Mittag–Leffler collocation optimization method for studying a physical problem in fluid flow with fractional derivatives

The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration.

Inverse design of microchannel fluid flow networks using Turing pattern dehomogenization

Structural and Multidisciplinary Optimization ◽

10.1007/s00158-020-02580-w ◽

2020 ◽

Vol 62 (4) ◽

pp. 2203-2210

Author(s):

Ercan M. Dede ◽

Yuqing Zhou ◽

Tsuyoshi Nomura

Keyword(s):

Porous Media ◽

Fluid Flow ◽

Large Scale ◽

Fluid Velocity ◽

Flow Distribution ◽

Reaction Diffusion ◽

Optimization Method ◽

Reaction Diffusion Equations ◽

Turing Pattern ◽

Space Filling

Abstract Microchannel reactors are critical in biological plus energy-related applications and require meticulous design of hundreds-to-thousands of fluid flow channels. Such systems commonly comprise intricate space-filling microstructures to control the fluid flow distribution for the reaction process. Traditional flow channel design schemes are intuition-based or utilize analytical rule-based optimization strategies that are oversimplified for large-scale domains of arbitrary geometry. Here, a gradient-based optimization method is proposed, where effective porous media and fluid velocity vector design information is exploited and linked to explicit microchannel parameterizations. Reaction-diffusion equations are then utilized to generate space-filling Turing pattern microchannel flow structures from the porous media field. With this computationally efficient and broadly applicable technique, precise control of fluid flow distribution is demonstrated across large numbers (on the order of hundreds) of microchannels.

Significances of exponential heating and Darcy's law for second grade fluid flow over oscillating plate by using Atangana-Baleanu fractional derivatives

Case Studies in Thermal Engineering ◽

10.1016/j.csite.2021.101266 ◽

2021 ◽

pp. 101266

Author(s):

Ying-Qing Song ◽

Ali Raza ◽

Kamel Al-Khaled ◽

Saadia Farid ◽

M. Ijaz Khan ◽

...

Keyword(s):

Fluid Flow ◽

Fractional Derivatives ◽

Darcy’S Law ◽

Darcy's Law ◽

Second Grade ◽

Second Grade Fluid ◽

Grade Fluid ◽

Oscillating Plate

Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects

Fractal and Fractional ◽

10.3390/fractalfract6010038 ◽

2022 ◽

Vol 6 (1) ◽

pp. 38

Author(s):

Ridhwan Reyaz ◽

Ahmad Qushairi Mohamad ◽

Yeou Jiann Lim ◽

Muhammad Saqib ◽

Sharidan Shafie

Keyword(s):

Fluid Flow ◽

Analytical Solution ◽

Thermal Radiation ◽

Fractional Derivative ◽

Chemical Reaction ◽

Fractional Derivatives ◽

Experimental Studies ◽

Fluid Flows ◽

Casson Fluid ◽

The Impact

Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations for fractional derivatives in the mechanics of fluid flows are yet to be discovered. Nonetheless, theoretical studies will be useful in facilitating future experimental studies. Therefore, the aim of this study is to showcase an analytical solution on the impact of the Caputo-Fabrizio fractional derivative for a magnethohydrodynamic (MHD) Casson fluid flow with thermal radiation and chemical reaction. Analytical solutions are obtained via Laplace transform through compound functions. The obtained solutions are first verified, then analysed. It is observed from the study that variations in the fractional derivative parameter, α, exhibits a transitional behaviour of fluid between unsteady state and steady state. Numerical analyses on skin friction, Nusselt number, and Sherwood number were also analysed. Behaviour of these three properties were in agreement of that from past literature.

Use of Fractional Derivatives for Fluid Flow in Heterogeneous Media

ECMOR III - 3rd European Conference on the Mathematics of Oil Recovery ◽

10.3997/2214-4609.201411072 ◽

1992 ◽

Cited By ~ 4

Author(s):

R. Lenormand

Keyword(s):

Fluid Flow ◽

Fractional Derivatives ◽

Heterogeneous Media

Caputo-Fabrizio Time Fractional Derivative Applied to Visco Elastic MHD Fluid Flow in the Porous Medium

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.30.28171 ◽

2018 ◽

Vol 7 (4.30) ◽

pp. 533

Author(s):

Salah Uddin ◽

M. Mohamad ◽

M. A. H. Mohamad ◽

Suliadi Sufahani ◽

M. Ghazali Kamardan ◽

...

Keyword(s):

Magnetic Field ◽

Fluid Flow ◽

Fractional Derivative ◽

Fractional Derivatives ◽

Hartmann Number ◽

Fluid Velocity ◽

Axially Symmetric ◽

Governing Equations ◽

Permeability Parameter ◽

The Magnetic Field

In this paper the laminar fluid flow in the axially symmetric porous cylindrical channel subjected to the magnetic field was studied. Fluidmodel was non-Newtonian and visco elastic. The effects of magnetic field and pressure gradient on the fluid velocity were studied by using a new trend of fractional derivative without singular kernel. The governing equations consisted of fractional partial differential equations based on the Caputo-Fabrizio new time-fractional derivatives NFDt. Velocity profiles for various fractional parameter a, Hartmann number, permeability parameter and elasticity were reported. The fluid velocity inside the cylindrical artery decreased with respect to Hartmann number, permeability parameter and elasticity. The results obtained from the fractional derivative model are significantly different from those of the ordinary model.

Spatial-fractional derivatives for fluid flow and transport phenomena

10.1016/b978-0-32-390089-8.00008-8 ◽

2022 ◽

pp. 69-96

Author(s):

Mohamed F. El-Amin

Keyword(s):

Fluid Flow ◽

Fractional Derivatives ◽

Transport Phenomena ◽

Flow And Transport

Exact analysis of MHD Walters’-B fluid flow with non-singular fractional derivatives of Caputo-Fabrizio in the presence of radiation and chemical reaction

Journal of Polymer Science and Engineering ◽

10.24294/jpse.v1i2.599 ◽

2018 ◽

Vol 1 (2) ◽

Author(s):

Muhammad Imran Asjad ◽

Maryam Aleem ◽

M. Bilal Riaz

Keyword(s):

Fluid Flow ◽

Newtonian Fluid ◽

Fractional Derivatives ◽

Fluid Model ◽

Governing Equations ◽

Laplace Transform Technique ◽

The Laplace Transform ◽

Derivatives Of ◽

Walters B Fluid ◽

Transfer Study

The present article reports the applications of Caputo-Fabrizio time-fractional derivatives. This article generalizes the idea of unsteady MHD free convective flow in a Walters.-B fluid with heat and mass transfer study over an exponential isothermal vertical plate embedded in a porous medium. The governing equations are converted into dimensionless form and extended to fractional model. The generalized Walters-B fluid model has been solved analytically using the Laplace transform technique. From the general solutions we reduce limiting solutions when to the similar motion for Newtonian fluid. The corresponding expressions for and Nusselt and Sherwood numbers are also assessed. Numerical results for velocity, temperature and concentration are demonstrated graphically for various factors of interest and discussed. As a result, we have plotted the influence of fractional parameter on fluid flow and drawn comparison between fractional Walters’-B and fractional Newtonian fluid and found that fractional Newtonian fluid is faster than fractional Walters’-B fluids.