scholarly journals Degenerate equations in a diffusion–precipitation model for clogging porous media

2019 ◽  
Vol 31 (6) ◽  
pp. 1050-1069
Author(s):  
RAPHAEL SCHULZ
Keyword(s):  
Transport Equation ◽  
Saturated Porous Medium ◽  
Coupled System ◽  
Weighted Norm ◽  
Weighted Function ◽  
Decay Behaviour ◽  
Variable Porosity ◽  
Weighted Function Space ◽  
Diffusion Precipitation ◽  
Change Of Porosity

In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.

Forced Convection in a Channel Filled With Porous Medium, Including the Effects of Flow Inertia, Variable Porosity, and Brinkman Friction

1987 ◽  
Vol 109 (4) ◽  
pp. 880-888 ◽  
Author(s):  
D. Poulikakos ◽  
K. Renken
Keyword(s):  
Porous Medium ◽  
Forced Convection ◽  
Flow Model ◽  
Saturated Porous Medium ◽  
Porous Matrix ◽  
Parallel Plates ◽  
Entry Length ◽  
Variable Porosity ◽  
Flow Inertia ◽  
Temperature And Flow Fields

This paper presents a series of numerical simulations which aim to document the problem of forced convection in a channel filled with a fluid-saturated porous medium. In modeling the flow in the channel, the effects of flow inertia, variable porosity and Brinkman friction are taken into account. Two channel configurations are investigated: parallel plates and circular pipe. In both cases, the channel wall is maintained at constant temperature. It is found that the general flow model predicts an overall enhancement in heat transfer between the fluid/porous matrix composite and the walls, compared to the predictions of the widely used Darcy flow model. This enhancement is reflected in the increase of the value of the Nusselt number. Important results documenting the dependence of the temperature and flow fields in the channel as well as the dependence of the thermal entry length on the problem parameters are also reported in the course of the study.

Weighted norm inequalities for positive operators restricted on the cone of λ-quasiconcave functions

2019 ◽  
Vol 150 (1) ◽  
pp. 17-39 ◽  
Author(s):  
Amiran Gogatishvili ◽  
Júlio S. Neves
Keyword(s):  
Sufficient Conditions ◽  
Type Inequality ◽  
Weighted Norm ◽  
Hardy Type Inequality ◽  
Weighted Function ◽  
Hardy Type ◽  
Quasiconcave Functions ◽  
Measurable Functions ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ρ be a monotone quasinorm defined on ${\rm {\frak M}}^ + $, the set of all non-negative measurable functions on [0, ∞). Let T be a monotone quasilinear operator on ${\rm {\frak M}}^ + $. We show that the following inequality restricted on the cone of λ-quasiconcave functions $$\rho (Tf) \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$where $1\les p\les \infty $ and v is a weighted function, is equivalent to slightly different inequalities considered for all non-negative measurable functions. The case 0 < p < 1 is also studied for quasinorms and operators with additional properties. These results in turn enable us to establish necessary and sufficient conditions on the weights (u, v, w) for which the three weighted Hardy-type inequality $$\left( {\int_0^\infty {{\left( {\int_0^x f u} \right)}^q} w(x){\rm d}x} \right)^{1/q} \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$holds for all λ-quasiconcave functions and all 0 < p, q ⩽ ∞.

Natural Convection in a Saturated Variable-Porosity Medium Due to Microwave Heating

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Watit Pakdee ◽  
Phadungsak Rattanadecho
Keyword(s):  
Porous Medium ◽  
Microwave Heating ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Saturated Porous Medium ◽  
Simple Algorithm ◽  
Heating Process ◽  
Variable Porosity ◽  
Dimensional Variation ◽  
Difference Time

Microwave heating of a porous medium with a nonuniform porosity is numerically investigated based on a proposed numerical model. A two-dimensional variation of porosity of the medium is considered. The generalized non-Darcian model developed takes into account the presence of a solid drag and the inertial effect. The transient Maxwell’s equations are solved by using the finite difference time domain method to describe the electromagnetic field in the waveguide and medium. The temperature profile and velocity field within a medium are determined by solution of the momentum, energy, and Maxwell’s equations. The coupled nonlinear set of these equations is solved using the SIMPLE algorithm. In this work, a detailed parametric study is conducted on heat transport inside a rectangular enclosure filled with a saturated porous medium of constant or variable porosity. The numerical results agree well with the experimental data. Variations in porosity significantly affect the microwave heating process as well as the convective flow pattern driven by microwave energy.

Natural Convection Experiments in a Stratified Liquid-Saturated Porous Medium

1986 ◽  
Vol 108 (3) ◽  
pp. 660-666 ◽  
Author(s):  
D. C. Reda
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
High Permeability ◽  
Transfer Rates ◽  
Variable Porosity ◽  
Radial Extent

Natural convection heat transfer from a constant-flux cylinder, immersed vertically through a stratified (two-layer) liquid-saturated porous medium, was investigated experimentally. Measured radial temperature profiles and heat transfer rates agreed well with numerical predictions based on the work of Hickox and Gartling. The 1:6 permeability-ratio interface existing between the two layers was found to effectively trap buoyancy-driven fluid motion within the high-permeability region, beneath the interface. Within this high-permeability region, Nusselt number versus Rayleigh number data were found to correlate with previously measured results, obtained for the same basic geometry, but with a fully permeable upper-surface hydrodynamic boundary condition. In both cases, the vertical and radial extent of the region under study were large compared to the radius of the heat source. Combined results indicate that, for a given Rayleigh number in the Darcy-flow regime, heat transfer rates from cylinders immersed vertically in uniform liquid-saturated porous media of large vertical and radial extent potentially approach limiting values. Variable-porosity effects which occur in unconsolidated porous media adjacent to solid boundaries were investigated numerically for cases where the particle-to-heater diameter ratio was small (≈ 10−2). Results showed variable-porosity effects to have a negligible influence on the thermal field adjacent to such boundaries under conditions of Darcy flow.

scholarly journals Characterization of weighted function spaces in terms of wavelet transforms

2018 ◽  
Vol 37 (4) ◽  
pp. 69-82
Author(s):  
Sanjay Sharma ◽  
Drema Lhamu ◽  
Sunil Kumar Singh
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Wavelet Transforms ◽  
Function Spaces ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Weighted Function Space

In this paper, we have characterized a weighted function space $ B_{\omega,\psi}^{p,q}, ~ 1\leq p,q<\infty$ in terms of wavelet transform and shown that the norms on the spaces $B_{\omega,\psi}^{p,q}$  and $\bigwedge_\omega^{p,q}$ (the space defined in terms of differences $\triangle_x$) are equivalent.

scholarly journals On the weighted function space in the non-Archimedean setting

2012 ◽  
Vol 23 (3) ◽  
pp. 571-588
Author(s):  
A.K. Katsaras ◽  
L.A. Khan ◽  
H.H. Alsulami
Keyword(s):  
Function Space ◽  
Weighted Function ◽  
Weighted Function Space

scholarly journals On approximation properties of a new construction of Baskakov operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Fuat Usta
Keyword(s):  
Asymptotic Formula ◽  
Degree Of Approximation ◽  
Approximation Properties ◽  
Weighted Modulus Of Continuity ◽  
Baskakov Operators ◽  
Weighted Function ◽  
Weighted Function Space ◽  
Weighted Modulus ◽  
New Construction ◽  
Selection Of

AbstractThe purpose of this research is to construct sequences of Baskakov operators such that their construction consists of a function σ by use of two function sequences, $\xi _{n} $ ξ n and $\eta _{n} $ η n . In these operators, σ not only features the sequences of operators but also features the Korovkin function set $\lbrace 1,\sigma ,\sigma ^{2} \rbrace $ { 1 , σ , σ 2 } in a weighted function space such that the operators fix exactly two functions from the set. Thereafter, weighted uniform approximation on an unbounded interval, the degree of approximation with regards to a weighted modulus of continuity, and an asymptotic formula of the new operators are presented. Finally, some illustrative results are provided in order to observe the approximation properties of the newly defined Baskakov operators. The results demonstrate that the introduced operators provide better results in terms of the rate of convergence according to the selection of σ.

Sign in / Sign up

Export Citation Format

Share Document