Fractional model for MHD flow of Casson fluid with cadmium telluride nanoparticles using the generalized Fourier’s law

The present work used fractional model of Casson fluid by utilizing a generalized Fourier’s Law to construct Caputo Fractional model. A porous medium containing nanofluid flowing in a channel is considered with free convection and electrical conduction. A novel transformation is applied for energy equation and then solved by using integral transforms, combinedly, the Fourier and Laplace transformations. The results are shown in form of Mittag-Leffler function. The influence of physical parameters have been presented in graphs and values in tables are discussed in this work. The results reveal that heat transfer increases with increasing values of the volume fraction of nanoparticles, while the velocity of the nanofluid decreases with the increasing values of volume fraction of these particles.


List of symbols µ
Dynamic viscosity e ij The i, j th component of deformation rate p y The yield stress of non-Newtonian fluid π = e ij e ij The product of the component of deformation rate itself π c The critical value of this product based on the non-Newtonian model µ γ The plastic dynamic viscosity u The fluid velocity in the x-direction The temperature ρ The fluid density γ C The material parameter of Casson fluid β � The thermal expansion coefficient g The acceleration due to gravity c p The specific heat capacity of fluids k The thermal conductivity In many engineering and industrial sectors, heat transport is an essential technical subject and becomes a challenge for engineers and manufacturers. To overcome this challenge, the one approach is to surge the available surface area of heat exchange. This technique leads to an unrealistic and unacceptable methodology for the increase in heat transfer in the heat managing systems. Engineers and industrialists face this issue due to the poor thermo-physical properties of conventional fluids such as water, alcohol, ethylene glycol, and oil. Therefore, there is an imperative demand to enhance the thermal conductivity of such fluids to overcome heat transport problems 1 . Nanofluids are used in different engineering and industrial sectors to overcome the heat transfer problems in the conventional fluids. Casson fluid is one on the important industrial fluids as it has tremendous properties and applications.

MHD flow of Casson fluid in a porous media. Computational analysis of temperature velocities and
skin friction coefficient for an incompressible Casson fluid was examined in a porous media with magnetohydrodynamic (MHD) boundary layer conditions. The temperature increases with an increase in the heat generation parameter, and the higher Casson fluid parameter is associated with the skin friction coefficient 14 . The mathematical investigation of Casson fluid's heat-absorbing and chemically reacting flow (clay or drilling mud) was examined both on a flat plate and vertical cone with a non-Darcy porous medium. The attributes of moving fluid were analyzed, which influence the concentration, velocities, temperature parameters, and average skinfraction values 15 . The behavior of the MHD flow of Casson fluid over a vertical plate was studied under constant temperature and wall shear conditions in the porous media. The velocity values are higher in the proximity of the plate with higher values of the Casson parameter while the velocity decreases away from the plate. The velocity has inversely related to shear wall stress, while it is directly related to the magnetic parameter 16 . The non-Newtonian flow of Casson fluid on an oscillating plate under the constant wall temperature by applying the Laplace transformation. The velocity field is reduced under the effect of the slip parameter 17 . The numerical analysis of heating and viscous effects under the constant temperature condition was discussed with homotopy solutions. The skin friction coefficient is inversely related to the Hartman number and the Casson parameter 18 . The homotopy analysis method (HAM) also verifies that the parameters of thermal conduction and viscosity for an incompressible Casson fluid are a linear function of temperature in the boundary layer conditions. An increase in the viscosity of Casson fluid results in a reduction of temperature with higher values of the velocity profile. The rate of heat transfer is significantly decreased in the presence of the magnetic field 19 . The phenomenon of heat generation in Casson fluid flow with temperature and concentration parameters was discussed by applying the fractional derivatives. The variation in velocity is associated with time values, as evident from exact solutions. Temperature and velocities are positively linked with the heat generation parameter, while fluid velocity is inversely related to the chemical reaction parameter 20 . The Lagrangian equation computationally analyzed the dynamics of submarine debris flow in viscoplastic fluids to compare various rheological models. The downslope movement of high-density fluid was discussed, keeping the fluid volume constant to describe the transition of fluid between viscous and plastic nature of flow 21 . The numerical analysis of viscosity and yield stress parameters for the rheology of submarine debris flows was incorporated using the plastic Bingham model. The yield surface is widely determined by the shear rate and viscosity of fluid 22 . The flow of MHD Casson fluid in a non-Darcy porous media was discussed with the transformation of the boundary layer equation to the differential equation. Magnetic and Casson parameters significantly influence the concentration, velocity, concentration, and skin friction. The higher values of temperature and concentration are directly related to the magnetic parameter. Casson parameter is directly associated with skin friction, while the magnetic parameter is inversely correlated with skin friction 23,24 . The flow of viscoelastic incompressible fluid through a uniform magnetic field over an infinite accelerated plate through a porous medium. Laplace transformation technique (LTT) was applied to study the velocity parameter and skin friction. The velocity of fluid has positively influenced by elasticity and permeability, while skin friction also increases with an increase in medium permeability 24  (a) Enhanced thermal conductivity and heat transfer was achieved by dispersing nanofluids in engine oil (b) The performance of the cooling system can be attained at a low volume fraction of nanoparticles (< 1%) 5 Zhang et al. 38 Nano-graphite (a) Nano-graphite was added to the heavy-duty diesel engine, and its performance was investigated (b) Around a 3% volume fraction of nano-graphite increased the cooling effect up to 15%  Qiu 44 Ni nanoparticles (a) The load-carrying capacity was improved by the addition of Ni nanoparticles (b) Lower concentrations of Ni particles gave a better anti-wear performance, below 1% (c) The value of the friction coefficient is smaller when the concentration is between 0.2 and 0.5 12 Wong and Leon 45 Al nanoparticles (a) The addition of nanoparticles with diesel fuel increased the total combustion heat (b) The concentration of smoke and nitrous oxide decreased in the emission 13 Asadi and Pourfattah 46 ZnO, MgO (a) The viscosity and thermal conductivity have been studied over the temperature range (15-55 °C) and concentration (0.125-1.5%) (b) Thermal conductivity and viscosity showed an increasing trend as the temperature and concentration increased (c) The maximum enhancement was 28% and 32% for ZnO and MgO, respectively (d) The increase in dynamic viscosity took place at 55 °C and 1.5% by just over 124% and 75% for ZnO and MgO, respectively (e) None of these fluids are suitable for the laminar flow regime 14 Hu et al. 47 Graphite nanoparticles (a) Three critical properties were studied, including temperature, particle volume fraction, and the shear rate (b) Temperature behaved as an essential factor affecting viscosity as compared to volume fraction (c) The nanofluid behaved as a Newtonian (constant viscosity) if the shear rate is 17-68 s −1 , but it gave non-linear behavior in the case of 667-3333 s −1 15 Soltani et al. 48  Fractional calculus. In the logic we differentiate or integrate a function once, twice, or whole number of times, differentiation and integration are normally considered as discrete processes in general. In some instances, though, the assessment of a non-integer order derivative is helpful. The definition of fractional computation is not new. In a letter to L'Hospital in 1695 Leibniz created an opportunity to generalize differentiation to noninteger order 54 . These, however, were Liouville, Abel, Heaviside, and Riemann's contributions which progressed fractional derivatives theory [55][56][57][58] . The fractional calculus provides more general and precise models of physical systems than ordinary calculus in many fields, for example chemistry, mechanics and biotechnology [59][60][61][62] . Fractional derivatives are also used for mathematical modeling of electric circuits, electromagnet theory and fractal theory [63][64][65] .

Mathematical modelling
We have considered the motion of Casson nanofluid is a vertical channel embedded in a porous media. The flow is assumed to be in the direction of x-axis while the y-axis is taken perpendicular to the plates. With ambient temperature 1 , both the fluid and plates are at rest when t ≤ 0 . At t = 0 + , the plate at y = d begin to move in its plane with velocity Uh(t) as shown in Fig. 1. At y = d , the plate temperature level raised to We suppose that the rheological equation for an incompressible Casson fluid is 66 The free convection flow of Casson nanofluid along with heat and mass transfer and using the well-known Boussinesq's approximation is governed by the following partial differential equations 68,69 : For the properties of the nanofluids with a subscript nf , refer to 70 . The thermophysical properties of nanoparticles and base fluid are given in Table 1.
In the dimensionless form the initial and boundary conditions are:  into Eqs. (2), (3), (4) and (5) we get: To obtain the more suitable form of the Eq. (15) we recall the time fractional integral operator This is the inverse operator of the derivative operator C ℘ α t (.) . Using the properties from Eq. (12) we have r y, t = η α * · r (t) = C ℘ α t r y, t , Eq. (15) can be written as:

Solution of the problem
The derived fractional is solved using the new defined mathematical setting and the integral transforms.
Energy field. Using the following transformation

Results and discussion
The exact solutions for the MHD flow of Casson nanofluid in a channel embedded in a porous media with heat transfer are obtained in this study. The associated energy equation is fractionalized using generalized Fourier's law. The obtained exact solutions are plotted through graphs, and the effects of different physical parameters on the flow and heat transfer are presented.
The variations in the nanofluid velocity for different values of the fractional parameter are displayed in Fig. 2. From this figure, it is noticed that four different velocity profiles are obtained for four different values of fractional parameter keeping all the other physical parameters constant. This shows that the fractional parameter significantly influences the obtained solutions, even this is not a physical parameter and is a purely mathematical parameter. These variations are due to the memory effect, which cannot be studied through integers order derivatives. These variations are also presented in Table 2 for the ease of numerical and experimental solvers.
An increasing trend is noticed in the velocity of the Casson nanofluid for increasing values of the Casson fluid parameter in Fig. 3 and Table 3. Physically, the viscosity of the fluid is increased for smaller values of the Casson fluid parameter. Another impressive result can be drawn through this graph that Casson fluid is more viscous than Newtonian fluid and when γ C → ∞ , the fluid behaves like a Newtonian fluid.
This study considered engine oil as a base fluid and Cadmium Telluride (CdTe) as nanoparticles. Figure 4 is drawn to show the effect of the volume fraction of nanoparticles on the fluid velocity. The fluid velocity is decreasing with the higher values of the volume fraction of nanofluid. This means the fluid will become more viscous with the addition of nanoparticles, and as a result, the lubrication of the engine oil will be improved. For the interest of the readers, Table 4 is also presented for the same phenomenon. Figure 5 and Table 5 are presented to show the influence of thermal Grashof number on the fluid velocity. Grashof number is the ratio of buoyancy forces to the viscous forces. The greater values of Grashof number means higher buoyancy forces, and hence the velocity is increasing with the higher values of Gr.
In this study, the MHD flow is considered. The velocity profile shows a decreasing trend for increasing vales of Hartman number in Fig. 6 and Table 6. Physically, higher values of M mean greater Lorentz forces flow opposing forces and control the flow of fluid. In Fig. 6, the velocity profile for non-MHD flow is also drawn when M = 0.
Prandtl number is the ratio of viscous forces to the thermal forces. The greater values of Prandtl number result in the higher viscous forces and weaker thermal forces and, as a result, decelerate the flow of the nanofluid. This phenomenon is described in Fig. 7 and Table 7.
The effect of the Darcey number (Permeability parameter) is shown on the velocity profile in Fig. 8 and Table 8. The velocity is increasing with the increasing values of K . The greater values of K means, the higher permeability of the media, and hence the media will allow the fluid to move fast and smoothly.
. Figure 2. Influence of the fractional parameter on the nanofluid velocity.  Fig. 9 and Table 9. From this figure, it is noticed that four different temperature profiles are obtained for four different values of fractional parameter keeping all the other physical parameters constant. This is showing that the fractional parameter has a significant influence on the obtained solutions; even this is not a physical parameter and is a purely mathematical parameter. These variations are due to the memory effect, which cannot be described through the integer order derivatives model.
Engine oil as a base fluid and Cadmium Telluride nanoparticles are considered in this analysis. Figure 10 is drawn to show the effect of the volume fraction of nanoparticles on the temperature profile. The temperature is     Table 10.