The Graetz Problem for the Ellis Fluid Model

2018 ◽  
Vol 74 (1) ◽  
pp. 15-24 ◽  
Author(s):  
N. Ali ◽  
M.W.S. Khan
Keyword(s):  
Energy Equation ◽  
Fluid Model ◽  
Separation Of Variables ◽  
Surface Heat ◽  
Thermal Boundary Conditions ◽  
Nusselt Numbers ◽  
Graetz Problem ◽  
Developing Flow ◽  
Sturm Liouville

AbstractThe determination of temperature and auxiliary quantities such as local and average Nusselt numbers for thermally developing flow is referred as the Graetz problem. In the classical Graetz problem, the fluid entering the tube or channel is Newtonian in nature. Here, an extension of the classical Graetz problem is presented by assuming that the fluid entering the tube or channel obeys the Ellis constitutive equation. The energy equation for the considered problem is solved using the separation of variables technique supplemented with the MATLAB routine bvp4c for computation of the eigenvalues and numerical solution of the associated Sturm-Liouville boundary value problem. The problem is solved for two types of thermal boundary conditions, namely, uniform surface temperature and uniform surface heat flux for both flat and circular geometries. Expressions for bulk mean temperature and local and average Nusselt numbers are presented and discussed through tables and graphs.

scholarly journals Natural convection air flow in vertical upright-angled triangular cavities under realistic thermal boundary conditions

2016 ◽  
Vol 20 (5) ◽  
pp. 1407-1420 ◽  
Author(s):  
Jaime Sieres ◽  
Antonio Campo ◽  
José Martínez-Súarez
Keyword(s):  
Natural Convection ◽  
Vertical Wall ◽  
Aperture Angle ◽  
Temperature Fields ◽  
Uniform Heat Flux ◽  
Thermal Boundary Conditions ◽  
Nusselt Numbers ◽  
Horizontal Wall ◽  
Rayleigh Numbers

This paper presents an analytical and numerical computation of laminar natural convection in a collection of vertical upright-angled triangular cavities filled with air. The vertical wall is heated with a uniform heat flux; the inclined wall is cooled with a uniform temperature; while the upper horizontal wall is assumed thermally insulated. The defining aperture angle ? is located at the lower vertex between the vertical and inclined walls. The finite element method is implemented to perform the computational analysis of the conservation equations for three aperture angles ? (= 15?, 30? and 45?) and height-based modified Rayleigh numbers ranging from a low Ra = 0 (pure conduction) to a high 109. Numerical results are reported for the velocity and temperature fields as well as the Nusselt numbers at the heated vertical wall. The numerical computations are also focused on the determination of the value of the maximum or critical temperature along the hot vertical wall and its dependence with the modified Rayleigh number and the aperture angle.

Analytical Solution of the Graetz Problem Extended to Slip-Flow in Microtube

2012 ◽  
Vol 134 (10) ◽  
Author(s):  
Fayao Xu ◽  
Hugen Ma
Keyword(s):  
Analytical Solution ◽  
Energy Equation ◽  
Temperature Jump ◽  
Separation Of Variables ◽  
Slip Flow ◽  
Power Series Expansion ◽  
Velocity Slip ◽  
Jump Condition ◽  
Dimensionless Temperature ◽  
Graetz Problem

The original Graetz problem is extended to slip-flow in microtube. The extended Graetz problem in slip-flow with the isothermal boundary condition on the wall was solved using conventional energy equation with both the velocity slip and the temperature jump condition of slip-flow. The analytical solution was obtained by solving the energy equation with the method of separation of variables. The accurate eigenvalues were obtained by solving the eigenfunction with the method of power series expansion. In the end, temperature distributions, local Nusselt number Nux, and dimensionless temperature jump θs were obtained. In addition, the influences of Knudsen number Kn on Nux, Nu∞, and θs were discussed.

A Complete Solution for Thermal-Elastohydrodynamic Lubrication of Line Contacts Using Circular Non-Newtonian Fluid Model

1992 ◽  
Vol 114 (3) ◽  
pp. 540-551 ◽  
Author(s):  
Hsing-Sen S. Hsiao ◽  
Bernard J. Hamrock
Keyword(s):  
Boundary Conditions ◽  
Newtonian Fluid ◽  
Energy Equation ◽  
Control Volume ◽  
Fluid Model ◽  
Complete Solution ◽  
Circular Model ◽  
Line Contacts ◽  
Stress Boundary Conditions ◽  
Stress Boundary

A complete solution is obtained for elastohydrodynamically lubricated conjunctions in line contacts considering the effects of temperature and the non-Newtonian characteristics of lubricants with limiting shear strength. The complete fast approach is used to solve the thermal Reynolds equation by using the complete circular non-Newtonian fluid model and considering both velocity and stress boundary conditions. The reason and the occasion to incorporate stress boundary conditions for the circular model are discussed. A conservative form of the energy equation is developed by using the finite control volume approach. Analytical solutions for solid surface temperatures that consider two-dimensional heat flow within the solids are used. A straightforward finite difference method, successive over-relaxation by lines, is employed to solve the energy equation. Results of thermal effects on film shape, pressure profile, streamlines, and friction coefficient are presented.

scholarly journals Leaky waves in planar dielectric waveguide

2019 ◽  
Vol 27 (4) ◽  
pp. 325-342
Author(s):  
Dmitriy V. Divakov ◽  
Alexandre A. Egorov ◽  
Konstantin P. Lovetskiy ◽  
Leonid A. Sevastianov ◽  
Andrey S. Drevitskiy
Keyword(s):  
Separation Of Variables ◽  
Liouville Problem ◽  
Initial Boundary ◽  
Eigenvalues And Eigenfunctions ◽  
Leaky Modes ◽  
Initial Boundary Problem ◽  
Dielectric Media ◽  
Complex Roots ◽  
Sturm Liouville ◽  
Waveguide Problem

A new analytical and numerical solution of the electrodynamic waveguide problem for leaky modes of a planar dielectric symmetric waveguide is proposed. The conditions of leaky modes, corresponding to the Gamow-Siegert model, were used as asymptotic boundary conditions. The resulting initial-boundary problem allows the separation of variables. The emerging problem of the eigen-modes of open three-layer waveguides is formulated as the Sturm-Liouville problem with the corresponding boundary and asymptotic conditions. In the case of guided and radiation modes, the Sturm-Liouville problem is self-adjoint and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the problem for leaky modes is not self-adjoint, so the eigenvalues are complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. To solve this problem, we used the method of minimizing the zero order. An analysis of the calculated distributions of the electric field strength of the first three leaky modes is given, showing the possibilities and advantages of our approach to the study of leaky modes.

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