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Magnetohydrodynamic peristalsis of variable viscosity Jeffrey liquid with heat and mass transfer
Accepted Manuscript Magnetohydrodynamic peristalsis of variable viscosity Jeffrey liquid with heat and mass transfer S. Farooq, M. Awais, Moniza Naseem, T. Hayat, B. Ahmad PII:
S1738-5733(17)30354-6
DOI:
10.1016/j.net.2017.07.013
Reference:
NET 395
To appear in:
Nuclear Engineering and Technology
Received Date: 12 June 2017 Revised Date:
5 July 2017
Accepted Date: 9 July 2017
Please cite this article as: S. Farooq, M. Awais, M. Naseem, T. Hayat, B. Ahmad, Magnetohydrodynamic peristalsis of variable viscosity Jeffrey liquid with heat and mass transfer, Nuclear Engineering and Technology (2017), doi: 10.1016/j.net.2017.07.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Magnetohydrodynamic peristalsis of variable viscosity Jeffrey liquid with heat and mass transfer
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S. Farooq a, 1, M. Awais b , Moniza Naseem b , T. Hayat a , c and B. Ahmad c a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan b Department of Mathematics, COMSATS Institute of Information Technology, Attock 43600, Pakistan c Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia. Abstract: Herein the mathematical aspects of Dufour and Soret phenomena on the peristalsis of magnetohydrodynamic (MHD) Jeffrey liquid in a symmetric channel are presented. Fluid viscosity is taken variable. Lubrication approach has been followed. Results for the velocity, temperature and concentration are constructed and explored for the emerging parameters entering into the present problem. The plotted quantities lead to comparative study between the constant and variable viscosities fluids. Graphical results indicates that for non-Newtonian materials pressure gradient is maximum, whereas pressure gradient is slow down for variable viscosity. Also both velocity and temperature in the case of variable viscosity are at maximum when compared with results for constant viscosity. Keywords: Jeffrey Fluid; Soret and Dufour effects; Variable viscosity; Magnetohydrodynamic;
Introduction
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Peristaltic transport of fluid frequently occurs in the physiological and industrial applications. In particular, peristalsis play a critical role in urine passage from kidney to bladder, in food bolus transport in the gastrointestinal tract, in lymph transport in lymphatic vessels, spermatozoa in the duct afferents, in finger and roller pumps, in sanitary fluid transport, in corrosive fluids transport, in locomotion of worms and in several other areas. Latham [1] and Shapiro et al. [2] initially addressed the peristalsis of viscous material in a channel. Afterwards the peristaltic transport of viscous and non-Newtonian fluids in flow configurations under different aspects and assumptions was studied extensively. Information on the topic is quite reasonable and researchers mention a few recent representative attempts and several useful references in their investigations [3-15]. It is also found that heat and mass transfer effects have significant role in peristalsis. Mention has been made of oxygenation, hemodialysis, cancer therapy and blood flow processes. Keeping these ideas in mind, the peristaltic transport of fluid has been examined in the presence of heat and mass transfer ( see [16-25] and some attempts therein). It is found from the previous literature that much past attention to peristalsis has been accorded to constant viscosity fluid. Such information can be further narrowed down when viscoelastic fluids of variable viscosity are taken into account. Further the peristaltic motion of viscoelastic fluids with variable viscosity and heat and mass transfer has scarcely been analyzed. To our knowledge, the peristaltic transport of variable viscosity Jeffrey fluid in the presence of Soret and Dufour effects has not been addressed so far. The purpose here is to make an attempt at such an analysis of 1
Tel.: + 92 51 90642172. e-mail address: [email protected] 1
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a magnetohydrodynamics (MHD) fluid. The relevant equation have been modeled by invoking conservation laws of mass, linear momentum, energy and concentration. Large wavelength and small Reynolds number analysis is carried out. Resulting equations are solved for the flow quantities of interest including stream function, temperature and concentration. Plots are presented and discussed. A comparative study between the cases of constant and variable viscosity fluids is executed.
Problem Statement
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Here we consider the flow of an incompressible magnetohydrodynamic Jeffrey liquid in a channel with width 2a . We take X and Y axes along and perpendicular to the channel walls respectively (see Fig. 1). The fluid is electrically conducting through an applied magnetic field B0 in a direction transverse to the flow.
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Fig. 1: Physical diagram.
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Sinusoidal wave propagation along channel walls with constant wave speed c are represented by: 2π (1) H ( X , t ) = a + b cos ( X − ct ) , λ in which c denotes the wave speed, b the wave amplitude, λ the wavelength and t the time. The velocity vector V and extra stress tensor S are given by (2) V = (U ( X , Y , t ),V ( X , Y , t ), 0), µ (Y ) d (3) S= ( A1 + λ2 A1 ). 1 + λ1 dt In Eq. (3) µ (Y ) represent the variable dynamic viscosity of the liquid, λ1 the ratio of relaxation to retardation times and λ2 the retardation time. Note that Eq. (3) recovers the viscous fluid when
λ1 = λ2 = 0 . Moreover [11,12,16]:
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A1 = (grad V ) + (grad V )∗ ,
(4) (5) (6)
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d ∂ A1 = ( A1 ) + (V.∇) A1. dt ∂t ∂U ∂V + = 0, ∂X ∂Y ∂ ∂ ∂P ∂S XY ∂S XX ∂ +V + + − σ B02U , ρ +U U = − ∂X ∂Y ∂Y ∂X ∂X ∂t ∂ ∂ ∂P ∂S XY ∂SYY ∂ +V + + − σ B02V , ρ +U V = − ∂X ∂Y ∂Y ∂X ∂Y ∂t
(7) (8)
∂ 2T ∂ 2T ∂ ∂ ∂V ∂ +U +V T = κ + ( SYY − S XY ) 2+ 2 ∂X ∂Y ∂Y ∂Y ∂t ∂X ρ DKT 2 2 2 + σ B0 (U + V ) + Cs
∂ 2C ∂ 2C 2+ , ∂Y 2 ∂X
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∂V ∂U + S XY + ∂X ∂Y
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ρC p
∂ 2C ∂ 2C DKT ∂ 2T ∂ 2T ∂ ∂ ∂ (10) +V 2 + 2 , +U C = D 2 + 2 + ∂X ∂Y ∂ X ∂ Y T ∂ X ∂ Y ∂t m in which P denotes the pressure, σ the electric conductivity, B0 the magnetic field strength, ρ the fluid density, C p the specific heat per unit, κ the thermal conductivity, D the mass diffusivity, KT the thermal diffusion ratio, Cs the concentration susceptibility, Tm the mean temperature, ( S XX , S XY , SYY ) the stress components T and C are the temperature and
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concentration fields. If (U , V ) and ( u , v ) are the velocity components in the laboratory
( X ,Y )
and wave ( x , y ) frames respectively, the transformations between laboratory and wave frames can be put into the forms x = X − ct , y = Y , u ( x , y ) = U ( X , Y , t ) − c, v ( x , y ) = V ( X , Y , t ), (11)
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T ( x , y ) = T ( X , Y , t ), C ( x , y ) = C ( X , Y , t ), p ( x , y ) = P ( X , Y , t ). Using above transformations and introducing the following variables x y u v a H a2 p µ( y) x= , y= , u= , v= , δ = , h= , p= , µ ( y) = , a c cδ a cµ0 λ λ λ µ0 1/ 2
σ DC0 KT ρ DKT T0 µ ct Du = , M = B0 a, Sr = , Sc = 0 , t = , Cs µ0T0 µ0TmC0 ρD λ µ0 µcp T − T0 C − C0 a c2 Θ= , φ= , S= S, Pr = , Ec = , Br = Pr Ec. T0 C0 k T0 c p µ0 c
(12)
∂ψ ∂ψ , v=− , (13) ∂y ∂x Consideration of large wavelength and low Reynolds assumptions are adequate for chyme movement through the small intestine [32]. Here wave speed is ( c = 2 ) cm/min, ( a = 1.25 ) cm u=
3
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and ( λ = 8.01) cm. Mathematically, it is justified that the half width of the intestine is very small
dp ∂ µ ( y ) ∂ 2ψ = dx ∂y 1 + λ1 ∂y 2
2 ∂ψ + 1 , −M ∂y
2 ∂ 2 µ ( y ) ∂ 2ψ 2 ∂ ψ − M = 0, ∂y 2 1 + λ1 ∂y 2 ∂y 2 2
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in comparison to the wavelength i.e. ( a / λ = 0.156 ) . Long wavelength and low Reynolds number limitations are highlighted quite impressively in several studies [33-35]. Therefore adopting low Reynolds number and large wavelength analysis, we see that Equation (6) is satisfied identically and Eqs. (7 − 10) yield (14)
(15)
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∂ 2 Θ Br µ ( y ) ∂ 2ψ ∂ 2φ (16) + Du 0= 2 + Pr , ∂y 1 + λ1 ∂y 2 ∂y 2 1 ∂ 2φ ∂ 2Θ (17) 0= , + Sr Sc ∂y 2 ∂y 2 where ψ is the stream function, M is the Hartman number, Pr is the Prandtl number, Ec is the Eckert number, p is the pressure, µ ( y ) is the variable viscosity, µ0 is the absolute viscosity, Du is the Dufour parameter, Sr is the Soret parameter, Sc is the Schmidt number, δ is the wave number, C0 and T0 are the concentration and temperature at the boundary, Θ is the dimensionless temperature, φ is the concentration and Br is the Brinkman number. The boundary conditions are ∂ 2ψ ∂Θ ∂φ (18) = 0, = 0, at y = 0, ψ = 0, 2 = 0, ∂y ∂y ∂y ∂ψ (19) = −1, Θ = 0, φ = 0, at y = h, ψ = F, ∂y h ∂ψ (20) h( x) = 1 + d cos ( 2π x ) , F = ∫ dy, 0 ∂y where h( x) is the dimensionless wall shape and F is the dimensionless flow rate in the wave frame. Pressure rise per wavelength ∆pλ is 1 dp (21) ∆pλ = ∫ dx. 0 dx The dimensionless expression of space dependent viscosity is [7]: (22) µ ( y ) = e −α y = 1 − α y, α 1, where "α " is the viscosity parameter. Results for constant viscosity are obtained for α = 0 . Moreover, the results for viscous fluid subject to long wavelength and low Reynolds number can be deduced for λ1 = 0 .
Solution methodology
The resulting differential system consists of non-linear differential equations (14 − 17) , subject to 4
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the boundary conditions (18) and (19) . In order to find the perturbed solution, we expand the quantities as follows: ψ = ψ 0 + αψ 1 + ...,
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dp dp dp0 = + α 1 ..., dx dx dx F = F0 + α F1 + ..., Θ = Θ0 + αΘ1 + ...,
φ = φ0 + αφ1 + ....
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Inserting the above equations into Eqs. (14 − 17) and then collecting the terms of like powers of α we have:
Zeroth order system
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∂ψ dp0 ∂ 3ψ 0 = − M 2 0 + 1 , 3 dx ∂y ∂y
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∂ 2ψ 0 ∂ 2Θ0 ∂ 2φ0 Br Du + + Pr = 0, 2 ∂y 2 ∂y 2 ∂y
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∂ 2φ0 ∂ 2Θ0 SrSc + = 0, ∂y 2 ∂y 2
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∂ 2ψ 0 ∂Θ0 ∂φ = 0, = 0, 0 = 0 at y = 0, 2 ∂y ∂y ∂y ∂ψ 0 = −1, Θ0 = 0, φ0 = 0 at y = h, ψ 0 = − F0 , ∂y The non-dimensional pressure rise per wavelength ( ∆Pλ0 ) at this order is
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ψ 0 = 0,
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First order system
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∂ 3ψ 0 ∂ 2ψ 0 dp1 1 ∂ 3ψ 1 ∂ψ 1 = − y − −M2 , 3 3 2 dx 1 + λ1 ∂y ∂y ∂y ∂y
∂ 2ψ 0 ∂ 2Θ1 Br ∂ 2ψ 1 ∂ 2ψ 1 + 2 − y 2 ∂y 2 1 + λ1 ∂y 2 ∂y 2 ∂y ∂ 2φ1 ∂ 2Θ1 + SrSc = 0, ∂y 2 ∂y 2
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∂ 2φ + Pr Du 21 = 0, ∂y
∂ 2ψ 1 ∂Θ1 ∂φ = 0, = 0, 1 = 0, at 2 ∂y ∂y ∂y ∂ψ 1 ψ 1 = − F1 , = −1, Θ1 = 0, φ1 = 0, ∂y
ψ 1 = 0,
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y = 0, at
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∆pλ1 = ∫ 0
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The perturbation expressions of φ , Θ, dp / dx and ∆pλ up to O(λ1 ) are denoted by φ (1) ,
φ (1) = φ0 + αφ1. Θ (1) = Θ 0 + αΘ1. dp (1) dp0 dp = +α 1 . dx dx dx (1) ∆pλ = ∆pλ0 + α∆pλ1 ,
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Θ(1) , ∆Pλ(1) and dp (1) / dx. These are given by
(36)
Results and discussion
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where the quantities for zeroth and first solutions are computed by solving zeroth and first order systems.
The main purpose of this section is to analyze the effects of different parameters on the longitudinal pressure gradient, pressure rise per wavelength, temperature, concentration and heat transfer coefficient. Here 2-D graphs are plotted for the analysis of longitudinal pressure gradient dp (1) dx
and pressure rise per wavelength ∆pλ(1) . The effects of different Deborah number λ1 on
longitudinal pressure gradient dpdx with ( d = 0.3 , M = 1.0 , θ = 0.3 , α = 0.5 ) are plotted in Fig. 2. It is seen that the magnitude of the longitudinal pressure gradient increases for the Jeffrey fluid ( λ1 ≠ 0 ) when compared with the viscous fluid ( λ1 = 0 ). Fig 3 depicts the variations of the longitudinal pressure gradient for different values of the Hartman number M when d = 0.3 ,
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θ = 0.3 , λ1 = 0.5 , α = 0.5 . It is observed that
dp (1) dx
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improves along with Hartman number M . It is also seen that the Hartman number in the Jeffrey fluid has a large impact on the longitudinal (1) pressure gradient. Fig. 4 shows the variations of dpdx for different values of the viscosity parameter when d = 0.3 , M = 1.0 , λ1 = 0.5 , θ = 0.3 . One can easily find that for constant viscosity ( α = 0 ) the pressure gradient is greater in magnitude compared with the variable viscosity case ( α ≠ 0 ). Figs. (5-7) illustrate the effects of different sundry parameters on the pressure rise per wavelength ∆pλ(1) versus mean flow rate θ . In Figs. 5 and 6 it can be seen that the pressure rise per wavelength ∆pλ(1) enhances via larger λ1 for d = 0.3 , M = 1.0 , α = 0.5 and M with d = 0.3 , λ1 = 0.5 , α = 0.5 . Fig. 7 shows that ∆pλ(1) decreases for different values of viscosity parameter α with d = 0.3 , λ1 = 0.5 , M = 1.0 . Effects of different sundry parameters on velocity, temperature, concentration and heat transfer coefficient are analyzed in a more feasible manner via 3-D graphs. The impacts of physical variables in the 3-D graphs are more visible. One main theme is to analyze the effects of variable viscosity ( α ≠ 0 ) in comparison to the constant viscosity α = 0 . In order to attain the desired results, 3-D plots are presented in such a way that all ( the " a " parts of the Figs. ) are for constant viscosity (i.e. for α = 0 ) and all ( the " b " parts of the Figs. ) are for variable viscosity (i.e. for α = 0.5 ). Figs. ( 8 and 9 ) show that the velocity has a maximum magnitude at the center of the channel for α = 0 . For ( α = 0.5 ) the 6
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velocity improves slowly in a smooth way when d = 0.3 , θ = 0.3 , M = 1.0 , x = 0 and d = 0.3 , θ = 0.3 , M = 1.0 , x = 0 respectively. Figs. ( 10-12 ) show the effect of different parameters of interest on the temperature profile Θ . These figures show that the temperature rises slowly in the case of constant viscosity when compared with the variable viscosity. It is noted that the temperature increases with an increase in M . It is because of the viscous dissipation and Ohmic heating that temperature increases with increasing M (see Fig. 10) with d = 0.3 , Br = 2.0 , Du = 0.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 . Fig. 11 shows the similar behavior of Du with temperature for M when d = 0.3 , Br = 2.0 , M = 2.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 . Fig. 12 illustrates the enhancing behavior of Br on the temperature both for the variable and the constant viscosity cases respectively. Here d = 0.3 , Du = 0.5 , M = 2.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 . Figs. ( 13-15 ) are provided to allow an analysis of the effects of variations of the embedded sundry parameters on the concentration profile . Decreasing behavior is observed in Fig. 13 for concentration when M increases with d = 0.3 , Br = 2.0 , Du = 0.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 . Fig. 14 shows that the concentration improves via Du when d = 0.3 ,
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Br = 2.5 , M = 2.0 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 . The increasing behavior of the concentration profile for larger Sr when d = 0.3 , Br = 2.0 , Du = 0.5 , Sc = 0.5 , M = 1.0 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 can be observed in Fig. 15. One can easily observe that for the constant viscosity case the concentration profile increases slowly in comparison to the case of variable viscosity fluid. Figs. ( 16-18 ) illustrate the effects of various parameters of interest on the heat transfer coefficient Z . All these figures show that in view of peristaltic motion due to a travelling wave Z has an oscillatory behavior because of peristaltic motion of the fluid (i.e. the phenomena of contraction and relaxation of mussels occurs). It is also observed that there is no variation in Z for amplitude ratio d between 0.0 and 0.5 . However Z increases with increasing M with d = 0.5 , Br = 2.5 , Du = 0.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , y = 1.4 , λ1 = 1.0 , θ = 0.3 in Fig.16. Fig. 17 shows that the heat transfer coefficient also increases for increasing values of Du with d = 0.5 , Br = 2.5 , M = 2.0 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , y = 1.4 , λ1 = 1.0 , θ = 0.3 . Heat transfer coefficient increases for different values of Br with d = 0.5 , M = 2.0 , Du = 0.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , y = 1.4 , λ1 = 1.0 , θ = 0.3 (see Fig. 18). In the case of a variable viscosity fluid (i.e. α = 0.5 ) the absolute value of Z is greater than it is in the case of constant viscosity of the fluid (i.e. α = 0.0 ).
Concluding remarks MHD peristaltic movement of variable viscosity Jeffrey fluid with heat and mass transfer is discussed here. The important results of the present analysis are listed below. • Pressure gradient is found to be at a maximum for constant viscosity when compared with case of variable viscosity. • Pressure gradient for viscous fluid (i.e. λ1 = 0 ) is minimum when compared with Jeffrey fluid. • Pressure rise per wavelength is an increasing function of Hartman number. • It is evident that velocity has small resistance in the case of variable viscosity. • Temperature is minimum for variable viscosity when compared with constant viscosity. • Opposite results for concentration and temperature are obtained for both variable and constant 7
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viscosity cases. Heat transfer coefficient behavior is found to be oscillatory. 0.0
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Fig. 4. Effects of ☺on pressure gradient
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M M M M
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Fig. 11. Effects of Du on temperature profile .
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Fig. 13. Effects of M on temperature profile .
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Fig. 15. Effects of Sr on concentration .
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Fig. 16. Effects of M on heat transfer coefficient Z.
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Fig. 17. Effects of Du on heat transfer coefficient Z.
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Fig. 18. Effects of Br on heat transfer coefficient Z.
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[23] A.M. Abd-Alla and S.M. Abo-Dahab, Rotation effect on peristaltic transport of a Jeffrey fluid in an asymmetric channel with gravity field, Alex. Eng. J. 55 (2016) 1725-1735. [24] D. Tripathi, A. Yadav and O.A. Bég, Electro-kinetically driven peristaltic transport of viscoelastic physiological fluids through a finite length capillary: Mathematical modeling, Math. Biosci. 283 (2017) 155-168. [25] K. Ramesh, Influence of heat and mass transfer on peristaltic flow of a couple stress fluid through porous medium in the presence of inclined magnetic field in an inclined asymmetric channel, J. Mol. Liq. 219 (2016) 256-271. [26] S.E. Ghasemi, Thermophoresis and Brownian motion effects on peristaltic nanofluid flow for drug delivery applications, J. Mol. Liq. 238 (2017) 115-121. [27] J.K. Grabski, J.A. Kołodziej, M. Mierzwiczak, Application of meshless procedure for the peristaltic flow analysis, Eng. Anal. Bound. Elem. 63 (2016) 125-133. [28] M.M. Bhatti, A. Zeeshan and R. Ellahi, Simultaneous effects of coagulation and variable magnetic field on peristaltically induced motion of Jeffrey nanofluid containing gyrotactic microorganism, Micro. Research. 110 (2017) 32-42. [29] M.M. Bhatti, A. Zeeshan, R. Ellahi and N. Ijaz, Heat and mass transfer of two-phase flow with Electric double layer effects induced due to peristaltic propulsion in the presence of transverse magnetic field, J. Mol. Liq. 230 (2017) 237-246. [30] M.M. Bhatti, A. Zeeshan and R. Ellahi, Endoscope analysis on peristaltic blood flow of Sisko fluid with Titanium magneto-nanoparticles, Comp. Bio. Medicine. 78 (2016) 29-41. [31] R. Ellahi, M.M. Bhatti and I. Pop, Effects of hall and ion slip on MHD peristaltic flow of Jeffrey fluid in a non-uniform rectangular duct, Int. J. Numer. Methods Heat Fluid Flow. 26 (2016) 1802-1820. [32] L.M. Srivastava and V.P. Srivastava, Peristaltic transport of a power law fluid: applications to the ductus efferentes of the reproductive tract, Rheologica Acta. 27 (1988) 428-433. [33] H.S. Lew, Y.C. Fung and C.B. Lowenstein, Peristaltic carrying and mixing of chime, J Biomech. 4 (1971) 297-315. [34] T. Hayat, S. Farooq, B.Ahmad and A.Alsaedi, Homogeneous-heterogeneous reactions and heat source/sink effects in MHD peristaltic flow of micropolar fluid with Newtonian heating in a curved channel, J. Mol. Liq. 223 (2016) 469-488. [35] T. Hayat, A. Bibi, H. Yasmin and B. Ahmad, Simultaneous effects of Hall current and homogeneous/heterogeneous reactions on peristalsis, J. Taiwan Inst. Chem. Eng. 58 (2016) 28-38.
17
S1738-5733(17)30354-6
DOI:
10.1016/j.net.2017.07.013
Reference:
NET 395
To appear in:
Nuclear Engineering and Technology
Received Date: 12 June 2017 Revised Date:
5 July 2017
Accepted Date: 9 July 2017
Please cite this article as: S. Farooq, M. Awais, M. Naseem, T. Hayat, B. Ahmad, Magnetohydrodynamic peristalsis of variable viscosity Jeffrey liquid with heat and mass transfer, Nuclear Engineering and Technology (2017), doi: 10.1016/j.net.2017.07.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Magnetohydrodynamic peristalsis of variable viscosity Jeffrey liquid with heat and mass transfer
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S. Farooq a, 1, M. Awais b , Moniza Naseem b , T. Hayat a , c and B. Ahmad c a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan b Department of Mathematics, COMSATS Institute of Information Technology, Attock 43600, Pakistan c Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia. Abstract: Herein the mathematical aspects of Dufour and Soret phenomena on the peristalsis of magnetohydrodynamic (MHD) Jeffrey liquid in a symmetric channel are presented. Fluid viscosity is taken variable. Lubrication approach has been followed. Results for the velocity, temperature and concentration are constructed and explored for the emerging parameters entering into the present problem. The plotted quantities lead to comparative study between the constant and variable viscosities fluids. Graphical results indicates that for non-Newtonian materials pressure gradient is maximum, whereas pressure gradient is slow down for variable viscosity. Also both velocity and temperature in the case of variable viscosity are at maximum when compared with results for constant viscosity. Keywords: Jeffrey Fluid; Soret and Dufour effects; Variable viscosity; Magnetohydrodynamic;
Introduction
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Peristaltic transport of fluid frequently occurs in the physiological and industrial applications. In particular, peristalsis play a critical role in urine passage from kidney to bladder, in food bolus transport in the gastrointestinal tract, in lymph transport in lymphatic vessels, spermatozoa in the duct afferents, in finger and roller pumps, in sanitary fluid transport, in corrosive fluids transport, in locomotion of worms and in several other areas. Latham [1] and Shapiro et al. [2] initially addressed the peristalsis of viscous material in a channel. Afterwards the peristaltic transport of viscous and non-Newtonian fluids in flow configurations under different aspects and assumptions was studied extensively. Information on the topic is quite reasonable and researchers mention a few recent representative attempts and several useful references in their investigations [3-15]. It is also found that heat and mass transfer effects have significant role in peristalsis. Mention has been made of oxygenation, hemodialysis, cancer therapy and blood flow processes. Keeping these ideas in mind, the peristaltic transport of fluid has been examined in the presence of heat and mass transfer ( see [16-25] and some attempts therein). It is found from the previous literature that much past attention to peristalsis has been accorded to constant viscosity fluid. Such information can be further narrowed down when viscoelastic fluids of variable viscosity are taken into account. Further the peristaltic motion of viscoelastic fluids with variable viscosity and heat and mass transfer has scarcely been analyzed. To our knowledge, the peristaltic transport of variable viscosity Jeffrey fluid in the presence of Soret and Dufour effects has not been addressed so far. The purpose here is to make an attempt at such an analysis of 1
Tel.: + 92 51 90642172. e-mail address: [email protected] 1
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a magnetohydrodynamics (MHD) fluid. The relevant equation have been modeled by invoking conservation laws of mass, linear momentum, energy and concentration. Large wavelength and small Reynolds number analysis is carried out. Resulting equations are solved for the flow quantities of interest including stream function, temperature and concentration. Plots are presented and discussed. A comparative study between the cases of constant and variable viscosity fluids is executed.
Problem Statement
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Here we consider the flow of an incompressible magnetohydrodynamic Jeffrey liquid in a channel with width 2a . We take X and Y axes along and perpendicular to the channel walls respectively (see Fig. 1). The fluid is electrically conducting through an applied magnetic field B0 in a direction transverse to the flow.
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Fig. 1: Physical diagram.
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Sinusoidal wave propagation along channel walls with constant wave speed c are represented by: 2π (1) H ( X , t ) = a + b cos ( X − ct ) , λ in which c denotes the wave speed, b the wave amplitude, λ the wavelength and t the time. The velocity vector V and extra stress tensor S are given by (2) V = (U ( X , Y , t ),V ( X , Y , t ), 0), µ (Y ) d (3) S= ( A1 + λ2 A1 ). 1 + λ1 dt In Eq. (3) µ (Y ) represent the variable dynamic viscosity of the liquid, λ1 the ratio of relaxation to retardation times and λ2 the retardation time. Note that Eq. (3) recovers the viscous fluid when
λ1 = λ2 = 0 . Moreover [11,12,16]:
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A1 = (grad V ) + (grad V )∗ ,
(4) (5) (6)
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d ∂ A1 = ( A1 ) + (V.∇) A1. dt ∂t ∂U ∂V + = 0, ∂X ∂Y ∂ ∂ ∂P ∂S XY ∂S XX ∂ +V + + − σ B02U , ρ +U U = − ∂X ∂Y ∂Y ∂X ∂X ∂t ∂ ∂ ∂P ∂S XY ∂SYY ∂ +V + + − σ B02V , ρ +U V = − ∂X ∂Y ∂Y ∂X ∂Y ∂t
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∂ 2T ∂ 2T ∂ ∂ ∂V ∂ +U +V T = κ + ( SYY − S XY ) 2+ 2 ∂X ∂Y ∂Y ∂Y ∂t ∂X ρ DKT 2 2 2 + σ B0 (U + V ) + Cs
∂ 2C ∂ 2C 2+ , ∂Y 2 ∂X
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∂V ∂U + S XY + ∂X ∂Y
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∂ 2C ∂ 2C DKT ∂ 2T ∂ 2T ∂ ∂ ∂ (10) +V 2 + 2 , +U C = D 2 + 2 + ∂X ∂Y ∂ X ∂ Y T ∂ X ∂ Y ∂t m in which P denotes the pressure, σ the electric conductivity, B0 the magnetic field strength, ρ the fluid density, C p the specific heat per unit, κ the thermal conductivity, D the mass diffusivity, KT the thermal diffusion ratio, Cs the concentration susceptibility, Tm the mean temperature, ( S XX , S XY , SYY ) the stress components T and C are the temperature and
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concentration fields. If (U , V ) and ( u , v ) are the velocity components in the laboratory
( X ,Y )
and wave ( x , y ) frames respectively, the transformations between laboratory and wave frames can be put into the forms x = X − ct , y = Y , u ( x , y ) = U ( X , Y , t ) − c, v ( x , y ) = V ( X , Y , t ), (11)
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T ( x , y ) = T ( X , Y , t ), C ( x , y ) = C ( X , Y , t ), p ( x , y ) = P ( X , Y , t ). Using above transformations and introducing the following variables x y u v a H a2 p µ( y) x= , y= , u= , v= , δ = , h= , p= , µ ( y) = , a c cδ a cµ0 λ λ λ µ0 1/ 2
σ DC0 KT ρ DKT T0 µ ct Du = , M = B0 a, Sr = , Sc = 0 , t = , Cs µ0T0 µ0TmC0 ρD λ µ0 µcp T − T0 C − C0 a c2 Θ= , φ= , S= S, Pr = , Ec = , Br = Pr Ec. T0 C0 k T0 c p µ0 c
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∂ψ ∂ψ , v=− , (13) ∂y ∂x Consideration of large wavelength and low Reynolds assumptions are adequate for chyme movement through the small intestine [32]. Here wave speed is ( c = 2 ) cm/min, ( a = 1.25 ) cm u=
3
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and ( λ = 8.01) cm. Mathematically, it is justified that the half width of the intestine is very small
dp ∂ µ ( y ) ∂ 2ψ = dx ∂y 1 + λ1 ∂y 2
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2 ∂ 2 µ ( y ) ∂ 2ψ 2 ∂ ψ − M = 0, ∂y 2 1 + λ1 ∂y 2 ∂y 2 2
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in comparison to the wavelength i.e. ( a / λ = 0.156 ) . Long wavelength and low Reynolds number limitations are highlighted quite impressively in several studies [33-35]. Therefore adopting low Reynolds number and large wavelength analysis, we see that Equation (6) is satisfied identically and Eqs. (7 − 10) yield (14)
(15)
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∂ 2 Θ Br µ ( y ) ∂ 2ψ ∂ 2φ (16) + Du 0= 2 + Pr , ∂y 1 + λ1 ∂y 2 ∂y 2 1 ∂ 2φ ∂ 2Θ (17) 0= , + Sr Sc ∂y 2 ∂y 2 where ψ is the stream function, M is the Hartman number, Pr is the Prandtl number, Ec is the Eckert number, p is the pressure, µ ( y ) is the variable viscosity, µ0 is the absolute viscosity, Du is the Dufour parameter, Sr is the Soret parameter, Sc is the Schmidt number, δ is the wave number, C0 and T0 are the concentration and temperature at the boundary, Θ is the dimensionless temperature, φ is the concentration and Br is the Brinkman number. The boundary conditions are ∂ 2ψ ∂Θ ∂φ (18) = 0, = 0, at y = 0, ψ = 0, 2 = 0, ∂y ∂y ∂y ∂ψ (19) = −1, Θ = 0, φ = 0, at y = h, ψ = F, ∂y h ∂ψ (20) h( x) = 1 + d cos ( 2π x ) , F = ∫ dy, 0 ∂y where h( x) is the dimensionless wall shape and F is the dimensionless flow rate in the wave frame. Pressure rise per wavelength ∆pλ is 1 dp (21) ∆pλ = ∫ dx. 0 dx The dimensionless expression of space dependent viscosity is [7]: (22) µ ( y ) = e −α y = 1 − α y, α 1, where "α " is the viscosity parameter. Results for constant viscosity are obtained for α = 0 . Moreover, the results for viscous fluid subject to long wavelength and low Reynolds number can be deduced for λ1 = 0 .
Solution methodology
The resulting differential system consists of non-linear differential equations (14 − 17) , subject to 4
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the boundary conditions (18) and (19) . In order to find the perturbed solution, we expand the quantities as follows: ψ = ψ 0 + αψ 1 + ...,
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dp dp dp0 = + α 1 ..., dx dx dx F = F0 + α F1 + ..., Θ = Θ0 + αΘ1 + ...,
φ = φ0 + αφ1 + ....
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Inserting the above equations into Eqs. (14 − 17) and then collecting the terms of like powers of α we have:
Zeroth order system
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∂ 2φ0 ∂ 2Θ0 SrSc + = 0, ∂y 2 ∂y 2
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∂ 2ψ 0 ∂Θ0 ∂φ = 0, = 0, 0 = 0 at y = 0, 2 ∂y ∂y ∂y ∂ψ 0 = −1, Θ0 = 0, φ0 = 0 at y = h, ψ 0 = − F0 , ∂y The non-dimensional pressure rise per wavelength ( ∆Pλ0 ) at this order is
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First order system
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∂ 3ψ 0 ∂ 2ψ 0 dp1 1 ∂ 3ψ 1 ∂ψ 1 = − y − −M2 , 3 3 2 dx 1 + λ1 ∂y ∂y ∂y ∂y
∂ 2ψ 0 ∂ 2Θ1 Br ∂ 2ψ 1 ∂ 2ψ 1 + 2 − y 2 ∂y 2 1 + λ1 ∂y 2 ∂y 2 ∂y ∂ 2φ1 ∂ 2Θ1 + SrSc = 0, ∂y 2 ∂y 2
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∂ 2ψ 1 ∂Θ1 ∂φ = 0, = 0, 1 = 0, at 2 ∂y ∂y ∂y ∂ψ 1 ψ 1 = − F1 , = −1, Θ1 = 0, φ1 = 0, ∂y
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∆pλ1 = ∫ 0
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The perturbation expressions of φ , Θ, dp / dx and ∆pλ up to O(λ1 ) are denoted by φ (1) ,
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Θ(1) , ∆Pλ(1) and dp (1) / dx. These are given by
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Results and discussion
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where the quantities for zeroth and first solutions are computed by solving zeroth and first order systems.
The main purpose of this section is to analyze the effects of different parameters on the longitudinal pressure gradient, pressure rise per wavelength, temperature, concentration and heat transfer coefficient. Here 2-D graphs are plotted for the analysis of longitudinal pressure gradient dp (1) dx
and pressure rise per wavelength ∆pλ(1) . The effects of different Deborah number λ1 on
longitudinal pressure gradient dpdx with ( d = 0.3 , M = 1.0 , θ = 0.3 , α = 0.5 ) are plotted in Fig. 2. It is seen that the magnitude of the longitudinal pressure gradient increases for the Jeffrey fluid ( λ1 ≠ 0 ) when compared with the viscous fluid ( λ1 = 0 ). Fig 3 depicts the variations of the longitudinal pressure gradient for different values of the Hartman number M when d = 0.3 ,
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dp (1) dx
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improves along with Hartman number M . It is also seen that the Hartman number in the Jeffrey fluid has a large impact on the longitudinal (1) pressure gradient. Fig. 4 shows the variations of dpdx for different values of the viscosity parameter when d = 0.3 , M = 1.0 , λ1 = 0.5 , θ = 0.3 . One can easily find that for constant viscosity ( α = 0 ) the pressure gradient is greater in magnitude compared with the variable viscosity case ( α ≠ 0 ). Figs. (5-7) illustrate the effects of different sundry parameters on the pressure rise per wavelength ∆pλ(1) versus mean flow rate θ . In Figs. 5 and 6 it can be seen that the pressure rise per wavelength ∆pλ(1) enhances via larger λ1 for d = 0.3 , M = 1.0 , α = 0.5 and M with d = 0.3 , λ1 = 0.5 , α = 0.5 . Fig. 7 shows that ∆pλ(1) decreases for different values of viscosity parameter α with d = 0.3 , λ1 = 0.5 , M = 1.0 . Effects of different sundry parameters on velocity, temperature, concentration and heat transfer coefficient are analyzed in a more feasible manner via 3-D graphs. The impacts of physical variables in the 3-D graphs are more visible. One main theme is to analyze the effects of variable viscosity ( α ≠ 0 ) in comparison to the constant viscosity α = 0 . In order to attain the desired results, 3-D plots are presented in such a way that all ( the " a " parts of the Figs. ) are for constant viscosity (i.e. for α = 0 ) and all ( the " b " parts of the Figs. ) are for variable viscosity (i.e. for α = 0.5 ). Figs. ( 8 and 9 ) show that the velocity has a maximum magnitude at the center of the channel for α = 0 . For ( α = 0.5 ) the 6
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velocity improves slowly in a smooth way when d = 0.3 , θ = 0.3 , M = 1.0 , x = 0 and d = 0.3 , θ = 0.3 , M = 1.0 , x = 0 respectively. Figs. ( 10-12 ) show the effect of different parameters of interest on the temperature profile Θ . These figures show that the temperature rises slowly in the case of constant viscosity when compared with the variable viscosity. It is noted that the temperature increases with an increase in M . It is because of the viscous dissipation and Ohmic heating that temperature increases with increasing M (see Fig. 10) with d = 0.3 , Br = 2.0 , Du = 0.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 . Fig. 11 shows the similar behavior of Du with temperature for M when d = 0.3 , Br = 2.0 , M = 2.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 . Fig. 12 illustrates the enhancing behavior of Br on the temperature both for the variable and the constant viscosity cases respectively. Here d = 0.3 , Du = 0.5 , M = 2.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 . Figs. ( 13-15 ) are provided to allow an analysis of the effects of variations of the embedded sundry parameters on the concentration profile . Decreasing behavior is observed in Fig. 13 for concentration when M increases with d = 0.3 , Br = 2.0 , Du = 0.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 . Fig. 14 shows that the concentration improves via Du when d = 0.3 ,
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Br = 2.5 , M = 2.0 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 . The increasing behavior of the concentration profile for larger Sr when d = 0.3 , Br = 2.0 , Du = 0.5 , Sc = 0.5 , M = 1.0 , Pr = 0.5 , x = 0.0 , λ1 = 1.0 , θ = 0.1 can be observed in Fig. 15. One can easily observe that for the constant viscosity case the concentration profile increases slowly in comparison to the case of variable viscosity fluid. Figs. ( 16-18 ) illustrate the effects of various parameters of interest on the heat transfer coefficient Z . All these figures show that in view of peristaltic motion due to a travelling wave Z has an oscillatory behavior because of peristaltic motion of the fluid (i.e. the phenomena of contraction and relaxation of mussels occurs). It is also observed that there is no variation in Z for amplitude ratio d between 0.0 and 0.5 . However Z increases with increasing M with d = 0.5 , Br = 2.5 , Du = 0.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , y = 1.4 , λ1 = 1.0 , θ = 0.3 in Fig.16. Fig. 17 shows that the heat transfer coefficient also increases for increasing values of Du with d = 0.5 , Br = 2.5 , M = 2.0 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , y = 1.4 , λ1 = 1.0 , θ = 0.3 . Heat transfer coefficient increases for different values of Br with d = 0.5 , M = 2.0 , Du = 0.5 , Sc = 0.5 , Sr = 0.5 , Pr = 0.5 , y = 1.4 , λ1 = 1.0 , θ = 0.3 (see Fig. 18). In the case of a variable viscosity fluid (i.e. α = 0.5 ) the absolute value of Z is greater than it is in the case of constant viscosity of the fluid (i.e. α = 0.0 ).
Concluding remarks MHD peristaltic movement of variable viscosity Jeffrey fluid with heat and mass transfer is discussed here. The important results of the present analysis are listed below. • Pressure gradient is found to be at a maximum for constant viscosity when compared with case of variable viscosity. • Pressure gradient for viscous fluid (i.e. λ1 = 0 ) is minimum when compared with Jeffrey fluid. • Pressure rise per wavelength is an increasing function of Hartman number. • It is evident that velocity has small resistance in the case of variable viscosity. • Temperature is minimum for variable viscosity when compared with constant viscosity. • Opposite results for concentration and temperature are obtained for both variable and constant 7
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References
[1] T.W. Latham, Fluid motion in peristaltic pump, M.S. Thesis, MIT Cambridge, MA (1966). [2] A.H. Shapiro, M.Y. Jafferin and S.L. Weinberg, Peristaltic pumping with long wavelength at low Reynolds number, J. Fluid Mech. 37 (1969) 799-825. [3 ]D. Tripathi, S.K. Pandey, S. Das, Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel, App. Math. Comp. 215 (2010) 3645-3654. [4] A. Ebaid, Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel, Comp. Math. Applic. 68 (2014) 77-85. [5] K. Ramesh and M. Devakar, Effect of heat transfer on the peristaltic transport of a MHD 15
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