Influence of nonlinear thermal radiation on rotating flow of Casson nanofluid

1 Introduction

An essential heat transfer enhancement in fluids through nanoparticle additives can enhance and improve the thermal conductivity of fluid and are known as nanofluid [1]. These are being increasingly used in heat exchange applications to enhance the heat transport properties of the fluid by order of magnitude. Based on the applications of nanofluid, many researchers have studied this fluid with different formulations and boundary conditions. Boundary layer flow of nanofluid in the presence of buoyancy force was discussed by Gorla and Chamkha [2]. Shehzad et al. [3] have taken distinct kinds of nanoparticles to study the Peristaltic transport of the nanofluid. Moreover, they have provided two models called Maxwell and Hamilton-Crosser to compare the study. Sheikholeslami et al. [4, 5, 6, 7, 8] analyzed the flow and heat transfer characteristics of a nanofluid in different geometries under various boundary conditions. Hayat et al. [9] initiated the Newtonian mass flux condition in the flow problem of nanofluid along a cylinder. Moreover, nowadays many investigators are incorporating magnetohydrodynamic flow in their research due its numerous industrial applications in metallurgy and chemical engineering process such as, drawing, annealing and thinning of copper wire. As the quality of the manufactured product can controlled by better cooling rate, an electrically polymeric liquid seems to be a good candidate for such applications of polymer and metallurgy because here the flow can be controlled by an applied magnetic field. Meanwhile, interesting investigations on MHD flows and heat transfer can be seen in the references [10, 11, 12, 13] which involve suspended nanoparticle in porous cavity, porous lid driven cubic cavity, cavity with tilted elliptic inner cylinder, porous cylinder, porous curved enclosure.

The study of rotating fluid flow which initiates the Coriolis force has significant applications in various disciplines such as astrophysics, oceanography, geophysical circumstances etc., Furthermore, this type of flow over a stretching surface is implementing in many fields essentially in fibre spinning, food processing, plastic sheets extrusion, glass blowing process etc., two dimensional stretchable surface was considered by Wang [14] to examine the rotating fluid flow problem. Additionally, when the rotating parameter was greater than unity he obtained an accurate solution using analytic method as compared with the numerical method. Zaimi et al. [15] have considered a non-Newtonian viscoelastic fluid to scrutinize the rotating flow using Keller-box method. Rashidi et al. [16] utilized the second law of thermodynamics to report the entropy generation for the rotating nanofluid flow problem with the consideration of different kinds of nanoparticles. Nadeem et al. [17] discussed the rotating fluid flow problem comprising of copper and titanium oxide nanoparticles and they found that, varying values of nanoparticle volume fraction decreases the velocity for nanoparticles. Mabood et al. [18] have analysed the effect of Brownian motion and thermophoresis on rotating nanofluid flow under the influence of magnetic, radiation, heat source and viscous dissipation effects. Das et al. [19] examined the combined effect of magnetic field and rotation in transient hydromagnetic Couette flow of viscous fluid and significant modification of fluid velocity was obtained due to these effects. Ali et al. [20] have considered different kinds of nanoparticle to scrutinize the influence of magnetic and viscous dissipation on hydromagnetic Couette flow in a rotating system.

In the nature there are many fluids which exhibit the nonlinear relationship between the stress and the deformation rate and are known as non-Newtonian fluids. The study of phenomena of transport of these type of fluid have been attracted by many researchers because of its wide applications in various fields such as crude oil extraction from petroleum products, processing of food stuffs, production of paper and fiber coating. As a single constitutive equation can’t describe the characteristics of all non-Newtonian fluids due to its versatility in nature, several models have been developed. Among them, Casson model is one of the most important and it describes the yield stress. This model was first introduced by Casson [21] to predict the flow characteristics of pigment oil suspensions of the printing ink type. In Casson fluid, it is assumed to have zero and infinite viscosity for an infinite and zero rate of shear respectively and have a yield stress below which no flow occurs. Nadeem et al. [22] have obtained the series solution for heat and mass transfer of a stagnation point flow problem of non-Newtonian Casson nanofluid. They found that, positive and negative value of Casson parameter varies the position of stagnation point with respect to the origin. Butt et al. [23] examined the heat transfer of Casson rotating fluid over a stretching surface. Ramesh et al. [24] have inspected the impact of variable thickness of a non-isothermal stretching sheet in Casson fluid flow. Ibrahim and Makinde [25] employed a well known numerical procedure Runge-Kutta-Fehlberg method to investigate the solution of MHD flow of Casson nanofluid in the presence of slip and convective condition at the boundary. A numerical investigation was carried out by Ullah et al. [26] for the Casson fluid flow over a moving wedge in the presence of viscous dissipation. Eegunjobi and Makinde [27] have presented the thermodynamics analysis in Casson fluid flow in channel having porous media.

Role of radiation heat transfer is superficial in many engineering processes which occur at high temperature. A substantial number of experimental and theoretical studies have been carried out by numerous researchers on radiation effect [28, 29, 30, 31]. In most of the above cited literature, radiation effect was inspected by implementing the linearized Rosseland approximation. This type of approximation involves the dimensionless parameters called Radiation parameter and Prandtl number which are sustainable if the temperature difference between the plate and ambient fluid is small. But, for the larger temperature difference nonlinearized Rosseland approximation is valid. Hayat et al. [32] have implemented a new category of boundary condition which is called zero nanoparticle mass flux to analyze the effect of nonlinear thermal radiation in three dimensional flow of viscoelastic fluid with suspended nanoparticles. Stagnation point flow of Casson fluid under the influence of induced magnetic field and nonlinear thermal radiation has been examined by Ananth et al. [33]. Mustafa et al. [34] have considered a deformable surface to investigate the heat transfer features in ferrofluid flow with water as base fluid.

Most of the works in the foregoing articles were done on nanofluid but, as nanofluid with non-Newtonian modelling has become influential in view of its various applications. Hence, we have considered Casson nanofluid rotating flow in the presence of magnetic field, nonlinear thermal radiation, viscous dissipation and Joule heating effect. The numerical solution was retrieved using Runge-Kutta-Fehlberg fourth-fifth order method along with shooting technique. Several previously published articles for certain cases are accomplished for analogy and the obtained results are found to be in excellent agreement. Effects of several dimensionless parameters have been described in detail through the graphical and tabular representation.

2 Mathematical Formulation

Consider a steady, laminar flow of an incompressible Casson nanofluid bounded by an infinite long stretching surface. The fluid is rotating with an angular velocity Ω about z– axis and the flow takes place in the domain z ≥ 0 (see Figure 1). The velocity of the stretching sheet is assumed to be U_w(X) = ax. Let u, v, w are the velocity components along the x, y, z directions respectively. Let T_w and C_w be the constant values of the temperature and nanoparticle concentration respectively and these constant values are assumed to be larger than the ambient temperature and nanoparticle concentration and are represented by T_∞ and C_∞ respectively. A constant magnetic field of strength B₀ is applied in the z-direction. Brownian motion, thermophoresis, nonlinear thermal radiation, viscous dissipation and Joule heating effects are incorporated in the energy equation.

Fig. 1

Schematic representation of the flow

The rheological equation of state for an isotropic and incompressible Casson fluid is given by

Here, π = e_ije_ij and e_ij is the (i, j)^th component of the deformation rate, π is the product of the component of the deformation rate with itself, π_c is a critical value of this product based on the non-Newtonian model, μ_B is the plastic dynamic viscosity of the non-Newtonian fluid, and py=μB2π/β is the yield stress of the fluid.

The governing boundary layer equations for the above defined formulation can be written as,

where v=μρf is the kinematic viscosity of the fluid, μ is the coefficient of fluid viscosity, β is the Casson parameter, ρ_f is the fluid density, σ is the electrical conductivity of the fluid, α = k/(ρc_p)_f is the thermal diffusivity of the fluid, k is the thermal conductivity, c_f and c_p are the specific heat coefficients of fluid and nanoparticle respectively, τ=(ρcp)p(ρcp)f is the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid, q_r is the radiative heat flux, D_B is the Brownian diffusion coefficient, D_T is the thermophoretic diffusion coefficient, T and C are the temperature and the nanoparticle concentration of the fluid respectively.

The boundary conditions for the present flow analysis are,

Unlike the linearized Rosseland approximation, we use nonlinear Rosseland diffusion approximation from which one can obtain results for both small and large differences between T_w and T_∞. Using Rosseland approximation for radiation, the radiative heat flux is simplified as,

where σ^* and k^* are the Stefan-Boltzmann constant and mean absorption coefficient respectively.

For a boundary layer flow over hot surface (Pantokratoras and Fang [35]), from equation (7) we get,

In view of equation (8), the energy equation (5) can be rewritten as,

Introduce the non-dimensional similarity variables as,

where θw=TwT∞,θw>1 being the temperature ratio parameter.

Using equation (10), equation (1) is instinctively satisfied and the nonlinear partial differential equations (2, 3, 9 and 5) reduced into a set of nonlinear ordinary differential equations as:

Corresponding boundary conditions are,

where γ₁ is the rotation parameter, M is the magnetic parameter, Rd is the radiation parameter, Pr is the Prandtl number, N_b is the Brownian motion parameter, N_t is the thermophoresis parameter, Ec is the Eckert number, Sc is the Schmidt number.

These parameters can be defined as,

The local skin friction coefficient (C_f) along x and y directions and local Nusselt number (Nu_x), local Sherwood number (Sh_x) are given by,

where the shear stress along the wall τ_w and wall heat flux q_w are given by,

Substituting equation (18) into the equation (17), we have

where Rex=Uw2av is the local Reynolds number.

3 Method of Solution

The system of coupled nonlinear ordinary differential equations (11–14) along with the boundary conditions (15) and (16) are solved numerically using Runge–Kutta–Fehlberg fourth-fifth order method (RKF45 Method) along with shooting technique. The equations are first reduced into system of first order simultaneous equations. For this, first let us consider, f=y1,y1′=y2,y2′=y3,g=y4,y4′=y5,y5′=y6,θ=y7,y7′=y8,y8′=y9,ϕ=y10,y10′=y11 then, we obtain the reduced equations as follows:

with boundary conditions as,

With the aid of shooting technique, omitted initial conditions are predicted. This technique is an iterative algorithm to determine the appropriate initial conditions for a relevant initial value problem that provides the solution to the original boundary value problem. We have considered infinity condition at a large but finite value of η.

After fixing finite value for η_∞, integration is carried out with the help of Runge-Kutta-Fehlberg-45(RKF-45) method. This method has a procedure to determine an accurate solution if the proper step size h is being used. At each step, two different approximations for the solution are made and compared. If the two answers are in close agreement, the approximation is accepted otherwise the step size is reduced until to get the required accuracy. For the present problem, we took step size Δη = 0.001, η_∞ = 6 and accuracy to the fifth decimal place.

To have an assessment on the precision of the numerical approach employed, we have manipulated f″(0), g(0) and θ′(0) that are carried out for viscous fluid for various values of rotating parameter and Prandtl number and are compared with the available published results of Wang [15] in table 1 and they are found to be in excellent agreement.

4 Results and Discussion

Numerical computation is carried out for several set of values of the Casson parameter (β), magnetic parameter (M), rotation parameter (γ₁), Prandtl number (Pr), radiation parameter (Rd), Brownian motion parameter (Nb), thermophoresis parameter (Nt), temperature ratio parameter (θ_w), Eckert number (Ec) and Schmidt number (Sc). In order to analyze the salient features of the problem, the numerical results are presented in Figures 2–17.

Fig. 2

Influence of Casson parameter on f′(η)

Fig. 3

Influence of Casson parameter on g(η)

Fig. 4

Influence of Magnetic parameter on f′(η)

Fig. 5

Influence of Magnetic parameter on g(η)

Fig. 6

Influence of Rotation parameter on f′(η)

Fig. 7

Influence of Rotation parameter on g(η)

Fig. 8

Influence of magnetic parameter on θ(η)

Fig. 9

Influence of Brownian motion parameter on θ(η)

Fig. 10

Influence of thermophoresis parameter on θ(η)

Fig. 11

Influence of rotation parameter on θ(η)

Fig. 12

Influence of Prandtl number on θ(η)

Fig. 13

Influence of Radiation parameter on θ(η)

Fig. 14

Influence of temperature ratio parameter on θ(η)

Fig. 15

Influence of Eckert number on θ(η)

Fig. 16

Influence of Brownian motion parameter on ϕ(η)

Fig. 17

Influence of Thermophoresis parameter on ϕ(η)

From the Figure 2, it is clearly observed that, velocity curve f′(η) and the corresponding momentum boundary layer thickness decreases with increasing values of Casson parameter. Physically, Casson parameter depends on the yield stress and this stress produces a resisting force which makes the velocity curve to decrease for the increasing values of β. The description of the same parameter for the lateral velocity profile g(η) along y direction can be viewed through Figure 3. It is evident from this figure that, the velocity profile g(η) is an increasing function of β. Here, the function g(η) has a parabolic profile and it reveals that, occurrence of flow is along negative direction as it consists of negative values. Variations of velocity profiles f′(η) for the varying values of magnetic parameter are displayed in the Figure 4. Here, increase in the magnetic parameter corresponds to decrease in velocity profiles f′(η). This is due to the fact that, applied magnetic field produces a drag in the form of Lorentz force which is generated due to motion of charges which decreases the magnitude of velocity along x direction. But, opposite behavior can be observed for the lateral velocity which is along y direction and this can be viewed from the Figure 5. Figure 6 exhibits the variation of f′(η) for the rotation parameter. It is found that, degradation of velocity along x direction can be seen for the increment of rotation parameter. Physically, the larger values of this parameter implies the smaller values of stretching rate along the x direction, this parameter makes the velocity to decrease along x direction. While, from the Figure 7 uplifting of lateral velocity can be observed in the region (2 ≤ η ≤ 6) for the same parameter. This result agrees with the result obtained by (Wang [14]).

Variation of temperature curve for the incrementing values of magnetic parameter is rendered in Figure 8. It illustrates that, increase of this parameter makes the temperature profile to increase because of the Lorentz force produced due to the presence of magnetic field. Figures 9 and 10 are plotted to show the role of Brownian motion and thermophoresis parameter on temperature profiles. These parameters represent the presence of nanoparticle in the fluid. As these particles enhance the thermal conductivity of the fluid, it leads to higher temperature and thicker thermal boundary layer. Figure 11 portraits that, the temperature profile is an increasing function of rotation parameter. It is clear from the Figure 12 that, thickness of the thermal boundary layer and temperature are decreasing functions of Pr. This is because as the Prandtl number increases, thermal diffusivity decreases there by decrease the temperature. Figure 13 exhibits the variation of temperature profile for the radiation parameter. As the radiation parameter releases the heat energy into the flow, with an increase in radiation parameter, temperature profile increases. Figure 14 illustrates that, the temperature profile is an increasing function of temperature ratio parameter. This phenomenon occurs because temperature ratio parameter describes the thermal state of the fluid and with the increase of this parameter, temperature also increases. Figure 15 illustrates the role of Eckert number on temperature of the fluid. Larger value of Eckert number causes an increase in temperature profile. This circumstance is due to dependence of this number on kinetic energy which releases heat energy into the fluid there by increases the temperature.

From Figure 16, we assert that the concentration profile is a decreasing function of Brownian motion parameter. Brownian motion takes place in the fluid because of the nanoparticles presence and with the increase in Nb, this motion is affected and consequently the nanoparticles concentration boundary layer thickness decreases. With the enhancement of Thermophoresis parameter, nanoparticles concentration increases and it can be analyzed through Figure 17. Impact of Schmidt number on concentration profile can be viewed through Figure 18. Since Schmidt number depends on the Brownian diffusion coefficient, larger values of Schmidt number makes the Brownian diffusion coefficient as lower, which shows a weaker nanoparticle concentration. Figure 19 represents the impact of Brownian motion with Thermophoresis parameter on Sherwood number. In this graph it can be seen that, enhancement of Brownian motion parameter decreases the Sherwood number whereas this number is increased by increasing the thermophoresis parameter.

Fig. 18

Influence of Schmidt number parameter on ϕ(η)

Fig. 19

Influence of Brownian motion and thermophoresis parameter on Sherwood number

Table 2 presents the values of local skin friction coefficient along x and y directions as well as local Nusselt number for various values of existing parameters in the flow problem. It is recognized that, local skin friction coefficient increases in both the direction for the Casson parameter but it decreases for the rotating parameter while, opposite variations can be observed for the magnetic parameter. Values of local Nusselt number decreases for theses parameters. The numerical values of local Nusselt number for various values of the parameters Rd, θ_w, Pr, Nb, Nt, Ec, Sc are manifested in Table 3. For the varying values of Pr, Nb, Nt, Ec, Sc, the rate of heat transfer at the surface decreases while increases for Rd and θ_w.

5 Conclusion

An instant analysis has been undertaken to scrutinize the rotating flow of a non-Newtonian Casson fluid with suspended nanoparticles in the presence of magnetic effect and nonlinear thermal radiation. The foremost perceived results are summarized as below:

1. Velocity profile f′(η) decreases for the increasing values of Casson, magnetic and rotating parameter, whereas lateral velocity profile g(η) exhibits the opposite behaviour for the same parameters.

2. Thermal boundary layer is thicker for the effect of magnetic parameter, Brownian motion parameter, thermophoresis parameter, rotation parameter, radiation parameter, temperature ratio parameter and Eckert number.

3. Temperature profile decreases with increasing values of Prandtl number.

4. Concentration boundary layer thickness is a decreasing function of Brownian motion parameter, Schmidt number whereas increasing function for thermophoresis parameter.