scholarly journals On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Abdon Atangana ◽  
Adem Kilicman
Keyword(s):  
Mass Transport ◽  
Fractional Derivative ◽  
Numerical Simulations ◽  
Transport Equation ◽  
Stability And Convergence ◽  
Hydrodynamic Dispersion ◽  
The Stability ◽  
Mass Transport Equation ◽  
Modified Equation ◽  
Convergence Of The Scheme

The hydrodynamic dispersion equation was generalized using the concept of variational order derivative. The modified equation was numerically solved via the Crank-Nicholson scheme. The stability and convergence of the scheme in this case were presented. The numerical simulations showed that, the modified equation is more reliable in predicting the movement of pollution in the deformable aquifers, than the constant fractional and integer derivatives.

L[sup 1]-Uniqueness of Weak Solution for One-Dimensional mass Transport Equation

2009 ◽  
Author(s):  
Ludovic Dan Lemle ◽  
Tudor Bi^nzar ◽  
Flavius Pater ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
Keyword(s):  
Weak Solution ◽  
Mass Transport ◽  
Transport Equation ◽  
One Dimensional ◽  
Mass Transport Equation

scholarly journals Some Second-Order σ Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 187
Author(s):  
Yaxin Hou ◽  
Cao Wen ◽  
Hong Li ◽  
Yang Liu ◽  
Zhichao Fang ◽  
...  
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Mixed Finite Element ◽  
Second Order ◽  
Stability And Convergence ◽  
Numerical Examples ◽  
The Stability ◽  
High Order Time ◽  
Distributed Order ◽  
Convergence Of The Scheme

In this article, some high-order time discrete schemes with an H 1 -Galerkin mixed finite element (MFE) method are studied to numerically solve a nonlinear distributed-order sub-diffusion model. Among the considered techniques, the interpolation approximation combined with second-order σ schemes in time is used to approximate the distributed order derivative. The stability and convergence of the scheme are discussed. Some numerical examples are provided to indicate the feasibility and efficiency of our schemes.

scholarly journals Stability and Convergence of a Time-Fractional Variable Order Hantush Equation for a Deformable Aquifer

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
S. C. Oukouomi Noutchie
Keyword(s):  
Partial Derivative ◽  
Mathematical Formulation ◽  
Stability And Convergence ◽  
Order Derivative ◽  
Variable Order ◽  
Leaky Aquifer ◽  
The Stability ◽  
Nicolson Scheme ◽  
Variable Order Derivative ◽  
Modified Equation

The medium through which the groundwater moves varies in time and space. The Hantush equation describes the movement of groundwater through a leaky aquifer. To include explicitly the deformation of the leaky aquifer into the mathematical formulation, we modify the equation by replacing the partial derivative with respect to time by the time-fractional variable order derivative. The modified equation is solved numerically via the Crank-Nicolson scheme. The stability and the convergence in this case are presented in details.

Modelling of Double Diffusion in Turbulent Mass Transport in Porous Media

Volume 1 ◽  
2004 ◽  
Author(s):  
Marcelo J. S. de Lemos
Keyword(s):  
Mass Transport ◽  
Transport Equation ◽  
Fluid Phase ◽  
Double Diffusion ◽  
Time Averaging ◽  
Turbulent Regime ◽  
Transport Mechanisms ◽  
Driving Mechanisms ◽  
Mass Fluxes ◽  
Mass Transport Equation

This work presents derivations of macroscopic heat and mass transport equations for turbulent flow in permeable structures. Two driving mechanisms are considered to contribute to the overall momentum transport, namely temperature driven and concentration driven mass fluxes. Double-diffusive natural convection mechanism is investigated for the fluid phase in turbulent regime. Equations are presented based on two distinct procedures. The first method considers time averaging of the local instantaneous mass transport equation before the volume average operator is applied. The second methodology employs both averaging operators but in a reverse order. This work is intended to demonstrate that additional transport mechanisms are mathematically derived if temperature, concentration and velocity present simultaneously time fluctuations and spatial deviations within the domain of analysis. A modeled form for the final mass transport equation is presented where turbulent transfer is based on a macroscopic version of the k-ε model.

scholarly journals Effects of convection and diffusion of the vapour in evaporating liquid films

2013 ◽  
Vol 732 ◽  
pp. 128-149 ◽  
Author(s):  
Kentaro Kanatani
Keyword(s):  
Boundary Layer ◽  
Mass Transport ◽  
Liquid Film ◽  
Transport Equation ◽  
Time Derivative ◽  
Vertical Direction ◽  
Liquid Films ◽  
Pure Liquid ◽  
Vapour Diffusion ◽  
Mass Transport Equation

AbstractWe propose a novel model of a pure liquid film evaporating into an inert gas, taking into account an effect of convection of the vapour by the evaporation flow of the gas. For the liquid phase, the long-wave approximation is applied to the governing equations. Assuming that fluctuations of the vapour concentration in the gas phase are localized in the vicinity of the liquid–gas interface, we consider only the limit of the mass transport equation at the interface. The diffusion term in the vertical direction of the mass transport equation is modelled by introducing the concentration boundary layer above the liquid film and solving the stationary convection–diffusion equation for the concentration inside the boundary layer. We investigate the linear stability of a flat film based on our model. The effect of vapour diffusion along the interface mitigates the Marangoni effect for short-wavelength disturbances. The local variation of vertical advection is found to be negligible. A critical thickness above which the film is stable exists under the presence of gravity. The effect of fluctuation of mass loss of the liquid induced by horizontal vapour diffusion becomes the primary instability mechanism in a very thin region. The effects of the resistance of phase change and the time derivative of the interface concentration are also examined.

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