Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet

1 Introduction

Heat, mass and momentum transfer in the laminar boundary layer flow over a stretching sheet is relevant to several industrial and engineering processes in the field of metallurgy and chemical engineering processes. These applications involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. Flow past a stretching surface with different stretching velocity, namely linear[1], exponential[2], non-linear[3], quadratic[4], hyperbolic[5], radially[6] and even oscillatory[7] has also been addressed previously. However studies on heat and mass transfer in the laminar boundary layer flow over a non- linear stretching sheet is very less. It is well known to the all research community that the stretching is not necessarily linear in many industrial applications. The problem of non-linear stretching sheet for different cases of fluid flow has also been analyzed by different researchers. Rana and Bhargava[8] numerically investigated the steady, laminar boundary fluid flow which results from the non-linear stretching of a flat surface in a nanofluid. Mondal et al.[9] carried out an analysis to study the mixed convective heat and mass transfer of nanofluids over a non-linear stretching sheet under the influence of suction/injection parameter, magnetic parameter and thermophoresis parameter. Mustafa et al.[10] obtained the analytical and numerical solutions for axisymmetric flow of nanofluid due to a non-linearly stretching sheet. Effects of nanoparticle migration and asymmetric heating on forced convective heat transfer of alumina/water nanofluid in microchannels in presence of a uniform magnetic field was theoretically presented by Malvandi and Ganji[11]. Khan et al.[12] studied the three-dimensional flow of nanofluid over an elastic sheet stretched non-linearly in two lateral directions and the results of their study reveal that penetration depths of temperature and nanoparticle volume fraction are decreasing functions of the power-law index. Dhanai et al.[13] numerically investigated the problem of magneto-hydrodynamic boundary layer flow of nanofluid induced by a power-law stretching/shrinking permeable sheet using shooting method, by taking the effect of viscous dissipation. Mabood et al.[14] obtained numerical results for MHD laminar boundary layer flow with heat and mass transfer of an electrically conducting water-based nanofluid over a nonlinear stretching sheet with viscous dissipation effect. Recently, Malvandi et al.[15] numerically investigated the thermal performance of hydromagnetic alumina/water nanofluid inside a vertical microannular tube by considering the different modes of nanoparticle migration. Malvandi[16] examined the anisotropic behavior of thermal conductivity and its effects on flow field and heat transfer characteristics at film boiling of MNFs over a vertical cylinder in the presence of a uniform variable-directional magnetic field.

Applying no-slip boundary condition, many authors have obtained both numerical and analytical solutions to heat, mass, and momentum transfer in the laminar boundary layer flow over a stretching surface. No-slip condition is inadequate for most non-Newtonian liquids, as some polymer melt often shows microscopic wall slip and that has a controlling influence by a nonlinear and monotone relation between the slip velocity and the traction. Partial velocity slip may occur on the stretching boundary when the fluid is particulate such as emulsions, suspensions, foams and polymer solutions. In various industrial processes, slip effects can arise at the boundary of pipes, walls, curved surfaces etc. A usual approach in studying slip phenomena is the Navier velocity slip condition. Boundary layer slip flow problems arises in polishing of artificial heart valves and internal cavities. Recently many authors obtained analytical and numerical solutions for boundary layer flow and heat transfer due to a stretching sheet with slip boundary conditions. Ibrahim and Shanker[17] commented on MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions. Malvandi and Ganji[18] theoretically investigated the laminar flow and convective heat transfer of alumina/water nanofluid inside a circular microchannel in the presence of a uniform magnetic field. Nadeem et al.[19] analyzed combined effects of magnetic field and partial slip on obliquely striking rheological fluid over a stretching convective surface. Das[20] performed a numerical investigation, to study the problem of boundary layer flow of a nanofluid over non-linear permeable stretching sheet at prescribed surface temperature in the presence of partial slip. The effects of velocity slip, thermal slip and magnetic field on MHD boundary layer mixed convection flow and heat transfer of an incompressible viscous fluid over a vertical plate of very small thickness and much larger breadth in the presence of suction/blowing are presented by Mukhopadhyay and Mondal[21]. The investigation on developing the transport phenomenon of the nanofluids falling condensate film, taking into account the effects of nanoparticle migration has been presented by Malvandi et al.[22].

In the context of space technology and in the processes involving high temperatures, the effect of radiation plays a vital role. Yazdi et al.[22] presented numerical solution for MHD liquid flow and heat transfer over a non-linear permeable stretching surface in the presence of chemical reactions and partial slip. Pal and Talukdar[24] analyzed the combined effect of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid past a vertical permeable surface embedded in a porous medium. Hady et al.[25] studied the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation which is included in the energy equation, and variable wall temperature. Pal and Mandal[26] obtained the numerical solution by fifth-order Runge—Kutta—Fehlberg method with shooting technique for magnetohydrodynamic boundary layer flow of an electrically conducting convective nanofluids induced by a non-linear vertical stretching/shrinking sheet with viscous dissipation, thermal radiation and Ohmic heating.

Over the last few years a considerable amount of experimental and theoretical research has been carried out to determine the role of natural convection in the kinetics of heat transfer accompanied with melting or solidification effect. Processes involving melting heat transfer in non-Newtonian fluids have promising applications in thermal engineering such as oil extraction, magma solidification, melting of permafrost, geothermal energy recovery, thermal insulation, etc. Roberts[27] was the first to describe the melting phenomena of ice placed in a hot stream of air at a steady state. Epstein and Cho[28] studied melting heat transfer in steady laminar flow over a flat plate. Krishnamurthy et al.[29] presented the numerical results to study the effect of chemical reaction on MHD boundary layer flow and melting heat transfer of Williamson nanofluid in porous medium.

Based on the aforementioned literature survey, we intended to perform a theoretical investigation of MHD velocity slip boundary layer flow and melting heat transfer of nanofluids induced by a nonlinear stretching sheet under the influence of thermal radiation and chemical reaction. The Brownian motion and thermophoresis effects are considered in the present flow analysis. Reduced governing nonlinear ordinary differential equations are solved numerically by means of Runge-Kutta-Fehlberg-45 order method. The effects of different flow parameters on flow fields are elucidated through graphs and tables.

2 Mathematical Formulation

Consider a steady two dimensional flow of water-based nanofluid induced by a nonlinear stretching surface. The sheet is extended with velocity uwx=axn with fixed origin location, where n is a nonlinear stretching parameter, a is a constant and x is the coordinate measured along the stretching surface. The flow is along x-axis and y is normal to it as shown in the Fig.1. A magnetic field of strength B(x) is applied normal to the stretching sheet.

Fig. 1

Schematic representation of the flow diagram.

The wall temperature T_w and the nanoparticle fraction C_w are assumed as constant at the stretching surface. When y tends to infinity, the ambient values of temperature and nanoparticle fraction are denoted by T_∞ and C_∞, respectively. Further, let the temperature of the melting surface as T_m and temperature in the free-stream condition as T_∞, where T_∞>T_m. The considered physical system is of importance in modern nano-technological fabrication and thermal materials processing. It is important to note that the constant temperature and nanoparticle fraction for the stretching surface T_m and C_w are assumed to be greater than the ambient temperature and nanoparticle fraction T_∞, C_∞, respectively.

The governing equations of momentum, thermal energy and concentration can be written as, Khan and Pop[30],

We assume that the variable magnetic field B(x) and is of the form Bx=B0xn−12 [30—33].

The boundary conditions for velocity, temperature and nanoparticle fraction are considered as[8]:

K1=A2νfxuwx(n+1) is the velocity slip factor and A is the dimensionless velocity slip parameter. Using Rosseland approximation for radiation, the radiative heat flux is simplified as,

where σ^* is Stefan-Boltzmann constant and k^* the mean absorption coefficient. The temperature differences within the flow are assumed to be enough small so that T⁴ may be expressed as a linear function of temperature T using a truncated Taylor series about the free stream temperature T_∞ and neglecting the higher order terms, we get,

Substituting (7) and (8) in (3), we have

Rana and Bhargava[8] introduced the following transformations so that equation (1) is satisfied identically.

The governing equations (2), (4) and (9) are reduced into the following set of equations with the aid of equation (10) as follows:

The transformed boundary conditions are

where primes denote differentiation with respect to η, A is the velocity slip parameter and the involved physical parameters are defined as:

which is a combination of the Stefan number cfT∞−Tmλ and cfT∞−T0λ for the liquid and solid phases, respectively. Here, Pr, Le, Nb, Nt, H, Ec, Nr, y and M denote the Prandtl number, the Lewis number, the Brownian motion parameter, the thermophoresis parameter, magnetic parameter, Eckert number, Radiation parameter, chemical reaction parameter and melting parameter, respectively. This boundary value problem reduces to a classical flow problem of heat and mass transfer due to a stretching surface in a viscous fluid when n=1 and Nb=Nt=0 in equations (12) and (13).

The quantities of practical interest, in this study, are the local skin friction coefficient C_fx, Nusselt number Nu_x and the Sherwood number Sh_x and are defined as,

where k is the thermal conductivity of the nanofluid, and q_w, q_m are the heat and mass fluxes at the surface, respectively and are given by

Using (10) and (15) in (14) , we obtain

where Rex=uwxνf is the local Reynolds number.

3 Method of Solution and Validation

The system of non-linear ordinary differential equations (11)-(13) with boundary conditions (14) have been solved using Runge-Kutta-Fehlberg fourth-fifth order method along with Shooting technique. In this system of differential equations f is of third order and θ and ϕ are of second order. In the first step, f, ∞ and ϕ are reduced to a system of seven simultaneous differential equations of first order as follows:

where f1=f,f2=f′,f3=f″,f4=θ,f5=θ′,f6=ϕ,f7=ϕ′ and prime denotes a derivative of the function with respect to η.

The corresponding boundary conditions will become

The shooting technique is employed to find the values of f₃(0), f₅(0) and f₇(0), since they are unknown. We start with some initial guess for some particular set of parameters to obtain f₃(0), f₅(0) and f₇(0). The procedure is repeated with another value of η_∞ until two successive values of f₃(0), f₅(0) and f₇(0) are differ only by the desired accuracy. We have compared the calculated values of f^',θ & ϕ at η =10 with the given boundary conditions and adjusting the estimated values of f^''(0), θ^'(0)& ϕ^'(0) to give a better approximation. One has to take infinity condition at a large but finite value of η where negligible variation in velocity, temperature and so on occurs. Our bulk computations are considered with the value at η_∞ 8 or 10, which is sufficient to achieve the far field boundary conditions asymptotically for all values of the parameters considered.

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 the 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 Δ_η = 0.001, η_∞=8 or 10 and accuracy to the fifth decimal places. To have a check on the accuracy of the numerical procedure used, first test computations for θ '(0) are carried out for viscous fluid for various values of Pr and compared with the available published results of Nadeem and Hussain[34], Gorla and Sidawi[35] and Goyal and Bhargava[36] in Table 1 and they are found to be in excellent agreement.

4 Results and discussion

A theoretical investigation of MHD velocity slip boundary layer flow and melting heat transfer of nanofluids over a nonlinear stretching sheet under the influence of thermal radiation and chemical reaction has been performed. The governing differential equations of the present formulation were solved numerically by means of Runge-Kutta-Fehlberg-45 order method. In order to study the behavior of velocity, temperature distribution and nanoparticle volume fraction profile for slip parameter (A), Eckert number (Ec), magnetic parameter (H), chemical reaction parameter y, nonlinear stretching parameters (n), melting parameter (M), Radiation parameter (Nr), Brownian motion (Nb) and thermophoresis parameter (Nt), graphs are plotted and physical reasons behind the trend of the graphs are discussed.

The variations in velocity field and temperature distribution for various values of the slip parameter (A) are presented in Figs.2(a) and2(b). These figures shows that the effect of increasing value of A is to reducing thickness of the momentum boundary layer. Therefore, the effect of slip parameter is to decrease the boundary layer velocity while the temperature increased with increase in the slip parameter.

Fig. 2

(a) Effect of A on dimensionless velocity; (b) Effect of A on dimensionless temperature.

Fig.3 shows the effect of dimensionless temperature profiles for Eckert number (Ec).This figure shows that the temperature profile increases for increasing values of Ec. Figs.4(a) and4(b) depicts the effect of magnetic parameter (H) on dimensionless velocity and temperature distributions, respectively. It is clear from these Figs. that the velocity decreases, whereas the temperature increases with increase in the magnetic parameter. This is due to the fact that the application of transverse magnetic field in an electrically conducting fluid produces a resistive force known as Lorentz force, which slow down the motion of the fluid in the boundary layer and thus reduces the velocity. The additional work done in dragging the conducting nanofluid against the action of the magnetic field, B_o is manifested as thermal energy. Thus, the presence of magnetic field decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness. The warming of the boundary layer therefore also aids in nanoparticle diffusion which causes a rise in nanoparticle volume fraction.

Fig. 3

Effect of Ec on dimensionless temperature.

Fig. 4

Effect of H on dimensionless (a) velocity, (b) temperature.

The effect of chemical reaction parameter y on nanoparticle volume fraction profile is depicted in Figs.5(a) and5(b) for species consumption and generation cases. It is observed that the nanoparticles volume fraction decreases for constructive chemical reaction parameter and increases for destructive chemical reaction parameter on melting surface.

Fig. 5

(a) Effect of on constructive dimensionless concentration. (b) Effect of on destructive dimensionless concentration.

Figs.6(a) and6(b) reveals that velocity and temperature distributions for different values of melting parameter (M). It is observed that for increasing values of $M$, the velocity and the boundary layer thickness increases and decreases the temperature distribution. This is because, an increase in M will increase the intensity of melting which act as blowing boundary condition at the stretching surface and hence tends to thicken the boundary layer.

Fig. 6

Effect of M on dimensionless (a) velocity, (b) temperature.

Figs.7(a) and7(b) exhibits the effect of nonlinear stretching parameters (n) on the dimensionless velocity and temperature distributions. It is observed that the velocity profile of the nanofluid is insignificantly reduces and temperature profile increases with increasing values of n. The effect of Lewis number (Le) on temperature and concentration profiles are illustrated as in Figs.8(a) and 8(b). It is clearly observed that the temperature profiles decreases and the nanoparticles volume fraction as well as its boundary-layer thickness increases considerably as Le increases. Physically, Le expresses the relative contribution of thermal diffusion rate to species diffusion rate in the boundary layer regime. An increase in Le values will reduce thermal boundary layer thickness and will be accompanied with a decrease in temperature and mass transfer rate increases as Le increases. It also reveals that the concentration gradient at surface of the plate increases.

Fig. 7

Effect of n on dimensionless (a) velocity, (b) temperature.

Fig. 8

Effect of Le on dimensionless (a) temperature, (b) concentration.

Fig.9 explains the effect of radiation parameter (Nr) on temperature profiles. It is observed that, the temperature profile decreases for increasing values of Nr. This is because, an increase in the radiation parameter Nr leads to decrease in the boundary layer thickness and enhance the heat transfer rate on melting surface in the presence of constructive chemical reaction parameter. The radiation parameter $Nr$ being the reciprocal of the Stark number (also known as Stephan number) is the measure of relative importance of the thermal radiation transfer to the conduction heat transfer. Thus larger values of Nr show a dominance of the thermal radiation over conduction. Consequently larger values of Nr are indicative of larger amount of radiative heat energy being poured into the system, causing a rise in θ (η). Fig.10 depicts the effect of Prandtl number (Pr) on temperature profiles. In the presence of melting parameter, an increase in Pr, increases the temperature profiles. Physically, larger Prandtl number possess weaker thermal diffusivity and smaller Prandtl number have stronger thermal diffusivity. This change in thermal diffusivity creates a reduction in the temperature and thermal boundary layer thickness.

Fig. 9

Effect of Nr on dimensionless temperature.

Fig. 10

Effect of Pr on dimensionless temperature.

The effects of Brownian motion (Nb) and thermophoresis parameter (Nt) on temperature and concentration profiles are depicted in Figs.11(a) and 11(b), 12(a) and 12(b) respectively. From these plots, it is observed that the effect of increasing the values of Nb increases temperature profiles and decrease the concentration profiles whereas increasing values of Nt increases both the profiles.

Fig. 11

Effect of Nb on dimensionless (a) temperature, (b) concentration.

Fig. 12

Effect of Nt on dimensionless (a) temperature, (b) concentration.

Fig.13 shows the effect of magnetic parameter (H) with velocity slip parameter (A) in the presence and absence of melting effect and nonlinear stretching parameter on skin friction coefficient. From this figure, it is observed that the skin friction coefficient increases for increasing values of magnetic parameter, while it decreases with velocity slip parameter. This tendency is observed in the presence and absence of melting parameter and nonlinear stretching parameter.

Fig. 13

Effect of H and A on skin friction coeffcient.

The effect of Brownian motion (Nb) with thermophoresis parameter (Nt) in the presence and absence of melting parameter on Nusselt number is illustrated respectively in the Figs.14(a) and 14(b). From these Figs. we observed that the Nusselt number decreases with Nt and Nb on both melting and non-melting surfaces. Further, we can observe that the variations in Nusselt number is more in the presence of melting parameter when compared to its absence.

Fig. 14

Variation of heat transfer rate with Nb and Nt when (a) M = 0, (b) M = 0.5.

Figs.15 and 16 shows the effect Eckert number (Ec) and radiation parameter (Nr) with Prandtl number (Pr) on heat transfer rate. From these Figs. we observed that the Nusselt number decreases for increasing values of Ec and increases for increasing values of Nr with Pr when M = 0.5, n = 2 and M = n = 0.

Fig. 15

Variation of heat transfer rate with Ec and Pr.

Fig. 16

Variation of heat transfer rate with Nr and Pr.

The effect of chemical reaction parameter y with Lewis number (Le) in the presence and absence of melting and nonlinear stretching parameter on Sherwood number is shown in the Fig.17. From these figures it is observed that the concentration gradient increases when M = 0.5, n = 2 and M = n = 0 for increasing values of y.

Fig. 17

Variation of concentration gradient with and Le.

5 Conclusion

MHD boundary layer slip flow and melting heat transfer of a water-based nanofluid induced by a nonlinear stretching sheet with thermal radiation, chemical reaction and viscous dissipation has been investigated numerically. The effects of governing parameters on the flow, concentration and melting heat transfer characteristics are presented through graphs and tables. The main observations of the present study are as follows:

1. The velocity decreases and the temperature will increase for velocity slip parameter.

2. For increasing values of melting parameter, the velocity increases and temperature decreases.

3. An increase in Brownian motion parameter and thermophoresis parameter is to increase the temperature in the thermal boundary layer which consequently reduces the heat transfer rate at the surface when both M = 0 and M = 0.5.

4. Both in the presence and absence of nonlinear stretching parameter and melting parameter, the skin friction coefficient increases, where as the reduced Nusselt numbers decrease for increasing values of magnetic parameter and velocity slip parameter.

5. The Nusselt number decreases for Eckert number and increases for radiation parameter when M = 0.5, n = 2 and M = n = 0.

6 Nomenclature