Influences of rotation and thermophoresis on MHD peristaltic transport of Jeffrey fluid with convective conditions and wall properties

Journal of Magnetism and Magnetic Materials 410 (2016) 89–99

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Influences of rotation and thermophoresis on MHD peristaltic transport of Jeffrey fluid with convective conditions and wall properties T. Hayat a,b, M. Rafiq a,n, B. Ahmad b a b

Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan Nonlinear and Applied Mathematics (NAAM) Research Group,Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

art ic l e i nf o

a b s t r a c t

Article history: Received 22 January 2016 Received in revised form 25 February 2016 Accepted 1 March 2016 Available online 7 March 2016

This article aims to predict the effects of convective condition and particle deposition on peristaltic transport of Jeffrey fluid in a channel. The whole system is in a rotating frame of reference. The walls of channel are taken flexible. The fluid is electrically conducting in the presence of uniform magnetic field. Non-uniform heat source/sink parameter is also considered. Mass transfer with chemical reaction is considered. Relevant equations for the problems under consideration are first modeled and then simplified using lubrication approach. Resulting equations for stream function and temperature are solved exactly whereas mass transfer equation is solved numerically. Impacts of various involved parameters appearing in the solutions are carefully analyzed. & 2016 Elsevier B.V. All rights reserved.

Keywords: Rotating frame Peristalsis Convective conditions Thermophoresis Chemical reaction

1. Introduction Studies pertaining to the flow of fluid in a flexible pipe or channel have garnered the attention of many researchers due to its extensive application in technological as well as in biophysical flows. Progressive waves generated due to contraction and expansion of flexible tube play key role in mixing and transporting the fluid. This mechanism of peristalsis can be seen in the gastrointestinal tract, bile ducts, the esophagus, the ureter and blood transport in the extracorporeal circulation etc. Engineers have adopted the peristaltic process in propelling the corrosive materials and fluids which are to be kept away from the pumping machinery. The phenomenon of peristalsis takes place in many practical appliances including roller and finger pumps, heart-lung machines, blood pump machines, dialysis machine etc. Due to its wide range of applications several experimental and theoretical researchers have been put forward after the initial attempts of Latham [1]. Theoretical study presented by Shapiro et al. [2] is found to be in good agreement with the results of experimental work done in Ref. [1]. Magnetohydrodynamic flows also have key importance in many practical applications which include materials processing, Magneto Hydro Dynamic (MHD) energy generators, cancer therapy [3] and biomedical flow control and separation n

Corresponding author. E-mail address: [email protected] (M. Rafiq).

http://dx.doi.org/10.1016/j.jmmm.2016.03.001 0304-8853/& 2016 Elsevier B.V. All rights reserved.

devices [4]. Its application in biomedical engineering includes hyperthermia regulation in the cardiovascular system by magnetic induction [5], MHD drug targeting [6], magnetofluid rotary blood pumps, MHD bio-micro-fluidic device design and micro-circulation flows [7,8] etc. Some researches on peristalsis with magnetohydrodynamics can be seen in the Refs. [9–19]. The study of heat transfer have significant applications in industry and medicine. Especially heat transfer in human body is an important area of research. Bioheat transfer in tissues has attracted the attention of biomedical engineers in view of thermotherapy and the human thermoregulation system. The heat transfer in humans takes place as conduction in tissues, perfusion of the arterial–venous blood through the pores of the tissue, metabolic heat generation etc. The other applications are destruction of undesirable cancer tissues, dilution technique in examining blood flow and vasodilation. In addition mass transfer also occurs when nutrients diffusion out from the blood to neighboring tissues, membrane separation process, reverse osmosis, distillation process, combustion process and diffusion of chemical impurities. The combined heat and mass transfer effects can be seen in processes like drying, evaporation, thermodynamics at the surface of a water body and oxygenation etc. With this view point the recent researchers have made efforts for peristalsis with combined effect of heat and mass transfer [see 20–30]. It is noticed that the peristalsis in the past mostly dealt with the heat transfer effect either through prescribed surface temperature or prescribed heat flux at the channel wall. Only limited information is available about

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peristalsis with convective boundary conditions [31–35]. Thermophoresis is the phenomenon involving the migration of colloidal particle in response to macroscopic temperature. When temperature gradient is established in the gas, small suspended particles move in the direction of decreasing temperature. This phenomenon occurs due to different average velocities of particles on either sides. The common example is the blackening of the glass globe of a kerosene lantern. The main objective here is to examine thermophoresis effect in peristaltic transport of an incompressible Jeffrey liquid in a channel with convective heat and mass conditions. We believe that there is not a single attempt available in the literature which analyzes the peristalsis through the combined effects of convective conditions and thermophoresis particle deposition via chemical reaction in the presence of nonuniform heat source/sink parameter. It is observed that energy and mass transfer for flowing fluid not only involves simple conduction/diffusion but bulk motion also in such transfer phenomenon. Long wavelength and low Reynolds number approximations are utilized. Closed form solutions for stream function and velocity have been obtained whereas numerical results are discussed for the temperature and concentration profiles using NDSolve in mathematica. Impacts of different parameters appearing in solutions are analyzed in detail. Trapping phenomenon is examined for the pertinent variables.

2. Modeling Here we investigate the problem for two-dimensional peristaltic flow of Jeffrey liquid in a symmetric channel of width 2d. The channel walls are convectively heated. A uniform magnetic field of strength B0 is applied. Electric field effects are taken zero and induced magnetic field is neglected due to small magnetic Reynolds number. Thermal radiation and non-uniform heat source/sink effects are present. The whole system is in a rotating frame of reference with constant angular velocity Ω. Flow configuration is presented in Fig. 1. Flow inside the channel is induced due to propagation of sinusoidal waves of wavelength λ along the flexible walls of the channel with constant speed c. The geometries of the wall surfaces are described by

⎡ ⎛ 2π ⎞⎤ z= ± η(x, t) = ± ⎢ d + a sin ⎜ (x – ct) ⎟ ⎥, ⎝ λ ⎠⎦ ⎣

(1)

where t and a represent the time and wave amplitude respectively. Here þ and  signs designate the upper and lower wall of the

channel. The constitutive equations for Jeffrey fluid are represented by [12, 27]:

τ= − pI +S,

S=

(2)

d⎞ μ ⎛ ⎜ 1 + λ2 ⎟ A1, 1 + λ1 ⎝ dt ⎠

A1 = (grad V) + (grad V)transpose ,

(3)

in which p shows the pressure, I the identity tensor, S the extra stress tensor, A1 the first Rivlin Ericksen tensor, τ the Cauchy stress tensor, μ the dynamic viscosity, λ1 the ratio of relaxation to retardation times and λ2 the retardation time. In rotating frame the set of pertinent field equations governing the flow are

∂u ∂v ∂w + + =0 ∂x ∂y ∂z

(4)

⎡ du ⎤ ∂p^ ∂Sxx ∂Sxy ∂Sxz ρ⎢ + + + −σB02 u, ⎥−2ρΩv= − ⎣ dt ⎦ ∂x ∂x ∂y ∂z

(5)

⎡ dv ⎤ ∂p^ ∂Syx ∂Syy ∂Syz ρ ⎢ ⎥+2ρΩv= − + + + −σB02 v, ⎣ dt ⎦ ∂y ∂x ∂y ∂z

(6)

⎡ dw ⎤ ∂p^ ∂Szx ∂Szy ∂Szz ρ⎢ + + + , ⎥= − ⎣ dt ⎦ ∂z ∂x ∂y ∂z

(7)

ρCp

ρCp

⎡ ∂ 2T ⎛ ∂u dT ∂ 2T ∂ 2T ⎤ ∂u ∂w ⎞⎟ ∂w = κ⎢ 2 + + 2 ⎥ + Sxx + Sxz ⎜ + + Szz ⎝ ∂z dt ∂x ∂x ⎠ ∂z ⎣ ∂x ∂y 2 ∂z ⎦ ∂qr − + Q 0 ( T − Ta ), (8) ∂z

⎡ ∂ 2C ∂ 2C ∂ 2C ⎤ ∂ VT ( C − C0 ) dC =D ⎢ 2 + 2 + 2 ⎥− −k1 ( C−C0 ), ⎣ ∂x ∂y ∂z ⎦ ∂z dt

(

)

(9)

1 ρΩ (x2+y2 ) 2

in which p^ =p − represents the modified pressure. Here ρ is the fluid density, Ω the angular velocity, Cp the specific heat at constant volume, κ the thermal conductivity, T the temperature of fluid, Ta the ambient temperature, Q0 the non-uniform heat source/sink parameter, D the coefficient of mass diffusivity, C the concentration of fluid, C0 the concentration at lower wall and d/dt the material time differentiation. Utilizing Rosselands approximation for radiative heat flux we have [24]:

qr =

4σ ⁎ ∂T 4 , 3k ⁎ ∂z

where sn and kn denote the Stefan–Boltzman and Rosseland mean absorption coefficient respectively. We assume the temperature variations in such a way that Taylors series expansions of T4 about Tm (mean temperature) can be obtained. Ignoring higher order terms we have

T4≈4T3mT − 3T 4m.

(10)

Thermophoretic velocity VT can be defined in the form [33]:

VT =k ⁎⁎v Fig. 1. : Geometry of the problem.

∇T k ⁎⁎v ∂T =− , Tr Tr ∂z

(11)

T. Hayat et al. / Journal of Magnetism and Magnetic Materials 410 (2016) 89–99

in which Tr is the reference temperature, v is the fluid kinematic viscosity and knn is the thermophoretic coefficient. Invoking Eqs. (10) and (11) into Eqs. (8) and (9) we get

ρCp

⎡ ∂ 2T ⎛ ∂u ∂ 2T ∂ 2T ⎤ ∂u ∂w ⎞⎟ ∂w dT = κ⎢ 2 + + 2 ⎥ + Sxx + Sxz ⎜ + + Szz 2 ⎝ ∂ ∂ ∂x ⎠ ∂z dt x z ⎣ ∂x ∂y ∂z ⎦ +

ρCp

16σ⁎T3m ∂ 2T + Q 0 ( T − Ta ), 3k ⁎ ∂z2

⎡ ∂ 2C ∂ 2C ∂ 2C ⎤ k ⁎v ⎡ ∂ ⎛ ∂T ⎞⎤ dC =D ⎢ 2 + 2 + 2 ⎥+ ⎢ ⎜ (C−C0 ) ⎟ ⎥−k1 ( C−C0 ). ⎠⎦ ⎣ ∂x ∂y ∂z ⎦ Tr ⎣ ∂z ⎝ ∂z dt

0 = (1 + Rd)

⎛ ∂ 2ψ ⎞2 ∂ 2θ + S θ + Br ⎜ 2⎟ , ⎝ ∂z ⎠ ∂z2

(23)

0=

∂θ ∂φ ⎤ 1 ∂ 2φ ⎡ ∂ 2θ −τ ⎢ 2 φ + ⎥−γφ, 2 ⎣ ∂z ∂z ∂z ⎦ Sc ∂z

(24)

Sxz =

1 ∂ 2ψ , (1+λ1) ∂z2

(25)

Syz =

1 ∂v , (1+λ1) ∂z

(26)

(12)

(13)

Governing equation for flexible wall satisfies

L (η)=p−p0 ,

91

(14)

With

L= − τ ′

∂2 ∂2 ∂ +m 2 +d′ . ∂x2 ∂t ∂t

(15)

(27)

Szz =0,

Boundary conditions for the flow under consideration are defined as follows:

where continuity Eq. (4) is identically satisfied. The boundary conditions in non-dimensionalized form are as follows:

u=0, v=0, at z= ± η,

ψz =0, v=0, at z = ± η,

(28)

⎡ ∂3 ∂S xz ∂ψ ∂3 ∂2 ∂5 ∂ ⎤ ⎢ E1 + E2 + E3 + E4 + E5 ⎥ η = − M2 + 2T ′v at z = ± η, 2 5 ⎢⎣ ∂x 3 ∂ x ∂ t ∂ x ⎥⎦ ∂z ∂z ∂x∂t ∂x

(29)

∂θ ±Biθ =0, at z= ± η, ∂z

(30)

∂Sxy ∂Sxz ∂ ∂p ∂S du + L (η)= = xx + −σB02 u+2ρΩv−ρ at z= ± η, ∂x ∂x ∂x ∂y ∂z dt

κ

κ

(16)

(17)

∂T = − η1 ( T −Ta ) at z= + η, ∂z

∂T = − η1 ( Ta−T ) at z= − η, ∂z

⎧ 1⎫ φ=⎨ ⎬ ⎩ 0⎭

at

z = ± η,

η = 1 + ϵ sin [2π (x − t )]. In above expressions

⎧ C1 ⎫ φ=⎨ ⎬ at z= ± η, ⎩ C0 ⎭

(31)

(18) M2 (

=

B02 d2σ /μ)

T ′(=

Re Ωd ) c

is the Taylor number,

the Hartman number, Br ( = c 2μ/κTa ) the Brinkman

c T − T0 C − C0 cd d , φ= , Re= , δ= , u=ψz , w = − δψx. λ2⁎= λ2, θ = d T1 − T0 C1 − C0 ν λ

number, Rd ( = 16σ ⁎T13/3kk⁎) the radiation parameter, S ( = Q 0 d2/k ) the non-uniform heat source/sink parameter, Sc ( = v/D) the Schmidt number, τ ( = − k⁎⁎Ta/Tr ) the thermophoretic parameter, γ ( = k1d2/ν ) the chemical reaction parameter (here γ < 0 shows the generative chemical reaction and γ > 0 for destructive chemical reaction), Bi ( = η1d/κ ) the Biot number, ϵ( = a/d ) the amplitude ratio and E1 ( = − τd3/λ3μc ), E2 ( = m1cd3/λ3μ), E3 ( = d′d3/λ2μ), E4 ( = bd3/λ5μc ) 4 and E5 ( = k2 d3/λμc ) the non-dimensional elasticity parameters. Eqs. (22) and (27) show that pressure is not a function of z. Further we can neglect the pressure term in Eq. (21) as secondary flow is the result of rotation [19–30]. In view of these facts, we can write Eqs. (22) and (23) in the forms

Utilizing the above mentioned variables and applying lubrication approach [13–22] to Eqs. (5)–(7), (12) and (13) give

2T ′vz +

(19)

here η1 stands for heat transfer coefficient and C1/C0 are the concentration at the upper/lower walls respectively. We consider the non-dimensional variables as follows:

x y z d2p ⁎ ct ⁎ u ⁎ v w x⁎= , y⁎ = , z ⁎= , p⁎= , t = , u = , v = , w ⁎= , λ λ λ d cμλ c c c S⁎=

dS ⁎ η ,η= , μc d

∂p ∂Sxz −2T ′v= − + −M 2ψz , ∂x ∂z

(20)

1

( 1+λ1) ∂p ∂Sxz 2T ′ψz = − + −M 2v, ∂y ∂z

(21)

∂p ∂Szz , = ∂z ∂z

(22)

1

( 1+λ1)

ψzzzz −M 2ψzz =0,

vzz −M 2v−2T ′ψz =0.

(32)

(33)

The Eqs. (23), (32) and (33) are solved exactly with corresponding boundary conditions (28)–(30). The exact solutions are given by

ψ = B13 z + B14 sinh ⎡⎣ B1 z⎤⎦+B15 sinh ⎡⎣ B2 z⎤⎦,

(34)

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T. Hayat et al. / Journal of Magnetism and Magnetic Materials 410 (2016) 89–99

v = B21 + B22 cosh ⎡⎣ B1z ⎤⎦ + B23 cosh ⎡⎣ B2 z ⎤⎦

(

× (cosh2 ⎡⎣ − sinh2 ⎡⎣

(

(

B1 + B1 +

3.1. Axial and secondary velocities

)

B2 z⎤⎦

)

B2 z⎤⎦ ),

)

(35)

θ = D1 cos ⎡⎣ A1z ⎤⎦ + D2 cos ⎡⎣ 2 B1z ⎤⎦ 2 + D3 cosh ⎡⎣ B1z ⎤⎦ cosh ⎡⎣ B2 z ⎤⎦ × D4 cosh ⎡⎣ B2 z ⎤⎦ 2 + D5 sinh ⎡⎣ B1z ⎤⎦ sinh ⎡⎣ B2 z ⎤⎦ + D6 sinh ⎡⎣ B2 z ⎤⎦ + D7 ,

(36)

where Bi (i¼ 1  23) and Dj (j ¼ 1  7) have been computed algebraically. The Eq. (24) is solved numerically using NDSolve in mathematica. Graphical analysis of the numerical data is presented in the next section.

3. Discussion Note that Figs. 2 and 3 depict the behavior of axial (u) and secondary (v) velocities whereas Figs. 4 and 5 show the results for temperature (θ) and concentration (ϕ). The heat transfer coefficient Z and streamlines ψ are displayed in the Figs. 6 and 7.

Fig. 2(a) studies the influence of wall parameters on axial velocity. The parameters E1 and E2 show elastic nature of the walls whereas E3 represents dissipative property. Here E4 and E5 are rigidity and stiffness parameters respectively. It is noteworthy that for E3 ¼ 0 the wall moves up and down with no damping force on it. The velocity enhances for increasing the values of elastance parameters (i.e. E1 and E2) and it shows opposite behavior for E3, E4 and E5. In fact wall damping is a kind of resistive force which decreases the velocity. This damping force has similar effect on velocity in the presence of rigidity and stiffness. We notice from Fig. 2(b) that by increasing rotation parameter T’ the velocity decreases in axial direction. It can also be observed that axial velocity is greater in the absence of rotation. Decrease in velocity is seen for increasing values of λ1 (see Fig. 2c). Imposing magnetic field in normal direction to flow produces drag or resistive force that has tendency to suppress the movement of fluid which in turn reduces the axial velocity (Fig. 2d). The rotation of channel about z-axis induces a secondary flow v in y-direction. Figs. 3(a–d) are prepared to analyze the effects of involved parameters on v. Fig. 3(a) explains the influence of wall parameters on v. It is revealed that there is a decrease in E2. However secondary velocity enhances for larger values of E3, E4 and E5. Effect of Taylor number T′ on v is presented through Fig. 3 (b). It is pertinent to mention here that secondary velocity enhances when rotation parameter increases. There is no secondary velocity for T′ ¼0. Increasing behavior is noticed for velocity for

Fig. 2. a: Variation of wall properties on u when T’ ¼0.1, λ1 ¼ M¼ 0.5, x¼ ε¼ 0.2 and t ¼0.1. b: Variation of T’ on u when E1 ¼ E2 ¼ 3.0, E3 ¼ 0.01, E4 ¼E5 ¼0.1, λ1 ¼ M ¼0.5, x ¼ε ¼0.2 and t ¼0.1. c: Variation of λ1 on u when E1 ¼ E2 ¼ 3.0, E3 ¼ 0.01, E4 ¼E5 ¼ 0.1, T' ¼M ¼ 0.5, x ¼ε ¼0.2 and t¼ 0.1. d: Variation of M on u when E1 ¼ E2 ¼ 3.0, E3 ¼0.01, E4 ¼ E5 ¼ 0.1, T'¼ 0.5, λ1 ¼ M ¼ 0.5, x¼ ε¼ 0.2 and t¼ 0.1.

T. Hayat et al. / Journal of Magnetism and Magnetic Materials 410 (2016) 89–99

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Fig. 3. a: Variation of wall properties on v when T' ¼λ1¼M ¼ 0.5, x ¼ ε¼ 0.2 and t¼ 0.1. b: Variation of T' on v when E1 ¼ E2 ¼3.0, E3 ¼ 0.01, E4 ¼ E5 ¼ 0.1, λ1 ¼ M ¼0.5, x¼ ε¼ 0.2 and t ¼0.1. c: Variation of λ1 on v when E1 ¼ E2 ¼ M ¼0.5, E3 ¼ 0.01, T' ¼E4 ¼0.1, E5 ¼0.6, x ¼ ε¼ 0.2 and t ¼ 0.1. d: Variation of M on v when λ1 ¼ E1 ¼ E2 ¼ 0.5, E3 ¼0.01, T' ¼ E4 ¼ 0.1, E5 ¼0.6, x ¼ ε¼ 0.2 and t¼ 0.1.

larger values of λ1 (see Fig. 3c). It is depicted from Fig. 3(d) that secondary velocity shows similar behavior for increasing Hartman number M as we observed for an axial velocity. 3.2. Temperature profile Figs. 4(a–h) are displayed in order to study the behavior of temperature for the involved parameters. Fig. 4(a) is prepared to study wall properties. These results indicate that temperature starts increasing as we increase the values of E1 and E2. On the other hand the temperature shows decreasing behavior for E3, E4 and E5. Temperature is defined as an average kinetic energy of particles. Therefore as velocity increases for E1 and E2 then temperature rises. Similar behavior is noticed for E3, E4 and E5 corresponding to velocity. We noticed from Fig. 4(b) that θ decreases as we increase the rotation parameter T′. According to Fig. 4(c), θ decreases for larger radiation parameter Rd. The internal friction produced by shear in the flow generates the heat which in turn rises the temperature of fluid. Hence there is an enhancement of temperature for increasing values of Br (Fig. 4d). We have observed from Fig. 4(e) that temperature is greater for Jeffrey fluid when compared with the viscous case i.e. λ1 ¼0. Effect of Biot number is shown in Fig. 4(f)The Biot number is taken larger than one due to non-uniform temperature fields inside the fluid. However problems dealing with small Biot number are thermally simple due to uniform temperature distribution within the fluid. The obtained result shows decrease in temperature. Decreased thermal conductivity with an increase in

Biot number justifies the temperature drop. Fig. 4(g) studies the influence of Hartman number M on temperature. Rise of temperature is noticed for increasing values of M. Effect of heat generation/ absorption coefficient S is shown in Fig. 4h. This Fig. indicates that for S40 (heat generation) the temperature increases whereas it decreases for So0 (heat absorption). 3.3. Concentration profile Effect of several controlling parameters on dimensionless concentration ϕ is discussed in this subsection through Figs. 5(a–j). It is depicted from Fig. 5(a) that concentration field decreases for the increasing values of E1 and E2 whereas it increases for E3, E4 and E5. Fig. 5(b) illustrates the influence of Taylor number T' on ϕ. It is noticed that concentration is decreased when we increase the value of T′. Here it can be seen that for increased Schmidt number Sc the concentration get decreased (Fig. 5c). Since Sc is defined as the ratio of momentum diffusivity (viscosity) and mass diffusivity therefore it is used to characterize the fluid flows in which there are simultaneous momentum and mass diffusion convection processes. Influence of chemical reaction parameter γ on concentration field cannot be ignored when discussing mass transfer. Chemical reaction increases the rate of interfacial mass transfer. Due to which local concentration is decreased thus increasing its concentration gradient and flux when we have constructive chemical reaction. That is why ϕ decreases for constructive chemical reaction (γ 40) and it increases for destructive chemical reaction (γ o0) (Fig. 5d).

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Fig. 4. a: Variation of wall properties on θ when Rd ¼ λ1 ¼M ¼ 0.8, T' ¼ Br¼ 2.0, Bi¼ 10.0, S ¼0.5, x¼ ε¼ 0.2 and t ¼0.1. b: Variation of T' on θ when E1 ¼ E2 ¼ 0.1, E3 ¼ E4 ¼0.01, E5 ¼ 0.6, Rd ¼ λ1 ¼M ¼ 0.8, Br¼ 2.0, Bi¼10.0, S¼ 0.5, x¼ ε¼ 0.2 and t¼ 0.1. c: Variation of Rd on θ when E1 ¼E2 ¼ 0.1, E3 ¼ E4 ¼0.01, E5 ¼ 0.6, λ1 ¼ M ¼0.8, T' ¼Br ¼2.0, Bi¼ 10.0, S ¼0.5, x¼ ε¼ 0.2 and t ¼0.1. d: Variation of Br on θ when E1 ¼ E2 ¼0.1, E3 ¼ E4 ¼ 0.01, E5 ¼0.6, Rd ¼ λ1 ¼ M ¼ 0.8, Ti'¼ 2.0, Bi¼10.0, S¼ 0.5, x¼ ε¼ 0.2 and t ¼0.1. e: Variation of λ1 on θ when E1 ¼ E2 ¼0.1, E3 ¼ E4 ¼0.01, E5 ¼0.6, Rd ¼ M ¼0.8, T' ¼ Br¼ 2.0, Bi¼ 10.0, S ¼ 0.5, x¼ ε¼ 0.2 and t¼ 0.1. f: Variation of Bi on θ when E1 ¼E2 ¼ 0.1, E3 ¼ E4 ¼ 0.01, E5 ¼ 0.6, Rd ¼ λ1 ¼ M¼ 0.8, T’Br ¼2.0, Bi¼ 10.0, S ¼0.5, x ¼ ε¼ 0.2 and t¼ 0.1. g: Variation of M on θ when E1 ¼ E2 ¼0.1, E3 ¼ E4 ¼ 0.01, E5 ¼0.6, Rd ¼ λ1 ¼0.8, T'Br ¼2.0, Bi¼ 10.0, S¼ 0.5, x¼ ε¼ 0.2 and t¼ 0.1. h: Variation of S on θ when E1 ¼ E2 ¼ 0.1, E3 ¼ E4 ¼ 0.01, E5 ¼0.6, Rd ¼M ¼ λ1 ¼0.8, T’Br ¼2.0, Bi¼ 10.0, x¼ ε¼ 0.2 and t ¼0.1.

T. Hayat et al. / Journal of Magnetism and Magnetic Materials 410 (2016) 89–99

95

Fig. 5. a: Variation of wall properties on ϕ when τ¼ 0.1, Rd ¼ λ1 ¼ M¼ 0.8, T' ¼Br ¼ 2.0, Bi¼10.0, S¼ 0.5, Sc ¼ γ¼ 1.0, x¼ ε¼ 0.2 and t¼ 0.1. b: Variation of T' on ϕ when τ¼ E1 ¼E2 ¼ 0.1, E3 ¼E4 ¼0.01, E5 ¼ 0.6, Rd ¼λ1 ¼ M ¼0.8, Br ¼2.0, Bi¼ 10.0, S ¼0.5, Sc ¼ γ ¼1.0, x ¼ ε¼ 0.2 and t¼ 0.1. c: Variation of Sc on ϕ when τ¼ E1 ¼E2 ¼0.1, E3 ¼ E4 ¼0.01, E5 ¼0.6, Rd ¼ λ1 ¼M ¼ 0.8, T'Br¼ 2.0, Bi¼10.0, S¼ 0.5, γ ¼1.0, x ¼ ε¼ 0.2 and t¼ 0.1. d: Variation of γ on ϕ when τ ¼E1 ¼ E2 ¼ 0.1, E3 ¼E4 ¼ 0.01, E5 ¼ 0.6, Rd ¼λ1 ¼ M ¼0.8, T’' ¼Br ¼2.0, Bi¼ 10.0, S ¼0.5, Sc ¼ 1.0, x¼ ε¼ 0.2 and t¼ 0.1. e: Variation of λ1 on ϕ when τ ¼ E1 ¼ E2 ¼0.1, E3 ¼E4 ¼ 0.01, E5 ¼0.6, Rd ¼ M ¼ 0.8, T’'¼ 2.0, Bi¼ 10.0, S ¼ 0.5, Sc ¼γ ¼ 1.0, x ¼ε ¼0.2 and t¼ 0.1. f: Variation of M on ϕ when τ ¼ E1 ¼ E2 ¼ 0.1, E3 ¼ E4 ¼ 0.01, E5 ¼ 0.6, Rd ¼λ1 ¼ 0.8, T’'¼ Br ¼2.0, Bi¼ 10.0, S¼ 0.5, Sc ¼γ ¼ 1.0, x ¼ε ¼0.2 and t ¼0.1. g: Variation of S on ϕ when τ ¼ E1 ¼ E2 ¼0.1, E3 ¼ E4 ¼ 0.01, E5 ¼0.6, Rd ¼ M ¼λ1 ¼ 0.8, T’'¼Br ¼ 2.0, Bi ¼10.0, Sc ¼γ ¼ 1.0, x¼ ε¼ 0.2 and t ¼ 0.1. h: Variation of τ on ϕ when E1 ¼ E2 ¼0.1, E3 ¼ E4 ¼ 0.01, E5 ¼ 0.6, Rd ¼ M ¼λ1 ¼ 0.8, T' ¼ Br¼ 2.0, Bi¼10.0, S¼ 0.5, Sc ¼γ ¼ 1.0, x¼ ε¼ 0.2 and t ¼0.1.

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Fig. 6. a: Variation of wall properties on Z when Rd ¼ λ1 ¼M ¼ 0.8, Br ¼ 2.0, Ti'¼ 10.0, S ¼0.5, ε¼ 0.2 and t ¼0.1. b: Variation of T' on Z when E1 ¼ E2 ¼ 0.1, E3 ¼E4 ¼0.01, E5 ¼0.6, Rd ¼ λ1 ¼ M ¼0.8, Br ¼2.0, Bi¼ 10.0, S¼ 0.5, ε¼ 0.2 and t¼ 0.1. c: Variation of Bi on Z when E1 ¼ E2 ¼ 0.1, E3 ¼E4 ¼ 0.01, E5 ¼ 0.6, λ1 ¼ M ¼0.8, Br ¼2.0, T’'¼ 0.0, S ¼0.5, ε¼ 0.2 and t¼ 0.1. d: Variation of S on Z when E1 ¼ E2 ¼ 0.1, E3 ¼E4 ¼ 0.01, E5 ¼ 0.6, Rd ¼ λ1 ¼M ¼ 0.8, Br ¼2.0, T'¼ Bi¼ 10.0, ε¼ 0.2 and t¼ 0.1. e: Variation of Rd on Z when E1 ¼ E2 ¼ 0.1, E3 ¼E4 ¼ 0.01, E5 ¼ 0.6, λ1 ¼M ¼ 0.8, Br ¼ 2.0, T ¼ Br¼ 10.0, S¼ 0.5, ε ¼0.2 and t¼ 0.1.

Effects of λ1 and M on concentration field are displayed through Figs. 5(e) and (f). Effect of heat generation/absorption coefficient on ϕ is demonstrated in Fig. 5(g). It is observed that concentration is increased when we consider heat generation. We have sketched Fig. 5h to analyze the effect of thermophoresis parameter τ on ϕ. It can be seen that concentration is decreased upon increasing the value of thermophoretic parameter. 3.4. Heat transfer coefficient The effects of wall properties, T', Bi, S and Rd on the rate of heat transfer are plotted in Figs. 6(a–e). The heat transfer coefficient at

the wall is denoted by Z(x) ¼ ηx θz (η). It is noticed from Fig. 6 (a) that rate of heat transfer enhances for increasing values of E1 and E2 but it has opposite behavior for E3, E4 and E5. Fig. 6 (b) indicates a decrease in heat transfer between wall and fluid when rotation increases. Similar effect of heat transfer is observed for Biot number Bi (Fig. 6c). Fig. 6(d) illustrates the influence of heat generation/absorption coefficient S on heat transfer coefficient. This Figure shows that the rate of heat transfer is higher for heat generation (S40). The behavior of heat transfer coefficient for different values of Rd can be observed through Fig. 6(e). Decrease in the heat transfer rate is seen for increasing values of radiation parameter.

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Fig. 7. Streamlines for wall properties when λ1 ¼ M ¼0.8, T’ ¼0.1, ε¼ 0.2, t¼ 0.0 (a) E1 ¼0.1, E2 ¼ 0.01, E3 ¼0.01, E4 ¼ 0.1, E5 ¼0.1 (b) E1 ¼ 0.3 (c) E2 ¼ 0.03 (d) E2 ¼0.03 (e) E4 ¼0.3 (f) E4 ¼ 0.3.

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Fig. 8. Streamlines for T' when E1 ¼ 0.1, E2 ¼ E3 ¼0.01, E4 ¼ E5 ¼ 0.1 λ1 ¼M ¼ 0.8, ε ¼0.2, t ¼0.0 (a) T' ¼ 0.1 (b) T'¼ 0.2.

3.5. Trapping In general the shape of streamlines is same as that of a boundary wall in the wave frame. Nevertheless some of streamlines split and enclose a bolus under certain conditions and this bolus moves as a whole with the waves. This phenomenon is known as trapping. Figs. 7(a–f) are plotted to study the influence of wall parameters on streamlines. It is noticed that size of trapping bolus decreases for increasing values of E1, E2 and E5 (see Fig. 7b, c and f). From Fig. 7(d) it is seen that by increasing the values of E3 the size of bolus decreases. The size and number of streamlines decrease as we increase E4 (see 7e). The streamlines for different values of rotation parameter T' are shown in the Fig. 8 (a) and (b). It is depicted from Fig. that size of streamlines are decreased for increasing values of T'.

4. Conclusions Here a mathematical model to study the peristaltic transport of an electrically conducting Jeffrey fluid in a convectively heated channel is developed in presence of thermal deposition of particles and chemical reaction. The key findings of the presented attempt are as follows: 1. Axial velocity decreases when we increase the rotation whereas secondary velocity shows opposite behavior. 2. Wall parameters (E1, E2, E3, E4 and E5) show opposite behavior for the axial and secondary velocities. 3. Temperature enhances by increasing the rotation parameter T' and Biot number Bi. 4. Temperature is higher for heat generation coefficient (S 40) when compared with case of absorption (So 0). 5. Concentration is decreasing function of rotation parameter T' and Schmidt number Sc. 6. Generative chemical reaction (γ 40) results in decreased concentration whereas for destructive chemical reaction (γ o0) the concentration shows reverse behavior.

7. Impact of particle deposition is decreasing on concentration. 8. Rate of heat transfer is less in the case of absorption (S o0) than generation (S 40). 9. Decrease in the rate of heat transfer is noticed for Bi, Rd and T'. 10. Upon increasing rotation the size of streamlines get decreased.

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