Conjugate Effects of Radiation Flux on Double Diffusive MHD Free Convection Flow of a Nanofluid over a Power Law Stretching Sheet

Abstract

This study theoretically investigates the conjugate effects of radiation flux and magnetohydrodynamic (MHD) on free convection boundary layer flow of a nanofluid over a nonlinear stretching sheet. It is assumed that the magnetic Reynolds number is small enough and the sheet is stretched with a power law velocity under the effects of the magnetic field, the buoyancy parameter, and the solutal buoyancy parameter. The model used for the nanofluid incorporates the effects of Rosseland approximation, Brownian motion, and thermophoresis parameters. By using appropriate similarity transformations, the governing nonlinear partial differential equations are transformed into dimensionless form and numerically solved using an implicit finite difference scheme known as the Keller-box method. It is found that the variations of magnetic field, buoyancy parameter, solutal buoyancy parameter, and the power law velocity parameter have strong influence on the motion.

1. Introduction

Heat and mass transfer (double-diffusion) phenomenon on free convection is driven by two density gradients which have different rates of diffusion and currently is an important fluid dynamics topic. A common example of double diffusive convection appears in oceanography, where heat and salt concentrations exist with different gradients and diffuse at differing rates. Double diffusive convection is also important in understanding the evolution of a number of systems that have multiple causes for density variations. These include convection in the earth’s oceans, in magma chambers, and in the sun where heat and helium diffuse at differing rates [1]. Double diffusive convection flows for Newtonian and non-Newtonian fluids are extensively studied. However, for nanofluids such studies are scarce due to their complicated nature [2, 3]. Moreover, recent developments in the field of fluid dynamics and nanotechnology confirm that nanofluids are industrially more important than other available fluids [4]. There are numerous biomedical applications that involve nanofluids such as magnetic cell separation, drug delivery, hyperthermia, and contrast enhancement in magnetic resonance imaging [5]. Hence, this is the motivation for considering nanofluids in the present work.

On the other hand, stretching sheet problems with double diffusion are important in extrusion process, glass fibber, paper production, hot rolling, wire drawing, electronic chips, crystal growing, plastic manufactures, food processing, and movement of biological fluids [6]. Khan and Pop [7] investigated the laminar flow of a nanofluid on a stretching flat surface by incorporating the effects of Brownian motion, thermophoresis and reported to be the pioneer for this study of stretching sheet in nanofluid. Rana and Bhargava [8] discussed the flow and heat transfer of a nanofluid over a nonlinearly stretching sheet. Furthermore, the MHD flow of nanofluid over a power law stretching sheet plays an important role in various industrial applications including magnetic control of molten iron flow in the steel industry and liquid metal cooling in nuclear reactors [9]. In addition, when thermal radiation is considered, such studies have useful chemical processing applications [10, 11].

The aim of the present work is to numerically investigate the thermal radiation effects for double diffusive MHD free convection flow of a nanofluid over a power law stretching sheet using Keller-box method [12, 13].

2. Problem Formulation

The unsteady two-dimensional boundary layer flow of a nanofluid past a stretching surface is considered. The stretching velocity is assumed to be 𝑢_𝑤(𝑥)=𝑎𝑥^𝑚, where 𝑎 is constant, 𝑚(𝑚≥0) is the power law exponent parameter, and 𝑥 is the coordinate measured along the stretching surface. The flow takes place above the stretching surface at 𝑦≥0. Here 𝑦 is the coordinate axis measured normal to the stretching surface. A uniform stress leading to equal and opposite forces is applied along the 𝑥-axis, so that the sheet is stretched keeping the origin fixed. Following the work of Hayat and Qasim [6] and Afify [14], we assume that a uniform transverse magnetic field of strength 𝐵 is imposed parallel to the 𝑦-axis and the induced magnetic field due to the motion of the electrically-conducting fluid is negligible. We also assume that the external electrical field is zero and the electric field due to the polarization of charges is negligible. Moreover, it is further supposed that at the stretching surface, the temperature 𝑇 and the nanoparticle fraction 𝐶 take constant values 𝑇_𝑤 and 𝐶_𝑤, whereas the ambient values of temperature 𝑇_∞ and the nanoparticle fraction 𝐶_∞are attained as 𝑦 tends to infinity. The Oberbeck-Boussinesq approximation is employed to the field equations. The governing boundary layer equations that are based on the balance laws of mass, linear momentum, energy, and concentration species for the present problem are given as follows [3, 15]: 𝜌∇⋅𝐕=0,_𝑓𝜕𝐕+()𝐕𝜕𝑡𝐕⋅∇=−∇𝑝+𝜇∇²𝐕−𝜎𝐵²(𝑥)𝐕+1−𝐶_∞𝜌_𝑓∞𝛽_𝑇𝑇−𝑇_∞+𝜌_𝑝−𝜌_𝑓∞𝛽_𝐶𝐶−𝐶_∞𝐠,𝜌𝑐_𝑓𝜕𝑇𝜕𝑡+𝐕⋅∇𝑇=𝑘∇²𝑇+𝜌𝑐_𝑝𝐷_𝐵𝐷∇𝐶⋅∇𝑇+^𝑇𝑇_∞∇𝑇⋅∇𝑇−∇⋅𝑞_𝑟,𝜕𝐶𝜕𝑡+𝐕⋅∇𝐶=𝐷_𝐵𝐷∇𝐶+^𝑇𝑇_∞∇²𝑇, (1) where 𝐕 is the velocity vector, 𝜎 is the electrical conductivity, 𝐠is the acceleration due to gravity, 𝜇 is the viscosity, 𝜌_𝑓 is the density of the base fluid, 𝜌_𝑝 is the density of the nanoparticle, 𝛽_𝑇 is the coefficient of volumetric thermal expansion, 𝛽_𝐶is the coefficient of volumetric concentration expansion, 𝐷_𝐵 is the Brownian diffusion coefficient, 𝐷_𝑇 is the thermophoretic diffusion coefficient, 𝑘is the thermal conductivity, (𝜌𝑐)_𝑝is the heat capacitance of the nanoparticles, (𝜌𝑐)_𝑓 is the heat capacitance of the base fluid, and 𝑞_𝑟 is the radiation flux.

In the following we adopt the Rosseland approximation 𝑞_𝑟=−4𝜎^∗3𝑘^∗𝜕𝑇⁴,𝜕𝑦 (2) where 𝜎^∗ is the Stefan-Boltzmann constant and 𝑘^∗ is the mean absorption coefficient. Assuming that the temperature difference between the free stream 𝑇_∞ and the fluid temperature 𝑇 is small enough, expanding 𝑇⁴ in a Taylor series about 𝑇_∞, and neglecting higher order terms, it results that 𝑇⁴=4𝑇³_∞𝑇−3𝑇⁴_∞. (3)

The two-dimensional steady-state problem under the assumptions that the external pressure in 𝑥-direction having diluted nanoparticles is constant; that is, 𝑝=𝑝₀ yields to 𝜕𝑝/𝜕𝑥=0 and Oberbeck-Boussinesq approximation is valid; (1) are reduced to 𝜕𝑢+𝜕𝑥𝜕𝑣𝜕𝜕𝑦=0,0=𝜇²𝑢𝜕𝑦²−𝜌_𝑓𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦−𝜎𝐵²(𝑥)𝑢+1−𝐶_∞𝜌_𝑓∞𝑔𝛽_𝑇𝑇−𝑇_∞−𝜌_𝑝−𝜌_𝑓∞𝑔𝛽_𝐶𝐶−𝐶_∞,𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦=𝛼∇²𝑇+𝜏𝐷_𝐵𝜕𝐶𝜕𝑦𝜕𝑇+𝐷𝜕𝑦𝑇^𝑇_∞𝜕𝑇𝜕𝑦²−1𝜌𝑐_𝑓𝜕𝑞_𝑟,𝑢𝜕𝑦𝜕𝐶𝜕𝑥+𝑣𝜕𝐶𝜕𝑦=𝐷_𝐵𝜕²𝐶𝜕𝑦²+𝐷_𝑇𝑇_∞𝜕²𝑇𝜕𝑦², (4) where 𝑘𝛼=𝜌𝑐_𝑓,𝜏=𝜌𝑐_𝑝𝜌𝑐_𝑓. (5)

Here 𝑢 and 𝑣 are the velocity components in the 𝑥 and 𝑦-directions, respectively, 𝛼 is the thermal diffusivity parameter and 𝜏 is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid.

The associated boundary conditions are 𝑢=𝑢_𝑤(𝑥)=𝑎𝑥^𝑚,𝑣=0,𝑇=𝑇_𝑤,𝐶=𝐶_𝑤at𝑦=0,𝑢→0,𝑣→0,𝑇→𝑇_∞,𝐶→𝐶_∞as𝑦→∞. (6) Using the stream function 𝜓=𝜓(𝑥,𝑦), the components of velocity are defined as 𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓.𝜕𝑥 (7) By using the following similarity transformations [14], 𝜓=2𝜈𝑎𝑥^𝑚+1𝑓𝜂𝜂=𝑚+1,𝜃𝑇−𝑇_∞𝑇_𝑤−𝑇_∞,𝜙𝜂=𝐶−𝐶_∞𝐶_𝑤−𝐶_∞,𝜂=𝑦(𝑚+1)𝑎𝑥^𝑚−1,2𝜈 (8) into (4), we get 𝑓^′′′+𝑓𝑓^′′−2𝑚𝑓𝑚+1^′2+2𝑀𝑚+1²𝑓^′1+𝜆𝜃−𝛿𝜙=0,()𝜃Pr1+𝑁^′′+𝑓𝜃^′+𝑁𝑏𝜙^′𝜃^′+𝑁𝑡𝜃^′2𝜙=0,_′′+Le𝑓𝜙^′+𝑁𝑡𝜃𝑁𝑏^′′=0, (9) where 𝑀²=𝜎𝐵²₀𝑎𝜌_𝑓,𝜆=𝐺𝑟Re_𝑥^3/2,𝛿=𝐺𝑚Re_𝑥^3/2,𝜈Pr=𝛼,𝑁=16𝜎^∗𝑇³_∞3𝑘𝑘^∗𝜈,Le=𝐷_𝐵,𝜇𝜈=𝜌_𝑓,𝑁𝑏=𝜏𝐷_𝐵𝐶_𝑤−𝐶_∞𝜈,𝑁𝑡=𝜏𝐷_𝑇𝑇_𝑤−𝑇_∞𝜈𝑇_∞,Re_𝑥=𝑢_𝑤(𝑥)𝑥𝜈,𝐺𝑟=1−𝐶_∞𝜌_𝑓∞/𝜌_𝑓𝑔𝑛𝑇_𝑤−𝑇_∞𝜈²Re_𝑥^1/2,𝜌𝐺𝑚=^𝑝−𝜌_𝑓∞/𝜌_𝑓𝑔𝑛₁𝐶_𝑤−𝐶_∞𝜈²Re_𝑥^1/2. (10)

Here 𝑀is the magnetic parameter called Hartmann number, 𝜆 is the buoyancy parameter, 𝛿 is the solutal buoyancy parameter, Pr is the Prandtl number, 𝑁 is the radiation parameter, Le is the Lewis number, 𝜈 is the kinematic viscosity of the fluid, 𝑁𝑏 is the Brownian motion parameter, 𝑁𝑡 is the thermophoresis parameter, Re_𝑥is the local Reynolds number based on the stretching velocity, 𝐺𝑟 is the local thermal Grashof number, 𝐺𝑚 is the local concentration Grashof number, and 𝑓,  𝜃, and 𝜙 are the dimensionless stream functions, temperature, and rescaled nanoparticle volume fraction, respectively. Further, in order to eliminate the dependence of 𝑀 on 𝑥, the magnetic field strength has been taken proportional to 𝑥, that is, 𝐵(𝑥)=𝐵₀𝑥^(𝑚−1)/2 where 𝐵₀ is a constant [14]. Also, 𝛽_𝑇 and 𝛽_𝐶 are proportional to 𝑥⁻³, that is, 𝛽_𝑇=𝑛𝑥⁻³,  𝛽_𝐶=𝑛₁𝑥⁻³,  where 𝑛 and 𝑛₁are constants of proportionality [16].

The corresponding boundary conditions are transformed to 𝑓=0,𝑓^′𝑓=1,𝜃=1,𝜙=1at𝜂=0,_′→0,𝜃→0,𝜙→0as𝜂→∞. (11) In the present case, we define Nusselt number, Sherwood number, and skin-friction as follows: 𝑞Nu=^𝑤𝑥𝑘𝑇_𝑤−𝑇_∞𝑞,Sh=^𝑚𝑥𝐷_𝐵𝐶_𝑤−𝐶_∞,𝐶_𝑓=𝜏_𝑤(1/2)𝜌𝑈², (12) where 𝑞_𝑤=−𝑘𝜕𝑇𝜕𝑦,𝑞_𝑚=−𝐷_𝐵𝜕𝐶𝜕𝑦,𝜏_𝑤=𝜇𝜕𝑢,𝜕𝑦at𝑦=0. (13)

The associated expressions of the dimensionless reduced Nusselt number −𝜃^′(0), reduced Sherwood number −𝜙^′(0), and skin-friction coefficient𝐶_𝑓𝑥 are defined as Nu_𝑟=Nu((𝑚+1)/2)Re_𝑥,Sh_𝑟=Sh((𝑚+1)/2)Re_𝑥,𝐶_𝑓𝑥=𝐶_𝑓22𝑚+1Re_𝑥. (14)

The transformed nonlinear ordinary differential (9) with the boundary conditions (11) are numerically solved by means of the Keller-box method [12, 13].

3. Results and Discussion

Numerical results for some physical parameters of interest are shown in Tables 1 and 2 and in Figures 1–9. In Table 1, the results for reduced Nusselt number −𝜃^′(0) and reduced Sherwood number −𝜙^′(0), when the Hartmann number 𝑀, the buoyancy parameter 𝜆, the solutal buoyancy parameter 𝛿, and the radiation parameter 𝑁 are zero and the power law velocity parameter 𝑚=1, are compared with those obtained by Khan and Pop [7]. We found that, this comparison shows an excellent agreement for the involved flow parameters and in general their solutions appeared as the special cases of our general solutions. Similarly if we take 𝑁𝑏=0.5,𝑁𝑡=0.5,Pr=2.0,Le=2.0, and 𝑚=2.0, the results of Rana and Bhargava [8] can also be obtained as a special case. Hence we are quite confident that our results are accurate and are more general than the existing studies in the literature.

Table 1 

Comparison of reduced Nusselt number and reduced Sherwood number.

Table 2 

Variations of reduced Nusselt number, reduced Sherwood number, and skin-friction coefficient.

Figure 1 

Variations of velocity profiles with 𝜂 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

Figure 2 

Variations of temperature profiles with 𝜂 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

Figure 3 

Variations of concentration profiles with 𝜂 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

Figure 4 

Variations of reduced Nusselt number with 𝜆 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

Figure 5 

Variations of reduced Sherwood number with 𝜆 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

Figure 6 

Variations of skin-friction coefficient with 𝜆 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

Figure 7 

Variations of reduced Nusselt number with 𝛿 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

Figure 8 

Variations of reduced Sherwood number with 𝛿 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

Figure 9 

Variations of skin-friction coefficient with 𝛿 for different values of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚.

The variations of reduced Nusselt number −𝜃^′(0), reduced Sherwood number −𝜙^′(0), and the skin-friction coefficient 𝐶_𝑓𝑥for different values of 𝑁𝑏, 𝑁𝑡, Pr, Le, 𝑀,  𝜆, 𝛿, 𝑁, and 𝑚 are shown in Table 2. It is observed that −𝜃^′(0) is a decreasing function of 𝑁𝑏, 𝑁𝑡, 𝑀, 𝛿, 𝑁, and 𝑚, whereas −𝜃^′(0) increases with increasing values of Pr, Le, and 𝜆. However, it is found that −𝜙^′(0) decreases for large values of 𝑁𝑡, Pr, 𝑀, 𝛿, and 𝑚 and increases for increasing values of 𝑁𝑏, Le, 𝜆, and 𝑁. It is further observed from this table that 𝐶_𝑓𝑥 is an increasing function of 𝑁𝑡, Pr, 𝑀, 𝛿, and 𝑚 and a decreasing one with respect to 𝑁𝑏, Le, 𝜆, and 𝑁.

In order to get physical insight into the problem, convergence of plots for dimensionless velocity, temperature and concentration profiles for different flow parameters are provided in Figures 1–3. Furthermore, the variations of the reduced Nusselt number, the reduced Sherwood number, and skin-friction coefficient as a function of 𝜆and 𝛿 are shown in Figures 4–9. All these graphs are plotted for fixed values of 𝑁𝑏=𝑁𝑡=0.1and Le=10.

It is evident from Figure 1 that an increase in Prresults in a decrease in the velocity, whereas velocity increases for the increasing values of 𝜆and 𝑁. The effect of increasing values of 𝑀 results in a decrease of momentum boundary layer thickness which reduces the dimensionless stream function and the velocity of nanofluid. Physically, it is possible due to the fact that an increase in Pr makes the fluid to be more viscous, which leads to decrease in the velocity for increasing values of Hartmann number and increase the frictional force which reduce the fluid velocity to slow down the fluid motion. Further, it is found that velocity decreases with increasing values of 𝛿 and 𝑚.

Figure 2 is plotted to analyze the heat transfer for combined effects of Pr, 𝑀, 𝜆, 𝛿, 𝑁, and 𝑚 on the temperature profiles. This figure indicates that an increase in Pr and 𝜆 results in a decrease in the temperature profiles, whereas these profiles increase for the increasing values of 𝛿, 𝑁, and 𝑚. The radiation parameter 𝑁 is very effective for the thickening of the thermal boundary layer which releases the heat energy from the flow region and causes the cooling of the system. Physically, it is true due to the fact that temperature increases by increasing the Rosseland approximation. Increasing values of 𝑀cause the temperature to rise up. This is due to the fact that nanoparticles dissipate energy in the form of heat which causes the thermal boundary layer thickness to increase in the case of nanofluids and ultimately a localized rise in temperature of the fluid occurs.

Figure 3 is prepared to study the effects of incorporated different flow parameters on concentration profiles. It results that the increasing values of 𝜆 and 𝑁 cause a decrease in the mass transfer. However, the concentration increases for large values of Pr, 𝑀, 𝛿, and 𝑚. Physically, the variations of Pr, 𝑀, 𝜆, 𝛿, and 𝑚 are important in the boundary layer flow and play a significant role in shortening the concentration boundary layer for the mass fraction. All profiles discussed above descend smoothly in the free stream satisfying boundary conditions. This ensures the accuracy of the obtained numerical results.

The variations of the reduced Nusselt number, reduced Sherwood number, and skin-fiction coefficient as functions of 𝜆 for the different values of Pr, 𝑀, 𝛿, 𝑁, and 𝑚 are shown in Figures 4–6. From Figure 4, we observed that the reduced Nusselt number increases for increasing values of Pr and decreases for increasing values of 𝑀, 𝛿, 𝑁, and 𝑚. This increase in the heat transfer rate at the surface of the sheet is due to the reason that nanofluids with high viscosities have large values of Prandtl number which results for relatively low thermal conductivities to reduce the conduction phenomenon [7]. Physically, this behaviour is meaningful due to the fact that with increasing viscosity of the base fluid, the thermal boundary layer thickness decreases because of less colloidal forces between the base fluid and suspended nanoparticles. So the thermal conductivities of nanofluids play an important role to increase the heat transfer coefficient between the heat transfer medium and heat transfer surface. Further, Figure 5 shows that the reduced Sherwood number decreases for increasing values of Pr  but increases for increasing values of 𝑁, 𝑀, 𝛿, and  𝑚. It is clearly seen from Figure 6 that the dimensionless skin-friction coefficient along 𝜆 increases with increasing values of Pr, 𝑀, 𝛿, and 𝑚but decreases with increasing values of 𝑁.

Finally, Figures 7–9 show the variation in the reduced Nusselt number, the reduced Sherwood number, and skin-fiction coefficient along 𝛿 for different values of Pr, 𝑀, 𝜆, 𝑁, and 𝑚. It is found from Figure 7 that the reduced Nusselt number decreases for increasing values of 𝑀, 𝑁, and 𝑚 and increases for increasing values of Pr and 𝜆. The reduced Sherwood number increases for large values of 𝜆 and 𝑁 but decreases when the values of Pr, 𝑀, and 𝑚 are increased as shown in Figure 8. Further, Figure 9 shows that skin-friction coefficient increases for the larger values of Pr, 𝑀, and 𝑚and decreases with regard to 𝜆and 𝑁. It is clearly seen that the influence of these parameters on the local Sherwood number and skin-fiction coefficient with 𝜆 and 𝛿 is quite opposite to each other.

4. Conclsions

In the present study we have investigated the conjugate effects of thermal radiation flux on double diffusive MHD free convection boundary layer flow of a nanofluid over a nonlinear stretching sheet. The model used here for nanofluid incorporates the effects of thermophoresis parameter 𝑁𝑏, Brownian motion parameter 𝑁𝑡, Prandtl number Pr, Lewis number Le, Hartmann number 𝑀, buoyancy parameter 𝜆, solutal buoyancy parameter 𝛿, radiation parameter 𝑁, and nonlinear stretching parameter 𝑚. The governing nonlinear partial differential equations are numerically solved using implicit finite difference scheme known as Keller-box method. Numerical solutions depending upon all parameters are presented in tables and figures. The physical interpretations provided by the authors reflect some important characteristics of free convection flow of a nanofluid over a power law stretching sheet and play a significant role in the concentration boundary layer for the mass fraction. The following main results are concluded from this study.(1)The reduced Nusselt number −𝜃^′(0) is a decreasing function of 𝑁𝑏, 𝑁𝑡, 𝑀, 𝛿, 𝑁, and 𝑚whereas increases with increasing values of Pr, Le, and 𝜆.(2)The reduced Sherwood number −𝜙^′(0) decreases for large values of𝑁𝑡, Pr, 𝑀, 𝛿, and 𝑚and increases for increasing values of 𝑁𝑏, Le, 𝜆, and 𝑁.(3)The skin-friction coefficient 𝐶_𝑓𝑥 is an increasing function of 𝑁𝑡, Pr, 𝑀, 𝛿, and 𝑚and a decreasing one with respect to 𝑁𝑏, Le, 𝜆, and 𝑁.(4)The velocity of nanofluid decreases for increasing Pr whereas increases for the increasing values of 𝜆 and 𝑁.(5)The temperature of nanofluid decreases when  Pr  and 𝜆 are increased whereas increases for large values of 𝑀, 𝛿, and 𝑚.(6)The species concentration of nanofluid decreases for large values of 𝜆and 𝑁 whereas increases for increasing values of Pr, 𝑀, 𝛿, and 𝑚.(7)The reduced Nusselt number versus  𝜆  increases for  Pr  and decreases for 𝑀, 𝛿, 𝑁, and 𝑚 whereas the reduced Nusselt number against 𝛿 decreases for 𝑀, 𝑁, and 𝑚 while increases for Pr and 𝜆. However, the reduced Sherwood number and skin-friction coefficient versus 𝜆 and 𝛿 are found to be opposite in nature.

Acknowledgments

The authors would like to acknowledge the financial support received from MOHE, Research Management Centre-RMC (4F109), UTM, Malaysia, and University Malaysia Pahang (RDU110108).