Meccanica (2014) 49:521–534 DOI 10.1007/s11012-013-9809-5
Influence of induced magnetic field on the peristaltic flow of nanofluid M. Mustafa · S. Hina · T. Hayat · B. Ahmad
Received: 22 October 2012 / Accepted: 3 September 2013 / Published online: 28 September 2013 © Springer Science+Business Media Dordrecht 2013
Abstract This article investigates the effects of an induced magnetic field on the mixed convection peristaltic motion of nanofluid in a vertical channel. Transport equations involve the combined effects of Brownian motion and thermophoretic diffusion of nanoparticles. Analysis has been addressed subject to long wavelength and low Reynolds number assumptions. Explicit expressions of stream function, magnetic force function, temperature and nanoparticles concentration are developed. Analytic expressions are validated with the obtained numerical solutions. Peristaltic pumping rate is found to increase upon increasing the strengths of electric and magnetic fields and the buoyancy force due to temperature gradient. Moreover
M. Mustafa (B) Research Centre for Modeling and Simulation (RCMS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad 44000, Pakistan e-mail:
[email protected] S. Hina Department of Mathematical Sciences, Fatima Jinnah Women University, Rawalpindi 46000, Pakistan T. Hayat Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan T. Hayat · B. Ahmad Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80207, Jeddah 21589, Saudi Arabia
temperature rises and nanoparticles concentration decreases with an intensification in the Brownian motion effect. Keywords Nanofluid · Nanoparticle · Peristalsis · Induced magnetic field · Homotopy analysis method · Mixed convection
1 Introduction Peristaltic transport is the form of fluid transport due to the wave travelling along the walls of the tube/channel. Peristaltic motion has applications in physiology like transport of urine, transport of food bolus through gastrointestinal tract, transport of blood through small blood vessels etc. In industry, this phenomenon occur in the roller and finger pumps to pump the sanitary and noxious fluids, in mechanical instruments such as heart lung machine, blood pump machine and dialysis process. Peristaltic transport with heat transfer has a key role in tissues such as heat conduction in tissues, heat convection due to the blood flow through the pores of the tissues and radiation between surface and its environment. Peristaltic transport with heat transfer also occurs in hemodialysis and oxygenation processes. Peristaltic flow was firstly analyzed by Latham [1]. Mixed convective heat and mass transfer along vertical wavy surface is discussed by Jang and Yan [2]. Eldabe et al. [3] studied the mixed convective heat and mass transfer effects on peristaltic
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transport of non-Newtonian fluid with temperaturedependent viscosity. Influence of wall properties on the MHD peristaltic transport with heat transfer and porous medium is analyzed by Kothandapani and Srinivas [4]. Mekheimer and elmaboud [5] studied the effects of heat transfer and magnetic field on peristaltic flow in a vertical annulus. Srinivas and Kothandapani [6] addressed heat and mass transfer effects on MHD peristaltic flow through a porous space with compliant walls. Muthuraj and Srinivas [7] presented the mixed convective heat and mass transfer in a vertical wavy channel with travelling waves and porous medium. Srinivas et al. [8] extended the work of Muthuraj and Srinivas [7] for an asymmetric channel. Srinivas and Muthuraj [9] discussed the effects of chemical reaction and space porosity on MHD mixed convective peristaltic flow in a vertical asymmetric channel. Hayat et al. [10] discussed the effect of heat and mass transfer on peristaltic transport of second grade fluid with wall properties. Nanofluid is a liquid filled with nanometer-sized particles with diameter less than 100 nm called nanoparticles. These particles are made up of metals such as (Al, Cu), oxides (Al2 O3 ), carbides (SiC), nitrides (AlN, SiN) or nonmetals (Graphite, carbon nanotubes). Choi [11] experimentally verified that addition of small amount of these particles in the base fluid results in the appreciable increase in the effective thermal conductivity of the base fluid. Recently, the researchers have used this concept of nanofluid as a route to enhance the performance of heat transfer rate in liquids. Non-homogeneous equilibrium model proposed by Buongiorno [12] reveals that this abnormal increase in the thermal conductivity occurs due to the presence of two main effects namely the Brownian motion and thermophoretic diffusion of nanoparticles. Excellent reviews on the flows of nanofluids have been conducted by Daungthongsuk and Wongwises [13], Wang and Mujumdar [14, 15] and Kakaç and Pramuanjaroenkij [16]. Boundary layer flow of nanofluid over a flat plate has been analyzed by Kuznetsov and Nield [17]. In another paper, Nield and Kuznetsov [18] addressed the Cheng-Mincowcz problem for flow of nanofluid through a porous medium. Flow of nanofluid over a moving flat plate with uniform free stream has been investigated by Bachok et al. [19]. Recently, various attempts dealing with the boundary layer flow of nanofluid over stationary or moving surfaces have been made (see Khan and Pop [20], Rana and Bhargava [21], Yacob et al. [22], Makinde and Aziz [23]
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and Mustafa et al. [24] etc.). On the other hand the peristaltic mechanism of nanofluids has been scarcely studied. Endoscopic effects on the peristaltic motion of nanofluid have been examined by Akbar and Nadeem [25]. Peristaltic transport of nanofluid in an asymmetric channel with slip conditions has been addressed by Akbar et al. [26]. Mustafa et al. [27] studied the effects of wall properties on the peristaltic flow of nanofluid. Peristaltic motion of a magnetohydrodynamic fluid has many physiological applications. For example, cell membrane, intercellular protein and haemoglobin makes the blood as a biomagnetic fluid. Magnetic resonance imaging (MRI), magnetic devices and magnetic particles used as drug carriers are some applications of hydromagnetic peristaltic flow. The effect on induced magnetic field on the peristaltic flow is not yet properly addressed. Peristaltic flow of magneto-micropolar and couple stress fluids with induced magnetic field has been investigated by Mekheimer [28, 29]. Shit et al. [30] extended the work of Mekheimer [28] for an asymmetric channel. Influence of induced magnetic field on the peristaltic flow of Carreau fluid is examined by Hayat et al. [31]. Recently, peristaltic transport of pseudoplastic fluid under induced magnetic field is studied by Hayat et al. [32]. There is not a single attempt in the literature that considers the induced magnetic field effects on peristaltic motion of nanofluids. Therefore we present a mathematical model regarding the influence of induced magnetic field in the peristaltic flow of nanofluid through a vertical channel. The viscous dissipation effects are also taken into consideration. The developed coupled nonlinear differential system has been solved by homotopy analysis method (HAM) [33–38]. The validation of homotopy solutions is provided through the exact solutions provided in [29] and [32] for viscous fluid case. In addition the numerical solutions are obtained using the symbolic computational software Mathematica 8. The behaviors of interesting parameters on the stream function, axial induced magnetic field, temperature and nanoparticles concentration are discussed by plotting graphs.
2 Mathematical model We consider the peristaltic flow of an incompressible nanofluid in a two-dimensional vertical channel of width 2d1 (see Fig. 1). The X and Y axes
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(ii) Continuity equation ∇ · V = 0.
(3)
(iii) Momentum equation ρf
dV = −∇p + div S + (1 − C0 )ρf gα(T − T0 ) dt μe ∇H +2 + (ρp − ρf )gβ(C − C0 ) − 2 − μe H+ · ∇ H+ . (4)
(iv) Energy equation κ dT = ∇ 2T dt (ρc)f DT + τ DB (∇C · ∇T ) + (∇T · ∇T ) Tm +
1 S · L. (ρc)f
(5)
(v) Equation of concentration Fig. 1 Physical configuration and coordinate system
are taken along and perpendicular to the channel walls respectively. Further, constant magnetic field H0 acts in the Y -direction. The total magnetic field in the presence of induced magnetic field becomes H+ (hX (X, Y, t), hY (X, Y, t) + H0 , 0). The flow is created due to the following sinusoidal wave 2π (X − ct) . (1) η¯ = d1 + a sin λ In above expression c is the wave speed, a the wave amplitude, λ is the wavelength. Furthermore T0 and C0 are the temperature and concentration at both the walls, respectively. In the present analysis the velocity and gravitational fields are V ≡ (U (X, Y, t), V (X, Y, t), 0) and g ≡ (g, 0, 0) respectively. The fundamental equations governing the present flow problem are (see Mekheimer [29] and Hayat et al. [32]): (i) Maxwell’s equations ∇ · E = 0,
∇ · H = 0,
∂H ∇ × E = −μe , ∇ × H = J, ∂t J = σ E + μe V × H + ,
(2)
dC DT 2 = DB ∇ 2 C + ∇ T. dt Tm
(6)
(vi) Induction equation 1 ∂H+ = ∇ × V × H+ + ∇ 2 H+ . ∂t ς
(7)
Here L = grad V, S = μ(L + LT ), ∂ d ∂ ∂ = +U +V , dt ∂t ∂X ∂Y ∇2 =
∂2 ∂2 + , ∂X 2 ∂Y 2
U and V are the velocity components in the laboratory frame (X, Y ), J the current density, E the electric field intensity, H the magnetic field strength, p the pressure, ρp the density of nanoparticles, ρf the density of fluid, μ the dynamic viscosity, ν the kinematic viscosity, ς(= σ μe ) the inverse of magnetic diffusivity, σ the electrical conductivity, μe the magnetic permeability, κ the thermal conductivity, C the concentration and T temperature of the fluid, α the coefficient of linear thermal expansion of the fluid, β the coefficient
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of expansion with concentration, g the acceleration due to gravity, DB the Brownian motion coefficient, DT the thermophoretic diffusion coefficient, Tm the mean fluid’s temperature and τ (= (ρc)p /(ρc)f ) the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid. If (x, y) and (u, v) are the coordinates and velocity components in the wave frame then x = X − ct, v = V,
y = Y,
u = U − c,
p(x) = P (X, t).
(8)
Defining the following non-dimensional quantities x∗ =
x ; λ
y∗ =
y ; d1
u∗ =
u ; c
v∗ =
v ; cδ
δ=
d1 ; λ
p∗ =
d12 p ; cμλ
η=
η¯ ; d1
E∗ =
=
a ; d1
φ=
Nb =
τ D B C0 ; ν
pm = p +
Re =
θ=
ν(ρc)f ; k
Ec =
c2 ; c f T0
T − T0 ; T0
Gc =
(ρp − ρf )βgC0 d12 ; cμ
τ D T T0 ; Tm ν H0 μe S= ; c ρf
Nt =
ρf cd1 ; μ
hx = ωy ;
+ Nt
2
∂θ + ∂y
2 (12)
,
∂ψ ∂ω ∂ω ∂ψ ∂ω ∂ψ −δ + + ∂y ∂x ∂y ∂x ∂x ∂y 2 2 ∂ ω ∂ ω 1 δ2 2 + 2 + Rm ∂x ∂y
(13)
E =1+
Rm = σ μe cd1 ; ϕ ; H0 d1
hy = −ωx ,
Eqs. (3)–(7) reduce to ∂u ∂v + = 0, ∂x ∂y
∂θ δ ∂x
∂φ ∂φ Re Sc uδ +v ∂x ∂y Nt 2 ∂ 2 θ ∂ 2φ ∂ 2φ ∂ 2θ . δ = δ2 2 + 2 + + Nb ∂x ∂y ∂x 2 ∂y 2
(1 − C0 )αgT0 d12 ; cν
ω=
(11)
∂v ∂u 2 ∂u 2 + + δ + Ec 4 δ ∂x ∂x ∂y ∂θ ∂φ 2 ∂φ ∂θ + Nb δ + ∂x ∂x ∂y ∂y
M 2 = Re Rm S 2 ; Gr =
∂v ∂v +v δ 3 Re u ∂x ∂y ∂pm ∂ 2v ∂ 2v =− + δ2 δ2 2 + 2 ∂y ∂x ∂y ∂ω ∂ ∂ω ∂ ∂ω 2 2 − δ Re S δ + 1−δ , ∂y ∂x ∂x ∂y ∂x ∂θ ∂θ Re uδ +v ∂x ∂y 1 2 ∂ 2θ ∂ 2θ δ = + Pr ∂x 2 ∂y 2
μe H +2 1 Re δ ; 2 ρf c 2
Pr =
∂ 2u ∂ 2u ∂pm + δ 2 2 + 2 + Grθ + Gcφ ∂x ∂x ∂y ∂ω ∂ ∂ω ∂ ∂ω + Re S 2 δ + 1−δ , (10) ∂y ∂x ∂x ∂y ∂y
=−
−E ; cH0 μe
C − C0 ; C0
∂u ∂u +v δ Re u ∂x ∂y
(9)
(14)
where asterik has been suppressed, Re the Reynolds number, δ the wave number, S the Strommer’s number, M the Hartman number, Rm the magnetic Reynolds number, Nb the Brownian motion parameter, Nt the thermophoresis parameter, Gr the thermal buoyancy parameter, Gc the concentration buoyancy parameter,
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Pr the Prandtl number, Ec the Eckert number, θ , φ, ω the non-dimensional temperature, concentration and magnetic force functions respectively. The velocity components u and v in terms of stream function are given by u=
∂ψ , ∂y
v=−
∂ψ . ∂x
Combining Eqs. (21)–(23) one can see that Θ = F + 1, η ∂ψ dy. F= 0 ∂y
(24) (25)
The dimensionless boundary conditions in the wave frame are given by (see Mekheimer [29] for details)
Using above relations in Eqs. (10)–(14) after long wave length and low Reynolds number approximation, we get the following differential equations ∂pm ∂ 3 ψ ∂ 2ω − + + Grθ + Gcφ + Re S 2 2 = 0, (15) 3 ∂x ∂y ∂y
∂ψ = −1, ∂y
ψ = F, φ = 0,
ω = 0,
ψ = 0,
∂ 2ψ = 0, ∂y 2
θ = 0,
at y = η, (26)
∂θ = 0, ∂y
∂pm = 0, ∂y
(16)
∂φ = 0, ∂y
∂ 2ω ∂ψ E − 1 + , = R m ∂y ∂y 2
(17)
and the expressions for pressure rise per wavelength, axial induced magnetic field and current density are
∂ 2θ
1 ∂θ ∂θ ∂φ + Nt + Nb 2 Pr ∂y ∂y ∂y ∂y
2
+ Ec
∂ 2ψ 2 ∂y 2
∂ω = 0, ∂y
= 0,
Δpλ = 0
1
dp dx, dx
at y = 0
hx =
∂ω , ∂y
jz = −
∂hx . ∂y (27)
(18) 3 Homotopy analytic solutions
∂ 2φ Nt ∂ 2 θ + = 0. Nb ∂y 2 ∂y 2
(19)
Eliminating the pressure gradient from Eqs. (15) and (16) we obtain
Rule of solution expression and the involved boundary conditions direct us to choose the initial guesses and the auxiliary linear operators as follows
2 ∂φ ∂ 4ψ ∂θ 2∂ ψ + Gc − M + Gr = 0. ∂y ∂y ∂y 4 ∂y 2
ψ0 =
(20)
The volume flow rate in laboratory and wave frames are η¯ η Q= U (X, Y, t)dY, q= u(x, y)dy. (21) 0
0
At a fixed position X, the time averaged flow over a period T¯ is written as ¯= 1 Q T¯
T¯
(22)
Qdt. 0
Defining Θ and F as the dimensionless mean flows in the laboratory and wave frames by Θ=
Q¯ , cd1
F=
q . cd1
(23)
−Fy 3 − y 3 η + 3Fyη2 + yη3 , 2η3
(28)
ω0 = θ0 = φ0 = 0, Lψ (ψ) =
d 4ψ , dy 4
d 2θ Lθ (θ ) = 2 , dy
Lω (ω) =
d 2ω , dy 2
d 2φ Lφ (φ) = 2 . dy
(29)
If p ∈ [0, 1] is an embedding parameter and ψ , ω , θ and φ are the non-zero auxiliary parameters then the generalized homotopic equations corresponding to Eqs. (17)–(20) are
(1 − p)Lψ Ψ (y; p) − ψ0 (y)
= pψ Nψ Ψ (y; p), Θ(y; p), Φ(y; p) , (30)
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(1 − p)Lω Ω(y; p) − ω0 (y)
= pω Nω Ψ (y; p), Ω(y; p) ,
(1 − p)Lθ Θ(y; p) − θ0 (y)
= pθ Nθ Ψ (y; p), Θ(y; p), Φ(y; p) ,
(1 − p)Lφ Φ(y; p) − φ0 (y)
= pφ Nφ Θ(y; p), Φ(y; p) , Ψ (y; p) = F, Θ(η; p) = 0, Ω(η; p) = 0,
∂Ψ (y; p) ∂y
Expanding Ψ , Ω, Θ and Φ using the Maclaurin’s series about p we get (31) Ψ (y; p) = (32)
y=η
Θ(y; p) =
∂Θ(y; p) ∂y
(34) = 0, y=0
∂Ω(y; p) |y=0 = 0, ∂y
+ Pr Ec
∂y 2
+ Pr Nb
=
Nt ∂ 2 Θ(y, p) ∂ 2 Φ(y, p) + . 2 Nb ∂y ∂y 2
θm (y)p m , (40)
1 ∂ m Θ(y, p) , θm (y) = m! ∂p m p=0 ∞
φm (y)p m , (41)
1 ∂ m Φ(y, p) , φm (y) = m! ∂p m p=0
∂ 2 Ψ (y, p) 2
∂Θ(y, p) ∂Φ(y, p) ∂y ∂y ∂Θ(y, p) 2 , + Pr Nt ∂y
Nφ Θ(y; p), Φ(y; p)
∞
m=0
∂Φ(y, p) ∂ 2 Ψ (y, p) − M2 , (35) ∂y ∂y 2
Nω Ψ (y; p), Ω(y; p) ∂Ψ (y, p) ∂ 2 Ω(y, p) E − 1 + , − R = m ∂y ∂y 2
Nθ Ψ (y; p), Θ(y; p), Φ(y; p) ∂y 2
(39)
m=0
Φ(y; p) =
+ Gc
=
ωm (y)p m ,
1 ∂ m Ω(y, p) , ωm (y) = m! ∂p m p=0
Ψ (0; p) = 0,
∞
m=0
= −1,
∂Θ(y, p) ∂ 4 Ψ (y, p) + Gr 4 ∂y ∂y
∂ 2 Θ(y, p)
(38)
1 ∂ m Ψ (y, p) , ψm (y) = m! ∂p m p=0 Ω(y; p) =
Nψ Ψ (y; p), Θ(y; p), Φ(y; p) =
ψm (y)p m ,
m=0
(33)
Φ(η; p) = 0,
∂ 2 Ψ (y; p) = 0, ∂y 2 y=0 ∂Φ(y; p) = 0, ∂y y=0
∞
(36)
(37)
and the final solutions are retrieved at p = 1. The functions ψm , ωm , θm and φm can be determined from the deformation of Eqs. (30)–(37). Explicitly mth-order deformation problems corresponding to Eqs. (30)– (37) are
f Lψ ψm (y) − χm ψm−1 (y) = ψ Rm (y),
Lω ωm (y) − χm ωm−1 (y) = ω Rωm (y),
Lθ θm (y) − χm θm−1 (y) = θ Rθm (y),
Lφ φm (y) − χm φm−1 (y) = φ Rφm (y), ψm = 0,
∂ψm = 0, ∂y
φm = 0,
ωm = 0,
ψm = 0,
∂ 2 ψm = 0, ∂y 2
∂φm = 0, ∂y
(42) (43) (44) (45)
θm = 0, at y = η,
∂ωm = 0, ∂y
∂θm = 0, ∂y at y = 0,
(46)
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Fig. 2 -curves for the functions ψ , ω, θ and φ when = 0.2, Θ = 2 and x = 0.1
f Rm (y)
∂φm−1 ∂ 4 ψm−1 ∂θm−1 + Gc = + Gr ∂y ∂y ∂y 4 − M2
Rωm (y) =
∂ 2 ψm−1 , ∂y 2
χm = (47)
∂ψm−1 ∂ 2 ωm−1 + Rm 2 ∂y ∂y − Rm (E − 1)(1 − χm ),
(48)
∂ 2 ψm−1−k ∂ 2 ψk 1 ∂ 2 θm−1 + Ec Pr ∂y 2 ∂y 2 ∂y 2 m−1
Rθm (η) =
0,
m = 1,
1,
m > 1.
Thus nonlinear boundary value problem (BVP) is reduced to several linear Eqs. (42)–(51) which can be easily solved by using computational software Mathematica. To seek the permissible values of auxiliary parameters, -curves for the functions ψ, ω, θ and φ at 15th-order of approximations have been given in Fig. 2a–2d. Here the interval of convergence lie in the flat portion of these curves. It is noticed that admissible values of ψ , ω , θ and φ are −0.9 ≤ ψ ≤ −0.35,
k=0
(52)
−1.6 ≤ ω ≤ −0.4,
(49) + Pr Nb
m−1
k=0
m−1
∂θm−1−k ∂θk , ∂y ∂y
(50)
1 ∂ 2 φm−1 Nt ∂ 2 θm−1 + , 2 Sc ∂y Nb ∂y 2
(51)
+ Pr Nt
k=0
Rφm (η) =
−0.8 ≤ θ ≤ −0.45 and
∂θm−1−k ∂φk ∂y ∂y
− 0.8 ≤ φ ≤ −0.45
respectively. The homotopy solutions are compared with the numerical solutions computed by employing the built in routine for solving nonlinear boundary value problems via shooting method through the computational software Mathematica. The numerical solutions are found in excellent agreement with the analytic solutions for all the values of embedding parameters.
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Fig. 3 Comparison of homotopy solutions with the exact solutions given in [29] and [32]. Points: Exact solutions, Lines: 10th-order HAM solutions at f = θ = φ = −0.8
Fig. 4 Comparison of numerical and homotopy solutions. Points: Numerical solutions, Lines: 10th-order HAM solutions at f = θ = φ = −0.8
4 Numerical results and discussion The behaviors of different parameters on the stream function ψ , axial induced magnetic field hx , current density jz , temperature θ and concentration φ have been depicted through graphical results. The series solutions are validated with the exact solutions for the viscous fluid presented in [29] and [32] and the numerical solutions obtained through the computational software Mathematica. It is clear from Figs. 3 and 4 that homotopy solutions at only 10th-order approximations are in agreement with the exact and numerical solutions for different values of the parameters. Pressure rise against the flow rate Θ for various values of parameters has been plotted in Fig. 5. Numerical integration through the software Mathematica has been performed to compute the pressure rise. We observed from Fig. 5a that maximum pressure rise against which peristalsis works as a pump (i.e. Δpλ for Θ = 0) increases with an increase in Gr or equivalently the buoyancy force due to temperature gradient.
The pressure rise continue to increase with an increase in Gr as the prescribed flow rate increases. The freepumping flux (Θ for Δpλ = 0) also increases with an increase in Gr. In the co-pumping region (Δpλ < 0, Θ > 0), for arbitrarily chosen Δpλ the flow rate Θ increases when Gr is increased. From physical point of view, it shows that the buoyancy forces due to the temperature gradient increases the pressure rise. Figure 5b shows that Δpλ is inversely proportional to Gc (for fixed flow rate) in the pumping region. Moreover the pumping rate decreases with an increase in Gc in the free pumping region (Δpλ = 0). Figure 5c elucidates that pumping rate significantly increases when M is increased in the pumping region (Δpλ > 0). The pumping rate is found to increase upon increasing M in the free-pumping (Δpλ = 0). Pumping rate is also an increasing function of M in the co-pumping region (Δpλ < 0). Figure 5d indicates that as the electric field strength intensifies this corresponds to an increase in the pressure rise for a fixed flow rate. Moreover the behavior of amplitude ratio on Δpλ is similar to that
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Fig. 5 Pressure rise for different values of the parameters: (a) = 0.2, Θ = 2, Gc = Nb = Nt = 0.1, Pr = Rm = 1, Ec = 1, M = 2, E = 0.5; (b) = 0.2, Θ = 2, Gr = Nb = Nt = 0.1, Pr = Rm = 1, Ec = 1, M = 2, E = 0.5; (c) = 0.2, Θ = 2, Gc = Gr = Nb = Nt = 0.1, Pr = Rm = 1, Ec = 1, E = 0.5; (d) = 0.2, Θ = 2, Gc = Gr = Nb = Nt = 0.1, Pr = Rm = 1, Ec = 1, M = 2; (e) Θ = 2, Gc = Gr = Nb = Nt = 0.1, Pr = Rm = 1, Ec = Sc = 1, M = 2, E = 0.5
accounted for the magnetic field in a qualitative sense (see Fig. 5e). Figure 6 is displayed to perceive the effects of Hartman number M, magnetic Reynolds number Rm and the electric field intensity E on the axial induced magnetic field hx and the current density jz . We noticed that hx is symmetric about the channel and decreases when the strengths of electric and magnetic fields are increased. On the other hand the axial induced magnetic field is an increasing function of Rm . Physically, by increasing magnetic Reynolds number the magentic permeability increases and increase of magnetic permeability, increase the induced magnetic field. It
can be seen further that the profiles for hx have one direction in the half region (y > 0) and, in the other half (y < 0), these are in the opposite direction. It is noticed that the current density distribution jz increases near the channel wall and decreases near the centre of the channel with an increase in the Hartman number M and electric field E. Figure 7 presents the temperature profiles for different values of Nb, Nt, Gr, Gc and M. It is quite obvious that in a viscous fluid flow the viscosity of the fluid will take kinetic energy from the motion of the fluid and transform it into internal energy of the fluid that heats up the fluid. This process is partially irreversible and is referred to as viscous dis-
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Fig. 6 Influence of parameters on axial induced magnetic field hx and current density jz : (a) and (d) = 0.2, x = 0.1, Θ = 2, Gr = Gc = 1, Nb = Nt = 0.1, Pr = Ec = Rm = E = 1; (b) and (e) = 0.2, x = 0.1, Θ = 2, Gr = Gc = 1, Nb = Nt = 0.1, Pr = Ec = Rm = M = 1, (c) and (f) = 0.2, x = 0.1, Θ = 2, Gr = Gc = 1, Nb = Nt = 0.1, Pr = Ec = M = E = 1
sipation. The presence of viscous dissipation increases the fluid’s temperature in the central part of the channel as can be seen from Figs. 7a–7f. It is seen that local temperature of the fluid rises when the Brownian motion parameter Nb is increased for a relatively weaker thermophoretic effect and one would expect that this increase becomes pronounced for sufficiently stronger thermophoretic effect. However there is slight reduction in the temperature function when Nt is increased. This observation leads to the conclusion that simultaneous increase in the Brownian motion and thermophoresis effects corresponds to an increase in the temperature θ (see Figs. 7a and 7b). A similar conclusion was drawn in the previous studies [20–24].
Figure 7c shows the influence of Gr on the temperature θ . We should notice that Gr represents the volumetric expansion capability of the fluid. Buoyancy force acts like a favorable pressure gradient and accelerates the fluid. In other words the gradual increase in Gr accompanies with a stronger buoyancy force which in turn rises the fluid’s temperature. However a minor decrease in the temperature θ is found when Gc is increased (see Fig. 7d). Thus we conclude that temperature increases as the strengths of buoyancy forces due to temperature and concentration differences are simultaneously increased. Magnetic field effects on the temperature θ are observed in the Fig. 7e. Here we found that temperature θ decreases with an increase
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Fig. 7 Temperature profiles for different values of parameters: (a) = 0.2, x = 0.1, Θ = 2, Gc = Gr = 5, Nt = 0.1, Pr = 1, Ec = M = 2; (b) = 0.2, x = 0.1, Θ = 2, Gc = Gr = 1, Nb = 0.1, Pr = 1, Ec = M = 2; (c) = 0.2, x = 0.1, Θ = 2, Gc = Nb = Nt = 0.1, Pr = 1, Ec = M = 2; (d) = 0.2, x = 0.1, Θ = 2, Gr = Nb = Nt = 0.1, Pr = 1, Ec = M = 2; (e) = 0.2, x = 0.1, Θ = 2, Gc = Nb = Nt = 0.1, Pr = 1, Ec = 2; (f) = 0.2, Θ = 2, Gc = Gr = 1, Nb = Nt = 0.1, Pr = 1, Ec = M = 2
in M. This decrease is prominent near the center of the channel. Temperature distributions at different cross sections of the channel are given in Fig. 7f. This figure shows the similar trend of temperature function as observed in previous graphs. Figures 8a–8f plot the nanoparticles concentration profiles for different values of parameters. With an increase in Nb, Brownian diffusion decreases which forces the concentration to decrease. Moreover the deviation in the profiles only occur for small values of Nb. In fact the concentration function φ is not affected for the values of Nb beyond 3. In contrast to the behavior of Nb on φ, there is a significant increase in nanoparticles concen-
tration when Nt is increased. Moreover the influences of rest of the parameters on the concentration φ are similar to those accounted for the temperature function. Although we have not included the plots for heat transfer coefficient at the wall Z(x) = ηx θy (η), it is visualizeable from the Figs. 7a and 7b that profiles become increasingly steeper near the wall when both Nb and Nt are simultaneously increased. As a consequence, the heat transfer coefficient, which is directly proportional to the slope at the wall, increases with an increase in Nb and Nt. This shows that performance of heat transfer rate is effectively improved through
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Fig. 8 Concentration profiles for different values of parameters: (a) = 0.2, x = 0.1, Θ = 2, Gc = Gr = 5, Nt = 0.1, Pr = 1, Ec = M = 2; (b) = 0.2, x = 0.1, Θ = 2, Gc = Gr = 1, Nb = 0.1, Pr = 1, Ec = M = 2; (c) = 0.2, x = 0.1, Θ = 2, Gc = Nb = Nt = 0.1, Pr = 1, Ec = M = 2; (d) = 0.2, x = 0.1, Θ = 2, Gr = Nb = Nt = 0.1, Pr = 1, Ec = M = 2; (e) = 0.2, x = 0.1, Θ = 2, Gc = Nb = Nt = 0.1, Pr = 1, Ec = 2; (f) = 0.2, Θ = 2, Gc = Gr = 1, Nb = Nt = 0.1, Pr = 1, Ec = M = 2
the addition of nanoparticles as already pointed out by Choi [11]. Figures 9 and 10 elucidate the behaviors of parameters on the stream function ψ . Generally the shape of streamlines is analogous to the wave travelling along the walls of the channel. Under certain conditions these streamlines split and enclose a bolus which moves along with the wave across the channel. Figure 10 indicates that size of trapped bolus increases and the bolus circulates faster when Gr is increased. However there is a decrease in the size of the trapped bolus with an increase in Gc for an arbitrarily chosen flow rate Θ. We also observed that Brownian motion and thermophoresis effects negligibly affect the stream function ψ . An increase in
the strength of magnetic field reduces the size and circulation of the trapped bolus. In fact the bolus disappears when sufficiently large values of M are chosen.
5 Conclusions A mathematical model is developed to study the effects of an induced magnetic field on the peristaltic motion of nanofluid in a vertical channel. Series solutions of the arising nonlinear coupled differential equations have been obtained by homtopy analysis method (HAM). The analytic solutions are found in
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Fig. 9 Streamlines for different values of Gr and Gc when = 0.2, Θ = 0.7, Nb = Nt = 0.5, Pr = Ec = 1, M = 0.5: (a) Gr = Gc = 1, (b) Gr = 2, Gc = 0.1 and (c) Gr = 1, Gc = 3
Fig. 10 Streamlines for different values of M when = 0.2, Θ = 0.7, Gc = Gr = 1, Nb = Nt = 0.5, Pr = Ec = 1: (a) Du = Sr = 1, (b) Du = 2.5, Sr = 1 and (c) Du = 1, Sr = 4
excellent agreement with the numerical results for all the values of the embedding parameters. It is found that temperature significantly rises as the Brownian motion effects strengthens. Axial induced magnetic field, temperature and concentration are decreasing functions of Hartman number. Moreover the size and circulation of the trapped bolus decrease with an increase in the magnetic field strength. The current analysis for the horizontal channel (which is not yet reported) can be recovered from the presented series solutions when Gr = Gc = 0. The results are compared with the exact solutions provided in Mekheimer [29] and Hayat et al. [32] for the Newtonian fluid case and found in excellent agreement. Acknowledgements We are grateful to the reviewer for his constructive and valuable suggestions. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), under grant no. (25-130/1433HiCi). The authors, therefore, acknowledge technical and financial support of KAU.
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