Caputo Fabrizio Time-fractional Derivative Research Articles

Exploring the couple stress flow of sodium alginate-based casson tetra hybrid nanofluid between inclined parallel microplates using Fourier analysis and Fick's laws

Compared to a regular fluid like water, the Couple Stress Fluid (CSF) has an additional material constant that manages both couple stress and lubricant viscosity. However, because this material constant creates a fourth-order derivative term in the momentum equation, there has been little research done on this type of fluid, even for traditional fluid problems. In this article, we analyze the flow of a tetra hybrid nanofluid consisting of Casson (sodium alginate-based) couple stress between two parallel microplates that are inclined infinitely. Fourier and Fick's laws are used to investigate the flow, where one plate remains stationary and the other oscillates at a constant speed. The flow is driven by the buoyant force generated by heat transfer and free convection, which results in the uniform dispersion of all spherical dust particles throughout the fluid. The system of partial differential equations (PDEs) is expanded using the Caputo-Fabrizio (CF) fractional derivative to simulate fluid flow. The problem is tackled by using finite sine Fourier and Laplace transforms to obtain closed-form solutions for temperature and velocity profiles. CF time fractional derivatives are used to provide CSF (Couette flow with suction or injection) results for various stimulating fluid parameters and examine their impact on the system. Graphical representations of the CSF, dust particle, and temperature profiles are presented, and skin friction and Nusselt number are calculated using Mathcad-15. It is found that an increase in the couple stress parameter leads to a retardation of both velocity profiles.

Exploring the potential of heat transfer and entropy generation of generalized dusty tetra hybrid nanofluid in a microchannel

Caputo–Fabrizio time-fractional derivatives are the subject of this paper. This article generalizes the concept of dusty Tetra hybrid nanofluid moving freely via convection between infinite vertical parallel static plates. Free convection and buoyant force cause the flow and transmit the heat. In addition, there is a consistent distribution of spherical dust particles over the whole flow. It is the temperature difference between the two regions that sets off free convection. Free convection takes heat transfer into account. The dust Tetra hybrid nanofluid classical model employs non-dimensional variables to achieve a dimensionless form. We also convert the dimensionally-free model into a fractional generalized dusty Tetra hybrid nanofluid model. In this paper, we use the finite sine approach to analytically solve the governing equations of the generalized Dusty Tetra hybrid nanofluid model. In this article, we generalize the concept of a dust-filled Tetra hybrid nanofluid freely flowing between infinite vertical parallel plates. We found an analytical solution to the governing equations for the generalized dusty Tetra hybrid nanofluid by combining the Finite Sine Fourier and Laplace transforms. Understanding the mechanics of velocity and temperature profiles requires the use of numerical computation for a variety of embedded factors. In-depth statistical analysis and charting of data are features of this investigation. Using Mathcad-15, we plot the profiles of the Tetra hybrid nanofluid, dust particles, and temperatures to see the findings physically. Also determined are the skin friction and Nusselt number. The rate of heat transfer decreases with time, as seen in Table 1. Similarly, as seen in Table 2, raising the fractional parameter results in a higher skin friction. In addition, the energy profile of both velocities increases with increasing tetra hybrid nano fluid volume percent, albeit the fraction's contribution decreases with time. Since the fractional models are more accurate, they also provide more potential outcomes. When all the facts are considered, these choices may out to be the best.

A time fractional model of a Maxwell nanofluid through a channel flow with applications in grease

Several scientists are interested in recent developments in nanotechnology and nanoscience. Grease is an essential component of many machines and engines because it helps keep them cool by reducing friction between their various elements. In sealed life applications including centralized lubrication systems, electrical motors, bearings, logging and mining machinery, truck wheel hubs, construction, landscaping, and gearboxes, greases are also utilized. Nanoparticles are added to convectional grease to improve its cooling and lubricating properties. More specifically, the current study goal is to investigate open channel flow while taking grease into account as a Maxwell fluid with MoS2 nanoparticles suspended in it. The Caputo-Fabrizio time-fractional derivative is used to convert the issue from a linked classical order PDE to a local fractional model. To determine the precise solutions for the velocity, temperature, and concentration distributions, two integral transform techniques the finite Fourier sine and the Laplace transform technique are jointly utilized. The resultant answers are physically explored and displayed using various graphs. It is important to note that the fractional model, which offers a variety of integral curves, more accurately depicts the flow behavior than the classical model. Skin friction, the Nusselt number, and the Sherwood number are engineering-related numbers that are quantitatively determined and displayed in tabular form. It is determined that adding MoS2 nanoparticles to grease causes a 19.1146% increase in heat transmission and a 2.5122% decrease in mass transfer. The results obtained in this work are compared with published literature for the accuracy purpose.

Heat Transfer Analysis of Generalized Nanofluid with MHD and Ramped Wall Temperature Using Caputo–Fabrizio Derivative Approach

The aim of the present work is to apply the fractional derivative to the heat transformation of a nanofluid along with ramped wall temperature. The flow is analyzed under the effect of magnetohydrodynamic together with heat transfer. A nanofluid under the application of fractional‐order differential equations by Caputo–Fabrizio derivatives with respect to time has the ability to explain the behavior of nanofluid under the influence of memory concept. For the same purpose, Caputo–Fabrizio time‐fractional derivative is applied to investigate the behavior of nanoparticles on the thermal conductivity of a fluid. Appropriate nondimensional variables are engaged in the equation which governs the problem and guides us to obtain the exact solutions for the fields of velocity and temperature. These obtained solutions for the nondimensional set of governing equations are found by extracting them from the governing equations by applying Laplace transform techniques along with Caputo–Fabrizio time‐fractional derivative. The influence of the fractional variable on the velocity, temperature, and Nusselt number is graphically exposed and discussed. The velocity for the state of wall temperature as ramped falls down with the enlarging values of the fractional parameter. Variation in Nusselt number is shown in the tabular form. Solutions are visualized graphically to make an analysis of how the variation is taking place in the physical behavior of the nanofluid flow with respect to the change in distinct physical parameters. The obtained results here will have useful industrial and engineering implementations. It is found that fluid velocity in the flow direction decreases with the increase in the magnetic parameter. The relationship of fractional parameter with the velocity and temperature of the nanofluid is found as direct proportional for a smaller time. However, this direct proportionality converts into inverse proportionality for larger values of time. It is observed that the increase in the nanoparticles volume fraction causes an increase in temperature distribution. It is due to lower specific heat of nanoparticles and its higher thermal conductivity than that of the base fluid.

Heat transfer analysis in a non-Newtonian hybrid nanofluid over an exponentially oscillating plate using fractional Caputo–Fabrizio derivative

In this paper, we have been study a hybrid nanofluid over an exponentially oscillating vertical flat plate. Therefore the fractional derivatives definition of Caputo–Fabrizio approach is applied to transform the classical model for this hybrid nanofluid to fractional model. Together with an oscillating boundary motion, therefore the heat transfer is cause as a result of the buoyancy force produce due temperature differences between the plate and the fluid. The dimensionless classical model is generalized by transforming it to the time fractional model using Caputo–Fabrizio time fractional derivative. Exact analytical solutions are obtained by using Laplace transform method to the set of dimensionless fractional governing equations, containing the momentum and energy equations subjected to the boundary and initial conditions. Numerical computations and graphical illustrations are used to checked the results of the Caputo–Fabrizio time-fractional parameter, the second-grade parameter, the magnetic parameter and the Grashof numbers on the velocity field. An assessment for time spin-off is shown graphically of integer order versus fractional-order for these non-Newtonian hybrid nanofluid through Mathcad software. The fluid velocity increases for increasing the value of the fractional parameter, second-grade parameter and Grashof number. Also for increasing the values of the MHD parameter the fluid velocity decreases.

Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem.

In this work we approximate the source for a non homogeneous fractionaldiffusion equation in 1D, from measurements of the concentration at a finite number ofpoints. We use Caputo-Fabrizio time fractional derivative to model anomalous diffusion.Separating variables, we arrive to a linear system which provides approximate values forthe Fourier coefficients of the unknown source. Numerical examples show the efficiency ofthe method, as well as some of its practical limitations.

Unsteady MHD Flow for Fractional Casson Channel Fluid in a Porous Medium: An Application of the Caputo-Fabrizio Time-Fractional Derivative

Theoretically, this work describes the exact solutions of fractional Casson fluid through a channel under the effect of MHD and porous medium. The unsteady fluid motion of the bottom plate, which is confined by parallel but perpendicular sidewalls, supports the flow. By introducing the dimensionless parameters and variables, the momentum equation, as well as the initial and boundary conditions, has been transformed to a dimensionless form. A mix of Laplace and Fourier transformations is used to get the exact solution for the momentum equation. The constitutive equations for Caputo-Fabrizio’s time-fractional derivative are also incorporated for recovering the exact solutions of the flow problem under consideration. After recovering the exact solutions for flow characteristics, three different cases at the surface of the bottom plate are discussed, by addressing the limiting cases under the influence of the side walls. Moreover, these solutions are captured graphically, and the effects of the Reynolds number Re , fractional parameter α , effective permeability K eff , and dimensionless parameter for Casson fluid β on the fluid’s motion are observed.

Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

<abstract><p>In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 < \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.</p></abstract>

Effects of fractional derivative and heat source/sink on MHD free convection flow of nanofluids in a vertical cylinder: A generalized Fourier's law model

This research looked at the unsteady free convection flows of an incompressible viscous fluid with heat/sink in a vertical cylinder containing a mixture of 47 nm alumina nanoparticles in water. The flow direction is subjected to a perpendicular magnetic field. The generalization entails taking into account a new version of the constitutive equation for thermal flux, known as the generalized Atangana-Baleanu derivative, which is based on the generalized time-fractional derivative with Mittag-Leffler kernel. Using the Laplace transform and the finite Hankel transform, closed forms of analytical solutions for temperature and velocity fields, represented with Bessel and generalized G–function of Lorenzo and Hartley functions, are determined. The generalized solutions are appropriate for particularizations to yield solutions corresponding to fractional derivatives with power-law kernel and exponential kernel. The Mittag-Leffler function is a one-parametric function. It is also possible to acquire the usual situation, which corresponds to classical Fourier's law. To compare models based on generalized Atangana-Baleanu, Atangana-Baleanu, Caputo, and Caputo-Fabrizio time fractional derivatives, numerical simulations produced with the program Mathcad are carried out and visually depicted.

Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach

It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and non-slippage assumptions for Oldroyd-B fluid with magnetized permeability. An innovative definition of Caputo-Fabrizio time fractional derivative is implemented to hypothesize the constitutive energy and momentum equations. The exact solutions of presented problem, are determined by using mathematical techniques, namely Laplace transform with slipping boundary conditions have been invoked to tackle governing equations of velocity and temperature. The Nusselt number and limiting solutions have also been persuaded to estimate the heat emission rate through physical interpretation. In order to provide the validation of the problem, the absence of retardation time parameter led the investigated solutions with good agreement in literature. Additionally, comprehensively scrutinize the dynamics of the considered problem with parametric analysis is accomplished, the graphical illustration is depicted for slipping and non-slipping solutions for temperature and velocity. A comparative studies between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently.

Exact Solution of the Startup Electroosmotic Flow of Generalized Maxwell Fluids in Triangular Microducts

Abstract The unsteady electroosmotic flow of generalized Maxwell fluids in triangular microducts is investigated. The governing equation is formulated with Caputo–Fabrizio time-fractional derivatives whose orders are distributed in the interval [0, 1). The linear momentum and the Poisson–Boltzmann equations are solved analytically in tandem in the triangular region with the help of the Helmholtz eigenvalue problem and Laplace transforms. The analytical solution developed is exact. The solution technique used is new, leads to exact solutions, is completely different from those available in the literature, and applies to other similar problems. The new expression for the velocity field displays experimentally observed ‘velocity overshoot’ as opposed to existing analytical studies none of which can predict the overshoot phenomenon. We show that when Caputo–Fabrizio time-fractional derivatives approach unity the exact solution for the classical upper convected Maxwell fluid is obtained. The presence of elasticity in the constitutive structure alters the Newtonian velocity profiles drastically. The influence of pertinent parameters on the flow field is explored.

Soret and Radiation Effects on Mixture of Ethylene Glycol-Water (50%-50%) Based Maxwell Nanofluid Flow in an Upright Channel

In this article, ethylene glycol (EG) + waterbased Maxwell nanofluid with radiation and Soret effects within two parallel plates has been investigated. The problem is formulated in the form of partial differential equations. The dimensionless governing equations for concentration, energy, and momentum are generalized by the fractional molecular diffusion, thermal flux, and shear stress defined by the Caputo–Fabrizio time fractional derivatives. The solutions of the problems are obtained via Laplace inversion numerical algorithm, namely, Stehfest’s. Nanoparticles of silver (Ag) are suspended in a mixture of EG + water to have a nanofluid. It is observed that the thermal conductivity of fluid is enhanced by increasing the values of time and volume fraction. The temperature and velocity of water-silver nanofluid are higher than those of ethylene glycol (EG) + water (H2O)-silver (Ag) nanofluid. The results are discussed at 2% of volume fraction. The results justified the thermo-physical characteristics of base fluids and nanoparticles shown in the tables. The effects of major physical parameters are illustrated graphically and discussed in detail.

Analysis of tumor cells in the absence and presence of chemotherapeutic treatment: The case of Caputo-Fabrizio time fractional derivative

In this work, a study of the mathematical model of uncontrolled growth of tumor cells in the presence of chemotherapy is proposed. The fractional form of the model with the non-singular exponential kernel is considered. This model consists of four coupled partial differential equations (PDEs) which depict the relationship among the normal, tumor, immune cells, and the chemotherapy parameter. The purpose is to investigate the behavior of all types of cells with a change in the fractional order parameter and to show the effect of chemotherapeutic treatment on tumor cells with different levels of immune system. Before applying the proposed numerical method, an approximate expression for the fractional-order Caputo-Fabrizio (C-F) derivative of polynomial function xk is derived. A shifted Chebyshev operational matrix of fractional order derivative is deduced by using an approximation of the C-F fractional derivative of the function and some properties of the orthogonal polynomials. The system of four coupled fractional order PDEs is studied by using the operational matrix method. The dynamics of all the aforementioned cells with respect to different fractional order derivatives are derived and computed numerically for the prescribed values of parameters. These are depicted through graphs to study the diffusive nature of cells and the effect of chemotherapy on all types of cells, before and after applying the therapy. This study shows that tumor cell growth decreases with time when chemotherapy treatment is started. The concentration of tumor cells is more in the invasive fronts of the tumor site as compared to the center of the tumor. It is concluded that the growth of tumor cells is less due to chemotherapy treatment for a person with a strong immune system.

Analysis of Caputo-Fabrizio fractional order semi-linear parabolic equations via effective amalgamated technique

The significance of semi-linear parabolic equations in various fields of physics and chemistry is perpetual. Literature is enriched with the modeling and numerical investigations of their various paradigms. In this paper, a class of semi-linear diffusion equations is considered as prototypical semi-linear parabolic equation. The equations are reformulated to fractional order derivative by applying Caputo-Fabrizio time fractional derivative (CFTFD). Moreover, an amalgamated technique, that is, a semi-analytical technique is also established, which is combination of Laplace transform and Picard’s iterative method (LTPIM). Specifically, it is designed to effectively simulate the governing semi-linear diffusion equations. In addition, the stability analysis of this amalgamated technique is also carried out through comparison with Banach fixed point theorem and H-stable mapping. The obtained results are illustrated graphically and in tabulated form, which evidently validates the proficiency of this technique for semi-linear parabolic equations.

Numerical simulation of advection–diffusion equation with caputo-fabrizio time fractional derivative in cylindrical domains: Applications of pseudo-spectral collocation method

The study of the unsteady fractional advection–diffusion equation (ADE) is carried out in cylindrical geometry along with time-exponential concentration on a cylindrical surface. We have used the Caputo-Fabrizio time-fractional derivative for the fractional model of the advection–diffusion. The analytical solutions for the solute concentration are determined by using integral transformations. For comparison, we also present a numerical scheme to get a numerical solution for different parameters' values. For time ordinary derivative approximation, we use the finite difference method. For space derivatives, we use a pseudo-spectral collocation method for higher-order accuracies. The advection-diffusion’ classical model is obtained by taking the fractional parameter to be unit. The influence of the memory, namely, the Caputo-Fabrizio time-fractional derivative on the solute concentration is studied and compared with the ordinary case. Also, the impact of the drift velocity is analyzed by employing the Peclet number. It has been found that the concentration is decreasing with the Peclet number and is increasing with the radial coordinate. The present investigation will be helpful in future research to use a higher-order approximation for ordinary derivatives. As the derivatives in space are ordinary derivatives, we use highly accurate pseudo-spectral collocation approximation for them.

Electroosmotic flow of generalized Burgers’ fluid with Caputo–Fabrizio derivatives through a vertical annulus with heat transfer

Applying the electric field to a fluid confined between capillary surfaces is the most recent technique used for studying the fluid movement. This technique is known as electroosmotic flow (EOF). The problem of the Caputo–Fabrizio time-fractional derivative of the electroosmotic generalized Burgers’ fluid through a vertical annulus with free convection heat transfer has been investigated. The annulus walls kept at constant values of a zeta potential. The dimensionless governing equations have been solved with the help of the Laplace and finite Hankel transforms. The inversion of Laplace for the transformed functions is calculated numerically. The essential features of the electroosmotic flow (EOF) and the related thermal characteristics are clearly illustrated by the variations in the velocity, the flow rate, the temperature and the Nusselt number. Moreover, comparisons with other non-Newtonian fluids have been discussed. It was found that the presence of the electric double layer (EDL) accelerates the fluid near the walls of the annulus, while in the core region, the reverse flow is noticed.

MHD effects on the channel flow of a fractional viscous fluid through a porous medium: An application of the Caputo-Fabrizio time-fractional derivative

This research article considers the exact solutions and theoretical aspects of the channel flow of a fractional viscous fluid which is electrically conducting and flowing through a porous medium. Joint Laplace and Fourier transform techniques are used to solve the momentum equation. The Caputo-Fabrizio time fractional derivative is used in the constitutive equations. Exact solutions for an arbitrary velocity are obtained, and then in the limiting cases over a bottom plate three types of flow are considered: that is, the impulsive, accelerating and oscillating motion of the fluid. The case where the flow of the fractional fluid is unaffected by the side walls, is correspondingly taken into account. For oscillating flow the solutions are separated into steady and transient parts for both sine and cosine oscillations. Moreover these solutions are captured graphically, and the effect of the Reynolds number “Re”, fractional parameter “α”, effective permeability “Keff” and the time “t”, on the fluid's motion are observed.

A Time Fractional Model of Generalized Couette Flow of Couple Stress Nanofluid With Heat and Mass Transfer: Applications in Engine Oil

The aim of this study is to obtain the closed form solutions for the laminar and unsteady couple stress fluid flow. The fluid is allowed to flow between two infinite parallel plates separated by distance l. Moreover, we have considered that the lower plate is moving with uniform velocity U 0 and upper plate is stationary. For this purpose, engine oil is taken as a base fluid and to enhance the efficiency of lubricating oil, Molybdenum disulphide nanoparticles are dispersed uniformly in the engine oil. The flow is formulated mathematically in terms of partial differential equations of order four. Furthermore, the derived system of partial differential equations are fractionalized by using the mostly used definition of Caputo-Fabrizio time fractional derivative. The more general exact solutions for velocity, temperature and concentration distributions are obtained by using the joint applications of Fourier and the Laplace transforms. The effect of different parameters of interest of the obtained general solutions are discussed by sketching graphs. Furthermore, substituting favorable limits of different parameters, four different limiting cases are recovered from our obtained general solutions i.e. (a) Couette flow (b) Classical couple stress fluid (c) Newtonian viscous fluid and (d) in the absence of thermal and concentration. Moreover, the effect of different physical parameters on the velocity, temperature and concentration distributions are discussed graphically. It is worth noting that couple stress parameter corresponds to a decrease in the velocity profile. In order to observe the differences clearly, all the figures are compared for integer order and fractional order which provide a more realistic approach as compared to the classical model. Additionally, skin friction is calculated at lower as well as upper plate. Nusselt number and Sherwood number are also tabulated. It is noticed that the rate of heat transfer of engine oil can be enhanced up to 12.38% and decrease in mass transfer up to 2.14% by adding Molybdenum disulphide nanoparticles in regular engine oil.

The effects of coupled heat and mass transfer in the fractional Jeffrey fluid over inclined plane

The flow of fractional Jeffrey fluid due to time-dependent motion of plate has been investigated. An inclined plate is considered, and thermal effects with mass diffusivity in flow are also taken into account. The set of governing equations is operated by integral transform namely Laplace transform technique. Caputo–Fabrizio time-fractional derivative is considered in energy and diffusion equations. Moreover, exact analytical solutions are achieved. In the limiting cases, the general solutions for accelerating and oscillating motion of Jeffrey fluid are obtained. Effects of involved physical parameters, fractional parameter $$\alpha$$, inclination of plate $$\theta$$, Prandtl number Pr and Schmidt number Sc on the flow, are graphically detected.

Two-Dimensional Advection–Diffusion Process with Memory and Concentrated Source

Two-dimensional advection–diffusion processes with memory and a source concentrated in the symmetry center of the domain have been investigated. The differential equation of the studied model is a fractional differential equation with short-tail memory (a differential equation with Caputo–Fabrizio time-fractional derivatives). An analytical solution of the initial-boundary value problem has been determined by employing the Laplace transform and double sine-Fourier transforms. A numerical solution of the studied problem has been determined using finite difference approximations. Numerical simulations for both solutions have been carried out using the software Mathcad.