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.

Transport processes in dilute gases over the whole range of Knudsen numbers. Part 1. General theory

1979 ◽  
Vol 93 (3) ◽  
pp. 585-607 ◽  
Author(s):  
L. C. Woods
Keyword(s):  
Free Path ◽  
Mean Free Path ◽  
Transport Processes ◽  
Burnett Equations ◽  
Flux Vector ◽  
Dilute Gases ◽  
Knudsen Numbers ◽  
Heat Flux Vector ◽  
The Mean ◽  
The One

The mean-free-path approach to kinetic theory, initiated by Maxwell, and largely abandoned after the Chapman-Enskog success with Boltzmann's equation, is revised and considerably extended in order to find expressions for the heat flux vector q and pressure tensor p, valid (it is hoped) for all Knudsen numbers, K. These expressions (equations (2.24) and (2.26)) are integrals taken over the whole volume of the fluid plus surface integrals taken over the solid boundaries. The one phenomenological element is the mean free path λ, which takes different values according to whether it is mass, momentum or energy that is transported by the molecules. The need for such an approach is evidenced by the existence of critical values of K, above which the Chapman-Enskog expansion in powers of K, truncated after a finite number of terms, fails to yield a solution. For example with the Burnett equations, which are correct to O(K2), the critical K in a shock wave is only 0·2 based upon the upstream λ.

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

Non-linear thermo-mechanical cylindrical bending of functionally graded plates

2008 ◽  
Vol 222 (3) ◽  
pp. 305-318 ◽  
Author(s):  
F Fallah ◽  
A Nosier
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Functionally Graded ◽  
Power Law Distribution ◽  
Cylindrical Bending ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Fg Plates ◽  
Non Linear ◽  
The One

Based on the first-order non-linear von Karman theory, cylindrical bending of functionally graded (FG) plates subjected to mechanical, thermal, and combined thermo-mechanical loadings are investigated. Analytical solutions are obtained for an FG plate with various clamped and simply-supported boundary conditions. The closed form solutions obtained are very simple to be used in design purposes. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. The effects of non-linearity, material property, and boundary conditions on various response quantities are studied and discussed. It is found that linear analysis is inadequate for analysis of simply-supported FG plates even in the small deflection range especially when thermal load is present. Also it is shown that bending—extension coupling can not be seen in response quantities of clamped FG plates. Also an exact solution is developed for the one-dimensional heat conduction equation with variable heat conductivity coefficient.

Numerical Study on Momentum Transfer Between Droplets and Carrier-Phase in Wave-Type Plate Separators

2017 ◽  
Author(s):  
Li Yuzheng ◽  
Liu Qianfeng ◽  
Bo Hanliang
Keyword(s):  
Momentum Transfer ◽  
Carrier Phase ◽  
Volume Fraction ◽  
Numerical Study ◽  
Euler Method ◽  
Coupling Method ◽  
Low Velocity ◽  
Wave Type ◽  
The One ◽  
Different Diameters

The steam flow is simulated by FLUENT. The Lagrange-Euler method is used to simulate the droplet-laden flow in wave-type separators. Two-way coupling method is used to study the influence of the momentum transfer between droplets and carrier-phase in wave-type plate separators. A group of the trajectories of droplets with different diameters are performed in wave-type plate separator flow field. The result shows that the momentum transfer has tiny impact on the behaviors of droplets in a low velocity flow. However, the momentum transfer affects the behaviors of droplets more significantly with rising flow velocity. The one-way coupling method overestimates the diffusion of droplets. In addition, the momentum transfer affects the total pressure loss more significantly with rising volume fraction. The conclusion verifies the importance of the momentum transfer in droplet-laden flows, which could be used to simulate the behavior of droplets moving in a separator.

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.

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