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

2020 ◽  
Vol 98 (1) ◽  
pp. 89-101 ◽  
Author(s):  
Nauman Raza ◽  
Muhammad Asad Ullah
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Maxwell Fluid ◽  
Heat Transfer Analysis ◽  
Flow Parameters ◽  
Fluid Parameter ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Temperature And Velocity Fields

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.

scholarly journals Heat Transfer Analysis for Viscous Fluid Flow with the Newtonian Heating and Effect of Magnetic Force in a Rotating Regime

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Rashid Ayub ◽  
Shahzad Ahmad ◽  
Muhammad Imran Asjad ◽  
Mushtaq Ahmad
Keyword(s):  
Viscous Fluid ◽  
Fluid Motion ◽  
Velocity Fields ◽  
Heat Transfer Analysis ◽  
Convection Flow ◽  
Newtonian Heating ◽  
Flow Parameters ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Temperature And Velocity Fields

In this article, an unsteady free convection flow of MHD viscous fluid over a vertical rotating plate with Newtonian heating and heat generation is analyzed. The dimensionless governing equations for temperature and velocity fields are solved using the Laplace transform technique. Analytical solutions are obtained for the temperature and components of velocity fields. The obtained solutions satisfy the initial and boundary conditions. Some physical aspects of flow parameters on the fluid motion are presented graphically.

A Note on Exact Solutions for the Unsteady Free Convection Flow of a Jeffrey Fluid

2015 ◽  
Vol 70 (6) ◽  
pp. 397-401 ◽  
Author(s):  
Ilyas Khan
Keyword(s):  
Shear Stress ◽  
Free Convection ◽  
Exact Solutions ◽  
Newtonian Fluid ◽  
Numerical Solutions ◽  
Convection Flow ◽  
Jeffrey Fluid ◽  
Free Convection Flow ◽  
Laplace Transform Technique ◽  
The Laplace Transform

AbstractIn this note, we investigate the unsteady free convection flow of a Jeffrey fluid past an infinite isothermal vertical plate. Exact solutions are obtained using the Laplace transform technique. These solutions are expressed in terms of exponential and complementary error functions, and satisfy all imposed initial and boundary conditions as well as the governing equations. The expression for the shear stress is also evaluated. The corresponding solutions for a Newtonian fluid can be easily obtained as a special case. It is found from the velocity and shear stress solutions that they strongly depend on the material parameters of a Jeffrey fluid. The exact solutions obtained here can be used as a benchmark for checking the correctness of other approximate or numerical solutions. In addition, this note will help in understanding the characteristics of non-Newtonian fluid flows that are subject to free convection due to buoyancy force.

scholarly journals Heat Transfer Analysis For Oscillating Flow of Magnetized Fluid By Using The Modified Prabhakar-Like Fractional Derivatives

2021 ◽  
Author(s):  
Ali Raza ◽  
Sami Ullah Khan ◽  
M. Ijaz Khan ◽  
Essam Roshdy El-Zahar
Keyword(s):  
Heat Transfer ◽  
Fractional Derivatives ◽  
Velocity Fields ◽  
Heat Transfer Analysis ◽  
Solution Approach ◽  
Flow Parameters ◽  
And Behavior ◽  
Temperature And Velocity Fields ◽  
Oscillating Plate ◽  
With Memory

Abstract In this analysis, an unsteady and incompressible flow of magnetized fluid in presence of heat transfer has been presented with fractional simulations. The oscillating plate with periodically variation has induced the flow. The model is formulated in terms of partial differential equations (PDE’s). The traditional PDEs cannot analyze and examine the physical behavior of flow parameters with memory effects. On this end, the solution approach is followed with the efficient mathematical fractional technique namely Prabhakar fractional derivative. The non-dimensional leading equations are transformed into the fractional model and then solved with the help of the Laplace transformation scheme. The effects and behavior of significant physical and fractional parameters are analyzed graphically and numerically. As a result, we have concluded that the temperature and velocity profiles decrease with the enhancement of fractional parameters. Furthermore, with time both (temperature and velocity fields)decreasing away from the plate and asymptotically increases along y-direction, which also satisfies the corresponding conditions.

Exact Solutions for Unsteady Magnetohydrodynamic Oscillatory Flow of a Maxwell Fluid in a Porous Medium

2013 ◽  
Vol 68 (10-11) ◽  
pp. 635-645 ◽  
Author(s):  
Ilyas Khan ◽  
Farhad Ali ◽  
Sharidan Shafie ◽  
Keyword(s):  
Steady State ◽  
Exact Solutions ◽  
Mhd Flow ◽  
Maxwell Fluid ◽  
Transform Method ◽  
Transient Solutions ◽  
The Laplace Transform ◽  
Porous Half Space ◽  
The Given ◽  
Steady State Solutions

In this paper, exact solutions of velocity and stresses are obtained for the magnetohydrodynamic (MHD) flow of a Maxwell fluid in a porous half space by the Laplace transform method. The flows are caused by the cosine and sine oscillations of a plate. The derived steady and transient solutions satisfy the involved differential equations and the given conditions. Graphs for steady-state and transient velocities are plotted and discussed. It is found that for a large value of the time t, the transient solutions disappear, and the motion is described by the corresponding steady-state solutions.

scholarly journals Enhanced heat transfer in working fluids using nanoparticles with ramped wall temperature: Applications in engine oil

2019 ◽  
Vol 11 (11) ◽  
pp. 168781401988098 ◽  
Author(s):  
Muhammad Arif ◽  
Farhad Ali ◽  
Nadeem Ahmad Sheikh ◽  
Ilyas Khan
Keyword(s):  
Molybdenum Disulfide ◽  
Wall Temperature ◽  
Base Fluid ◽  
Maxwell Fluid ◽  
Graphical Analysis ◽  
Engine Oil ◽  
Mathematical Expressions ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Graphene Nanoparticles

The purpose of this article is to investigate the flow of Maxwell fluid with nanoparticles, that is, molybdenum disulfide and graphene with ramped temperature condition at the boundary, and engine oil is considered as base fluid. Furthermore, molybdenum disulfide and graphene nanoparticles are uniformly distributed in the base fluid. The problem is modeled in terms of partial differential equations with physical initial and boundary conditions. To make the system of governing equations dimensionless, we introduced some suitable non-dimensional variables. The obtained dimensionless system of equations is solved using the Laplace transform technique. From graphical analysis, it can be noticed that the velocity is high with isothermal wall temperature and lower for ramped wall temperature. These solutions are verified by comparing with the well-known published results. In addition, the physics of all parameters of interest is discussed through graphs. The mathematical expressions for skin friction and Nusselt number are mentioned and the obtained results are presented in tabular form. Finally, the effect of molybdenum disulfide and graphene nanoparticles is briefly discussed for the flow and heat profiles for Maxwell nanofluid.

scholarly journals On applications of Caputo k-fractional derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ghulam Farid ◽  
Naveed Latif ◽  
Matloob Anwar ◽  
Ali Imran ◽  
Muhammad Ozair ◽  
...  
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Chebyshev Inequality ◽  
Absolutely Continuous ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Type Variable

Abstract This research explores Caputo k-fractional integral inequalities for functions whose nth order derivatives are absolutely continuous and possess Grüss type variable bounds. Using Chebyshev inequality (Waheed et al. in IEEE Access 7:32137–32145, 2019) for Caputo k-fractional derivatives, several integral inequalities are derived. Further, Laplace transform of Caputo k-fractional derivative is presented and Caputo k-fractional derivative and Riemann–Liouville k-fractional integral of an extended generalized Mittag-Leffler function are calculated. Moreover, using the extended generalized Mittag-Leffler function, Caputo k-fractional differential equations are presented and their solutions are proposed by applying the Laplace transform technique.

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

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.

scholarly journals Semianalytical Solutions of Some Nonlinear-Time Fractional Models Using Variational Iteration Laplace Transform Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Javed Iqbal ◽  
Khurram Shabbir ◽  
Liliana Guran
Keyword(s):  
Laplace Transform ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Fractional Partial Differential Equations ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Variational Iteration ◽  
The Laplace Transform ◽  
Fractional Models ◽  
Iteration Technique

In this work, we combined two techniques, the variational iteration technique and the Laplace transform method, in order to solve some nonlinear-time fractional partial differential equations. Although the exact solutions may exist, we introduced the technique VITM that approximates the solutions that are difficult to find. Even a single iteration best approximates the exact solutions. The fractional derivatives being used are in the Caputo-Fabrizio sense. The reliability and efficiency of this newly introduced method is discussed in details from its numerical results and their graphical approximations. Moreover, possible consequences of these results as an application of fixed-point theorem are placed before the experts as an open problem.

New Exact Solutions for Stokes First Problem of a Generalized Jeffreys Fluid in a Porous Half Space

2013 ◽  
Vol 477-478 ◽  
pp. 246-253
Author(s):  
Xiaoyi Guo
Keyword(s):  
Exact Solutions ◽  
Half Space ◽  
Fluid Model ◽  
Maxwell Fluid ◽  
Constitutive Relationship ◽  
New Exact Solutions ◽  
Fourier Sine Transform ◽  
The Laplace Transform ◽  
Porous Half Space ◽  
Relationship Of

The fractional calculus approach has been taken into account in the Darcys law and the constitutive relationship of fluid model. Based on a modified Darcys law for a viscoelastic fluid, Stokes first problem is considered for a generalized Jeffreys fluid in a porous half space. By using the Fourier sine transform and the Laplace transform, two forms of exact solutions of Stokes first problem for a generalized Jeffreys fluid in the porous half space are obtained in term of generalized Mittag-Leffler function, and one of them is presented as the sum of the similar Newtonian solution and the corresponding non-Newtonian contributions. As the limiting cases, solutions of the Stokes first problem for the generalized second fluid, the fractional Maxwell fluid and the Newtonian fluid in the porous half space are also obtained.

Applications of fractional derivatives to nanofluids: exact and numerical solutions

2018 ◽  
Vol 13 (1) ◽  
pp. 2 ◽  
Author(s):  
Sidra Aman ◽  
Ilyas Khan ◽  
Zulkhibri Ismail ◽  
Mohd Zuki Salleh
Keyword(s):  
Nusselt Number ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Volume Fraction ◽  
Physical Processes ◽  
Carrier Fluid ◽  
Laplace Transform Technique ◽  
Classical Models ◽  
Nanoparticles Volume Fraction ◽  
Graphene Nanoparticles

In this article the idea of time fractional derivatives in Caputo sense is used to study memory effects on the behavior of nanofluids because some physical processes complex visco-elasticity, behavior of mechatronic and rheology are impossible to described by classical models. In present attempt heat and mass transfer of nanofluids (sodium alginate (SA) carrier fluid with graphene nanoparticles) are tackled using fractional derivative approach. Exact solutions are determined for temperature, concentration and velocity field, and Nusselt number via Laplace transform technique. The obtained solutions are then expressed in terms of wright function or its fractional derivatives. Numerical solutions for velocity, temperature, concentration and Nusselt number are obtained using finite difference scheme. It is found that these solutions are significantly controlled by the variations of parameters including thermal Grashof number, fractional parameter and nanoparticles volume fraction. It is observed that rate of heat transfer increases with increasing nanoparticles volume fraction and Caputo time fractional parameters.

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