Time fractional analysis of channel flow of couple stress Casson fluid using Fick’s and Fourier’s Laws

This study aim to examine the channel flow of a couple stress Casson fluid. The flow is generated due to the motion of the plate at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document}y=0, while the plate at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=d$$\end{document}y=d is at rest. This physical phenomenon is derived in terms of partial differential equations. The subjected governing PDE’s are non-dimensionalized with the help of dimensionless variables. The dimensionless classical model is generalized by transforming it to the time fractional model using Fick’s and Fourier’s Laws. The general fractional model is solved by applying the Laplace and Fourier integral transformation. Furthermore, the parametric influence of various physical parameters like Casson parameter, couple stress parameter, Grashof number, Schmidt number and Prandtl number on velocity, temperature, and concentration distributions is shown graphically and discussed. The heat transfer rate, skin friction, and Sherwood number are calculated and presented in tabular form. It is worth noting that the increasing values of the couple stress parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ deaccelerate the velocity of Couple stress Casson fluid.

www.nature.com/scientificreports/ considered the Newtonian heating effect. Idowu and Falodun 28 discussed heat and mass transfer processes on Walters-B viscoelastic and Casson fluids. Casson Fluid is one of the non-Newtonian fluid 29 . A few of the Casson liquid comprises of ketchup, molten polymers, blood, jelly, toothpaste, etc. Casson fluids are widely used in paints, drilling, metallurgy, china clay, food processing, synthetic lubricants, and bioengineering operations 30,31 . Ismael 32 investigated the impact of energy transfer with the internal heat source on Carreau-Casson Fluids flow and used the numerical scheme RK − 4 method to find the numerical solution of governing equations. Raju et al. 33 developed a mathematical model of Casson and Williamson fluid flow to investigate the flow, heat, and mass transfer over a stretching surface. Loganathan and Deepa 34 investigated the flow of convective Casson fluid paste a Riga plate. Rafiq 35 studied Casson fluid flow generated by the non-coaxial rotation of a disk. Exact solutions of BVP are obtained by using Laplace transform technique. Model et al. 36 used variational principle to optimize the total stress of the casson and Ree-Eyring fluids flow. Nadeem et al. 37 use fractional derivatives to develop the generalized Casson fluid model. Hussain et al. 38 discussed the behavior of Casson fluid flow under the influence of magnetic field and obtained the numerical solutions using shooting method of the involved equations.
Applications of CSF are discussed in 39,40 . Tripathi et al. 41 , used MHD flow of a CS Nanofluid over a convective wall have discussed. Reddy et al. 42 studied the hydromagnetic peristaltic motion of CS fluid in a slanted channel keeping the wavelength long and Reynolds number small. Farooq et al. 43 51 , considered the flow of laminar and unsteady couple stress fluid between infinite plates. They considered engine oil as a base fluid and exact solutions are obtained by using Laplace and Fourier transforms. They concluded that the engine oil efficiency has been improved by 12.8 percent by adding nanoparticles.
In the above literature no one has considered couple stress Casson fluid in a channel using Fick's and Fourier's laws to find the closed form solutions. In this article, we considered the flow of couple stress Casson fluid in a channel and the flow is generated due to the motion of the plate at y = 0 . The governing PDE's are nondimensionalized using suitable dimensionless variables and the energy and mass equations are transformed to fractional model using Fick's and Fourier's Laws. Caputo definition is used for the fractional model and then these fractional PDE's are solved by using the joint application of Laplace and Fourier transforms. The obtained results are depicted in the form of figures. The impact of different parameters on Nusselt and Sherwood numbers and skin fraction are shown in tables.

Mathematical formulation
Consider the motion of couple stress Casson fluid between in a infinite parallel plates.The flow is considered in x direction. Initially, (t ≤ 0) the fluid as well as both the plates are at rest having ambient temperature T d and concentration C d . After some time t (t > 0) , the plate at y = 0 is dragged with constant velocity u 0 H(t) where u 0 is the characteristic velocity and second plate remains a rest. The temperature and concentration of the moving plate raise to T 1 and C 1 respectively and then remains constant as shown in the Fig. 1.
For the velocity v = (u(y, t), 0, 0) , the equation of continuity is identically satisfied and using the well known Boussinesq's approximation, the governing equations for unsteady couple stress Casson fluid flow between infinite parallel plates 52 are given by: The momentum equation is: The Fourier's law: The Thermal balance equation: The Fick's law: (2) (ρC p ) ∂T(y, t) ∂t = − ∂q(y, t) ∂y .
(3) q(y, t) = −k ∂T -(y, t) ∂y . Here u, Tand C are the velocity, temperature and concentration of the fluid respectively. ρ , (ρCp) , µ , η , K, , g, and D is the density, specific heat, Dynamic viscosity, couple stress parameter, thermal conductivity, gravitational acceleration and mass diffusivity respectively. To obtain Dimensionless system of PDE's, we Introduce the following dimensionless variables: By using these dimensionless variables and dropping the * signs, the Eqs. (1-8) becomes: .
∂y .  Here Here δ(t) is the Dirac's delta function. Using Eq. (16) and the above properties given in (17), we can write: Using the definition of Caputo fractional operator in equation (16), and also from Eqs. (9,11,14,15), we can write: To find the simplify form of Eqs. (19) and (20), we consider the time fractional integral operator: The Eq. (21) is the inverse operator of the fractional derivative C D α t defined in Eq. (16). Using the properties defined in 52 , the Eqs. (19) and (20) can be written as: Solution of the problem Solution of energy field. Using Laplace transform technique (LT) to Eq. (22), we have: and the transformed initial and boundary conditions are given by:  .

Pr
.sin(kπy).   Figure 2 shows the impact of fractional parameter α on the velocity distribution. From the figure, it is evident that unlike the classical model, multiple integral curves of velocity are obtained. These multiple integral curves may be best fitted with the experimental results or real data. Figures 3 and 4 show the influence of thermal Grashof number Gr and mass Grashof number Gm on the velocity of couple stress Casson fluid. These Portrayed figures exhibit that velocity is the increasing function of these numbers. It is physically true because the increase in Gr and Gm increase the buoyancy forces that decrease the viscosity of the fluid and hence increase in velocity occurs. In Fig. 5, the effect of the Prandtl number on the velocity. As the Prandtl number is the ratio of viscous forces to thermal forces. Thus rise in Prandtl number means that the viscous forces become dominant www.nature.com/scientificreports/ over thermal forces, which causes to decrease the velocity. Figure 6 captured the impact of Schmidt number. As Schmidt number is the ratio of viscous forces to mass diffusion, therefore an increase in Schmidt number causes to increase viscous forces and decrease mass diffusion and thus, the velocity of the fluid decreases. Figure 7 captured the impact of Casson parameter β on the velocity profile. It is clear from the graph that by increasing the value of Casson parameter β , the velocity of the flow decreases. The physics of this behavior is, by increasing the value of Casson parameter increases the viscous forces that producing resistance to the flow and retards the flow. The impact couple stress parameter on the velocity is shown in Fig. 8, which shows that velocity profile is the decreasing function of couple stress parameter . The physics of this behavior is, by increasing the Couple stress parameter, the viscosity increases which reduce the fluid velocity. Figure 9 captures the temperature profile for different values of fractional parameter α . From the figure, it is can be easily seen that it has dual effect. For small time i.e. (t = 0.2) the effect is quite opposite to that for large time t = 2 . The impact of Prandtl number Pr on temperature profile is displayed in Fig. 10. It shows that the temperature profile is the decreasing function of Pr. Physically it is true because Prandtl number is the ratio of viscosity to the thermal diffusivity, so by increasing the Prandtl number the viscosity of the fluid also increases and thus rise in temperature occurs. Concentration profile for different values of fractional parameter α is presented in Fig. 11. Here the behavior of concentration distribution is similar to temperature distribution. Figure 12 exhibited concentration distribution for different values of Schmidt number Sc which shows that concentration distribution behavior is unchanged for small time as well for large time. As concentration depends on viscosity, an increase in Schmidt number increase viscosity, and thus concentration profile increases. Table 1 shows the variation in skin friction due to the change in values of different parameters. Skin friction is very important in different field of engineering specially civil engineering.The increasing values of the Casson parameter β , increases the viscous forces and hence the skin friction increases. Also if we increase the Schmidt number Sc, the viscous forces increases that causes to increase the skin friction. As Prandtl number Pr is proportional to viscous forces, so increase in Pr increases the viscous forces that causes to increases the skin friction. By increasing the couple stress parameter , the viscosity of the fluid increases which increases the skin  Table 1. Finally, by increasing thermal Grashof number G r and mass Grashof number G m , the skin friction also decreases. The physics of this behavior is, by increasing in G r and G m the buoyancy forces increases that decreases the viscosity and hence the skin friction decreases. Table 2 shows Nusselt number. As Prandtl number Pr is the ratio of momentum diffusivity to thermal diffusivity, so increase in Pr increase the momentum diffusion which reduces the thermal boundary layer thickness that decreases the Nusselt numbes. Table 3 shows Sherwood number. As we increase the time , the Sherwood number decrease and by increasing Schmidt number, the sherwood number increases. This physics is that, Shmidt number is the ratio of viscous forces to the mass diffusion. Increasing Schmidt number increases viscous forces and decreases the mass diffusion and hence the Sherwood number increases.

Conclusion
This manuscript deals with the modern approach of Fick's and Fourier's laws to transform the classical model to time-fractional model using the definition of Caputo. The integral transforms (Laplace and Fourier ) are used to find exact solutions. The impact of various embedded parameters on velocity, temperature, and concentration