Caputo Fabrizio Time-fractional Derivative Research Articles

The effect of slip on electro-osmotic flow of a second-grade fluid between two plates with Caputo–Fabrizio time fractional derivatives

This work aims to probe the slip flow of second-grade fluid. The impetus of the flow is taken to be the electro-osmosis and the pressure gradient. The flow is considered to be in a thin channel-like passage formed by two parallel plates. The potential difference existing between the surface of the solid and fluid is taken to be non-symmetric. The governing equations are formed for the second-grade fluid with the Caputo–Fabrizio fractional derivative. The Laplace transform is used for transforming the problem into space parameters after introducing the dimensionless variables. Instead of developing an analytical expression for inverse Laplacian, the numerical Stehfest algorithm is used. A tabular comparison of the obtained results by two different methods (Stehfest and Tzou) is given and the conformity of the two ensures the validity of our obtained results. The results are also pictured in terms of graphs and carry the information of the slip flow effect. Furthermore, the effect of the fractional parameter on velocity has also been tabulated using different values of fractional parameter.

A comparative study of heat transfer analysis of fractional Maxwell fluid by using Caputo and Caputo–Fabrizio derivatives

A comparative analysis is carried out to study the unsteady flow of a Maxwell fluid in the presence of Newtonian heating near a vertical flat plate. The fractional derivatives presented by Caputo and Caputo–Fabrizio are applied to make a physical model for a Maxwell fluid. Exact solutions of the non-dimensional temperature and velocity fields for Caputo and Caputo–Fabrizio time-fractional derivatives are determined via the Laplace transform technique. Numerical solutions of partial differential equations are obtained by employing Tzou’s and Stehfest’s algorithms to compare the results of both models. Exact solutions with integer-order derivative (fractional parameter α = 1) are also obtained for both temperature and velocity distributions as a special case. A graphical illustration is made to discuss the effect of Prandtl number Pr and time t on the temperature field. Similarly, the effects of Maxwell fluid parameter λ and other flow parameters on the velocity field are presented graphically, as well as in tabular form.

Response functions in linear viscoelastic constitutive equations and related fractional operators

This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

Effect of fractional derivatives on transient MHD flow and radiative heat transfer in a micro-parallel channel at high zeta potentials

By using the Caputo–Fabrizio time fractional derivative, unsteady flows of an incompressible viscous through a circular tube are developed. Flows are generated due to the combination of electroosmotic, pressure and magnetic forces. The derived fractional momentum equation incorporating the electromagnetic body force resulting from the electric double layer (EDL) field was solved using the Laplace transform technique, Riemann-sum approximation method, and the Stehfest’s algorithm. The solution obtained is validated by assenting comparisons between the Riemann-sum approximation method and Stehfest’s algorithm. Based upon the velocity field solution, the energy equation is analyzed by taking the viscous dissipation, thermal radiative, the volumetric heat generation due to Joule heating effect and electromagnetic couple effect into account. Numerical simulations and graphical illustrations were carried out using the software package Mathcad. The results highlights that, fractional models of fluid-flow and heat transfer with fractional derivatives bring significant differences compared to the ordinary model. The variations of fluid velocity, fluid temperature, skin friction and Nusselt number with related parameters are interpreted and the results are discussed.

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

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.

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

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.

Blood flow problem in the presence of magnetic particles through a circular cylinder using Caputo-Fabrizio fractional derivative

In this paper, the flow of blood mixed with magnetic particles subjected to uniform transverse magnetic field and pressure gradient in an axisymmetric circular cylinder is studied by using a new trend of fractional derivative without singular kernel. The governing equations are fractional partial differential equations derived based on the Caputo-Fabrizio time-fractional derivatives NFDt. The current result agrees considerably well with that of the previous Caputo fractional derivatives UFDt.

Magnetohydrodynamic electroosmotic flow of Maxwell fluids with Caputo–Fabrizio derivatives through circular tubes

Unsteady flows of an incompressible Maxwell fluid with Caputo–Fabrizio time-fractional derivatives through a circular tube are studied. Flows are generated by an axial oscillating pressure gradient. The influence of a magnetic field, perpendicular on the flow direction, and of an axial electric field are considered. Solutions for the velocity and temperature fields are obtained by combining the Laplace transform with respect to the time variable t, and the finite Hankel transform with respect to the radial variable r. Influences of the order of Caputo–Fabrizio fractional time-derivative and the pertinent system parameters on the fluid flow and heat transfer performance were analyzed numerically by using the Mathcad software. Results show that the fluid velocity and the associated heat transfer modeled by fractional derivatives are quite distinct from those of the ordinary fluids. The fluid velocity and the thermal performance in cylindrical tubes can be controlled by regulating the fractional derivative parameter.

Heat transfer analysis of fractional second-grade fluid subject to Newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: A comparison

The present study is a comparative analysis of unsteady flows of a second-grade fluid with Newtonian heating and time-fractional derivatives, namely, the Caputo fractional derivative (singular kernel) and the Caputo-Fabrizio fractional derivative (non-singular kernel). A physical model for second-grade fluids is developed with fractional derivatives. The expressions for temperature and velocity fields in dimensionless form as well as rates of heat transfer are determined by means of the Laplace transform technique. Solutions for ordinary cases corresponding to integer order derivatives are also obtained. Numerical computations for a comparison between the solutions of the problem with the Caputo time-fractional derivative, problem with Caputo-Fabrizio time-fractional derivative and of the ordinary fluid problem were made. The influence of some flow parameters and fractional parameter $\alpha$ on temperature field as well as velocity field was presented graphically and in tabular forms.

Modeling electro-magneto-hydrodynamic thermo-fluidic transport of biofluids with new trend of fractional derivative without singular kernel

This paper investigates the electro-magneto-hydrodynamic flow of the non-Newtonian behavior of biofluids, with heat transfer, through a cylindrical microchannel. The fluid is acted by an arbitrary time-dependent pressure gradient, an external electric field and an external magnetic field. The governing equations are considered as fractional partial differential equations based on the Caputo–Fabrizio time-fractional derivatives without singular kernel. The usefulness of fractional calculus to study fluid flows or heat and mass transfer phenomena was proven. Several experimental measurements led to conclusion that, in such problems, the models described by fractional differential equations are more suitable. The most common time-fractional derivative used in Continuum Mechanics is Caputo derivative. However, two disadvantages appear when this derivative is used. First, the definition kernel is a singular function and, secondly, the analytical expressions of the problem solutions are expressed by generalized functions (Mittag-Leffler, Lorenzo–Hartley, Robotnov, etc.) which, generally, are not adequate to numerical calculations. The new time-fractional derivative Caputo–Fabrizio, without singular kernel, is more suitable to solve various theoretical and practical problems which involve fractional differential equations. Using the Caputo–Fabrizio derivative, calculations are simpler and, the obtained solutions are expressed by elementary functions. Analytical solutions of the biofluid velocity and thermal transport are obtained by means of the Laplace and finite Hankel transforms. The influence of the fractional parameter, Eckert number and Joule heating parameter on the biofluid velocity and thermal transport are numerically analyzed and graphic presented. This fact can be an important in Biochip technology, thus making it possible to use this analysis technique extremely effective to control bioliquid samples of nanovolumes in microfluidic devices used for biological analysis and medical diagnosis.

Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model

The present article reports the applications of Caputo-Fabrizio time-fractional derivatives. This article generalizes the idea of free convection flow of Jeffrey fluid over a vertical static plate. The free convection is caused due to the temperature gradient. Therefore, heat transfer is considered for free convection. The classical model for Jeffrey fluid is written in dimensionless form with the help of non-dimensional variables. Furthermore, the dimensionless model is converted into a fractional model called as a generalized Jeffrey fluid model. The governing equations of generalized Jeffrey fluid model have been solved analytically using the Laplace transform technique. The recovery of existing solutions in the open literature is possible through this work in terms of classical Jeffrey fluid, fractional Newtonian fluid as well as classical Newtonian fluid. For various embedded parameters, the physics of velocity and temperature profiles is studied by means of numerical computation. This report provides a detailed discussion as well as a graphical representation of the obtained results.

Steady-state heat conduction in a medium with spatial non-singular fading memory: Derivation of Caputo-Fabrizio space-fractional derivative from Cattaneo concept with Jeffrey`s Kernel and analytical solutions

Starting from the Cattaneo constitutive relation with a Jeffrey's kernel the derivation of a transient heat diffusion equation with relaxation term expressed through the Caputo-Fabrizio time fractional derivative has been developed. This approach allows seeing the physical back ground of the newly defined Caputo-Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories.

A modern approach of Caputo–Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium

The present analysis represents the concept of the Caputo–Fabrizio derivatives of fractional order to MHD flow of a second-grade fluid together with radiative heat transfer. The fluid flow is subjected to an infinite oscillating vertical plate embedded in a saturated porous media. The fluid starts motion due to an oscillating boundary and temperature difference between the plate and the fluid. The problem is modeled in terms of partial differential equations, which consist of momentum equation and heat equation. The Laplace transform method is used to obtain the closed-form solutions for velocity and temperature profiles. In order to understand the physics of the problem under consideration, numerical results are obtained using Mathcad software and brought into light through graphical representations. The influence of various physical parameters is studied and displayed in various figures. The corresponding skin friction coefficient and Nusselt number are provided in tables. A graphical comparison is provided showing a strong agreement with the published results in the open literature.

Fundamental solutions to advection–diffusion equation with time-fractional Caputo–Fabrizio derivative

The advection–diffusion equation with time-fractional derivatives without singular kernel and two space-variables is considered. The fundamental solutions in a half-plane are obtained by using the Laplace transform with respect to temporal variable t and Fourier transform with respect to the space coordinates x and y. The Dirichlet problem and the source problem are investigated. The general solution for the fractional case is particularized for the ordinary case of the normal advection–diffusion phenomena and for the fractional/normal diffusion process. It should be noted that, it is more advantageous to use the time-fractional derivative without singular kernel (Caputo–Fabrizio time-fractional derivative), instead of Caputo time-fractional derivative. The advantages are reflected both in the handling of computations, but especially in expressing convenient of solutions. Some numerical calculations are carried out and the results are discussed and illustrated graphically.

Flows between two parallel plates of couple stress fluids with time-fractional Caputo and Caputo-Fabrizio derivatives

The present work provides a comparative study of the unsteady flows between two parallel plates of a couple stress fluid with two different time-fractional derivatives, namely, Caputo time-fractional derivative (derivative with singular kernel) and Caputo-Fabrizio time-fractional derivative (derivative without singular kernel). The solutions to flows of the ordinary couple stress fluid are obtained as limiting cases, using the properties of the time-fractional derivatives. The analysis result shows that it is more advantageous to use the time-fractional derivatives without singular kernel. Advantages consist both in simpler calculations, and, especially, in the final expressions of solutions which are more appropriate for numerical computations. The solutions of the studied problems are obtained by means of the Laplace transform with respect to the time variable t and the finite Fourier transform with respect to the y-variable. It should be noted that by convenient manipulations of the inverse integral transforms, fluid velocity expressions are written as the sum between the steady-state solution (post-transient solution) and the transient solution. Some numerical calculations are carried out in order to study the influence of the time-fractional derivative order on the fluid velocity, shear stresses and couple stress. Also, the critical time at which the steady flow is obtained was numerically determined. Numerical results are illustrated graphically.

Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model

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.

Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey’s Kernel to the Caputo-Fabrizio time-fractional derivative

Starting from the Cattaneo constitutive relation with a Jeffrey’s kernel the derivation of a transient heat diffusion equation with relaxation term expressed through the Caputo-Fabrizio time fractional derivative has been developed. This approach allows seeing the physical background of the newly defined Caputo-Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories.