Abstract
In many flow phenomenons of fluid with medium molecular weight, the energy flux is effected due to the inhomogeneity of concentration of mass. This contribution of the concentration to the energy flux is considered as diffusion-thermo effect or Dufour effect. In this research article the diffusion-thermo effect is addressed for the magnetohydrodynamics (MHD) flow of Jeffrey’s fractional fluid past an exponentially accelerated vertical plate with generalized thermal and mass transports through a porous medium. For the generalization of the thermal and mass fluxes the constant proportional Caputo (CPC) fractional derivative is utilized. The governing of this generalized flow are reduced to non-dimensional forms and then solved semi analytically by Laplace transform. In additions the physical aspects of flow and material parameters especially the effect of and fractional parameters are discussed by sketching the graphs. From the graphical illustration, it is concluded that in the presence of Dufour effect flow speeds up. Moreover, a comparison between fractionalized and ordinary velocity fields is also drawn and it is also observed that fractional model with constant proportional derivative is of the more decaying nature as compare to the model contracted with classical Caputo and Caputo fractional derivatives.
1. Introduction
The non-Newtonian fluids posses the diverse nature from Newtonian fluids due to their complex rheological properties. Now a days the study of non-Newtonian becomes a popular research area due to its scientific and technological applications in the processing industry, and biological sciences, like making of plastic sheets, lubricant’s performance and motion of biological fluid. There are several non-Newtonian fluid models have been presented to demonstrate the distinction between Newtonian and non-Newtonian fluids but the Jeffrey fluid model is more efficient to demonstrate attribute of stress relaxation for memory time scale (the relaxation of time). Mohd-Zin et al. [1] studied the porosity effect on unsteady MHD free convection flow of Jeffrey fluid past an oscillating vertical plate with ramped wall temperature. Bajwa et al. [2] solved a problem of transient flow of Jeffrey fluid semi-analytically over permeable wall. Asgir et al. [3] discussed the heat transfer analysis of channel flow of MHD Jeffrey fluid with porosity. The model on Jeffrey fluid be simplest and most popular. Some of the work on Jeffrey fluid are of Das [4] and Qasim [5]. Ali et al. [6] discussed the magnetohydrodynamic fluctuating free convection flow of incompressible electrically conducting viscoelastic fluid in a porous medium in the presence of a pressure gradient. Shah et al. [7] worked on the new semi analytical technique for the solution of fractional Order Navier Stokes equation.
The fractional derivative is the generalization of ordinary derivative by taking the non-integer order of differentiation. Due its generalized property the fractional derivative becomes a potent tool to describe the heat and mass transfer phenomenons and has attained the attention by researchers. Chandra [8] present the MHD flow for Jeffrey fluid past an inclined porous plate by applying Laplace transform method. Jameel et al. [9] obtain the analytic solutions for the incompressible unsteady flow of fractionalized MHD Jeffrey fluid over an accelerating porous plate with linear slip effect by using Caputo fractional derivative. Abro et al. [10] obtain the solution of Jeffrey fluid flow acquired by non-singular kernel (Caputo-Fabrizio) using integral transform technique. During the last decade, different generalized fractional derivatives have appeared in the literature that are derivatives of Caputo, Caputo-Fabrizio, Atangana-Baleanu [11, 12]. Jawad et al. [13] observed the behavior of Caputo time fractional model based on generalized Fourier’s and Fick’s laws for Jeffrey Nano fluid. Siddique et al. [14] studied the unsteady double convection flow of a magnetohydrodynamics (MHD) differential-type fluid flow in the presence of Dufour effect, Newtonian heating, and heat source over an infinite vertical plate with fractional mass diffusion and thermal transports.
Sandeep et al. [15] discussed the behavior of momentum and heat transfer on Jeffrey, Maxwell and Oldroyd-B nanofluids over a stretching surface in the presence of transverse magnetic field, non-uniform heat source/sink. A magnetohydrodynamic Jeffrey fluid flow with thermal and mass transfer on an infinitely rotating upright cone investigated by Saleem et al. [16]. Sulochana et al. [17] solved a problem of a MHD radiative Casson fluid flow numerically over a wedge to analyze the heat and molecular transfer. A bio convective nanoliquid flow with the effect of second order slip and chemical reaction between the parallel plates studied by Acharya et al. [18]. The radiative couple stress fluid and chemically reactive nanofluid over a stretched cylinder with magnetic effect analyzed by Acharya et al. [19, 20]. Riaz et al. [21] obtained the solution of peristaltic flow of Prandtl fluid by using homotopy perturbation method. The role of hybrid nanoparticles in thermal performance of peristaltic flow of Eyring-Powell fluid model have discussed by Riaz et al. [22].
Different types of fractional operators has discussed by Yuri-Luchko [23]. Baleanu et al. [24] presented a new fractional operator combining proportional Caputo and solve different kind of example with CPC derivative. Chu et al. [25] worked on fractional model of Second grade fluid induced by generalized thermal and molecular fluxes with constant proportional Caputo. Siddique et al. [26] analyzed the blood liquor model fractionalized with constant proportional Caputo fractional derivative and Dolat et al. [27] investigate a fractional model of MHD viscous fluid with heat transfer by using the constant proportional Caputo fractional derivative.
The purpose of this article is to utilized the Constant proportional Caputo fractional derivative for the investigation of diffusion-thermo effect on MHD flow of Jeffrey fluid past an exponentially accelerated vertical plate in the presence of chemical reaction, and heat generation through a porous medium with generalized heat and mass fluxes. Constant proportional Caputo fractional derivative is considered for the generalization of constitutive equations for heat and mass fluxes. Initially, the proposed governing equations are reduced to non-dimensional form then solved semi analytically via Laplace transform. The physical aspects of flow regarding involved parameters are also discussed by sketching some graphs for velocity field of Jeffrey fluid. Moreover the obtained result for velocity with CPC will compare with ordinary result by sketching the graphs.
2. Mathematical Model
The flow of Jeffrey fluid is studied in the presence of MHD by considering the diffusion-thermo effect through a porous media. The rectangular coordinates system is oriented in such a way that the -axis is pointing along the plate in the vertical direction and -axis is pointing normal to the plate as shown in Figure 1. At the time the system with the plate and fluid is suppose to be at rest, at the temperature with concentration . However, for the time , the plate suppose to be moves with exponential velocity in its own plane. The plate’s temperature as well as concentration of the fluid raised to and respectively. Magnetic field effect is also considered normally with a constant strength . Energy flux due to concentration gradient is also considered. In view of above assumptions, the Jeffrey fluid flow model with Boussinesq’s approximation appears in the following form [3, 13, 28]
Thermal Eq. is
Fourier’s Law states that [29]
Diffusion Eq. is
Fick’s Law states that, is written as [30]
The initial as well as boundary conditions of the flow problem are [20]
The dimensionless form of the flow parameters are
Using non-dimensional variables of equation (9) into equations (1-8), we have
where Pr, , K, Sc, , and represent the Prandtl number, Jeffrey parameter, non-dimensional permeability, Schmidt number, chemical reaction, and magnetic parameter respectively.
3. Generalization of Model
Eq. (10) can be written as
Fractional form of Fourier’s law [25, 31] is used to generalize Eq. (3) in dimensionless form
Fractional form of Fick’s Law [25] is used to generalize Eq. (5) in dimensionless form
Using equation (12) into (2), and (3) into (3), we get
Equations (14) and (15) can be written aswhere indicates the CPC fractional derivative of [24] as
4. Solution of Problem
The formulated initial and boundary value value problem can be solved by means of the Laplace transform method.
4.1. Concentration Field
(17) be solved via Laplace transform for concentration species as
with
Put (20) into (19), and result is
This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (21).
4.2. Sherwood Number
The local coefficient of the rate of mass transfer is define in term of Sherwood number, and define by the following relation
4.3. Temperature Field
(16) be solved via Laplace transform for temperature profile as
with
Put (24) into (23), and result is
This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (25).
4.4. Nusselt Number
The local coefficient of the rate of heat transfer is define in term of Nusselt number as
4.5. Velocity field
(11) be solved via Laplace transform for velocity field as
with
This is complicated to solve analytically. Algorithm [32, 33] is used to derive the numerical result of (29).
4.6. Skin Friction
The expression of skin friction is define by using (29) as
5. Results and Discussion
In this article the diffusion-thermo effect is investigated for the magnetohydrodynamics (MHD) flow of Jeffrey’s fractional fluid past an exponentially accelerated vertical plate with generalized thermal and mass transports through a porous medium. The semi-analytical results for the concentration, temperature, and velocity fields are obtained. Moreover to see the physical effect of involved parameters, the concentration, temperature, and velocity fields are postured by some graphs.
Velocity profile is sketched in the Figure 2(a) to twilight the influence magnetic field over the fluid velocity and it is observed that fluid slows down for greater values of . In the presence of magnetic field a retarding force is created which oppose the fluid motion as depicted in Figure 2(a). The effect of porosity parameter on fluid flow is defected in Figure 2(b) and it is noted that the fluid motion enhance with increasing value of . The larger value of refers to the less Darcy resistance to the flow that is why flow speeds up with increasing value of . The influence of mass on fluid velocity is illustrate in Figures 3(a) and 3(b). It is noted that fluid motion increases as values of increasing. Physically higher values of refers the higher concentration gradients and greater the buoyancy force and more current in flow domain so velocity increases. Figure 4(a) represent the result of on fluid velocity . The fluid motion rises up with maximizing the values of , and it represents the more impact of thermal buoyancy force as compare to the viscous effect. Therefore maximizing the values of refers the higher temperature gradient due to which velocity field rises. The impact of for velocity field is presented in Figure 4(b).
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The diffusion-thermo or Dufour effect (Du) over the velocity field are discussed in the Figures 5(a) and 5(b). An enhancing flow pattern is observed due to the increasing variations of the Du. The concentration gradient give a contribute to the temperature gradient which creates an additional energy flux in the flow domain. Therefore an increasing value of the Du refer to the more energy flux which generates the more flow current that is why the flow speeds up with increasing value of Du. Moreover the negative values refer to the opposite direction of the additional energy flux therefore more negativity of Du generates the more opposite flow current in the flow domain.
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Figures 6(a) and 6(b) are drawn to see impact of Jeffrey parameters and over velocity field respectively. As parameter refer to the viscosity of the fluid therefore fluid velocity falls due to the increasing value of and the parameter refers to the elasticity of the fluid material so fluid speed up with increasing value . The influence of heat generation on is reported in Figure 7(a). An increasing value of decreases the fluid velocity as shown in Figure 7(b). The impact of on is illustrate in Figure 8(a). Here, maximizing the values of slow down the fluid motion due to decay of molecular diffusion. It is analyzed that motion of fluid increases with raising values of as depicted in Figure 8(b). The graphical behavior of , , , and on are shown in Figures 9(a), 9(b), 10(a) as well as 10(b) respectively. Figures illustrate that temperature reduces by exceeding the values of , and , and temperature increases by exceeding the values of and . The impacts of Schmidt number as well as chemical reaction on are present in Figures 11(a) as well as 11(b) respectively. Figures 12(a) and 12(b) represent the comparison of velocity and temperature distribution of present work with Naseem [20] respectively. If we take fractional parameters , and in Naseem [28], the velocity fields are identical that shows the validity of this present work. Moreover the comparison of present work with other fractional derivatives Caputo and Caputo Fabrizio used in Nazar et al. [34] and Nadeem et al. [35] respectively in the absence of , , , , are shown in Figure 12(c). If we put the fluid curves are identical as shown in Figure 12(d). From the figures, it is concluded that Jeffrey fluid with CPC fractional derivative is the best choice to enhance the fluid motion. Figures 13(a) and 13(b) represent the authenticity of inversion algorithms for temperature and concentration distributions. The velocity distributions overlap which shows the authenticity of inversion algorithms as presented in Figure 14. Skin friction, rate of heat and mass transfer can be enhanced by increasing the values of fractional parameter and presented in table 1.
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• With decreasing fractional parameter values, the velocity distribution slows down. • Thermal buoyancy forces accelerate the fluid velocity. • Fluid velocity reduces as Schmidt number Sc, chemical reaction , and magnetic parameter M rises. • The fluid velocity increased for larger values of Jeffrey parameter as well as fractional parameter. • The temperature distribution increases by the smaller values of Prandtl number Pr. •The concentration distribution decreases for greater values of Schmidt number Sc. • The concentration distribution decreases by exceeding the values of chemical reaction .
6. Conclusions
The Magnetohydrodynamics flow model of Jeffrey’s fractional fluid model has been considered with the effect of magnetic field in the porous regime. Heat generation, and radiation are also consider with effect of chemical reaction in the flow domain. This flow model is solved analytically and results for velocity, temperature, and concentration fields are constructed by using Laplace transform. Further different graphs of optimising fields are plotted to highlight the influence parameters. The key outcomes of this flow model are:
Nomenclature
| : | : fluid velocity, : Soret number |
| : | : Thermal Grashof number, : Mass Grashof number, : Nusselt number |
| : | : Schmidt number |
| : | : Jeffrey parameter |
| : | acceleration due to gravity, : Specific heat at a constant pressure, : Temperature of fluid, : Fluid temperature at plate, : Fluid temperature far away from the plate, : Concentration of fluid, : Concentration level at plate, : Concentration of the fluid far away from the plate, : Laplace transform parameter |
| : | Fluid density, : Fractional parameter |
| : | Kinematic viscosity, : Volumetric Coefficient of thermal expansion, : Volumetric Coefficient of mass expansion, . |
Data Availability
The data used to support the findings of this study are included within the article.
Additional Points
With decreasing fractional parameter values, the velocity distribution slows down. Thermal buoyancy forces accelerate the fluid velocity. Fluid velocity reduces as Schmidt number Sc, chemical reaction , and magnetic parameter M rise. The fluid velocity increased for larger values of Jeffrey parameter as well as fractional parameter. The temperature distribution increases by the smaller values of Prandtl number Pr. The concentration distribution decreases for greater values of Schmidt number Sc. The concentration distribution decreases by exceeding the values of the chemical reaction .
Conflicts of Interest
The authors declare that they have no conflicts of interest.