Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives
TL;DR: In this article, a Caputo-Fabrizio fractional derivatives approach to the thermal analysis of a second grade fluid over an infinite oscillating vertical flat plate is presented, where the heat transfer is caused by the buoyancy force induced by temperature differences between the plate and the fluid.
Abstract: This paper presents a Caputo–Fabrizio fractional derivatives approach to the thermal analysis of a second grade fluid over an infinite oscillating vertical flat plate. Together with an oscillating boundary motion, the heat transfer is caused by the buoyancy force induced by temperature differences between the plate and the fluid. Closed form solutions of the fluid velocity and temperature are obtained by means of the Laplace transform. The solutions of ordinary second grade and Newtonian fluids corresponding to time derivatives of integer and fractional orders are obtained as particular cases of the present solutions. Numerical computations and graphical illustrations are used in order to study the effects of the Caputo–Fabrizio time-fractional parameter \(\upalpha \), the material parameter \(\alpha _2 \), and the Prandtl and Grashof numbers on the velocity field. A comparison for time derivative of integer order versus fractional order is shown graphically for both Newtonian and second grade fluids. It is found that fractional fluids (second grade and Newtonian) have highest velocities. This shows that the fractional parameter enhances the fluid flow.
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TL;DR: In this article, Atangana and Baleanu (AB) were applied to free convection flow of generalized Casson fluid due to the combined gradients of temperature and concentration with heat generation and first order chemical reaction.
Abstract: Atangana and Baleanu (AB) in their recent work introduced a new version of fractional derivatives which uses the generalized Mittag-Leffler function as the non-singular and non-local kernel and accepts all properties of fractional derivatives. This articles aims to apply the AB fractional derivative to free convection flow of generalized Casson fluid due to the combined gradients of temperature and concentration with heat generation and first order chemical reaction. For the sake of comparison, this problem is also solved via Caputo-Fabrizio (CF) derivative technique. Exact solutions in both cases of AB and CF derivatives are obtained via Laplace transform and compared graphically as well as in tabular form. In the case of AB approach, the influence of pertinent parameters on velocity field is displayed in plots and discussed. It is found that for a unit time, the velocities obtained via AB and CF derivatives are identical. Velocities for the time less than 1 show little variation and for time bigger than 1, this variation increases.
188 citations
TL;DR: In this paper, the authors apply the idea of Caputo-Fabrizio time fractional derivatives to magnetohydrodynamics (MHD) free convection flow of generalized Walters'-B fluid over a static vertical plate.
Abstract: The present article applies the idea of Caputo-Fabrizio time fractional derivatives to magnetohydrodynamics (MHD) free convection flow of generalized Walters’-B fluid over a static vertical plate. Free convection is caused due to combined gradients of temperature and concentration. Hence, heat and mass transfers are considered together. The fractional model of Walters’-B fluid is used in the mathematical formulation of the problem. The problem is solved via the Laplace transform method. Exact solutions for velocity, temperature and concentration are obtained. The physical quantities of interest are examined through plots for various values of fractional parameter: $\alpha$
, Walters’-B parameter $\Gamma$
, magnetic parameter M , Prandtl number Pr, Schmidt number Sc, thermal Grashof number Gr and mass Grashof number Gm. As a special case, the published results from open literature are recovered.
164 citations
TL;DR: In this paper, Atangana-Baleanu fractional derivative has been applied to study heat transfer problem of magnetohydrodynamic (MHD) Maxwell fluid over a vertical plate embedded in a porous medium.
Abstract: Atangana-Baleanu fractional derivative has been applied to study heat transfer problem of magnetohydrodynamic (MHD) Maxwell fluid over a vertical plate embedded in a porous medium. The analytical solutions have been obtained for temperature distribution and velocity field by employing Laplace transforms technique for both sine and cosine oscillations of the plate. The general solutions have been expressed in terms of Fox-H function satisfying imposed conditions. The results are plotted graphically and discussed for embedded parameters such as magnetic field, Maxwell parameter, porous medium, Prandtl number and fractional parameter.
112 citations
TL;DR: In this article, a new fractional derivative involving the normalized sinc function without singular kernel is proposed, and the Laplace transform is used to find the analytical solution of the anomalous heat-diffusion problems.
Abstract: In this paper, a new fractional derivative involving the normalized sinc function without singular kernel is proposed. The Laplace transform is used to find the analytical solution of the anomalous heat-diffusion problems. The comparative results between classical and fractional-order operators are presented. The results are significant in the analysis of one-dimensional anomalous heat-transfer problems.
104 citations
Cites background from "Heat transfer analysis in a second ..."
...The heat transfer problem within the non-singular second grade fluid was discussed in [10]....
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01 Jan 2015
TL;DR: In this article, the authors present a new definition of fractional derivative with a smooth kernel, which takes on two different representations for the temporal and spatial variable, for which it is more convenient to work with the Fourier transform.
Abstract: In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.
1,972 citations
TL;DR: In this paper, the authors deal with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractional derivatives and integrals, and model of neurons in biology.
Abstract: This paper deals with recent applications of fractional calculus
to dynamical systems in control theory, electrical circuits with
fractance, generalized voltage divider, viscoelasticity,
fractional-order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons
in biology. Special attention is given to numerical computation of
fractional derivatives and integrals.
617 citations
TL;DR: In this article, a four-parameter Maxwell model with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition is used to determine the relaxation and retardation functions.
Abstract: A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition. This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leffler function. The retardation function is determined via Laplace transformation and is solely a power law type. The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles. This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value. In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function.
390 citations
TL;DR: The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with afractional derivative of order β ∋ (0, 1).
Abstract: The time fractional diffusion equation is obtained from the standarddiffusion equation by replacing the first-order time derivative with afractional derivative of order β ∋ (0, 1). From a physicalview-point this generalized diffusion equation is obtained from afractional Fick law which describes transport processes with longmemory. The fundamental solution for the Cauchy problem is interpretedas a probability density of a self-similar non-Markovian stochasticprocess related to a phenomenon of slow anomalous diffusion. By adoptinga suitable finite-difference scheme of solution, we generate discretemodels of random walk suitable for simulating random variables whosespatial probability density evolves in time according to this fractionaldiffusion equation.
361 citations
TL;DR: In this paper, the Laplace transform method and fractional calculus of arbitrary (noninteger) differentiation were applied to the solution of time-dependent, viscous-diffusion fluid mechanics problems.
Abstract: We present the application of fractional calculus, or the calculus of arbitrary (noninteger) differentiation, to the solution of time-dependent, viscous-diffusion fluid mechanics problems. Together with the Laplace transform method, the application of fractional calculus to the classical transient viscous-diffusion equation in a semi-infinite space is shown to yield explicit analytical (fractional) solutions for the shear-stress and fluid speed anywhere in the domain
329 citations