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Analytical study for unsteady nanofluid MHD Flow impinging on heated stretching sheet
Journal of Molecular Liquids 219 (2016) 216–223
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Analytical study for unsteady nanofluid MHD Flow impinging on heated stretching sheet F. Mabood a,⁎, W.A. Khan b a b
Department of Mathematics, University of Peshawar, Pakistan Department of Mechanical and Industrial Engineering, College of Engineering, Majmaah University, Majmaah 11952, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 15 November 2015 Received in revised form 2 January 2016 Accepted 22 February 2016 Available online xxxx Keywords: Unsteady boundary layer MHD nanofluid Thermal radiation HAM
a b s t r a c t An analysis is carried out to obtain analytical solution of an unsteady two-dimensional MHD nanofluid flow with heat and mass transfer over a heated surface. The governing partial differential equations are reduced to system of nonlinear ordinary differential equations using suitable transformations. The resulting nonlinear coupled system subject to the boundary conditions is solved using homotopy analysis method (HAM). Graphical and numerical demonstrations of the convergence of the HAM solutions are provided. A detailed study illustrating the influences of the magnetic, unsteady, suction/injection and nanofluid parameters, on the dimensionless velocity, temperature, concentration as well as on the skin friction coefficient, the reduced Nusselt and Sherwood numbers is conducted. It is found out that the flow field is substantially influenced due to unsteadiness, transpiration and magnetic field. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Magneto-fluid-dynamics analyzes the dynamics of electrically conducting fluids such as plasmas, liquid metals and salt water. The term magnetohydrodynamics (MHD) is a combination of magneto (magnetic field), hydro (liquid) and dynamics (movement of particles). The magnetic field induced current flows in a fluid and creates forces on the fluid. The combination of the Navier-Stokes equations of fluidmechanics and Maxwell's equations of electromagnetism consequently establishes MHD relations [1,2]. Due to promising potential for heat transfer applications, fluid heating and cooling have received significant importance in different industries such as electronics, power, manufacturing and transportation. Due to practical impact, modern cooling techniques are essential for cooling of high-energy devices. In many cases of natural convection, the conventional fluids such as water, engine oil, ethylene glycol etc. have a low thermal conductivity, which limits the heat transfer capabilities. However, the growing demand of advanced technology with respect to miniaturization of electronic devices requires further improvement of heat transfer from the point of view of point energy saving. To overcome this drawback, there is a need to develop a new heat transfer medium behaving like fluid with higher heat transfer capabilities to increase thermal characteristics which is known as nanofluid. Recently numerous numerical/analytical studies for the improvement of heat transfer of nanofluids have been published [3–14]. ⁎ Corresponding author. E-mail addresses: [email protected] (F. Mabood), [email protected] (W.A. Khan).
http://dx.doi.org/10.1016/j.molliq.2016.02.071 0167-7322/© 2016 Elsevier B.V. All rights reserved.
An excellent review of nanofluid physics and developments can be found in the review paper by Wang and Mujumdar [15]. Buongiorno [16] reported a complete review of convective transport in nanofluids by considering slip mechanisms which generate a relative velocity between the base fluid and nanoparticles. An overview on convective heat transfer in nanofluids and their applications were investigated experimentally and theoretically by Godson et al. [17]. Thermal radiation plays an important role on the flow and heat transfer in space technology. Radiation effects on convective MHD flow problems are also significant in electrical power generation, astrophysical ground, solar power technology, re-entry of space vehicle and other industrial areas. Hayat et al. [18] discussed the effects of thermal radiation on the MHD mixed convection flow over a stretching surface. Ibrahim [19] investigated the effects of mass transfer, radiation, Joule heating, and viscous dissipation on steady MHD Marangoni convection flow over a flat surface with suction and injection. Thermal radiation effects on mixed convection heat transfer for viscoelastic fluid flow over a porous wedge were reported by Rashidi et al. [20]. Rana and Bhargava [21] provided numerical solution for nanofluid flow over a nonlinear stretching sheet whilst Mabood et al. [22] studied the effect of thermal radiation on Casson fluid flow, heat and mass transfer around a circular cylinder in porous medium. Shateyi and Prakash [23] studied thermal radiation effects on MHD flow and heat transfer of nanofluids over a moving surface. Nanofluid flow and heat transfer characteristics were studied by numerous researchers including [24–26]. In this paper, our main objective is to investigate the unsteady flow of an electrically conducting nanofluid past a heated stretching sheet with thermal radiation. The problem is first modeled and then solved analytically by homotopy analysis method [27–29]. Convergence
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
region of the solutions is determined. The behavior of Brownian motion and thermophoretic diffusion of nanoparticles has been examined graphically. 2. Governing equations Consider an unsteady boundary layer flow of nanofluid past a permeable stretching surface in the presence of external magnetic ! field B and thermal radiation. The coordinate system is chosen in such a way that x-axis is measured along the sheet and y-axis is normal to it as shown in the Fig. 1. The flow is confined in the region y N 0. Two equal and opposite forces are spontaneously applied along the direction of x-axis so that the sheet is stretched with fixed origin. For time t N 0, it is understood that the flow is in steady state. The unsteady fluid flow starts at t = 0. The sheet is being stretched with velocity U w ðx; tÞ ¼ ax ð1−λtÞ
along x-axis where a is the stretching rate, λ is the positive
constant with the property λt b 1 and dimension of λ is (time)−1. We bx assume that ambient fluid moves with a velocity, U e ðx; tÞ ¼ ð1−λtÞ in the y-direction towards the stagnation point on the plate where b(N 0) is the strength of stagnation point flow. Also, the temperature of the sheet Tw(x, t) and the volume fraction of nanoparticles Cw(x, t) both vary with x and time t, T∞ and C∞ are the temperature and nanoparticles volume fraction of the fluid far away from the sheet. Keeping in view of thermal equilibrium, no slip occurs between base fluid and nanoparti! ! cles. As the fluid is electrically conducting, so the Lorentz force J B , ! ! where J is the electrical current density and B ¼ ð0; B; 0Þ is the transverse magnetic field, must be included in the momentum equation. In the boundary layer approximation the Lorentz force is simplified as −σB2(u − Ue), where σ is the electrical conductivity, u is B0 ffi is the magnetic field the x-component of the fluid velocity, B ¼ pffiffiffiffiffiffiffiffi 1−λt
imposed along y-axis and B0 is the intensity of the magnetic field. The governing equations for the continuity, momentum, energy, and concentration can be written: ∂u ∂v þ ¼ 0; ∂x ∂y
where x and y are the coordinates along and normal to the sheet, u and v are the components of the velocity in the x- and y- directions respectively, ρf is density of the base fluid, νf is kinematic viscosity of the base fluid, α is the thermal diffusivity, T is the temperature, C is the nanoparticles volume fraction, τ = (ρc)f/(ρc)p is the ratio of nanoparticles heat capacity and the base fluid heat capacity, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, σ⁎ is the Stefan Boltzmann constant, k⁎ is the mean absorption coefficient. The boundary conditions for the above described model are:
2
∂u ∂u ∂u ∂U e ∂U e ∂ u σ B ðu−U e Þ ; þu þv ¼ þ Ue þ νf 2 − ρnf ∂t ∂x ∂y ∂y ∂t ∂x ∂T ∂T ∂T 16T 3∞ σ þu þv ¼α 1þ ∂t ∂x ∂y 3k κ
!
ð2Þ
2 2 ∂C ∂C ∂C ∂ C DT ∂ T ; þu þv ¼ DB 2 þ T ∞ ∂y2 ∂t ∂x ∂y ∂y
ð5Þ
ax2 ð1−λtÞ−2 ; C w ðx; t Þ 2ν 2f ax ð1−λtÞ−2 ; ¼ C∞ þ C0 2ν f
T w ðx; t Þ ¼ T ∞ þ T 0
ð6Þ
where T0 , C0 are the positive reference temperature and the nanoparticles volume fraction respectively such that 0 ≤ T0 ≤ Tw and 0 ≤C0 ≤Cw. The above expressions are valid when (1− λt) N 0. As per usual, the stream function ψ is defined as u = ∂ ψ/∂ y and v = − ∂ ψ/∂ x so that Eq. (1) is satisfied. We introduce the following dimensionless quantities: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > aν f a > > y; ψ ¼ xf ðηÞ; η ¼ > < ð1−λtÞ ν f ð1−λtÞ ! ! −2 2 > ax2 ð1−λtÞ−2 > > T ¼ T þ T ax ð1−λtÞ > θ ð η Þ; C ¼ C ϕðηÞ: þ C ∞ ∞ 0 0 : 2ν f 2ν f ð7Þ The velocity components u and v are given as: u¼
ax 0 f ðηÞ; ð1−λtÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffi aν f f ðηÞ; 1−λt
v¼−
ð8Þ
where prime denotes differentiation with respect to η. Now substituting Eq. (7) into Eqs. (2)–(4), we get the following system of non-linear ordinary differential equations:
2
∂ T ∂y2 2 ! ∂C ∂T DT ∂T ; þ τ DB þ T∞ ∂y ∂y ∂y
y ¼ 0 : u ¼ U w ðx; t Þ; v ¼ vw ðx; t Þ; T ¼ T w ðx; t Þ; C ¼ C w ðx; t Þ; y→∞ : u→U e ðx; t Þ; T→T ∞ ; C→C ∞ ;
pffiffiffiffiffiffiffiffiffiffiffiffiffi where vw ¼ v0 = 1−λt is the suction/injection velocity. The temperature of the sheet Tw(x, t) and the volume fraction of nanoparticles Cw(x,t) at the surface are assumed in the form
ð1Þ 2
217
ð3Þ
ð4Þ
η ‴ ″ 02 ″ 0 0 f þ f f −f −δ f þ f þ M ε−f þ ε2 þ εδ ¼ 0; 2
ð9Þ
η 1 2 0 ð1 þ RÞθ″ þ f θ0 −2f θ−δ θ0 þ 2θ þ Nbθ0 ϕ0 þ Ntθ0 ¼ 0; Pr 2
ð10Þ
η Nt 0 ϕ″ þ Sc f ϕ0 −2 f ϕ −Sc δ ϕ0 þ 2ϕ þ θ″ ¼ 0; 2 Nb
ð11Þ
The transformed boundary conditions of the problem are:
0
f ð0Þ ¼ f w ; f ð0Þ ¼ 1; θð0Þ ¼ 1; ϕð0Þ ¼ 1; 0 f ð∞Þ ¼ ε; θð∞Þ ¼ 0; ϕð∞Þ ¼ 0;
ð12Þ
where δ ¼ λa is an unsteadiness parameter, ε ¼ ba is the stretching νf 16T 3∞ σ α is the Prandtl number, R ¼ 3k κ is thermal radiation w −T ∞ Þ parameter, Nb ¼ τDB ðCνwf−C ∞ Þ is Brownian motion number, Nt ¼ τDT ðT ν f T∞ ν is the thermophoresis number, Sc ¼ DBf is the Schmidt number, M ¼ qffiffiffiffiffiffiffi σ 0 pvffiffiffiffiffiffi ρnf aB0 is the magnetic field parameter, f w ¼ − aν f is the suction/
parameter, Pr ¼
Fig. 1. Schematic diagram.
injection parameter. Herefw = 0 represents the impermeable
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surface, fw N 0 represents suction and f w b 0 represents injection of the fluid through permeable surface. The skin friction coefficient, the local Nusselt number, local Sherwood number are given by ∂u ; Cf ¼ 2 ∂y ρ f Uw y¼0 μ
2 x 4σ 4κ ∂T Nu ¼ − κ ðT w −T ∞ Þ ∂y y¼0 3k
Sh ¼ −
ð13Þ
∂T 4 ∂y
!
3 5;
ð14Þ
y¼0
x ∂C : ðC w −C ∞ Þ ∂y y¼0
ð15Þ
Substitute Eq. (7) into Eqs. (13)–(15) to obtain the dimensionless form of Cfr,Nur and Shr.
Table 1 Convergence of HAM solutions for different order of approximations. when fw =R=ε=0.2,M=δ=0.5, Pr =Sc=1,Nb=Nt=0.1 and ℏf =ℏθ =ℏϕ = −0.6. Order of approximation
−f″(0)
−θ′(0)
−ϕ′(0)
1 6 11 14 20 30 40 Numerical
1.13000 1.26066 1.26163 1.26163 1.26163 1.26163 1.26163 1.26163
1.47250 1.50353 1.50373 1.50373 1.50373 1.50373 1.50373 1.50373
1.27253 0.94622 0.94148 0.94141 0.94141 0.94141 0.94141 0.94141
where ci (i = 1 − 7) are arbitrary constants. Let p ∈ [0, 1] represent an embedding parameter then the zeroth order deformation equations are constructed as follows: h i h i ð1−pÞL f ^f ðη; pÞ− f 0 ðηÞ ¼ p ℏ f N f ^f ðη; pÞ ;
ð21Þ
h i h i ð1−pÞLθ ^θðη; pÞ−θ0 ðηÞ ¼ p ℏθ N θ ^θðη; pÞ ;
ð22Þ
h i h i ^ ðη; pÞ−ϕ ðηÞ ¼ p ℏϕ Nϕ ϕ ^ ðη; pÞ ; ð1−pÞLϕ ϕ 0
ð23Þ
^f ð0; pÞ ¼ f ; fb0 ð0; pÞ ¼ 1; fb0 ð∞; pÞ ¼ ε; w
ð24Þ
^θð0; pÞ ¼ 1; ^θð∞; pÞ ¼ 0;
ð25Þ
3.1. Zeroth-order deformation problems
^ ð0; pÞ ¼ 1; ϕ ^ ð∞; pÞ ¼ 0; ϕ
ð26Þ
Initial approximations f0(η) , θ0(η) and ϕ0(η) and auxiliary linear operators Lf , L θ and L ϕ are taken in the forms:
where ℏf , ℏθ ,ℏϕ denote the nonzero auxiliary linear parameters and the nonlinear operators are defined as
f 0 ðηÞ ¼ f w þ εη þ ð1−ε Þð1−e−η Þ; θ0 ðηÞ ¼ e−η ;
!2 2^ h i ∂3 ^f ðη; pÞ ^ ^f ðη; pÞ ∂ f ðη; pÞ − ∂ f ðη; pÞ N f ^f ðη; pÞ ¼ þ ∂η3 ∂η2 ∂η ! ! 2^ ^ η ∂ f ðη; pÞ ∂ f ðη; pÞ ∂^f ðη; pÞ þ M ε− þ ε 2 þ εδ; −δ þ 2 ∂η2 ∂η ∂η
pffiffiffiffiffiffiffiffi 9 ″ C fr ¼ Rex C f ¼ f ð0Þ; > > > > Nu = Nur ¼ pffiffiffiffiffiffiffiffi ¼ −ð1 þ RÞθ0 ð0Þ; Re x > > > > > > Nu > > : ; Shr ¼ pffiffiffiffiffiffiffiffi ¼ −ϕ0 ð0Þ; Rex 8 > > > > <
ð16Þ
xU w ν
is the local Reynolds number, Cfr is the reduced skin where Rex ¼ friction, Nur is the reduced Nusselt number, Shr is the reduced Sherwood number. 3. Homotopy solutions
‴
ϕ0 ðηÞ ¼ e−η ;
0
ð17Þ
L f ¼ f − f ; L θ ¼ θ″ −θ; L ϕ ¼ ϕ″ −ϕ;
ð18Þ
H f ¼ Hθ ¼ H ϕ ¼ 1;
ð19Þ
ð27Þ with L f ðc1 þ c2 eη þ c3 e−η Þ ¼ 0; Lθ ðc4 eη þ c5 e−η Þ ¼ 0;
Lϕ ðc6 eη þ c7 e−η Þ ¼ 0;
ð20Þ
2^ h i ^ ^ ðη; pÞ ¼ 1 ð1 þ RÞ ∂ θðη; pÞ þ ^f ðη; pÞ ∂θðη; pÞ N θ ^f ðη; pÞ; ^θðη; pÞ; ϕ Pr ∂η ∂η2 ! ∂^f ðη; pÞ ^ η ∂^θðη; pÞ −2 θðη; pÞ−δ þ 2^θðη; pÞ 2 ∂η ∂η
ð28Þ !
2^ h i ^ ^ ^ ðη; pÞ ¼ ∂ ϕðη; pÞ þ Sc ^f ðη; pÞ ∂ϕðη; pÞ −2ϕ ^ ðη; pÞ ∂ f ðη; pÞ N ϕ ^f ðη; pÞ; ^θðη; pÞ; ϕ 2 ∂η ∂η ! ∂η 2^ ^ ðη; pÞ η ∂ϕ ^ ðη; pÞ þ Nt ∂ θðη; pÞ : −δSc þ 2ϕ 2 ∂η Nb ∂η2
ð29Þ
Table 2 Comparison of −f″(0) for several values of velocity ratio parameter when M=fw =δ=0. ε
Fig. 2. h-Curve of fʺ (0), θʹ (0) and ϕ′(0).
0.1 0.2 0.5 1 2 3
Mahparta & Gupta [12] 0.9694 0.9181 0.6673 – −2.0175 −4.7293
Nazar et al. [13]
Hayat el al. [14]
Present
0.9694 0.9181 0.6673 0.0000 −2.0175 −4.7296
0.96938 0.91810 0.66732 0.00000 −2.01750 −4.72928
0.96938 0.91811 0.66726 0.00000 −2.01750 −4.72928
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223 Table 3 Effects of M,fw ,R,Nb,Ntandδ on Cfr when Pr=1,ε=0.5. M 0 0.5 1 1
fw 0.2
−0.5 0 0.5 0.5
δ
R
0.1
0.2
0.2 0.5 0.8 0.8
0.4 0.7 1 1
Sc 0.5
1 5 10 1
Nb
Nt
Cfr
Nur
Shr
0.1
0.1
0.734887 0.819621 0.896241 0.722065 0.842809 0.981744 0.991636 1.020974 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799
1.681908 1.670330 1.660447 1.337748 1.560786 1.820905 1.881042 2.050242 2.205521 2.362104 2.575947 2.770383 2.752425 2.711920 2.697740 2.691147 2.572599 2.459260 2.429127 2.370830 2.315054
0.229195 0.225046 0.221835 0.328861 0.257436 0.161327 0.179449 0.228282 0.270752 0.397780 0.539672 0.644490 1.465806 5.321093 8.826713 1.811209 1.983437 2.040439 1.953431 1.789902 1.639440
0.2 0.4 0.6 0.6
0.2 0.4 0.6
m−1 η X ‴ ″ 0 0 ″ 0 R f ;m ðηÞ ¼ f m−1 þ f m−1−k f k − f m−1−k f k −δ f m−1 þ f m−1 2 k¼0 0 þ ε2 þ δε þ εM ð1−χ m−1 Þ −M f m−1 ;
ð34Þ Rθ;m ðηÞ ¼
The resulting problems at the m following form
th
order can be presented in the
m−1 η X 1 0 ð1 þ RÞθ″m−1 þ f m−1−k θ0k −2θm−1−k f k −δ θ0m−1 þ 2θm−1 Pr 2 k¼0 m−1 X þ Nb θ0m−1−k ϕ0k þ Nt θ0m−1−k θ0k ; k¼0
ð35Þ Rϕ;m ðηÞ ¼ ϕ″m−1 m−1 η X 0 þ Sc f m−1−k ϕ0k −2ϕm−1−k f k −δSc ϕ0m−1 þ 2ϕm−1 2 k¼0 Nt ″ ; θ þ Nb m−1 ð36Þ Xm ¼
3.2. mth order deformation problems
0; 1;
m≤1; mN1:
ð30Þ
Lθ ½θm ðη; pÞ−Xm θm−1 ðηÞ ¼ ℏθ Rθ;m ðηÞ;
ð31Þ
Lϕ ½ϕm ðη; pÞ−Xm ϕm−1 ðηÞ ¼ ℏϕ Rϕ;m ðηÞ;
The general solutions (fm, θm, ϕm) comprising the special solutions ðf m ; θm ; ϕm Þ can be written as
ð32Þ
f m ðηÞ ¼ f m ðηÞ þ c1 þ c2 eη þ c3 e−η ;
ð38Þ
θm ðηÞ ¼ θm ðηÞ þ c4 eη þ c5 e−η ;
ð39Þ
ϕm ðηÞ ¼ ϕm ðηÞ þ c6 eη þ c7 e−η ;
ð40Þ
where the constants ci (i = 1, 2, ... , 7) can be determined from the boundary conditions Eq. (32) have the values
c1 ¼ −c3 −f m ð0Þ; c3 ¼ 0
ð37Þ
L f ½ f m ðη; pÞ−Xm f m−1 ðηÞ ¼ ℏ f R f ;m ðηÞ;
0
219
f m ð0Þ ¼ f m ð0Þ ¼ f m ð∞Þ ¼ θm ð0Þ ¼ θm ð∞Þ ¼ ϕm ð0Þ ¼ ϕm ð∞Þ ¼ 0;
ð33Þ
∂f m ðηÞ
∂η
η¼0
¼ −ϕm ð0Þ; c2 ¼ c4 ¼ c6 ¼ 0:
Fig. 3. The effect of magnetic parameter on velocity (a) for suction (b) for injection.
; c5 ¼ −θm ð0Þ; c7 ð41Þ
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F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
Fig. 4. The effect of unsteadiness parameter on velocity (a) for suction (b) for injection.
of ℏ from this range. Further, the series solution converges in the whole region of η (0b η b ∞) when ℏf = ℏθ = ℏϕ = −0.6.
4. Convergence of the HAM Solution The analytical solutions of Eqs. (9)–(11) subject to the boundary conditions Eq. (12) are computed by means of homotopy analysis method. The convergence of the series solution depends strongly upon auxiliary parameters ℏf , ℏθ, and ℏϕ. These parameters provide convenience to adjust and control the convergence region and convergence rate of the series solution. So as to choose appropriate values for these auxiliary parameters, the so called ℏf , ℏθ,and ℏϕ curves are displayed at 12th order approximations (see Fig. 2). We can see that the admissible values of ℏf , ℏθ and ℏϕ are − 0.9 ≤ ℏf ≤ − 0.3, −0.8≤ ℏθ ≤ − 0.3 and −0.8 ≤ ℏϕ ≤ − 0.4. It is important to note that the correct result up to 5 decimal places is obtained by choosing the values
5. Results and discussion Computations were carried out using homotopy analysis method (HAM) for several non-dimensional parameters. Convergence of the series solution up to 40th order of approximations is presented in Table 1. It is observed that the convergence for the dimensionless velocity, temperature and concentration is achieved at 11th, 11th and 14th order of approximations respectively. We validate our model by comparing skin friction coefficients with Mahparta & Gupta [12], Nazar et al. [13] and Hayat el al. [14] which shows an excellent
Fig. 5. The effect of magnetic parameter on temperature (a) for suction (b) for injection.
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
221
Fig. 6. The effect of unsteadiness parameter on temperature (a) for suction (b) for injection.
agreement (see Table 2). Table 3 presents numerical values of skin friction coefficient, reduced Nusselt and Sherwood numbers for several values of the governing parameters. It is found that skin friction coefficient increases with M , fw and δ. It is also found that reduced Nusselt and Sherwood numbers decreases with M and Nt. Figs. 3 and 4 exhibit the dimensionless velocity profiles for different values of magnetic parameter M, unsteady parameter δ in the presence of suction/injection. The variation in the dimensionless velocity with an increase in magnetic parameter M can be seen for suction in Fig. 3(a) and for injection in Fig. 3(b). It is noticed that magnetic field reduces the dimensionless velocity for both cases. This is due to the fact that application of a transverse magnetic field to an electrically conducting fluid results in a resistive type force (Lorentz force) which tends to slow down the motion of the fluid in the boundary layer. It is further observed that for stronger suction the boundary layer thickness is reduced, while an opposite trend is noticed for injection. Influence of an unsteady parameter δ on the velocity profiles can be observed in Fig. 4(a) for suction and in Fig. 4(b) for injection. Increasing values of
an unsteady parameter δ corresponds to a smaller stretching rate in the x-direction and results in a slight reduction in the dimensionless velocity. Figs. 5 and 6 are plotted for the dimensionless temperature distribution θ(η) against various parameters including the magnetic parameter M, an unsteady parameter δ and the wall suction/injection parameter fw. A profound effect of the magnetic field on the dimensionless temperature can be noticed in Fig. 5(a) for suction and in 5(b) for injection. The effect of transverse magnetic field M is found to enhance the dimensionless temperature since M reduces the flow field in both cases. Fig. 6(a) depicts the influence of an unsteady parameter δ and radiation parameter R on the dimensionless temperature. As shown in Fig. 6(a), an increase in the unsteady parameter δ decreases the dimensionless temperature inside the thermal boundary layer and, as a result, the thermal boundary layer thickness decreases. Further, an increase in radiation enhances the heat flux from the sheet which gives rise to the fluid's temperature, therefore, the dimensionless temperature increase with an increase in radiation which can be observed from Fig. 6(a). On
Fig. 7. The effect of unsteadiness and nano particles parameters on concentration.
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F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
Fig. 8. The variation of skin friction coefficient against suction/injection, magnetic, and unsteadiness parameters.
Fig. 10. The variation of mass transfer rate against Schmidt number and nanoparticles parameters.
the other hand, Fig. 6(b) illustrates the effects of Brownian motion parameter Nb and thermophoresis parameter Nt on the dimensionless temperature. As, the random motion of nanoparticles increases, the collision of particles as well as kinetic energy increases. This kinetic energy is converted in to heat energy. Hence the dimensionless temperature increases with an increase in Brownian motion parameterNb. The similar behavior of the temperature can be seen for thermophoresis parameterNt. This is due to the fact that the thermophoretic force, generated by the temperature gradient, creates a fast flow away from the stretching surface. In this way more heated fluid is moved away from the surface, and consequently, as Nt increases, the temperature within the boundary layer increases as shown in Fig. 6(b). The effects of Brownian motion parameter Nb, thermophoresis parameter Nt, unsteady parameter δ and Schmidt number on dimensionless nanoparticles concentration for the fixed values of other parameters are shown in Fig. 7(a) and (b). Fig. 7(a) indicates that an increase in the Brownian motion parameter Nb decreases the
dimensionless concentration. It also shows that the dimensionless concentration increases when thermophoresis parameter Nt is increased. Fig. 7(b) shows that an increase in the unsteady parameter δ decreases the dimensionless concentration. The influence of Schmidt number on the dimensionless concentration is displayed in Fig. 7(b). We infer that the dimensionless concentration decreases with an increase in Schmidt number Sc. This is due to the fact that an increase in Schmidt number reduces the molecular diffusivity D and that results in a decrease in thickness of the concentration boundary layer. Figs. 8–10 show the variations of the skin-friction coefficient as well as the reduced Nusselt and Sherwood numbers. Fig. 8 demonstrates that increasing magnetic parameter M increases the skin friction coefficient monotonically at the wall. This is because of the presence of the magnetic field which reduces the dimensionless velocity and the momentum boundary layer thickness, and thus increases the velocity gradient. Further, it is observed that the skin friction coefficient is higher for suction as compared to injection. This observation is consistent with the trend
Fig. 9. The variation of heat transfer rate against magnetic, unsteadiness, suction/injection, nanoparticles and radiation parameters.
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
given in Table 3. The variation of the Nusselt number is shown for magnetic, suction/injection, unsteady, radiation and nanoparticles parameters in Fig. 9. Fig. 9(a) shows the behavior of heat transfer rate against magnetic parameter M for different values of suction/injection and unsteady parameters. It is found that the heat transfer rate increases with an increase in suction/injection and unsteady parameters. It is also found that heat transfer rate monotonically decreases with magnetic parameter. Fig. 9(b) depicts the variation of heat transfer rate with radiation and nanoparticles parameters. An increase in radiation and nanofluid parameters decreases the heat transfer rate. Finally, Fig. 10 illustrates the effects of nanofluids parameters and Schmidt number on Sherwood number. It is observed that Sherwood number increases monotonically with Schmidt number and Brownian motion parameter while reduces with increasing values of thermophoresis parameter Nt. 6. Conclusion MHD unsteady flow of nanofluid over a heated stretching sheet is studied. Effects of various parameters on the dimensionless velocity, temperature, concentration, skin friction coefficient and reduced Nusselt and Sherwood numbers are analyzed. The main observations of the present study are as follows: • The dimensionless velocity decreases as unsteady and suction parameters effects intensify. • The dimensionless temperature is higher in the presence of magnetic field. • The dimensionless concentration reduces with Schmidt number and Brownian motion parameter. • The skin friction coefficient increases, whereas the reduced Nusselt number decreases with magnetic parameter. • The skin friction coefficient, reduced Nusselt and Sherwood number are decreasing functions of unsteady parameter. • The reduced Nusselt number decreases but the reduced Sherwood number increases as the Brownian motion effects intensify.
References [1] B.K. Chakraborty, H.P. Mazumdar, MHD flow of Newtonian fluid over a stretching sheet, Approx. Theory Appl. 16 (2000) 32–41. [2] R. Tamizharasi, V. Kumaran, Pressure in MHD/Brinkman flow past a stretching sheet, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 4671–4681. [3] M.M. Rashidi, N.V. Ganesh, A.K.A. Hakeem, B. Ganga, Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation, J. Mol. Liq. 198 (2014) 234–238. [4] M. Sheikholeslami, R. Ellahi, H.R. Ashorynejad, G. Donairry, T. Hayat, Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium, J. Comput. Theor. Nanosci. 11 (2014) 486–496. [5] F. Mabood, W.A. Khan, A.I.M. Ismail, MHD boundary layer flow and heat transfer of nanofluids over a non-linear stretching sheet: a numerical study, J. Magn. Magn. Mater. 374 (2015) 569–576.
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[6] T. Hayat, M. Imtiaz, A. Alsaedi, Unsteady flow of nanofluid with double stratification and magnetohydrodynamics, Int. J. Heat Mass Transf. 92 (2016) 100–109. [7] J.A. Khan, M. Mustafa, T. Hayat, M.A. Farooq, A. Alsaedi, S.J. Liao, On model for threedimensional flow of nanofluid: an application to solar energy, J. Mol. Liq. 194 (2014) 41–47. [8] W. Ibrahim, O.D. Makinde, The effect of double stratification on boundary layer flow and heat transfer of nanofluid over a vertical plate, Comput. Fluids 86 (2013) 433–441. [9] O.D. Makinde, A. Aziz, Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, Int. J. Therm. Sci. 50 (2011) 1326–1332. [10] W.A. Khan, I. Pop, Boundary layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transf. 53 (2010) 2477–2483. [11] W.A. Khan, A. Aziz, Double-diffusive natural convective boundary layer flow in a porous medium saturated with a nanofluid over a vertical plate, prescribed surface heat, solute and nanoparticle fluxes, Int. J. Therm. Sci. 50 (2011) 2154–2160. [12] T.R. Mahapatra, A.S. Gupta, Heat transfer in stagnation-point flow towards a stretching sheet, Heat Mass Transf. 38 (2002) 517–521. [13] R. Nazar, N. Amin, D. Filip, I. Pop, Stagnation point flow of a micropolar fluid towards a stretching sheet, Int. J. Nonlinear Mech. 39 (2004) 1227–1235. [14] T. Hayat, T. Javed, Z Abbas, MHD flow of a microploar fluid near a stagnation-point towards a non-linear stretching surface, Nonlinear Anal. Real World Appl. 10 (2009) 1514–1526. [15] X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids a review, Int. J. Therm. Sci. 46 (2007) 1–19. [16] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [17] L.B. Godson, D. Raja, D. Mohan Lal, S. Wongwisesc, Enhancement of heat transfer using nanofluids—an overview, Renew. Sust. Energ. Rev. 14 (2010) 629–641. [18] T. Hayat, Z. Abbas, I. Pop, S. Asghar, Effects of radiation and magnetic field on the mixed convection stagnation point flow over a vertical stretching sheet in porous medium, Int. J. Heat Mass Transf. 53 (2010) 466–476. [19] S.M. Ibrahim, Effects of mass transfer, radiation, Joule heating, and viscous dissipation on steady MHD Marangoni convection flow over a flat surface with suction and injection, Int. J. Eng. Math. 2013 (2013) 903818 9 pp. [20] M.M. Rashidi, M. Ali, N. Freidoonimehr, B. Rostami, A. Hossian, Mixed convection heat transfer for viscoelastic fluid flow over a porous wedge with thermal radiation, Adv. Mech. Eng. 204 (2014) 735939. [21] P. Rana, R. Bhargava, Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: a numerical study, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 212–226. [22] F. Mabood, S. Shateyi, W.A. Khan, Effects of thermal radiation on Casson flow heat and mass transfer around a circular cylinder in porous medium, Eur. Phys. J. Plus. 130 (2015) 188. [23] S. Shateyi, J. Prakash, A new numerical approach for MHD laminar boundary layer flow and heat transfer of nanofluids over a moving surface in the presence of thermal radiation, Bound. Value Probl. 2014 (2014) 2, http://dx.doi.org/10.1186/16872770-2014-2. [24] M. Sheikholeslami, M.M. Rashidi, D.D. Ganji, Effect of non-uniform magnetic field on forced convection heat transfer of Fe3O4–water nanofluid, Comput. Methods Appl. Mech. Eng. 294 (2015) 299–312. [25] T. Hayat, F.M. Abbasi, Maryem Al-Yami, Shatha Monaquel, Slip and joule heating effects in mixed convection peristaltic transport of nanofluid with Soret and Dufour effects, J. Mol. Liq. 194 (2014) 93–99. [26] J. Serna, Heat and mass transfer mechanisms in nanofluids boundary layers, Int. J. Heat Mass Transf. 92 (2016) 173–183. [27] M.M. Rashidi, E. Erfani, A new analytical study of MHD stagnation-point flow in porous media with heat transfer, Comput. Fluids 40 (2011) 172–178. [28] F. Mabood, W.A. Khan, A.I.M. Ismail, Approximate analytical solution for influence of heat transfer on MHD stagnation point flow in porous medium, Comput. Fluids 100 (2014) 72–78. [29] A. Mastroberardino, Mixed convection in viscoelastic boundary layer flow and heat transfer over a stretching sheet, Adv. Appl. Math. Mech. 6 (2014) 359–375.
Contents lists available at ScienceDirect
Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Analytical study for unsteady nanofluid MHD Flow impinging on heated stretching sheet F. Mabood a,⁎, W.A. Khan b a b
Department of Mathematics, University of Peshawar, Pakistan Department of Mechanical and Industrial Engineering, College of Engineering, Majmaah University, Majmaah 11952, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 15 November 2015 Received in revised form 2 January 2016 Accepted 22 February 2016 Available online xxxx Keywords: Unsteady boundary layer MHD nanofluid Thermal radiation HAM
a b s t r a c t An analysis is carried out to obtain analytical solution of an unsteady two-dimensional MHD nanofluid flow with heat and mass transfer over a heated surface. The governing partial differential equations are reduced to system of nonlinear ordinary differential equations using suitable transformations. The resulting nonlinear coupled system subject to the boundary conditions is solved using homotopy analysis method (HAM). Graphical and numerical demonstrations of the convergence of the HAM solutions are provided. A detailed study illustrating the influences of the magnetic, unsteady, suction/injection and nanofluid parameters, on the dimensionless velocity, temperature, concentration as well as on the skin friction coefficient, the reduced Nusselt and Sherwood numbers is conducted. It is found out that the flow field is substantially influenced due to unsteadiness, transpiration and magnetic field. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Magneto-fluid-dynamics analyzes the dynamics of electrically conducting fluids such as plasmas, liquid metals and salt water. The term magnetohydrodynamics (MHD) is a combination of magneto (magnetic field), hydro (liquid) and dynamics (movement of particles). The magnetic field induced current flows in a fluid and creates forces on the fluid. The combination of the Navier-Stokes equations of fluidmechanics and Maxwell's equations of electromagnetism consequently establishes MHD relations [1,2]. Due to promising potential for heat transfer applications, fluid heating and cooling have received significant importance in different industries such as electronics, power, manufacturing and transportation. Due to practical impact, modern cooling techniques are essential for cooling of high-energy devices. In many cases of natural convection, the conventional fluids such as water, engine oil, ethylene glycol etc. have a low thermal conductivity, which limits the heat transfer capabilities. However, the growing demand of advanced technology with respect to miniaturization of electronic devices requires further improvement of heat transfer from the point of view of point energy saving. To overcome this drawback, there is a need to develop a new heat transfer medium behaving like fluid with higher heat transfer capabilities to increase thermal characteristics which is known as nanofluid. Recently numerous numerical/analytical studies for the improvement of heat transfer of nanofluids have been published [3–14]. ⁎ Corresponding author. E-mail addresses: [email protected] (F. Mabood), [email protected] (W.A. Khan).
http://dx.doi.org/10.1016/j.molliq.2016.02.071 0167-7322/© 2016 Elsevier B.V. All rights reserved.
An excellent review of nanofluid physics and developments can be found in the review paper by Wang and Mujumdar [15]. Buongiorno [16] reported a complete review of convective transport in nanofluids by considering slip mechanisms which generate a relative velocity between the base fluid and nanoparticles. An overview on convective heat transfer in nanofluids and their applications were investigated experimentally and theoretically by Godson et al. [17]. Thermal radiation plays an important role on the flow and heat transfer in space technology. Radiation effects on convective MHD flow problems are also significant in electrical power generation, astrophysical ground, solar power technology, re-entry of space vehicle and other industrial areas. Hayat et al. [18] discussed the effects of thermal radiation on the MHD mixed convection flow over a stretching surface. Ibrahim [19] investigated the effects of mass transfer, radiation, Joule heating, and viscous dissipation on steady MHD Marangoni convection flow over a flat surface with suction and injection. Thermal radiation effects on mixed convection heat transfer for viscoelastic fluid flow over a porous wedge were reported by Rashidi et al. [20]. Rana and Bhargava [21] provided numerical solution for nanofluid flow over a nonlinear stretching sheet whilst Mabood et al. [22] studied the effect of thermal radiation on Casson fluid flow, heat and mass transfer around a circular cylinder in porous medium. Shateyi and Prakash [23] studied thermal radiation effects on MHD flow and heat transfer of nanofluids over a moving surface. Nanofluid flow and heat transfer characteristics were studied by numerous researchers including [24–26]. In this paper, our main objective is to investigate the unsteady flow of an electrically conducting nanofluid past a heated stretching sheet with thermal radiation. The problem is first modeled and then solved analytically by homotopy analysis method [27–29]. Convergence
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
region of the solutions is determined. The behavior of Brownian motion and thermophoretic diffusion of nanoparticles has been examined graphically. 2. Governing equations Consider an unsteady boundary layer flow of nanofluid past a permeable stretching surface in the presence of external magnetic ! field B and thermal radiation. The coordinate system is chosen in such a way that x-axis is measured along the sheet and y-axis is normal to it as shown in the Fig. 1. The flow is confined in the region y N 0. Two equal and opposite forces are spontaneously applied along the direction of x-axis so that the sheet is stretched with fixed origin. For time t N 0, it is understood that the flow is in steady state. The unsteady fluid flow starts at t = 0. The sheet is being stretched with velocity U w ðx; tÞ ¼ ax ð1−λtÞ
along x-axis where a is the stretching rate, λ is the positive
constant with the property λt b 1 and dimension of λ is (time)−1. We bx assume that ambient fluid moves with a velocity, U e ðx; tÞ ¼ ð1−λtÞ in the y-direction towards the stagnation point on the plate where b(N 0) is the strength of stagnation point flow. Also, the temperature of the sheet Tw(x, t) and the volume fraction of nanoparticles Cw(x, t) both vary with x and time t, T∞ and C∞ are the temperature and nanoparticles volume fraction of the fluid far away from the sheet. Keeping in view of thermal equilibrium, no slip occurs between base fluid and nanoparti! ! cles. As the fluid is electrically conducting, so the Lorentz force J B , ! ! where J is the electrical current density and B ¼ ð0; B; 0Þ is the transverse magnetic field, must be included in the momentum equation. In the boundary layer approximation the Lorentz force is simplified as −σB2(u − Ue), where σ is the electrical conductivity, u is B0 ffi is the magnetic field the x-component of the fluid velocity, B ¼ pffiffiffiffiffiffiffiffi 1−λt
imposed along y-axis and B0 is the intensity of the magnetic field. The governing equations for the continuity, momentum, energy, and concentration can be written: ∂u ∂v þ ¼ 0; ∂x ∂y
where x and y are the coordinates along and normal to the sheet, u and v are the components of the velocity in the x- and y- directions respectively, ρf is density of the base fluid, νf is kinematic viscosity of the base fluid, α is the thermal diffusivity, T is the temperature, C is the nanoparticles volume fraction, τ = (ρc)f/(ρc)p is the ratio of nanoparticles heat capacity and the base fluid heat capacity, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, σ⁎ is the Stefan Boltzmann constant, k⁎ is the mean absorption coefficient. The boundary conditions for the above described model are:
2
∂u ∂u ∂u ∂U e ∂U e ∂ u σ B ðu−U e Þ ; þu þv ¼ þ Ue þ νf 2 − ρnf ∂t ∂x ∂y ∂y ∂t ∂x ∂T ∂T ∂T 16T 3∞ σ þu þv ¼α 1þ ∂t ∂x ∂y 3k κ
!
ð2Þ
2 2 ∂C ∂C ∂C ∂ C DT ∂ T ; þu þv ¼ DB 2 þ T ∞ ∂y2 ∂t ∂x ∂y ∂y
ð5Þ
ax2 ð1−λtÞ−2 ; C w ðx; t Þ 2ν 2f ax ð1−λtÞ−2 ; ¼ C∞ þ C0 2ν f
T w ðx; t Þ ¼ T ∞ þ T 0
ð6Þ
where T0 , C0 are the positive reference temperature and the nanoparticles volume fraction respectively such that 0 ≤ T0 ≤ Tw and 0 ≤C0 ≤Cw. The above expressions are valid when (1− λt) N 0. As per usual, the stream function ψ is defined as u = ∂ ψ/∂ y and v = − ∂ ψ/∂ x so that Eq. (1) is satisfied. We introduce the following dimensionless quantities: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > aν f a > > y; ψ ¼ xf ðηÞ; η ¼ > < ð1−λtÞ ν f ð1−λtÞ ! ! −2 2 > ax2 ð1−λtÞ−2 > > T ¼ T þ T ax ð1−λtÞ > θ ð η Þ; C ¼ C ϕðηÞ: þ C ∞ ∞ 0 0 : 2ν f 2ν f ð7Þ The velocity components u and v are given as: u¼
ax 0 f ðηÞ; ð1−λtÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffi aν f f ðηÞ; 1−λt
v¼−
ð8Þ
where prime denotes differentiation with respect to η. Now substituting Eq. (7) into Eqs. (2)–(4), we get the following system of non-linear ordinary differential equations:
2
∂ T ∂y2 2 ! ∂C ∂T DT ∂T ; þ τ DB þ T∞ ∂y ∂y ∂y
y ¼ 0 : u ¼ U w ðx; t Þ; v ¼ vw ðx; t Þ; T ¼ T w ðx; t Þ; C ¼ C w ðx; t Þ; y→∞ : u→U e ðx; t Þ; T→T ∞ ; C→C ∞ ;
pffiffiffiffiffiffiffiffiffiffiffiffiffi where vw ¼ v0 = 1−λt is the suction/injection velocity. The temperature of the sheet Tw(x, t) and the volume fraction of nanoparticles Cw(x,t) at the surface are assumed in the form
ð1Þ 2
217
ð3Þ
ð4Þ
η ‴ ″ 02 ″ 0 0 f þ f f −f −δ f þ f þ M ε−f þ ε2 þ εδ ¼ 0; 2
ð9Þ
η 1 2 0 ð1 þ RÞθ″ þ f θ0 −2f θ−δ θ0 þ 2θ þ Nbθ0 ϕ0 þ Ntθ0 ¼ 0; Pr 2
ð10Þ
η Nt 0 ϕ″ þ Sc f ϕ0 −2 f ϕ −Sc δ ϕ0 þ 2ϕ þ θ″ ¼ 0; 2 Nb
ð11Þ
The transformed boundary conditions of the problem are:
0
f ð0Þ ¼ f w ; f ð0Þ ¼ 1; θð0Þ ¼ 1; ϕð0Þ ¼ 1; 0 f ð∞Þ ¼ ε; θð∞Þ ¼ 0; ϕð∞Þ ¼ 0;
ð12Þ
where δ ¼ λa is an unsteadiness parameter, ε ¼ ba is the stretching νf 16T 3∞ σ α is the Prandtl number, R ¼ 3k κ is thermal radiation w −T ∞ Þ parameter, Nb ¼ τDB ðCνwf−C ∞ Þ is Brownian motion number, Nt ¼ τDT ðT ν f T∞ ν is the thermophoresis number, Sc ¼ DBf is the Schmidt number, M ¼ qffiffiffiffiffiffiffi σ 0 pvffiffiffiffiffiffi ρnf aB0 is the magnetic field parameter, f w ¼ − aν f is the suction/
parameter, Pr ¼
Fig. 1. Schematic diagram.
injection parameter. Herefw = 0 represents the impermeable
218
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
surface, fw N 0 represents suction and f w b 0 represents injection of the fluid through permeable surface. The skin friction coefficient, the local Nusselt number, local Sherwood number are given by ∂u ; Cf ¼ 2 ∂y ρ f Uw y¼0 μ
2 x 4σ 4κ ∂T Nu ¼ − κ ðT w −T ∞ Þ ∂y y¼0 3k
Sh ¼ −
ð13Þ
∂T 4 ∂y
!
3 5;
ð14Þ
y¼0
x ∂C : ðC w −C ∞ Þ ∂y y¼0
ð15Þ
Substitute Eq. (7) into Eqs. (13)–(15) to obtain the dimensionless form of Cfr,Nur and Shr.
Table 1 Convergence of HAM solutions for different order of approximations. when fw =R=ε=0.2,M=δ=0.5, Pr =Sc=1,Nb=Nt=0.1 and ℏf =ℏθ =ℏϕ = −0.6. Order of approximation
−f″(0)
−θ′(0)
−ϕ′(0)
1 6 11 14 20 30 40 Numerical
1.13000 1.26066 1.26163 1.26163 1.26163 1.26163 1.26163 1.26163
1.47250 1.50353 1.50373 1.50373 1.50373 1.50373 1.50373 1.50373
1.27253 0.94622 0.94148 0.94141 0.94141 0.94141 0.94141 0.94141
where ci (i = 1 − 7) are arbitrary constants. Let p ∈ [0, 1] represent an embedding parameter then the zeroth order deformation equations are constructed as follows: h i h i ð1−pÞL f ^f ðη; pÞ− f 0 ðηÞ ¼ p ℏ f N f ^f ðη; pÞ ;
ð21Þ
h i h i ð1−pÞLθ ^θðη; pÞ−θ0 ðηÞ ¼ p ℏθ N θ ^θðη; pÞ ;
ð22Þ
h i h i ^ ðη; pÞ−ϕ ðηÞ ¼ p ℏϕ Nϕ ϕ ^ ðη; pÞ ; ð1−pÞLϕ ϕ 0
ð23Þ
^f ð0; pÞ ¼ f ; fb0 ð0; pÞ ¼ 1; fb0 ð∞; pÞ ¼ ε; w
ð24Þ
^θð0; pÞ ¼ 1; ^θð∞; pÞ ¼ 0;
ð25Þ
3.1. Zeroth-order deformation problems
^ ð0; pÞ ¼ 1; ϕ ^ ð∞; pÞ ¼ 0; ϕ
ð26Þ
Initial approximations f0(η) , θ0(η) and ϕ0(η) and auxiliary linear operators Lf , L θ and L ϕ are taken in the forms:
where ℏf , ℏθ ,ℏϕ denote the nonzero auxiliary linear parameters and the nonlinear operators are defined as
f 0 ðηÞ ¼ f w þ εη þ ð1−ε Þð1−e−η Þ; θ0 ðηÞ ¼ e−η ;
!2 2^ h i ∂3 ^f ðη; pÞ ^ ^f ðη; pÞ ∂ f ðη; pÞ − ∂ f ðη; pÞ N f ^f ðη; pÞ ¼ þ ∂η3 ∂η2 ∂η ! ! 2^ ^ η ∂ f ðη; pÞ ∂ f ðη; pÞ ∂^f ðη; pÞ þ M ε− þ ε 2 þ εδ; −δ þ 2 ∂η2 ∂η ∂η
pffiffiffiffiffiffiffiffi 9 ″ C fr ¼ Rex C f ¼ f ð0Þ; > > > > Nu = Nur ¼ pffiffiffiffiffiffiffiffi ¼ −ð1 þ RÞθ0 ð0Þ; Re x > > > > > > Nu > > : ; Shr ¼ pffiffiffiffiffiffiffiffi ¼ −ϕ0 ð0Þ; Rex 8 > > > > <
ð16Þ
xU w ν
is the local Reynolds number, Cfr is the reduced skin where Rex ¼ friction, Nur is the reduced Nusselt number, Shr is the reduced Sherwood number. 3. Homotopy solutions
‴
ϕ0 ðηÞ ¼ e−η ;
0
ð17Þ
L f ¼ f − f ; L θ ¼ θ″ −θ; L ϕ ¼ ϕ″ −ϕ;
ð18Þ
H f ¼ Hθ ¼ H ϕ ¼ 1;
ð19Þ
ð27Þ with L f ðc1 þ c2 eη þ c3 e−η Þ ¼ 0; Lθ ðc4 eη þ c5 e−η Þ ¼ 0;
Lϕ ðc6 eη þ c7 e−η Þ ¼ 0;
ð20Þ
2^ h i ^ ^ ðη; pÞ ¼ 1 ð1 þ RÞ ∂ θðη; pÞ þ ^f ðη; pÞ ∂θðη; pÞ N θ ^f ðη; pÞ; ^θðη; pÞ; ϕ Pr ∂η ∂η2 ! ∂^f ðη; pÞ ^ η ∂^θðη; pÞ −2 θðη; pÞ−δ þ 2^θðη; pÞ 2 ∂η ∂η
ð28Þ !
2^ h i ^ ^ ^ ðη; pÞ ¼ ∂ ϕðη; pÞ þ Sc ^f ðη; pÞ ∂ϕðη; pÞ −2ϕ ^ ðη; pÞ ∂ f ðη; pÞ N ϕ ^f ðη; pÞ; ^θðη; pÞ; ϕ 2 ∂η ∂η ! ∂η 2^ ^ ðη; pÞ η ∂ϕ ^ ðη; pÞ þ Nt ∂ θðη; pÞ : −δSc þ 2ϕ 2 ∂η Nb ∂η2
ð29Þ
Table 2 Comparison of −f″(0) for several values of velocity ratio parameter when M=fw =δ=0. ε
Fig. 2. h-Curve of fʺ (0), θʹ (0) and ϕ′(0).
0.1 0.2 0.5 1 2 3
Mahparta & Gupta [12] 0.9694 0.9181 0.6673 – −2.0175 −4.7293
Nazar et al. [13]
Hayat el al. [14]
Present
0.9694 0.9181 0.6673 0.0000 −2.0175 −4.7296
0.96938 0.91810 0.66732 0.00000 −2.01750 −4.72928
0.96938 0.91811 0.66726 0.00000 −2.01750 −4.72928
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223 Table 3 Effects of M,fw ,R,Nb,Ntandδ on Cfr when Pr=1,ε=0.5. M 0 0.5 1 1
fw 0.2
−0.5 0 0.5 0.5
δ
R
0.1
0.2
0.2 0.5 0.8 0.8
0.4 0.7 1 1
Sc 0.5
1 5 10 1
Nb
Nt
Cfr
Nur
Shr
0.1
0.1
0.734887 0.819621 0.896241 0.722065 0.842809 0.981744 0.991636 1.020974 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799 1.049799
1.681908 1.670330 1.660447 1.337748 1.560786 1.820905 1.881042 2.050242 2.205521 2.362104 2.575947 2.770383 2.752425 2.711920 2.697740 2.691147 2.572599 2.459260 2.429127 2.370830 2.315054
0.229195 0.225046 0.221835 0.328861 0.257436 0.161327 0.179449 0.228282 0.270752 0.397780 0.539672 0.644490 1.465806 5.321093 8.826713 1.811209 1.983437 2.040439 1.953431 1.789902 1.639440
0.2 0.4 0.6 0.6
0.2 0.4 0.6
m−1 η X ‴ ″ 0 0 ″ 0 R f ;m ðηÞ ¼ f m−1 þ f m−1−k f k − f m−1−k f k −δ f m−1 þ f m−1 2 k¼0 0 þ ε2 þ δε þ εM ð1−χ m−1 Þ −M f m−1 ;
ð34Þ Rθ;m ðηÞ ¼
The resulting problems at the m following form
th
order can be presented in the
m−1 η X 1 0 ð1 þ RÞθ″m−1 þ f m−1−k θ0k −2θm−1−k f k −δ θ0m−1 þ 2θm−1 Pr 2 k¼0 m−1 X þ Nb θ0m−1−k ϕ0k þ Nt θ0m−1−k θ0k ; k¼0
ð35Þ Rϕ;m ðηÞ ¼ ϕ″m−1 m−1 η X 0 þ Sc f m−1−k ϕ0k −2ϕm−1−k f k −δSc ϕ0m−1 þ 2ϕm−1 2 k¼0 Nt ″ ; θ þ Nb m−1 ð36Þ Xm ¼
3.2. mth order deformation problems
0; 1;
m≤1; mN1:
ð30Þ
Lθ ½θm ðη; pÞ−Xm θm−1 ðηÞ ¼ ℏθ Rθ;m ðηÞ;
ð31Þ
Lϕ ½ϕm ðη; pÞ−Xm ϕm−1 ðηÞ ¼ ℏϕ Rϕ;m ðηÞ;
The general solutions (fm, θm, ϕm) comprising the special solutions ðf m ; θm ; ϕm Þ can be written as
ð32Þ
f m ðηÞ ¼ f m ðηÞ þ c1 þ c2 eη þ c3 e−η ;
ð38Þ
θm ðηÞ ¼ θm ðηÞ þ c4 eη þ c5 e−η ;
ð39Þ
ϕm ðηÞ ¼ ϕm ðηÞ þ c6 eη þ c7 e−η ;
ð40Þ
where the constants ci (i = 1, 2, ... , 7) can be determined from the boundary conditions Eq. (32) have the values
c1 ¼ −c3 −f m ð0Þ; c3 ¼ 0
ð37Þ
L f ½ f m ðη; pÞ−Xm f m−1 ðηÞ ¼ ℏ f R f ;m ðηÞ;
0
219
f m ð0Þ ¼ f m ð0Þ ¼ f m ð∞Þ ¼ θm ð0Þ ¼ θm ð∞Þ ¼ ϕm ð0Þ ¼ ϕm ð∞Þ ¼ 0;
ð33Þ
∂f m ðηÞ
∂η
η¼0
¼ −ϕm ð0Þ; c2 ¼ c4 ¼ c6 ¼ 0:
Fig. 3. The effect of magnetic parameter on velocity (a) for suction (b) for injection.
; c5 ¼ −θm ð0Þ; c7 ð41Þ
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Fig. 4. The effect of unsteadiness parameter on velocity (a) for suction (b) for injection.
of ℏ from this range. Further, the series solution converges in the whole region of η (0b η b ∞) when ℏf = ℏθ = ℏϕ = −0.6.
4. Convergence of the HAM Solution The analytical solutions of Eqs. (9)–(11) subject to the boundary conditions Eq. (12) are computed by means of homotopy analysis method. The convergence of the series solution depends strongly upon auxiliary parameters ℏf , ℏθ, and ℏϕ. These parameters provide convenience to adjust and control the convergence region and convergence rate of the series solution. So as to choose appropriate values for these auxiliary parameters, the so called ℏf , ℏθ,and ℏϕ curves are displayed at 12th order approximations (see Fig. 2). We can see that the admissible values of ℏf , ℏθ and ℏϕ are − 0.9 ≤ ℏf ≤ − 0.3, −0.8≤ ℏθ ≤ − 0.3 and −0.8 ≤ ℏϕ ≤ − 0.4. It is important to note that the correct result up to 5 decimal places is obtained by choosing the values
5. Results and discussion Computations were carried out using homotopy analysis method (HAM) for several non-dimensional parameters. Convergence of the series solution up to 40th order of approximations is presented in Table 1. It is observed that the convergence for the dimensionless velocity, temperature and concentration is achieved at 11th, 11th and 14th order of approximations respectively. We validate our model by comparing skin friction coefficients with Mahparta & Gupta [12], Nazar et al. [13] and Hayat el al. [14] which shows an excellent
Fig. 5. The effect of magnetic parameter on temperature (a) for suction (b) for injection.
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
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Fig. 6. The effect of unsteadiness parameter on temperature (a) for suction (b) for injection.
agreement (see Table 2). Table 3 presents numerical values of skin friction coefficient, reduced Nusselt and Sherwood numbers for several values of the governing parameters. It is found that skin friction coefficient increases with M , fw and δ. It is also found that reduced Nusselt and Sherwood numbers decreases with M and Nt. Figs. 3 and 4 exhibit the dimensionless velocity profiles for different values of magnetic parameter M, unsteady parameter δ in the presence of suction/injection. The variation in the dimensionless velocity with an increase in magnetic parameter M can be seen for suction in Fig. 3(a) and for injection in Fig. 3(b). It is noticed that magnetic field reduces the dimensionless velocity for both cases. This is due to the fact that application of a transverse magnetic field to an electrically conducting fluid results in a resistive type force (Lorentz force) which tends to slow down the motion of the fluid in the boundary layer. It is further observed that for stronger suction the boundary layer thickness is reduced, while an opposite trend is noticed for injection. Influence of an unsteady parameter δ on the velocity profiles can be observed in Fig. 4(a) for suction and in Fig. 4(b) for injection. Increasing values of
an unsteady parameter δ corresponds to a smaller stretching rate in the x-direction and results in a slight reduction in the dimensionless velocity. Figs. 5 and 6 are plotted for the dimensionless temperature distribution θ(η) against various parameters including the magnetic parameter M, an unsteady parameter δ and the wall suction/injection parameter fw. A profound effect of the magnetic field on the dimensionless temperature can be noticed in Fig. 5(a) for suction and in 5(b) for injection. The effect of transverse magnetic field M is found to enhance the dimensionless temperature since M reduces the flow field in both cases. Fig. 6(a) depicts the influence of an unsteady parameter δ and radiation parameter R on the dimensionless temperature. As shown in Fig. 6(a), an increase in the unsteady parameter δ decreases the dimensionless temperature inside the thermal boundary layer and, as a result, the thermal boundary layer thickness decreases. Further, an increase in radiation enhances the heat flux from the sheet which gives rise to the fluid's temperature, therefore, the dimensionless temperature increase with an increase in radiation which can be observed from Fig. 6(a). On
Fig. 7. The effect of unsteadiness and nano particles parameters on concentration.
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Fig. 8. The variation of skin friction coefficient against suction/injection, magnetic, and unsteadiness parameters.
Fig. 10. The variation of mass transfer rate against Schmidt number and nanoparticles parameters.
the other hand, Fig. 6(b) illustrates the effects of Brownian motion parameter Nb and thermophoresis parameter Nt on the dimensionless temperature. As, the random motion of nanoparticles increases, the collision of particles as well as kinetic energy increases. This kinetic energy is converted in to heat energy. Hence the dimensionless temperature increases with an increase in Brownian motion parameterNb. The similar behavior of the temperature can be seen for thermophoresis parameterNt. This is due to the fact that the thermophoretic force, generated by the temperature gradient, creates a fast flow away from the stretching surface. In this way more heated fluid is moved away from the surface, and consequently, as Nt increases, the temperature within the boundary layer increases as shown in Fig. 6(b). The effects of Brownian motion parameter Nb, thermophoresis parameter Nt, unsteady parameter δ and Schmidt number on dimensionless nanoparticles concentration for the fixed values of other parameters are shown in Fig. 7(a) and (b). Fig. 7(a) indicates that an increase in the Brownian motion parameter Nb decreases the
dimensionless concentration. It also shows that the dimensionless concentration increases when thermophoresis parameter Nt is increased. Fig. 7(b) shows that an increase in the unsteady parameter δ decreases the dimensionless concentration. The influence of Schmidt number on the dimensionless concentration is displayed in Fig. 7(b). We infer that the dimensionless concentration decreases with an increase in Schmidt number Sc. This is due to the fact that an increase in Schmidt number reduces the molecular diffusivity D and that results in a decrease in thickness of the concentration boundary layer. Figs. 8–10 show the variations of the skin-friction coefficient as well as the reduced Nusselt and Sherwood numbers. Fig. 8 demonstrates that increasing magnetic parameter M increases the skin friction coefficient monotonically at the wall. This is because of the presence of the magnetic field which reduces the dimensionless velocity and the momentum boundary layer thickness, and thus increases the velocity gradient. Further, it is observed that the skin friction coefficient is higher for suction as compared to injection. This observation is consistent with the trend
Fig. 9. The variation of heat transfer rate against magnetic, unsteadiness, suction/injection, nanoparticles and radiation parameters.
F. Mabood, W.A. Khan / Journal of Molecular Liquids 219 (2016) 216–223
given in Table 3. The variation of the Nusselt number is shown for magnetic, suction/injection, unsteady, radiation and nanoparticles parameters in Fig. 9. Fig. 9(a) shows the behavior of heat transfer rate against magnetic parameter M for different values of suction/injection and unsteady parameters. It is found that the heat transfer rate increases with an increase in suction/injection and unsteady parameters. It is also found that heat transfer rate monotonically decreases with magnetic parameter. Fig. 9(b) depicts the variation of heat transfer rate with radiation and nanoparticles parameters. An increase in radiation and nanofluid parameters decreases the heat transfer rate. Finally, Fig. 10 illustrates the effects of nanofluids parameters and Schmidt number on Sherwood number. It is observed that Sherwood number increases monotonically with Schmidt number and Brownian motion parameter while reduces with increasing values of thermophoresis parameter Nt. 6. Conclusion MHD unsteady flow of nanofluid over a heated stretching sheet is studied. Effects of various parameters on the dimensionless velocity, temperature, concentration, skin friction coefficient and reduced Nusselt and Sherwood numbers are analyzed. The main observations of the present study are as follows: • The dimensionless velocity decreases as unsteady and suction parameters effects intensify. • The dimensionless temperature is higher in the presence of magnetic field. • The dimensionless concentration reduces with Schmidt number and Brownian motion parameter. • The skin friction coefficient increases, whereas the reduced Nusselt number decreases with magnetic parameter. • The skin friction coefficient, reduced Nusselt and Sherwood number are decreasing functions of unsteady parameter. • The reduced Nusselt number decreases but the reduced Sherwood number increases as the Brownian motion effects intensify.
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