Heat Flux ◽  
Granular Materials ◽  
Flux Vector ◽  
Heat Flux Vector ◽  
Special Cases

Numerical Study of a Non-Linear Model for the Heat Flux Vector for Granular Materials

2012 ◽  
Author(s):  
Hyunjin Yang ◽  
Nadine Aubry ◽  
Mehrdad Massoudi
Keyword(s):  
Heat Flux ◽  
Granular Materials ◽  
Volume Fraction ◽  
Numerical Study ◽  
Constitutive Relations ◽  
Flux Vector ◽  
Heat Flux Vector ◽  
Non Linear ◽  
Effects Of Radiation ◽  
The One

The two important constitutive relations needed for the study of flow and heat transfer in granular materials, where the effects of radiation are ignored, are the stress tensor and the heat flux vector. Massoudi [1, 2] derived a constitutive model that reflects the dependence of the heat flux vector on the temperature gradient, the density gradient and the velocity gradient, in an appropriate frame-invariant formulation. In this paper we use a simplified version of this model and consider the one dimensional fully developed flow of granular materials down a heated inclined plane, subject to a constant temperature boundary condition. The equations are made dimensionless and a parametric study is performed in order to examine the effects of the additional parameters on the heat flux vector. The derived governing equations are coupled non-linear second order ordinary differential equations which are solved numerically and the results are shown for the temperature, volume fraction and velocity profiles.

PROPERTIES OF HEAT FLUX FUNCTIONS AND A LINEAR THEORY OF HEAT FLUX

1995 ◽  
Vol 09 (09) ◽  
pp. 1113-1122 ◽  
Author(s):  
LIQIU WANG
Keyword(s):  
Thermal Conductivity ◽  
Heat Flux ◽  
Temperature Gradient ◽  
Linear Function ◽  
Linear Theory ◽  
Strain Tensor ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Flux Vector ◽  
Heat Flux Vector

The symmetry and positive definiteness of thermal conductivity tensor K are used to derive some properties of heat flux functions ɸi (i=0, 1, 2). All ɸi are shown to be real-valued. Both ɸ0 and ɸ2 are found to be positive definite, and ɸ1 is constrained between −(ɸ0 + ɸ2) and (ɸ0 + ɸ2). By assuming heat flux vector q to be a linear function of temperature gradient ∇θ and velocity strain tensor D, ɸi reduce to three coefficients which are independent of D and ∇θ.

scholarly journals The Heat Flux Vector(s) in a Two Component Fluid Mixture

Fluids ◽  
2020 ◽  
Vol 5 (2) ◽  
pp. 77
Author(s):  
A. D. Kirwan ◽  
Mehrdad Massoudi
Keyword(s):  
Heat Flux ◽  
Interaction Force ◽  
Heat Fluxes ◽  
Fluid Mixture ◽  
Two Fluids ◽  
Heat Flux Vector ◽  
Special Cases ◽  
Different Temperatures ◽  
Phenomenological Coefficients ◽  
Kinematic Properties

Bulk kinematic properties of mixtures such as velocity are known to be the density weighed averages of the constituent velocities. No such paradigm exists for the heat flux of mixtures when the constituents have different temperatures. Using standard principles such as frame indifference, we address this topic by developing linear constitutive equations for the constituent heat fluxes, the interaction force between constituents, and the stresses for a mixture of two fluids. Although these equations contain 18 phenomenological coefficients, we are able to use the Clausius-Duhem inequality to obtain inequalities involving the principal and cross flux coefficients. The theory is applied to some special cases and shown to reduce to standard results when the constituents have the same temperature.

Heat flux vector in highly inhomogeneous nonequilibrium fluids

1995 ◽  
Vol 51 (5) ◽  
pp. 4362-4368 ◽  
Author(s):  
B. D. Todd ◽  
Peter J. Daivis ◽  
Denis J. Evans
Keyword(s):  
Heat Flux ◽  
Flux Vector ◽  
Heat Flux Vector

Hydromagnetic waves for a collisionless plasma in strong magnetic fields

1985 ◽  
Vol 34 (1) ◽  
pp. 67-76 ◽  
Author(s):  
S. Duhau ◽  
A. De La Torre
Keyword(s):  
Heat Flux ◽  
Collisionless Plasma ◽  
Phase Speed ◽  
Larmor Radius ◽  
Heat Conducting ◽  
Flux Vector ◽  
Heat Flux Vector ◽  
Vector Direction ◽  
Hydromagnetic Waves ◽  
Dynamic Variables

A hydrodynamic system of equations, valid in the limit in which the Larmor radius and the electron to ion mass ratio are both zero, and including the thermo-dynamic variables and the energy equation of the electrons, is used to investigate the propagation of small-amplitude waves in a collisionless heat-conducting plasma. The result is compared with that derived from the Chew, Goldberger & Low equations. It is found that for zero heat flux, the inclusion of the electron pressure does not change the number and characteristic of the modes but modifies the mirror stability criterion. In the general case, the phase speed is symmetric with respect to two axes: one parallel to the heat flux vector and the other normal to it. The heat flux generates a new mode and couples strongly the slow and fast magnetosonic modes whose wavenumber vectors have projections in the positive flux vector direction, giving rise to a new overstability whose existence does not depend on the ion anisotropy.

scholarly journals Heat Flux Distribution Near a Crevasse

1963 ◽  
Vol 4 (34) ◽  
pp. 461-465
Author(s):  
C. J. Pings
Keyword(s):  
Heat Flux ◽  
Heat Flow ◽  
Flux Distribution ◽  
Temperature Data ◽  
Flux Vector ◽  
Heat Flux Distribution ◽  
Flow Lines ◽  
Heat Flux Vector ◽  
Graphical Differentiation ◽  
Improved Accuracy

AbstractPreviously reported experimental temperature data were used to compute the two components of the heat flux vector in the ice body adjacent to a crevasse in a glacier of the ice sheet of northern Greenland. Graphical differentiation techniques were employed. The computed components were used to synthesize values of the beat flux vector, including magnitude and direction. Improved accuracy was achieved over the previously reported technique of sketching heat flow lines orthogonal to the isotherms.

scholarly journals A Constitutive Formulation for the Linear Thermoelastic Behavior of Arbitrary Fiber-Reinforced Composites

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Melek Usal
Keyword(s):  
Heat Flux ◽  
Free Energy ◽  
Composite Material ◽  
Series Expansion ◽  
Symmetric Tensor ◽  
Elastic Behavior ◽  
Free Energy Function ◽  
Flux Vector ◽  
Conservation Of Energy ◽  
Heat Flux Vector

The linear thermoelastic behavior of a composite material reinforced by two independent and inextensible fiber families has been analyzed theoretically. The composite material is assumed to be anisotropic, compressible, dependent on temperature gradient, and showing linear elastic behavior. Basic principles and axioms of modern continuum mechanics and equations belonging to kinematics and deformation geometries of fibers have provided guidance and have been determining in the process of this study. The matrix material is supposed to be made of elastic material involving an artificial anisotropy due to fibers reinforcing by arbitrary distributions. As a result of thermodynamic constraints, it has been determined that the free energy function is dependent on a symmetric tensor and two vectors whereas the heat flux vector function is dependent on a symmetric tensor and three vectors. The free energy and heat flux vector functions have been represented by a power series expansion, and the type and the number of terms taken into consideration in this series expansion have determined the linearity of the medium. The linear constitutive equations of the stress and heat flux vector are substituted in the Cauchy equation of motion and in the equation of conservation of energy to obtain the field equations.

Sign in / Sign up

Export Citation Format

Share Document