Peristaltic transport of tangent hyperbolic fluid with variable viscosity

Accepted Manuscript Peristaltic transport of tangent hyperbolic fluid with variable viscosity T. Hayat, Asma Riaz, Anum Tanveer, Ahmed Alsaedi PII: DOI: Reference:

S2451-9049(17)30272-X https://doi.org/10.1016/j.tsep.2018.04.002 TSEP 157

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Thermal Science and Engineering Progress

Please cite this article as: T. Hayat, A. Riaz, A. Tanveer, A. Alsaedi, Peristaltic transport of tangent hyperbolic fluid with variable viscosity, Thermal Science and Engineering Progress (2018), doi: https://doi.org/10.1016/j.tsep. 2018.04.002

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Peristaltic transport of tangent hyperbolic fluid with variable viscosity T. Hayata,b , Asma Riaza , Anum Tanveera , a b

1

and Ahmed Alsaedib

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia. Abstract: This investigation examines the influence of variable viscosity on peristaltic

transport of tangent hyperbolic fluid with heat and mass transfer. Viscous dissipation and Joule heating have been taken. Temperature dependent viscosity is considered. An electrically conducting fluid is taken in symmetric inclined channel. Effect of heat and mass transfer are outlined. Channel inclination dominates gravitational effects such that the mixed convection is not ignored. Soret and Dufour effects are highlighted. The mathematical expressions are subject to lubrication approach. Decline in velocity is noticed with an increase in magnetic field and gravitational effects. Whereas larger values of Soret and Dufour number raise the temperature and heat transfer rate. The graphs in present communication have been sketched directly using NDSolve built in routine of Mathematica. Keywords: Peristaltic motion, Mixed convection, Soret and Dufour effects, Tangent hyperbolic fluid, Inclined channel.

1

Introduction

Peristalsis is fluid transport which occurs due to the propagation of waves along the walls of channel. Such propagation of fluid is important due to its applications in engineering as well as in physiological processes. Physical-process like urine passage through the ureter, motion of the chyme, swallowing food, motion of lymph, spermatozoa motion and the ovum transport in the female fallopian tube works through peristalsis. Industrial applications of peristalsis covers designing of roller pumps, biomedical system and noxious fluid transport in nuclear industry. Processes of endoscope, hyperthermia, magnetotherapy, cancer therapy and arterial flow are designed on the principle of peristaltic pumping, peristalsis is hot topic of interest in recent time after theoretical attempts of Latham [1] and Shapiro et al. [2]. 1

Corresponding author. Tel.: + 92 51 90642172.

e-mail address: [email protected]

1

Different analysis in this direction have been cited in the refs. [3-11]. Analysis based on non-Newtonian fluids has been a significant role in research due to its involvement in engineering and industry. Coal-oil slurries, paints, clay, coating and suspensions, cosmetic products, grease, custard and physiological liquids are some particular examples of such fluids. Heat transfer and non-Newtonian fluids flow play an important role in petroleum production, food engineering, power engineering and in polymer solutions. Non-Newtonian liquids appear with different properties and therefore several representations of such fluids is offered. Further a relationship between the shear stress and rate of strain in fluids is non-linear. Thus non-Newtonian materials’s equations are nonlinear and higher order than the viscous materials. Tangent hyperbolic fluid is one such non-Newtonian models which has been used to describe the movements of normal stresses and shear thinning. Refs. [12-17] cover some recent attempts on flows of non-Newtonian fluids. The concept of an inclined channel are widely used in moving heavy loads over vertical obstacles, walking up a pedestrian ramp and automobile (see refs. [18-19]). Magnetohydrodynamics is significant in higher temperature apparatus like power generators and the aviation phenomenon by using an compass of airplane etc. Moreover the magnetic field impacts are important in arterial flow, magnetotherapy, cancer therapy and hyperthermia etc. Further, practical applications of smaller intensity and fields of frequency pulsating accurate the tissue and cell [20-21]. Prakash and Kothandapani [22] studied the peristaltic motion of tangent hyperbolic nanofluid with inclined magnetohydrodynamic in straight channel. Hayat et al. [23] calculated inclined magnetic field impact and inclined channel on peristaltic transport of Williamson fluid. Convective conditions are utilized in this attempt. Hayat et al. [24] calculated impact of magnetohydrodynamic on peristaltic transport of hyperbolic tangent nanofluid with Joule heating. Ramesh and Devakar [25] computed heat transfer effect on peristaltic motion of magnetohydrodynamic second grade fluid in an inclined channel. Interchange of thermal energy between the different segments of a physical system is referred as heat transfer. It depends on the temperature differences between components of system and surroundings. Mass transfer specifies the mass transport from one component to another. It occurs frequently in chemical engineering and reaction engineering. Combined influences of mass and heat transfer are important in geothermal and geophysical engineering, eradication of geothermal energy, movement of moisture in fibrous insulation and chemical pollutants in saturated soil. Refs. [26-28] cover some relevant works on peristalsis with heat 2

and mass transfer. Mixed convection effect perceives large applications in engineering for cooling of microelectronic devices, electronic devices, float glass production, food processing, nuclear reactors, coating and solar power. Having this fact in mind mixed convection has been discussed by the number of researchers (see refs. [29-31]). Soret effect is examined in combinations of mobile particles where the different particles show different reactions towards temperature gradient. Dufour effect is a phenomenon in which the energy flux due to a concentration gradient taking place as a coupled impact of irreversible processes. However less has been said about such effects in peristaltic flows (see refs. [32-35]). This paper discusses the peristaltic motion of hydromagnetic tangent hyperbolic fluid in an inclined channel. Mixed convection effects are retained in the flow governing equations. The viscous dissipation and Joule heating effects are also presented. The Dufour and Soret effects are highlighted. The resulting system is simplified by invoking long wavelength and low Reynolds number approximations. Graphical illustrations for several parameters are exhibited numerically.

2

Problem formulation

Fig. 1: Geometry of the problem Consider peristaltic transport of an incompressible hyperbolic tangent fluid in an inclined channel. Further, inclined channel has an angle α with the vertical axes. A uniform magnetic field with constant strength B0 is taken inclined at an angle β. The impacts of magnetic field are ignored by assuming smaller magnetic Reynolds number. The problem is not studied in the presence of electric field. Further the gravitational effects are not ignored.

3

The wall geometry is described in the following form   2π y = ±h(x, t) = ± a + b cos (x − ct) . λ

(1)

Here c denotes wave speed, a the half channel width, b the wave amplitude, λ the wavelength, t the time, h the displacement of upper wall and −h the displacement of lower wall. Continuity equation for incompressible fluid is taken by div V = 0.

(2)

The momentum equation is ρ

dV = −∇p + div S + J × B. dt

(3)

The equations for heat and mass transfer with viscous dissipation, Dufour and Soret effects are: ρ cp


dT DKT = k∇2 T + ∇2 C + τ · L, dt cs

(4)

L = [grad V + (grad V)T ],

(5)


dC DKT = D∇2 C + ∇2 T . dt Tm

(6)

In above expressions V is the velocity, ρ the density of fluid, T the fluid temperature, C the mass concentration, cs the concentration susceptibility, k the thermal conductivity, Tm the mean temperature of fluid, cp the specific heat at constant pressure, D the coefficient of mass diffusivity, cs the concentration susceptibility and KT the thermal diffusion ratio. Cauchy stress tensor τ for tangent hyperbolic fluid is [24]: τ = −pI + S,

(7)

S = [µ∞ + (µ + µ∞ ) tanh(Γγ) ˙ m ]A1 ,

(8)

where S is an extra stress tensor, µ∞ the infinite shear rate viscosity, Γγ˙ the material constant, µ the zero shear rate viscosity, m the power law index and A1 the first RivilinEreckson tensor. Here γ˙ is defined as: r γ˙ =

1P P γ˙ γ˙ , 2 i j ij ji r 1 γ˙ = Π, 2 4

(9) (10)

where Π = tr[grad V + (grad V)T ]2 .

(11)

Assume µ∞ = 0 and Γγ˙ ≺ 1 one obtains the stress tensor for hyperbolic tangent fluid by using (8) as: S = µ[1 + m(Γγ˙ − 1)]A1 .

(12)

Eq. (12) in component form yields ∂u , ∂x ∂u ∂v = µ[1 + m(Γγ˙ − 1)]( + ), ∂y ∂x ∂v = 2µ[1 + m(Γγ˙ − 1)] , ∂y

Sxx = 2µ[1 + m(Γγ˙ − 1)] Sxy Syy with  γ˙ =

2

∂u ∂x

2

 +

∂u ∂v + ∂y ∂x

2

 +2

∂v ∂y

(13)

2 ! 12 .

(14)

in which u, v the components of V in x and y directions respectively, µ the dynamic fluid viscosity, S the extra stress tensor, Sxx , Sxy and Syy the components of extra stress tensor. The governing expressions are ∂u ∂v + = 0, ∂x ∂y  ρ

du dt

 ρ



dv dt

∂p ∂Sxx ∂Sxy + + − σB02 cos β(u cos β − v sin β) + ρg sin α ∂x ∂x ∂y ∗ +ρgα (T − T0 ) sin α + ρgβ ∗ (C − C0 ) sin α,

(15)

= −

 =−

∂p ∂Sxy ∂Syy + + + σB02 sin β(u cos β − v sin β) − ρg cos α, ∂y ∂x ∂y

" (   2 )  2 # 2 ∂u ∂v ∂u ∂v ρcp = µ(1 + m(Γγ˙ − 1)) 2 + + + ∂x ∂y ∂y ∂x  2    ∂ T ∂ 2T DKT ∂ 2 C ∂ 2 C 2 2 +k + + σB0 (u cos β − v sin β) + + , ∂x2 ∂y 2 cs ∂x2 ∂y 2  2    dC ∂ C ∂ 2C DKT ∂ 2 T ∂ 2T =D + + + . dt ∂x2 ∂y 2 Tm ∂x2 ∂y 2 

dT dt

(16) (17)



(18)

(19)

In the above equations p is the pressure in the laboratory frame, g the gravitational acceleration, α∗ , β ∗ the coefficient of thermal expansion, T0 and C0 the temperature and concentration at the walls. 5

The assumptions of long wavelength is broadly used in the investigation of peristaltic flows. Such calculation takes the peristaltic wave wavelength significantly enormous when associated with the half width channel. These considerations have special relevance for motion of chyme in smaller intestine. Boundary conditions are: u = 0, T = T0 , C = C0 at y = −h,

(20)

u = 0, T = T0 , C = C0 at y = h,

(21)

Now, the stream function ψ (x, y, t) and dimensionless variables are: u=

x∗ = h∗ = ∗ Sxx =

θ = Gn =

∂ψ ∂ψ , v=− , ∂y ∂x

(22)

x ∗ y ∗ ct ∗ u ∗ v a a2 p γc ˙ , y = , t = , u = , v = , δ = , p∗ = , γ∗ = , λ a λ c cδ λ cλµ a h µ µ σ ρca Γc c2 , ν = , µ(θ) = , M 2 = B02 a2 , Re = , We = , Fr = , a ρ µ0 µ µ a ga µC a a a µ C − C0 ∗ ∗ Sxx , Sxy = Sxy , Syy = Syy , Pr = p , Sc = ,φ= , µc µc µc k Dρ C0 T − T0 ρDKT T0 D m KT C0 ρga∗ T0 a2 , Sr = , Du = , Gr = , T0 µTm C0 µcp cs T0 µc ρgβ ∗ C0 a2 c2 , Br = Pr E, E = . (23) µc cp T0

Here δ is the wave number, Br the Brinkman number, Sr the Soret number, M the Hartman number, W e the Weissenberg number, Sc the Schmidt number, Gr the heat transfer Grashof number, Du the Dufour number, Re the Reynolds number, E the Eckert number, P r the Prandtl number, F r the Froud number, Gn the mass transfer Grashof number, θ the temperature and φ the concentration. The non-dimensional components of extra stress tensor yield ∂ 2ψ , ∂x∂y ∂ 2ψ ∂ 2ψ = µ(θ)[1 + m(W eγ˙ − 1)]( 2 − δ 2 2 ), ∂y ∂x ∂ 2ψ = 2µ(θ)[1 + m(W eγ˙ − 1)] , ∂x∂y

Sxx = 2µ(θ)[1 + m(W eγ˙ − 1)] Sxy Syy

(24)

with γ˙ = [2(δ

∂ 2ψ 2 ∂ 2ψ ∂ 2ψ ∂ 2ψ 2 1 ) + ( 2 − δ 2 2 )2 + 2(δ ) ]2 . ∂x∂y ∂y ∂x ∂x∂y 6

(25)

Approximations of long wavelength and small Reynolds number after dropping asterisks yield the following expressions   ∂p ∂ ∂ 2ψ ∂ 2ψ ∂ψ = µ(θ) 2 (1 + m(W e 2 − 1)) − M 2 cos2 β ∂x ∂y ∂y ∂y ∂y Re sin α +Grθ sin α + Gnφ sin α + , Fr ∂p =0, ∂y  2 2   ∂ 2θ ∂ ψ ∂ 2ψ ∂ 2φ + Brµ(θ) 1 + m(W e − 1) + Pr Du ∂y 2 ∂y 2 ∂y 2 ∂y 2  2 ∂ψ +BrM 2 cos2 β =0, ∂y ∂ 2φ ∂ 2θ + ScSr =0. ∂y 2 ∂y 2

(26) (27)

(28) (29)

Temperature dependent viscosity is considered in the form µ(θ) = e−γθ ,

(30)

with γ is considered as viscosity parameter. For γ = 0 the expression transforms for the constant viscosity case [23]. We consider µ(θ) = 1 − γθ.

(31)

From Eq. (25) p 6= p(y), simplification of Eqs. (24) and (25) and invoking definition of µ(θ) from (29) one gets  4   ∂ ψ ∂ 2θ ∂ 2ψ ∂θ ∂ 3 ψ ∂ 2ψ (1 − γθ) − γ 2 2 − 2γ 1 + m(W e 2 − 1) ∂y 4 ∂y ∂y ∂y ∂y 3 ∂y   3 3 2 ∂ ψ ∂ ψ ∂θ ∂ ψ ∂ 2ψ ∂ 4ψ +2mW e 3 (1 − γθ) 3 − γ + (1 − γθ)mW e ∂y ∂y ∂y ∂y 2 ∂y 2 ∂y 4 ∂ 2ψ ∂θ ∂φ −M 2 cos2 β 2 + Gr sin α + Gn sin α = 0. ∂y ∂y ∂y

2.1

(32)

Rate of volume flow

The volume flow rate in fixed frame (x, y) is Zh Q=

u(x, y)dy, −h

7

(33)

whereas the volume flow rate in wave frame is Zh q=

u∗ (x, y)dy,

(34)

−h

Q = q + 2ch.

(35)

The time mean flow is defined as 1 Q∗ = ∗ T

ZT ∗ Qdt,

(36)

0

where T is time period Q∗ = q + 2ca.

(37)

Now if we consider non-dimensional mean flow η in laboratory frame and F in wave frame such that η=

Q∗ q , F = . ca ca

η = F + 2, Zh ∂ψ F = dy. ∂y

(38) (39) (40)

−h

The dimensionless boundary conditions are F ∂ψ ψ=− , = 0, θ = 0, φ = 0 at y = −h, 2 ∂y ψ=

F ∂ψ , = 0, θ = 0, φ = 0 at y = h, 2 ∂y

(41) (42)

with h = 1 + d cos[2π(x − t)].

3

(43)

Results and discussion

In this section the graphical representations have been obtained through NDSolve in Mathematica. The objective of this section is to examine the graphs of longitudinal velocity u, temperature θ and mass concentration φ. The results are shown and discussed through plots.

8

3.0.1

Velocity profile

The plot of velocity is observed parabolic with larger value near the center of the channel. The parameters deriving from mixed convection are effective in heating or cooling of channel walls to dissolve energy more dynamically than forced convection in case of laminar flow. The impact of velocity is increasing with increasing Grashof number Gr as seen in Fig. 2. The growing values of power law index m decrease the velocity in the particular part of the channel (see Fig. 3). The impact of velocity is decreasing with increasing Gn due to enhance in fluid concentration as shown in Fig. 4. The plot of velocity profile reduces in downward half channel and increases for upward half channel with growing values of W e which controls the shear thinning influences of the fluid (see Fig. 5). The impression is same for the Williamson fluid described by Hayat et al. [22] in a symmetric channel. Resistive property of Hartman number M on u is depicted in Fig. 6. Rising values of Hartman number decreases the velocity near the center line due to apply the magnetic field in transverse direction on the flow which perceives damping effect. Similar result for the hyperbolic tangent nanofluid is studied by Hayat et al. [14] in a symmetric channel. An increase in velocity is captured for viscosity parameter (see Fig. 7).

Graph of velocity u for Grashof number Gr

9

Graph of velocity u for power index m Fig. 2: Graph of velocity u for the Grashof number Gr when γ = 0.3, α = π4 , x = 0.3, M = 2.5, W e = 0.01, m = 0.1, Du = 0.5, Sc = 0.5, Gn = 0.5, Sr = 0.5, β = π4 , η = 0.3, Pr = 1.5, d = 0.3, Br = 0.5 and t = 0.2. Fig. 3: Graph of velocity u for power index m when γ = 0.6, α = π3 , d = 0.3, M = 3, x = 0.3, W e = 0.3, Du = 0.8, Sc = 0.7, Sr = 0.7, β = π4 , η = 0.3, Pr = 1.7, Br = 1, Gr = 2, Gn = 0.5 and t = 0.1.

Graph of velocity u for Grashof number Gn

10

Graph of velocity u for Weissenberg number W e Fig. 4: Graph of velocity u for Grashof number Gn when d = 0.5, t = 0.2, γ = 0.3, α = π4 , M = 2, m = 0.1, Du = 0.8, Sc = 0.5, Sr = 0.5, Gr = 0.7, β = π4 , η = 0.6, Pr = 1.7, Br = 0.5 and x = 0.5. Fig. 5: Graph of velocity u for Weissenberg number W e when γ = 0.5, d = 0.3, t = 0.2, Gn = 0.5, α = π4 , M = 2, m = 0.3, Du = 0.8, Gr = 0.7, Sc = 0.5, Sr = 0.5, β = π4 , η = 0.6, Pr = 1.7, Br = 0.6 and x = 0.3.

Graph of velocity u for Hartman number M

11

Graph of velocity u for viscosity parameter γ Fig. 6: Graph of velocity u for Hartman number M when d = 0.5, t = 0.1, γ = 0.2, α =

π , 4

W e = 0.01, m = 0.1, Du = 0.7, Sc = 0.5, Sr = 0.6, β =

π , 4

η = 0.6, Pr = 1.5,

Gn = 0.5, Br = 0.6, Gn = 0.5, Gr = 0.5 and x = 0.3. Fig. 7: Graph of velocity u for viscosity parameter γ when d = 0.5, t = 0.1, α = M = 0, W e = 0.01, m = 0.2, Du = 0.7, Sc = 0.5, Sr = 0.6, β =

π , 4

π , 4

η = 0.6, Pr = 1.5,

Gn = 0.5, Br = 0.6, Gr = 0.5 and x = 0.3.

3.1

Temperature profile

The impressions of different parameters γ, α, β, Pr, W e, M , Sr and Br on the temperature profile are analyzed in this subsection. It is examined that the profile of temperature rises for Dufour and Soret number. It is due to decline in viscosity of fluid (see Figs. 8 and 9). Inclination of channel decreases the temperature as depicted in Fig. 10. In presence of magnetic field , temperature profile enhances for growing value of the Hartman number due to influence of Joule heating (see Fig. 11). This investigation is analogous for the analysis attained by Reddy and Reddy [11]. Reduction in temperature profile is noticed for an increase in inclined magnetic field (see Fig. 12). Temperature is decreasing for viscosity of fluid (see Fig. 13). Also greater impact is noticed for constant viscosity (γ = 0). Brinkman number appears due to influence of the viscous dissipation and the temperature rises. Hence an enhancement in temperature is noticed towards growing values of Br (see Fig. 14). Similar impression has been achieved by Hayat et al. [22]. Weissenberg number is specified as the ratio of viscous to the elastic forces for growing W e cause development in the viscosity then flow becomes much resistive and therefore temperature drops (see Fig. 15). 12

Graph of temperature θ for Dufour number Du

Graph of temperature θ for Soret number Sr Fig. 8: Graph of temperature profile θ for Dufour number Du when d = 0.3, t = 0.1, Gn = 0.7, γ = 0.2, α =

π , 3

M = 0.5, W e = 0.01, m = 0.1, Sc = 0.5, Sr = 0.5, β =

π , 4

η = 0.3, Pr = 1.5, Br = 0.7, Gr = 0.5 and x = 0.3. Fig. 9: Graph of temperature profile θ for Soret number Sr when d = 0.3, t = 0.1, Gn = 0.7, γ = 0.2, α = π3 , M = 2, W e = 0.01, m = 0.3, Du = 0.3, Sc = 0.5, β = π3 , η = 0.6, Pr = 1.7, Br = 1.5, Gr = 0.7 and x = 0.3.

13

Graph of temperature θ for channel inclination α

Graph of temperature θ for Hartman number M Fig. 10: Graph of temperature profile θ for channel inclination α when d = 0.3, t = 0.1, Gn = 0.5, γ = 0.3, M = 0.5, W e = 0.03, m = 0.3, Du = 0.6, Sc = 1.5, Sr = 0.5, β = π4 , η = 0.7, Pr = 1.5, Br = 1.5, Gr = 1.5 and x = 0.3. Fig. 11: Graph of temperature profile θ for Hartman number M when d = 0.3, t = 0.1, Gn = 0.6, γ = 0.2, α =

π , 3

W e = 0.01, m = 0.2, Du = 0.8, Sc = 0.5, Sr = 0.5, β =

η = 0.3, Pr = 1.5, Br = 0.7, Gr = 0.5 and x = 0.3.

14

π , 4

Graph of temperature θ for inclined MHD β

Graph of temperature θ for variable viscosity γ Fig. 12: Graph of temperature profile θ for inclined MHD β when d = 0.3, t = 0.2, Gn = 0.6, γ = 0.2, α = π3 , M = 0.5, W e = 0.01, m = 0.2, Du = 0.8, Sc = 0.5, Sr = 0.5, η = 0.3, Pr = 1.5, Br = 0.7, Gr = 0.7 and x = 0.3. Fig. 13: Graph of temperature profile θ for variable viscosity parameter γ when d = 0.5, t = 0.1, Gn = 0.7, α =

π , 4

M = 1, W e = 0.01, m = 0.1, Du = 0.8, Sc = 0.5, Sr = 0.5,

β = π4 , η = 0.6, Pr = 1.7, Br = 2, Gr = 1.5 and x = 0.3.

15

Graph of temperature θ for Brinkman number Br

Graph of temperature θ for Weissenberg number W e Fig. 14: Graph of temperature profile θ for Brinkman number Br when d = 0.3, t = 0.1, Gn = 0.7, γ = 0.2, α = π3 , M = 0.5, W e = 0.01, m = 0.1, Du = 0.8, Sc = 0.5, Sr = 0.5, β = π4 , η = 0.3, Pr = 1.5, Gr = 1.5 and x = 0.3. Fig. 15: Graph of temperature profile θ for Weissenberg number W e when d = 0.3, t = 0.1, Gn = 0.7, γ = 0.2, α = π3 , M = 0.5, m = 0.2, Du = 0.8, Sc = 0.5, Sr = 0.5, β = π4 , η = 0.6, Pr = 1.5, Br = 0.5, Gr = 0.5 and x = 0.3.

3.1.1

Concentration profile

Fig. 16 shows the increasing outcome of concentration profile for inclination of channel α. Fig. 17 depicts that rising values of Dufour number causes a decline in the fluid viscosity. That is why the concentration profile decreases. The concentration profile rises when Weissenberg number W e takes higher values (see Fig. 18). As Schmidt number Sc states ratio 16

of momentum diffusivity to the mass diffusivity. Obviously in Fig. 19 that an enhancement in Sc tends to decline the mass diffusion and thus concentration distribution declines. Fig. 20 depicts that concentration distribution also decreases for Soret number Sr due to decay in viscosity. The observations of Schmidt number and Soret number are similar to that achieved by Hayat et al. [22].

Graph of concentration φ for channel inclination α Fig. 16: Graph of concentration φ for channel inclination α when d = 0.5, t = 0.1, γ = 0.3, M = 3, Gn = 0.6, W e = 0.03, m = 0.3, Du = 0.7, Sc = 0.5, Sr = 0.7, β = η = 0.6, Pr = 1.5, Br = 1.5, Gr = 1.1 and x = 0.3.

Graph of concentration φ for Dufour number Du

17

π , 4

Graph of concentration φ for Weissenberg number W e Fig. 17: Graph of concentration φ for Dufour number Du when d = 0.5, t = 0.1, Gn = 0.7, γ = 0.2, α = π3 , M = 3, W e = 0.01, m = 0.1, Sc = 0.5, Sr = 0.5, β = π4 , η = 0.3, Pr = 2, Br = 2, Gr = 0.7 and x = 0.3. Fig. 18: Graph of concentration φ for Weissenberg number W e when d = 0.5, t = 0.1, Gn = 0.7, γ = 0.2, α = π3 , M = 2, m = 0.3, Du = 0.8, Sc = 1, Sr = 0.8, β = π4 , η = 0.6, Pr = 1.0, Br = 0.9, Gr = 0.6 and x = 0.3.

Graph of concentration φ for Schmidt number Sc

18

Graph of concentration φ for Soret number Sr Fig. 19: Graph of concentration φ for Schmidt number Sc when d = 0.5, t = 0.1, Gn = 0.7, α = π3 , γ = 0.3, M = 2, W e = 0.01, m = 0.1, Du = 0.5, Sr = 0.7, β = π4 , η = 0.6, Pr = 0.8, Br = 1, Gr = 0.7 and x = 0.3. Fig. 20: Graph of concentration φ for Soret number Sr when d = 0.5, t = 0.1, Gn = 0.7, α = π3 , γ = 0.4, M = 3, W e = 0.01, m = 0.3, Du = 0.8, Sc = 0.5, β = π4 , η = 0.3, Pr = 2.0, Br = 2.0, Gr = 0.7 and x = 0.2.

3.2

Heat transfer coefficient

The present subsection has been made to investigate the influence of several parameters on heat transfer coefficient Z. Heat transfer coefficient is written as follows: Z = hx θy (h).

(44)

Heat transfer coefficient Z depicts oscillatory behavior for pertinent parameters. It is because of the contraction and expansion of peristaltic waves along the walls of channel. Decline in the heat transfer Z is noticed for growing values of inclined MHD β (see Fig. 20). The Dufour number Du enhances the absolute value of heat transfer coefficient Z (see Fig. 21). It is observed that mass transfer is very strong along right axis. Enlargement in rate of heat transfer is examined for Soret number Sr, Brinkman number Br and Prandtl number P r (see Figs. 22, 23 and 24). Similar observations have been reported by Hayat et al. [14] in their analysis of the hyperbolic tangent nanofluid in a symmetric channel.

19

Graph of heat transfer coefficient Z for inclined MHD β

Graph of heat transfer coefficient Z for Dufour number Du Fig. 21: Graph of heat transfer coefficient Z for inclined MHD β when t = 0.1, Gn = 0.5, γ = 0.5, α = π4 , M = 3, W e = 0.01, m = 0.1, Du = 0.5, Sr = 0.5, η = 0.6, Pr = 2, Gr = 0.6, d = 0.5 and Br = 2. Fig. 22: Graph of heat transfer coefficient Z for Dufour number Du when γ = 0.7, Gn = 0.5, t = 0.1, α = π3 , M = 3, W e = 0.01, m = 0.1, β = π3 , .Sr = 0.5, η = 0.6, Pr = 2, Br = 2, Gr = 0.5 and d = 0.5.

20

Graph of heat transfer coefficient Z for Soret number Sr

Graph of heat transfer coefficient Z for Brinkman number Br

Graph of heat transfer coefficient Z for Prandtl number P r Fig. 23: Graph of heat transfer coefficient Z for Soret number Sr when t = 0.1, Gn = 0.5, d = 0.5, α = π4 , M = 3, W e = 0.01, m = 0.1, β = π3 , Du = 0.5, η = 0.6, Pr = 2, Br = 2, Gr = 0.7 and γ = 0.6. 21

Fig. 24: Graph of heat transfer coefficient Z for Brinkman number Br when t = 0.1, Gn = 0.5, d = 0.6, α = π3 , M = 2, W e = 0.03, m = 0.1, β = π3 , Du = 0.5, η = 0.6, Pr = 2, Sr = 0.5, Gr = 0.6 and γ = 0.6. Fig. 25: Graph of heat transfer coefficient Z for Prandtl number P r when t = 0.1, Gn = 0.5, γ = 0.7, α = π4 , M = 2, W e = 0.01, m = 0.1, β = π3 , Du = 0.5, η = 0.6, Br = 2, Sr = 0.5, Gr = 0.6 and d = 0.5.

4

Concluding remarks

Here we examine the peristaltic motion of tangent hyperbolic fluid in an inclined channel. The magnetohydrodynamics and gravitational effects are presented. In addition Dufour and Soret effects are accounted. The main observations are given below: • Velocity profile declines for increasing Hartman number and mass transfer Grashof number Gn. • Impact of velocity profile increases for Brinkman number and heat transfer Grashof number Gr. • Temperature enhances for growing values of Hartman number and Dufour and Soret numbers. • Decline in temperature is noticed for an increase in inclined MHD. • Concentration profile rises when Weissenberg number is increased. • Concentration profile declines for larger value of Schmidt number and Soret number. • Heat transfer coefficient increases for Dufour and Soret numbers.

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