In this paper, finite difference method is used to study the combined effects of thermal radiation, inclined magnetic field and temperature-dependent internal heat generation on unsteady two-dimensional flow and heat transfer analysis of dissipative Casson-Carreau nanofluid over a stretching sheet embedded in a porous medium. In the study, kerosene is used as the base fluid which is embedded with the silver (Ag) and copper (Cu) nanoparticles. Also, effects of other pertinent parameters on the flow and heat transfer characteristics of the Casson-Carreau nanofluids are investigated and discussed. From the results, it is established that the temperature field and the thermal boundary layers of Ag-Kerosene nanofluid are highly effective when compared with the Cu-Kerosene nanofluid. Heat transfer rate is enhanced by increasing power-law index and unsteadiness parameter. Skin friction coefficient and local Nusselt number can be reduced by magnetic field parameter and they can be enhanced by increasing the aligned angle. Friction factor is depreciated and the rate of heat transfer increases by increasing the Weissenberg number. A very good agreement is established between the results of the present study and the previous results. The present analysis can help in expanding the understanding of the thermo-fluidic behaviour of the Casson-Carreau nanofluid over a stretching sheet.
The roles and importance of thermal radiation are evident in solar power technology, nuclear plants, propulsion devices for aircraft, combustion chambers, glass production, furnace design, and space technology applications, such as comical flight aerodynamics rocket, space vehicles, propulsion systems, plasma physics in the flow structure of atomic plants, combustion processes, internal combustion engines, ship compressors, solar radiations and in chemical processes and space craft re-entry aerodynamics which operates at high temperatures. Also, there are various engineering and industrial applications of magnetohydrodynamic (MHD) fluid behavior such as in the design of cooling system with liquid metals, accelerators, MHD generators, nuclear reactor, pumps, flow meters, study of crystal growth, metal casting, liquid metal cooling blankets for fusion reactors and blood flow. Additionally, recent research works in the past few decades have shown that the study of flow of non-Newtonian materials is a topic of great interest amongst the recent workers in the study of fluid dynamics. The flow applications of non-Newtonian fluids such as blood transport in micro-circulatory system, printer ink, paints, liquid detergents, multi grade oil, polymers, sauce, mud, apple sauce are evident in polymer devolatisation and processing, bubble absorptions, fermentation, plastic foam processing, bubble columns, composite processing, wire and fiber coating, heat exchangers, extrusion process, chemical processing equipment, etc. Also, the analysis of stretched flow with heat transfer is very significant in controlling the quality of the end product in the afore-mentioned areas of applications. Such processes have great dependence on the stretching and cooling rates [1]. Consequently, in the past few years, research efforts have been directed towards the analysis of this very important phenomenon of wide areas of applications. Moreover, the promising significance of magnetohydrodynamics (MHD) fluid behavior in various engineering and industrial applications (such as in the design of cooling system with liquid metals, accelerators, MHD generators, nuclear reactor, pumps, flow meters, study of crystal growth, metal casting, liquid metal cooling blankets for fusion reactors and blood flow) still provokes the continuous studies and interests of researchers.
Additionally, the study of thermal radiation is important in solar power technology, nuclear plants, and propulsion devices for aircraft, combustion chambers, glass production and furnace design, and also in space technology applications, such as comical flight aerodynamics rocket, space vehicles, propulsion systems, plasma physics in the flow structure of atomic plants, combustion processes, internal combustion engines, ship compressors, solar radiations and in chemical processes and space craft re-entry aerodynamics which operates at high temperatures. Therefore, the influences of external factors such as magnetic field and thermal radiation on the flow and heat transfer problem of Newtonian and non-Newtonian fluid have been widely analyzed in recent times. In an early study, MHD fluid flow over a stretching surface was carried out by Anderson et al. [2, 3]. The effect of unsteadiness parameter on the film thickness has been studied [2] and the effect of magnetic field on the flow characteristics of the fluid were explored numerically [3]. Few years later, Chen [4] investigated the power-law fluid film flow of unsteady heat transfer stretching sheet while Dandapat et al. [5, 6] analyzed the effect of variable viscosity and thermo- capillarity on the heat transfer of liquid film flow over a stretching sheet. Meanwhile, Wang [7] developed an analytical solution for the momentum and heat transfer of liquid film flow over a stretching surface. Also, Chen [8] and Sajid et al. [9] investigated the flow characteristics of a non-Newtonian thin film over an unsteady stretching surface considering viscous dissipation using homotopy analysis and homotopy perturbation methods. After a year, Dandapat et al. [10] presented the analysis of two-dimensional liquid film flow over an unsteady stretching sheet while in the same year, effect of power-law index on unsteady stretching sheet was studied by Abbasbandy et al. [11]. Santra and Dandapat [12] numerically studied the flow of the liquid film over an unsteady horizontal stretching sheet. A numerical approach was also used by Sajid et al. [13] to analyze the micropolar film flow over an inclined plate, moving belt and vertical cylinder. A year later, Noor and Hashim [14]1 investigated the effect of magnetic field and thermocapillarity on an unsteady flow of a liquid film over a stretching sheet while Dandapat and Chakraborty [15]) and Dandapat and Singh [16] presented the thin film flow analysis over a non-linear stretching surface with the effect of transverse magnetic field. Heat transfer characteristics of the thin film flows considering the different channels have also been analyzed by Abdel-Rahman [17], Khan et al. [18], Liu et al. [19] and Vajaravelu et al. [20] Meanwhile, Liu and Megahad [21] used homotopy perturbation method to analyze thin film flow and heat transfer over an unsteady stretching sheet with internal heating and variable heat flux. Effect of thermal radiation and thermocapillarity on the heat transfer thin film flow over a stretching surface was examined by Aziz et al. [22]. In their study on the numerical simulation of Eyring-Powell flow and unsteady heat transfer of a laminar liquid film over a stretching sheet using finite difference method, Khader and Megahed [23] established that increasing the Prandtl number reduces the temperature field across the thin film. Vajravelu et al. [24] analyzed the convective heat transfer over a stretching surface with applied magnetic field while Pop and Na [25] studied the influence of magnetic field flow over a stretching permeable surface. In another study, Xu et al. [26] presented a series solutions of the unsteady three-dimensional MHD flow and heat transfer over an impulsively stretching plate. Nazar et al. [27] examined the hydro magnetic flow and heat transfer over a vertically stretched sheet. The effect of MHD stagnation point flow towards a stretching sheet was investigated by Ishak et al. [28] while the influence of thermal radiation on heat transfer in an electrically conducting fluid at stretching surface was explored by Emad [29].
Also, Reddy [30] studied the thermal radiation boundary layer flow of a nanofluid past a stretching sheet under applied magnetic field. Effects of thermal radiation on convective heat transfer in an electrically conducting fluid over a stretching surface with variable viscosity and uniform free stream was examined by Abo-Eldahab and Elgendy [31]. In a recent study, Gnaneswara Reddy [32] investigated the thermal radiation and chemical reaction effects on MHD mixed convective boundary layer slip flow in a porous medium with heat source and Ohmic heating. In another study, Gnaneswara Reddy [33] studied the influence of thermophoresis, Viscous Dissipation and Joule Heating on Steady MHD Flow over an Inclined Radiative Isothermal Permeable Surface. The effect of thermal radiation on magnetohydrodynamics flow was examined by Raptis et al. [34] while Seddeek [35] investigated the impacts of thermal radiation and variable viscosity on magnetohydrodynamics in free convection flow over a semi-infinite flat plate. In another study, Mehmood et al. [36] analyzed unsteady stretched flow of Maxwell fluid in presence of nonlinear thermal radiation and convective condition. Hayat et al. [37] addressed the effects of nonlinear thermal radiation and magnetohydrodynamics on viscoelastic nanofluid flow. Effects of nonlinear thermal radiation on stagnation point flow Farooq et al. [38]. Also, Shehzad et al. [39] presented a study o MHD three-dimensional flow of Jeffrey nanofluid with internal heat generation and thermal radiation. In a recent study, Lin et al. [40] examined the effect of MHD pseudo-plastic nanofluid flow and heat transfer film flow over a stretching sheet with internal heat generation. Numerically, Raju and Sandeep [41] studied heat and mass transfer in MHD non-Newtonian flow while Tawade et al. [42] presented the unsteady flow and heat transfer of thin film over a stretching surface in the presence of thermal radiation, internal heating in the presence of magnetic field. Heat and mass transfer of MHD flows through different channels have been analyzed [43, 44, 45, 46, 47, 48]. Makinde and Animasaun [49] investigated the effect of cross diffusion on MHD bioconvection flow over a horizontal surface. In another study, Makinde and Animasaun [50] presented the MHD nanofluid on bioconvection flow of a paraboloid revolution with nonlinear thermal radiation and chemical reaction while Sandeep [51], Reddy et al. [52] and Ali et al. [53] studied the heat transfer behaviour of MHD flows. Maity et al. [54] analyzed thermocapillary flow of a thin Nanoliquid film over an unsteady stretching sheet.
The above studies have been the consequent of the various industrial and engineering applications of non-Newtonian fluids. Among the classes of non-Newtonian fluids, Carreau fluid which its rheological expressions were first introduced by Carreau [55], is one of the non-Newtonian fluids that its model is substantial for gooey, high and low shear rates [56]. On account of this headway, it has profited in numerous innovative and assembling streams [56]. Owing to these applications, different studies have been carried out to explore the characteristics of Carreau liquid in flow under different conditions. Kumar et al. [40] applied Runge-Kutta and Newton’s method to analyze the flow and heat transfer of electrically conducting liquid film flow of Carreau nanofluid over a stretching sheet by considering the aligned magnetic field in the presence of space and temperature dependent heat source/sink, viscous dissipation and thermal radiation. Hayat et al. [57] studied the influence of induced magnetic field and heat transfer on peristaltic transport of a Carreau fluid. Olajuwon [58] presented a study on MHD flow of Carreau liquid over vertical porous plate with thermal radiation. Hayat et al. [59] investigated the convectively heated flow of Carreau fluid while in the same year, Akbar et al. [60] analyzed the stagnation point flow of Carreau fluid. Also, Akbar [61] presented blood flow of Carreau fluid in a tapered artery with mixed convection. A year later, Mekheimer [62] investigated the unsteady flow of a carreau fluid through inclined catheterized arteries haveing a balloon with time-variant overlapping stenosis. Elmaboud et al. [63] developed series solution of a natural convection flow for a Carreau fluid in a vertical channel with peristalsis. Using a revised model, flow of Carreaunanoliquid in the presence of zero mass flux condition at the stretching sheet has been examined by Hashim and Khan [64]. The MHD flow of Carreau fluid with thermal radiation and cross diffusion effects was investigated by Machireddy and Naramgari [65]. Sulochana et al. [66] provided an analysis of magnetohydrodynamic stagnation-point flow of a Carreau nanofluid.
Another non-Newtonian fluid is Casson fluid. Casson fluid is a non-Newtonian fluid first invented by Casson in 1959 [67]. It is a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear [68, 69]. If yield stress is greater than the shear stress then it acts as a solid, whereas if yield stress lesser than the shear stress is applied then the fluid would start to move. The fluid is based on the structure of liquid phase and interactive behaviour of solid of a two-phase suspension. It is able to capture complex rheological properties of a fluid, unlike other simplified models like the power law [70] and second, third or fourth-grade models [71]. Some examples of Casson fluid are Jelly, honey, tomato sauce and concentrated fruit juices. Human blood is also treated as a Casson fluid in the presence of several substances such as fibrinogen, globulin in aqueous base plasma, protein, and human red blood cells. Concentrated fluids like sauces, honey, juices, blood, and printing inks can be well described using this model. It has various applications in fibrinogen, cancer homeo-therapy, protein and red blood cells form a chain type structure. Due to these applications many researchers are concentrating characteristics of Casson fluid. Application of Casson fluid for flow between two rotating cylinders is performed in [72]. The effect of magnetohydrodynamic (MHD) Casson fluid flow in a lateral direction past linear stretching sheet was explained by Nadeem et al. [73].
Numerical methods such as Euler and Runge–Kutta methods are limited to solving initial value problems. With the aid of shooting method, the methods could be carried out iteratively to solve boundary value problems. However, these numerical methods are only useful for solving ordinary differential equations i.e. differential equations with a single independent variable. On the other hand, numerical methods such as finite difference method (FDM), finite element methods (FEM) and finite volume method (FVM) can be adopted solve differential equations with single and multiple independent variables as they have been used to different linear and non-linear differential equations in literatures. The numerical solution of FDM represents an efficient way of obtaining temperature profile for the steady heat transfer processes. The FDM can be used for solving any complex body by breaking the body into small domains. Also, choice of finer grids which requires high computing capability can remove approximation errors to larger extent. Hence, in this work, finite difference method. Therefore, in this study, finite difference method is applied to analyze the combined influences of thermal radiation, inclined magnetic field and temperature-dependent internal heat generation on unsteady two-dimensional flow and heat transfer analysis of dissipative Casson-Carreau nanofluid over a stretching sheet embedded in a porous medium are examined. Using kerosene as the base fluid embedded with the silver (Ag) and copper (Cu) nanoparticles, the effects of other pertinent parameters on flow and heat transfer characteristics of the nanofluids are investigated and discussed.
Where \(\tau\) is the shear stress, \(\tau_0\) is the Casson yield stress, \(\mu\) is the dynamic viscosity, \(\dot{\sigma}\) is the shear rate, \(\pi = e_{ij}e_{ij}\) and \(e_{ij}\) is the \((i,j)\)th component of the deformation rate, \(\pi\) is the product of the component of deformation rate with itself, \(\pi_c\) is a critical value of this product based on the non-Newtonian model, \(\mu_B\) is the plastic dynamic viscosity of the non-Newtonian fluid, and \(p_y\) is the yield stress of the fluid. The velocity as well as the temperature are functions of \(y,t\) only.
The extra stress tensor for Carreau fluid is given as:
Where, \( \bar{\tau }_{ij} \) is the extra tensor, \( \eta_{o} \) is the zero shear rate viscosity, \( \Gamma \) is the time constant, \( n \) is the power-law index, and \( \bar{\dot{\gamma }} \) is defined as:
\[ \bar{\dot{\gamma }}=\sqrt{\frac{1}{2} \sum _{i}\sum _{j}\bar{\dot{\gamma }}_{ij} \bar{\dot{\gamma }}_{ji} } =\sqrt{\frac{1}{2} } \Pi \]
Where \( \Pi \) is the second invariant strain tensor.
Following the assumptions, the equations for continuity and motion for the flow analysis of Carreau and Casson fluids are:
Where:
Assuming no-slip condition, the appropriate boundary conditions are given as:
It should be stated at this juncture that the mathematical problem is implicitly formulated only for \(x \geq 0.\) In order to avoid the complications due to surface waves, a further assumption is made that the surface of the planar liquid film is smooth. Also, the influence of interfacial shear due to the quiescent atmosphere, i.e., the effect of surface tension, is assumed to be negligible. The viscous shear stress \( \tau =\mu \frac{\partial u}{\partial y} \) and the heat flux \( \dot{q”}=-k\left(\frac{\partial T}{\partial y} \right) \) vanish at the adiabatic free surface at \(y = h\).
It should be noted that:
The above boundary conditions are in line with the works of Kumar et al. [56].
The non-uniform heat generation/absorption \(q”’\) is taken as:
Where the surface temperature \(T_{s}\) of the stretching sheet varies with respect to distance \(x\) from the slit as:
And the stretching velocity varies with respect to \(x\) as:
On introducing the following stream functions:
And the similarity variables:
Substituting Equations (19) and (20) into Equations (5), (6), (13), and (14), we have a partially coupled third- and second-order ordinary differential equation:
Where:
And the boundary conditions become:
Equations (21) and (22) are systems of coupled non-linear ordinary differential equations that are to be solved using the boundary conditions (24). The exact solution is not possible for this set of equations. The use and accuracy of the finite difference method for analyzing nonlinear problems were pointed out by Han et al. (2005). Therefore, in this work, the finite difference method is used to discretize the governing equations combined with the boundary conditions.
The finite difference schemes for each derivative in the governing differential equations are given as:
\(f””=\frac{2f_{i+1} -9f_{i} +16f_{i-1} -14f_{i-2} +6f_{i-3} -f_{i-4} }{\left(\Delta \eta \right)^{4} }\)
\(f”’=\frac{3f_{i+1} -10f_{i} +12f_{i-1} -6f_{i-2} +f_{i-3} }{2\left(\Delta \eta \right)^{3} }\)
\(f”=\frac{f_{i+1} -2f_{i} -f_{i-1} }{\left(\Delta \eta \right)^{2} }\)
\(f’=\frac{f_{i+1} -f_{i-1} }{2\left(\Delta \eta \right)}\)
\(\theta ”=\frac{\theta _{i+1} -2\theta _{i} -\theta _{i-1} }{\left(\Delta \eta \right)^{2} }\)
\(\theta ‘=\frac{\theta _{i+1} -\theta _{i-1} }{2\left(\Delta \eta \right)}\)
Substituting the finite difference schemes into Equations (21) and (22), we obtain the equivalent finite difference schemes:
The boundary conditions are expressed in finite difference form as:
The Equations (25) and (26), along with the boundary conditions (27), are reduced into a block tri-diagonal system that is solved using the block elimination method.
Figures 14 and 15 demonstrated the effect of power law index on velocity and temperature fields. As the power index is increased, it was observed that the velocity profile increases while the temperature profile decreases. This is because, increasing value of the power law index, thickens the liquid film associated with an increase of the thermal boundary layer. An increase in the momentum boundary layer thickness and a decrease in thermal boundary layer thickness is observed for the increasing values of the power law index including~shear thinning~to~shear thickening~fluids. Also, it should be pointed out that an increase in Weissenberg number correspond a decrease in the local skin~friction~coefficient and the magnitude of the local Nusselt number s decreases when the Weissenberg number increases. The effects of nanoparticles volume fraction on the velocity and temperature profiles are depicted in Figures 16 and 17, respectively. The result shows that as the solid volume fraction of the film increases both the velocity and temperature field increases. This is because as the nanoparticle volume increases, more collision occurs between nanoparticles and particles with the boundary surface of the plate and consequently the resulting friction enhances the thermal conductivity of the flow and gives rise to increase the temperature within the fluid near the boundary region.
Figures 18 and 19 depict the influence of non-uniform heat source/sink parameter on the temperature field. It is revealed that increasing the non-uniform heat source/sink parameter enhances the temperature fields. It is observed in the analysis that the temperature and thermal boundary layer thickness is depressed by increasing the Prandtl number Pr. The effect of Eckert number on temperature profile is shown in Figure 20. It was established that as the values of Eckert number increases, the values of the temperature distributions in the fluid increases. This is because as Ec increases, heat energy is saved in the liquid due to the frictional heating.
The effect of nanoparticle volume fraction \(\mathrm{\phi}\) on the film thickness of the nanofluid is shown in Figure 21. It is evident from the figure that the film thickness is enhanced as the values of \(\mathrm{\phi}\) is increased. It can be inferred from Equation (15) that if nanoparticle volume fraction \(\mathrm{\phi}\) is increased, the nanofluid viscosity will increased as there exist a direct relationship or proportion between the two parameters. As a result, the increasing viscosity resists the fluid motion along the stretching direction leading to the slowdown of the film thinning process [54].
