scholarly journals Analysis of Brinkman-Forchheimer extended Darcy's model in a fluid saturated anisotropic porous channel

2012 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Timir Karmakar ◽  
Meraj Alam ◽  
G. P. Raja Sekhar
Keyword(s):  
Porous Medium ◽  
Approximate Solution ◽  
Existence And Uniqueness ◽  
Perturbation Expansion ◽  
Positive Semidefinite ◽  
Darcy Number ◽  
Momentum Equation ◽  
Matched Asymptotic Expansion ◽  
Regular Perturbation ◽  
Uniqueness Results

<p style='text-indent:20px;'>We present asymptotic analysis of Couette flow through a channel packed with porous medium. We assume that the porous medium is anisotropic and the permeability varies along all the directions so that it appears as a positive semidefinite matrix in the momentum equation. We developed existence and uniqueness results corresponding to the anisotropic Brinkman-Forchheimer extended Darcy's equation in case of fully developed flow using the Browder-Minty theorem. Complemented with the existence and uniqueness analysis, we present an asymptotic solution by taking Darcy number as the perturbed parameter. For a high Darcy number, the corresponding problem is dealt with regular perturbation expansion. For low Darcy number, the problem of interest is a singular perturbation. We use matched asymptotic expansion to treat this case. More generally, we obtained an approximate solution for the nonlinear problem, which is uniformly valid irrespective of the porous medium parameter values. The analysis presented serves a dual purpose by providing the existence and uniqueness of the anisotropic nonlinear Brinkman-Forchheimer extended Darcy's equation and provide an approximate solution that shows good agreement with the numerical solution.</p>

Numerical Investigation of Heat Transfer in Forced Convection in a Helical Pipe Filled With a Porous Medium

2005 ◽  
Author(s):  
Liping Cheng ◽  
Andrey V. Kuznetsov
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Saturated Porous Medium ◽  
Axial Flow ◽  
Darcy Number ◽  
Momentum Equation ◽  
Darcy Law ◽  
Helical Pipe ◽  
Fluid Saturated Porous Medium ◽  
Flow Inertia

This paper investigates numerically heat transfer in a helical pipe filled with a fluid saturated porous medium. The analysis is based on the full momentum equation for porous media that accounts for the Brinkman and Forchheimer extensions of the Darcy law as well as for the flow inertia. Numerical computations are performed in an orthogonal helical coordinate system. The effects of the Darcy number, the Forchheimer coefficient as well as the Dean and Germano numbers on the axial flow velocity, secondary flow, temperature distribution, and the Nusselt number are analyzed.

Low Rayleigh Number Thermal Convection in a Saturated Porous Medium Bounded by Two Horizontal, Eccentric Cylinders

1984 ◽  
Vol 106 (1) ◽  
pp. 166-175 ◽  
Author(s):  
H. H. Bau
Keyword(s):  
Porous Medium ◽  
Thermal Convection ◽  
Saturated Porous Medium ◽  
Fluid Motion ◽  
Perturbation Expansion ◽  
Infinite Medium ◽  
Regular Perturbation ◽  
Eccentric Annulus ◽  
Eccentric Cylinders ◽  
First Case

A two-term regular perturbation expansion is presented for two-dimensional, steady-state thermal convection in a fully saturated porous medium bounded by two horizontal, eccentric cylinders. Both cylinders are impermeable to fluid motion and are maintained at different, uniform temperatures. The complicated boundary conditions are handled through the use of bicylindrical coordinates. Three geometrical configurations are considered: an eccentric annulus; a pipe buried in a semi-infinite medium; and two cylinders, one outside the other, imbedded in an infinite medium. Detailed results, however, are presented only for the first case. It is demonstrated that eccentric insulations may be more effective under certain conditions and therefore more economical than the currently used concentric ones.

The Accuracy of Short Bearing Theory for Newtonian Lubricants

1986 ◽  
Vol 108 (1) ◽  
pp. 73-79 ◽  
Author(s):  
R. H. Buckholz ◽  
B. Hwang
Keyword(s):  
Aspect Ratio ◽  
Journal Bearing ◽  
Numerical Solutions ◽  
Load Capacity ◽  
Perturbation Expansion ◽  
Approximate Solutions ◽  
Matched Asymptotic Expansion ◽  
Eccentricity Ratio ◽  
Regular Perturbation ◽  
Slider Bearings

The accuracy of the short bearing approximation is analyzed in this discussion. The results apply to Newtonian lubricants, and they can also be extended to non-Newtonian power-law lubricants. Reynolds’ lubrication equation is first solved by applying a regular perturbation expansion in pressure to the π film journal bearing; after this, a matched asymptotic expansion is applied to linear slider bearings. Approximate solutions are then compared with numerical solutions, to estimate the accuracy of the short bearing approximation. Finally, the accuracy of fluid film pressures predicted via short bearing theory is shown to depend upon three factors: the bearing aspect ratio, eccentricity ratio, and the partial-arc extent. Ocvirk’s short bearing series approximation—for finite bearing aspect ratio—is shown to become singular in the limit as the eccentricity ratio approaches unity. The one term π film Ocvirk solution is shown to be a nonconservative approximation to the journal bearing load capacity. A method to extend the accuracy of the short bearing approximation for partial-arcs and slider bearings is then presented.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

scholarly journals BSDEs with polynomial growth generators

2000 ◽  
Vol 13 (3) ◽  
pp. 207-238 ◽  
Author(s):  
Philippe Briand ◽  
René Carmona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Polynomial Growth ◽  
Probabilistic Representation ◽  
Existence And Uniqueness Results ◽  
State Variable ◽  
Cahn Equation ◽  
Uniqueness Results ◽  
Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

Nontrivial solutions of non-autonomous dirichlet fractional discrete problems

2020 ◽  
Vol 23 (4) ◽  
pp. 980-995
Author(s):  
Alberto Cabada ◽  
Nikolay Dimitrov
Keyword(s):  
Existence And Uniqueness ◽  
Point Boundary ◽  
Nontrivial Solutions ◽  
Fractional Difference ◽  
Nonlinear Part ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Difference Equation ◽  
Discrete Problems ◽  
Series Of Functions

AbstractIn this paper, we introduce a two-point boundary value problem for a finite fractional difference equation with a perturbation term. By applying spectral theory, an associated Green’s function is constructed as a series of functions and some of its properties are obtained. Under suitable conditions on the nonlinear part of the equation, some existence and uniqueness results are deduced.

Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative

2019 ◽  
Vol 14 (3) ◽  
pp. 311 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Zakia Hammouch ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Numerical Scheme ◽  
Arbitrary Order ◽  
Fixed Point Theory ◽  
Hepatitis E ◽  
Point Theory ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Essential Properties

A virus that causes hepatitis E is known as (HEV) and regarded on of the reason for lever inflammation. In mathematical aspects a very low attention has been paid to HEV dynamics. Therefore, the present work explores the HEV dynamics in fractional derivative. The Caputo–Fabriizo derivative is used to study the dynamics of HEV. First, the essential properties of the model will be presented and then describe the HEV model with CF derivative. Application of fixed point theory is used to obtain the existence and uniqueness results associated to the model. By using Adams–Bashfirth numerical scheme the solution is obtained. Some numerical results and tables for arbitrary order derivative are presented.

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