MHD 3-dimensional nanofluid flow induced by a power-law stretching sheet with thermal radiation, heat and mass fluxes

Boundary layer flow over a stretching surface has various trade and technical applications, such as metal and polymer extrusion, paper production, and many others.

Also, it has important applications in engineering and manufacturing processes. Such processes are wire and fibre covering, foodstuff dispensation, heat-treated materials travelling between a feed roll and a wind-up roll or materials manufactured by extrusion, glass fibre production, cooling of metallic sheets or electronic chips, crystal growing, drawing of plastic sheets and so on. The quality of the final product with the desired features in these processes depends on the degree of freezing in the process and the process of stretching.

Flow past a linearly stretching sheet was first studied analytically by Crane (1970). Gupta and Gupta (1977) mentioned that in all realistic situations, the stretching mechanism is not linear. Later on, several investigators, Hamad and Pop (2010), Turkyilmazoglu (2013), Pal and Mandal (2015), Waqas et al. (2016), studied such type of flow problems either with heat or with mass transfer effects.

The main characteristic of nanofluid (a combination of nanoparticles having size 1–50 nm and liquid) is to improve the base fluid's thermal conductivity. Usually, ethylene glycol, oil, water, and so on are taken as base fluids that have limited heat transfer properties due to their low thermal conductivity. By adding nanoparticles into the base fluid, Masuda et al. (1993) first detected the change in viscosity and thermal conductivity of such fluids. The abnormal increase in the thermal conductivity occurs by the dispersion of nanoparticle, which was first observed by Choi and Eastman (1995). In case of nanofluids, Buongiorno (2006) studied the convective heat transfer and indicated that the Brownian motion, thermophoresis as the most important means for the unpredicted heat transfer augmentation. Later on, different aspects of nanofluid flow were addressed by several researchers. Considering heat and mass transfer, Das (2012) pointed out the slip effects in nanofluid flow. By the finite element method, Rana and Bhargava (2012) studied the nanofluid flow towards a non-linearly stretched surface. Nadeem and Lee (2012) studied the steady nanofluid flow over an exponentially stretched sheet. Considering the convective boundary condition (CBC), Mustafa et al. (2013) discussed nanofluid flow over an exponentially stretched surface. Using the Jeffrey's model for non-Newtonian fluid, Nadeem et al. (2014) investigated nanofluid flow over a stretching surface to obtain a numerical solution. Magnetohydrodynamics (MHD) plays a great role in several areas particularly in geophysics, medicine, engineering and many others [Hayat et al. (2016)]. Magneto nanofluids have significant applications in drug delivery, cancer therapy, enhancement of magnetic resonance and magnetic cell separation (Hayat et al. 2016). In 2013, Khan et al. analysed the effects of a magnetic field, applied externally on nanofluid flow due to a stretching sheet considering a non-similar solution of the problem. Influences of the magnetic field on Cu-water flow were analysed by Sheikholeslami et al. (2014). Using the Buongiorno's model, in 2016, Sheikholeslami and his co-authors described the unsteady flow of nanofluid in the presence of a magnetic field. Ma et al. (2019) studied the consequences of the magnetic field on forced convection Ag-MgO nanofluid flow in a channel with heat transfer. Many other research papers on nanofluid with magnetic field effects can be found in the studies conducted by Ma et al. (2019) and Bhatti et al. (2018).

Thermal radiation affects the heat transfer characteristics significantly particularly in control of heat transfer, space technology and high-temperature processes (Das et al. 2014). Hayat et al. (2016) reported the combined effects due to Joule heating and solar emission on MHD thixotropic nanofluid flow. Radiative flows in companies with the external magnetic fields have wider applications in nuclear engineering, power technology, astrophysics, power generation and so on (Hayat et al. 2016).

The investigations described above are two-dimensional. Three-dimensional problem being more realistic and has already drawn the attention of several researchers. Wang (1984) first addressed the three-dimensional flow due to a bidirectional stretching sheet. Ariel (2007) obtained the solution for a three-dimensional flow caused by a stretching surface using a perturbation technique. Liu and Andersson (2008) analysed the heat transport of a 3-D flow due to a stretching surface. Ahamad and Nazar (2011) discussed the MHD 3-D flow of a viscoelastic fluid due to a stretching surface. In 2013, Hayat and his co-authors considered 3-D Oldroyd-B fluid flow past a stretching surface with a CBC. Considering the exponential temperature distribution, 3-D flow and heat transfer of a viscous fluid were investigated by Liu et al. (2013). Nadeem et al. (2013) analysed the 3-D Casson fluid flow due to a linearly stretching sheet in the presence of a magnetic field. Khan et al. (2014) obtained analytical and numerical solutions for a three-dimensional flow caused by a non-linearly stretching sheet. Khan et al. (2015) investigated three-dimensional flow of a nanofluid past a non-linearly stretching sheet and highlighted its applications to the solar energy industry. Existence of two solutions was reported by Raju et al. (2016) for flow due to a non-linearly stretching sheet. Considering thermal radiation and heat generation/absorption, Shehzad et al. (2016) analysed 3-D MHD flow of a Jeffrey nanofluid over a bidirectional stretching sheet.

The combined effects of MHD and thermal radiation on three-dimensional non-linear convective Maxwell nanofluid flow owing to a stretching sheet were reported by Hayat et al. (2017). The literature review reveals the fact that there is not so much work has been done on three-dimensional nanofluid flow caused by a non-linearly stretching sheet.

Motivated by this, in this article, we investigate the radiative heat transport of the MHD flow of a nanofluid past a stretching sheet with power-law velocity in two directions. The mathematical representations of the nanofluid model proposed by Buongiorno (2006) reflecting the contribution of nanoparticle volume fraction were used. This model also includes the effects of thermophoresis and the Brownian motion. The main objective here is to analyse the heat transport features of the flow of a nanofluid owing to the collective influences of magnetic field and radiation. Two different cases are considered: In Case I, uniform surface temperature is considered. Due to thermophoresis, a flux of nanoparticles volume fraction at the sheet is assumed to be zero. This assumption controls the volume fraction of nanoparticles submissively rather than actively (Mansur and Ishak 2013; Nield and Kuznetsov 2014); on the other hand, in Case II, heat and mass fluxes are considered at the sheet. Suitable similarity transformations are considered to transform the highly non-linear partial differential equations into ordinary ones. Numerical results are obtained by a Runge–Kutta method with a shooting technique. For a clear understanding of the flow and heat transport characteristics, the obtained results are analysed in detail with their physical interpretation through Tables and Figures. It is observed that the physical parameters affect the heat and mass transport significantly.

Let us consider a 3-D steady boundary layer nanofluid flow past a non-linearly stretching surface. In a usual notation, the governing equations of the problem can be written as (1) ∂u∂x+∂υ∂y+∂w∂z=0 \frac{{\partial u}}{{\partial x}} + \frac{{\partial \upsilon }}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0 (2) u∂u∂x+υ∂u∂y+w∂u∂z=ν∂2u∂z2-σB02ρfu u\frac{{\partial u}}{{\partial x}} + \upsilon \frac{{\partial u}}{{\partial y}} + w\frac{{\partial u}}{{\partial z}} = \nu \frac{{{\partial ^2}u}}{{\partial{z^2}}} - \frac{{\sigma B_0^2}}{{{\rho _f}}}u (3) u∂υ∂x+υ∂υ∂y+w∂υ∂z=ν∂2υ∂z2-σB02ρfυ u\frac{{\partial \upsilon }}{{\partial x}} + \upsilon \frac{{\partial \upsilon}}{{\partial y}} + w\frac{{\partial \upsilon }}{{\partial z}} = \nu\frac{{{\partial ^2}\upsilon }}{{\partial {z^2}}} - \frac{{\sigma B_0^2}}{{{\rho_f}}}\upsilon (4) u∂T∂x+υ∂T∂y+w∂T∂z=α∂2T∂z2-1(ρc)f∂qr∂z+(ρc)p(ρc)f[DB∂C∂z∂T∂z+DTT∞(∂T∂z)2] u\frac{{\partial T}}{{\partial x}} + \upsilon \frac{{\partial T}}{{\partial y}}+ w\frac{{\partial T}}{{\partial z}} = \alpha \frac{{{\partial ^2}T}}{{\partial{z^2}}} - \frac{1}{{{{\left( {\rho c} \right)}_f}}}\frac{{\partial{q_r}}}{{\partial z}} + \frac{{{{(\rho c)}_p}}}{{{{(\rho c)}_f}}}\left[{{D_B}\frac{{\partial C}}{{\partial z}}\frac{{\partial T}}{{\partial z}} +\frac{{{D_T}}}{{{T_\infty }}}{{\left( {\frac{{\partial T}}{{\partial z}}}\right)}^2}} \right] (5) u∂C∂x+υ∂C∂y+w∂C∂z=DB∂2C∂z2+DTT∞∂2T∂z2 u\frac{{\partial C}}{{\partial x}} + \upsilon \frac{{\partial C}}{{\partial y}}+ w\frac{{\partial C}}{{\partial z}} = {D_B}\frac{{{\partial ^2}C}}{{\partial{z^2}}} + \frac{{{D_T}}}{{{T_\infty }}}\frac{{{\partial ^2}T}}{{\partial {z^2}}} where the components of velocity in the x,y and z directions are u, υ and w, respectively; kinematic viscosity is ν, electrical conductivity is σ, ρ_f is the density, T denotes the temperature of the fluid, α denotes the thermal diffusivity, B₀ = a/(x + y)¹⁻ⁿ represents the variable magnetic field, the effectual heat capability of the nanoparticles is (ρc)_p, and (ρc)_f is the effectual heat capability of nanofluid; D_B, D_T are the diffusion coefficients due to the Brownian motion and the thermophoresis, respectively; q_r represents the radiative heat flux, C is the nanoparticle volume fraction and T_∞ is the steady temperature of the inviscid free stream.

Using the Rosseland approximation, the expression for radiative heat flux can be written as (6) qr=-4σs3ke∂T4∂z {q_r} = - \frac{{4{\sigma _s}}}{{3{k_e}}}\frac{{\partial {T^4}}}{{\partial z}} where σ_s and k_e are the Stefan–Boltzmann constant and the mean absorption coefficient, respectively. This investigation is restricted to the optically bulky fluid. For sufficiently small temperature difference linearisation of Eq. (6) can be performed. Using the Taylor's series expansion for T⁴ about T_∞ and ignoring the higher-order terms, we get T4=4T∞3T-3T∞4. {T^4} = 4T_\infty ^3T - 3T_\infty ^4.

A. Boundary conditions for fluid flow

The boundary conditions can be written as (7) at z=0, u=uw=a(x+y)n,υ=υw=b(x+y)n,w=0 and as z→∞,u→0,υ→0 {\rm{at\;\; }}z = 0,\,\,u = {u_w} = a{\left( {x + y} \right)^n},\upsilon ={\upsilon _w} = b{\left( {x + y} \right)^n},w = 0\,{\rm{ \;\;and}}\,\,{\rm{as\;\;}}z \to\infty ,u \to 0,\upsilon \to 0 u_w and υ_w denote the velocities of stretching along the x and y directions, respectively, and n is a non-linearly stretching parameter and C_∞ is the uniform volume fraction for the nanoparticles in the inviscid free stream.

B. Boundary conditions for heat, mass transport

There are different types of heating process that specify the temperature distribution on the surface and ambient fluid. These are constant or prescribed wall temperature (CWT or PWT); constant or prescribed surface heat flux (CHF or PHF), CBC and Newtonian heating. In this problem, we have considered two cases of temperature distributions: CWT and CHF separately. Moreover, different methods are available for the active or passive control of nanoparticles, namely, constant or prescribed concentration at the wall (CHF or PWC); zero mass or normal flux of nanoparticles at the surface (ZMF); constant or prescribed mass flux (CMF or PMF). In this problem, two different boundary conditions, ZMF and CMF for mass transfer, are considered.

Therefore, the conditions at the boundary for temperature, volume fraction for nanoparticles are

Case I: (8a) T=Tw, DB∂C∂z+DTT∞∂T∂z=0 at z=0 (CWT,ZMF) & T→T∞, C→C∞ as z→∞ T = {T_w},\,{D_B}\frac{{\partial C}}{{\partial z}} + \frac{{{D_T}}}{{{T_\infty}}}\frac{{\partial T}}{{\partial z}} = 0\,{\rm{at}}\,\,z = 0\text{ (CWT, ZMF) & }T\to {T_\infty },\,C \to {C_\infty }\,\,{\rm{as}}\,\,z \to \infty

Let us define the similarity variable and similarity transformations as η=aν(x+y)n-12z, \eta = \sqrt {\frac{a}{\nu }} {\left( {x + y} \right)^{\frac{{n - 1}}{2}}}z, (9a) u=a(x+y)nf′(η), υ=a(x+y)ng′(η), w=-aν(x+y)n-12[n+12(f+g)+n-12η(f′+g′)], u = a{\left( {x + y} \right)^n}f'(\eta ),\,\upsilon = a{\left( {x + y}\right)^n}g'(\eta ),\,w = - \sqrt {a\nu } {\left( {x + y} \right)^{\frac{{n -1}}{2}}}\left[ {\frac{{n + 1}}{2}\left( {f + g} \right) + \frac{{n - 1}}{2}\eta\left( {f' + g'} \right)} \right], θ(η)=T-T∞Tw-T∞ \theta (\eta ) = \frac{{T - {T_\infty }}}{{{T_w} - {T_\infty }}} , φ=C-C∞C∞ \varphi = \frac{{C - {C_\infty }}}{{{C_\infty }}} (for Case-I) (9b) T=T∞+Twα(νa(n+1)(x+y)n-1)12θ(η), C=C∞+CwDB(νa(n+1)(x+y)n-1)12φ(η) T = {T_\infty } + \frac{{{T_w}}}{\alpha }{\left( {\frac{\nu }{{a(n + 1){{(x +y)}^{n - 1}}}}} \right)^{\frac{1}{2}}}\theta (\eta ),\,C = {C_\infty } +\frac{{{C_w}}}{{{D_B}}}{\left( {\frac{\nu }{{a(n + 1){{(x + y)}^{n - 1}}}}}\right)^{\frac{1}{2}}}\varphi (\eta ) (for Case-II).

Using these transformations in Eqs (1)–(5) and also using the relation given by Eq. (6), we get the following self-similar equations (11) f‴+n+12(f+g)f″-n(f′+g′)f′-Mf′=0 f''' + \frac{{n + 1}}{2}(f + g)f'' - n(f' + g')f' - {\rm{M}}f' = 0 (12) g‴+n+12(f+g)g″-n(f′+g′)g′-Mg′=0 g''' + \frac{{n + 1}}{2}(f + g)g'' - n(f' + g')g' - {\rm{M}}g' = 0 (13) 1Pr(1+43R)θ″+n+12(f+g)θ′+Nbθ′ϕ′+Ntθ′2=0 \frac{1}{{\Pr }}(1 + \frac{4}{3}R)\theta '' + \frac{{n + 1}}{2}(f + g)\theta ' +{N_b}\theta '\phi ' + {N_t}{\theta '^2} = 0 (14) ϕ″+(n+1)2Sc(f+g)ϕ′+NtNbθ″=0. \phi '' + \frac{{(n + 1)}}{2}Sc(f + g)\phi ' + \frac{{{N_t}}}{{{N_b}}}\theta ''= 0. Here, the magnetic parameter is M=σB02ρfa {\rm{M}} = \frac{{\sigma B_0^2}}{{{\rho_f}a}} , Prandtl number is Pr=να \Pr = \frac{\nu }{\alpha } , Nb=(ρc)pDBC∞(ρc)fν {N_b} = \frac{{{{({\rho _c})}_p}{D_B}{C_\infty }}}{{{{({\rho _c})}_f}\nu }} represents the Brownian motion parameter and Nt=(ρc)pDT(Tw-T∞)(ρc)fvT∞ {N_t} = \frac{{{{({\rho _c})}_p}{D_T}({T_w} - {T_\infty })}}{{{{({\rho _c})}_f}v{T_\infty }}} represents the thermophoresis parameter, radiation parameter is R=4σsT∞3ke(ρc)fα {\rm{R}} = \frac{{4{\sigma _s}T_\infty^3}}{{{k_e}{{(\rho c)}_f}\alpha }} , Sc=νDB Sc = \frac{\nu }{{{D_B}}} is the Schmidt number, λ=ba \lambda = \frac{b}{a} denotes the ratio of rate of stretching along y-direction to that of x-direction.

The boundary condition becomes (15a) f(η)=0, f′(η)=1 at η=0 and f′(η)→1 as η→∞ f(\eta ) = 0,\,f'(\eta ) = 1\,{\rm{at}}\,\eta = 0\,{\rm{and}}\,f'(\eta ) \to1\,{\rm{as}}\,\eta \to \infty (15b) g(η)=0, g′(η)=λ at η=0 and g′(η)→1 as η→∞ g(\eta ) = 0,\,\,g'(\eta ) = \lambda \,\,{\rm{at}}\,\,\eta =0\,\,{\rm{and}}\,\,\,g'(\eta ) \to 1\,\,{\rm{as}}\,\,\eta \to \infty θ(η)=1 and Nbϕ′(η)+Ntθ′(η)=0 at η=0 (for Case-I) \theta (\eta ) = 1\,{\rm{and }}{{\rm{N}}_{\rm{b}}}\phi '(\eta ) +{{\rm{N}}_{\rm{t}}}\theta '(\eta ) = 0\,{\rm{\;at\;}}\,\eta = 0\,{ (\text{for}\,\, \text{Case-I})} (15c) θ′(η)=-1,ϕ′(η)=-1 at η=0 (for Case-II) \theta'(\eta) = -1, \phi'(\eta) = -1\; at \; \eta =0\;\;\;\;\mathrm{(for\;\; Case-II) } (15d) θ(η)→0,ϕ(η)→0 as η→∞. \theta (\eta) \to 0 , \phi(\eta) \to 0 \;\; as \;\; \eta \to \infty. Dimensionless wall shear stresses, heat and mass fluxes are Cfx=τzxρfuw2, Cfy=τzyρfνw2, Nu=(x+y)qwk(Tw-T∞), Sh=(x+y)jwDB(Cw-C∞) Cf_x = \frac{\tau_{zx}}{\rho_f u_w^2}, \; Cf_y = \frac{\tau_{zy}}{\rho_f \nu_w^2} , \; Nu = \frac{(x+y)q_w}{k(T_w -T_\infty)} , \; Sh = \frac{(x+y)j_w}{D_B(C_w - C_\infty)} where τ_zx and τ_zy are shear stresses at the wall and q_w represents the heat flux at the wall, j_w denotes the mass flux which are described as τzx=μ(∂u∂z+∂w∂x)z=0,τzy=μ(∂ν∂z+∂w∂y)z=0,qw=-k(∂T∂y)z=0,jw=-DB(∂C∂y)z=0. \tau_{zx} = \mu \left( \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right)_{z=0},\tau_{zy} = \mu \left(\frac{\partial \nu}{\partial z} + \frac{\partial w}{\partial y} \right)_{z=0},q_w =-k \left( \frac{\partial T}{\partial y} \right)_{z=0},j_w = -D_B \left( \frac{\partial C}{\partial y}\right)_{z=0}. Finally, we get CfxRex1/2=f″(0), CfyRey1/2λ3/2=g″(0), NuRex-1/2=-θ′(0), ShRex-1/2=-ϕ′(0) Cf_x Re_x^{1/2} =f '' (0), \; Cf_y Re_y^{1/2} \lambda^{3/2} = g'' (0), \; NuRe_x^{-1/2} = - \theta'(0), \; ShRe_x^{-1/2} = - \phi'(0) where the Reynolds numbers (local) along the directions of x,y are Rex=uw(x+y)vf Re_x = \frac{u_w(x+y)}{v_f} and Rey=uw(x+y)vf Re_y = \frac{u_w(x+y)}{v_f} , respectively. The far field component of velocity may be written in the form w(x,t,∞)=-avy(x+y)n-1n+12[f(∞)+g(∞)] w(x,t,\infty) = - \sqrt{a v_y (x+y)^{n-1}} \frac{n+1}{2} [f(\infty) + g(\infty)]

The differential Eqs (11)–(14) are coupled and non-linear. Equations (11)–(14), together with the conditions (15a–15d), form a two-point boundary value problem. By converting it into an initial value problem, those equations are numerically solved using the Runge–Kutta method and the shooting technique by choosing an appropriate value for n_∞.

The current study provides solutions (numerical) for a steady MHD three-dimensional nanofluid flow past a bi-directional non-linearly stretching surface. Two types of conditions at the boundary for temperature and volume fraction for nanoparticles are considered. From the current investigation, the following observations are derived:

A rise in the magnetic strength causes a reduction in the resistance and the rate of heat transport.