ABD ELMABOUD Y.， MEKHEIMER Kh. S.， MOHAMED Mohamed S.1. Mathematics Department， Faculty of Science and Arts， Khulais， University Of Jeddah， Saudi Arabia2. Mathematics Department， Faculty of Science， Al-Azhar University （Assiut Branch）， Assiut， Egypt，E-mail： yass_math@yahoo.com3. Mathematics Department， Faculty of Science， Taif University Hawia， P.O. Box 888， Taif， Saudi Arabia4. Mathematics Department， Faculty of Science， Al-Azhar University， Nasr City， 11884 Cairo， Egypt

Series solution of a natural convection flow for a Carreau fluid in a vertical channel with peristalsis*

ABD ELMABOUD Y.1，2， MEKHEIMER Kh. S.3，4， MOHAMED Mohamed S.3，4
1. Mathematics Department， Faculty of Science and Arts， Khulais， University Of Jeddah， Saudi Arabia
2. Mathematics Department， Faculty of Science， Al-Azhar University （Assiut Branch）， Assiut， Egypt，E-mail： yass_math@yahoo.com
3. Mathematics Department， Faculty of Science， Taif University Hawia， P.O. Box 888， Taif， Saudi Arabia
4. Mathematics Department， Faculty of Science， Al-Azhar University， Nasr City， 11884 Cairo， Egypt

（Received August 31， 2014， Revised Ocotober 27， 2014）

An analysis has been achieved to study the natural convection of a non-Newtonian fluid （namely a Carreau fluid） in a vertical channel with rhythmically contracting walls. The Navier-Stokes and the energy equations are reduced to a system of nonlinear PDE by using the long wavelength approximation. The optimal homotopy analysis method （OHAM） is introduced to obtain the exact solutions for velocity and temperature fields. The convergence of the obtained OHAM solution is discussed explicitly. Numerical calculations are carried out for the pressure rise and the features of the flow and temperature characteristics are analyzed by plotting graphs and discussed in detail.

homotopy analysis method （HAM）， peristaltic transport， carreau fluid， heat transfer， natural convection flow

Introduction0F

Peristaltic transport is a physical mechanism for the flow induced by the traveling wave. This mechanism is found in the body of living creatures， and it frequently occurs in organs such as ureter， intestines and arterioles （small arteries）. The mechanism of peristaltic transport has also been found in the industrial. There are many industrial applications such as sanitary fluid transport， blood pumps in heart lung machine and transport of corrosive fluids where the contact of the fluid with the machinery parts is prohibited. The first attempt was done by Latham［1］. Following this experimental work， Barton and Raynor［2］established a mathematical model for homogeneous fluids in a channel idealized under the assumption of inertia due to an infinite train of peristaltic waves. Shapiro et al.［3］used infinite wavelength instead of the small-amplitude assumption. Recently， a considerable attention has been devoted to the problem of peristaltic transport with Newtonian or non-Newtonian fluid in channel or a tube［4-6］.

In recent years， the study of non-Newtonian fluids especially with peristaltic transport［7，8］has obtained great importance， because this class of fluid simulates the fluid found in living creatures. In this paper， we choose rheological constitutive equation of Carreau fluid. The Carreau model has a four parameter beside helpful properties of a truncated power law model that does not have a discontinuous first derivative. It possesses a shear thinning （i.e. the viscosity reduces by increasing shear rate）. El Shehawy et al.［9］investigated peristaltic transport of Carreau fluid through non-uniform channel. Recently， some contributions are made to the study of Carreau fluid with the effect of magnetic field［10，11］.

The study of the heat transfer problems draws the attention of researchers especially in biology， because the transport of heat plays a vital role in life processes. In convection， heat transport occurs in a fluid with a combination of molecular diffusion and the fluid's bulk motion or flow. Natural convection is a type of heat transfer wherein non-human forces influence thecooling and heating of fluids. The interaction between peristalsis and heat transfer has been investigated recently， Mekheimer and Abd elmaboud［12］studied the influence of heat transfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus. Srinivas and Kothandapani［13］have investigated the peristaltic transport in an asymmetric channel with heat transfer. Mekheimer et al.［14］discussed the effect of heat transfer on the peristaltic flow of a Newtonian fluid through a porous space in a vertical asymmetric channel.

The common perturbation methods have some limitations， and also depended on the existence of a small parameter. Recently， many different methods have been introduced to eliminate the small parameter；one of these methods is called the homotopy analysis method （HAM）. The homotopy analysis method（HAM） is a new analytical technique， which has attracted special attention of researchers as it is both flexible in applying and give sufficiently accurate results with modest effort. This method has been first introduced in 1992 by Liao［15-17］. Recently， many authors［18，19］have been using HAM in a wide variety of scientific and engineering applications to solve different types of differential equations： linear and nonlinear， homogeneous and non-homogeneous.

With the above discussion in mind， the goal of this investigation is to study the effect of heat transfer on peristaltic flow of a Carreau fluid in a two-dimensional vertical channel. The governing equations are modeled and then solved using the HAM. The analysis for the stream function， the axial pressure gradient，the pressure rise and the heat transfer across the channel have been discussed for various values of the problem parameters. Also， the pumping characteristics and the trapping phenomena are discussed in detail. Finally， the main conclusions are summarized in the last section.

Fig.1 The geometry of the walls surface

1. Mathematical formulation

Consider the peristaltic motion of a non-Newtonian fluid， modeled as a Carreau fluid in a twodimensional vertical channel， where 2a is the undeformed width of the channel and the channel is considered to be infinitely long，b represents the amplitude of the sinusoidal waves traveling along the channel at velocityc，λis the wavelength. A rectangular coordinate system is chosen for the channel with X along the centerline andY normal to it. The wall Y=-H（ X， t）is maintained at temperatures T1and for the wall Y=H（ X， t）， Newtonian cooling law is applied considering T0as the temperature outside the region， obtaining -k（dT/dy）=γ（T-T0）， wherekis the thermal conductivity andγis the heat transfer coefficient. Let （U， V）be the longitudinal and transverse velocity components， respectively. It is assumed that an infinite train of sinusoidal waves progresses along the walls in the Xdirection （see Fig.1）. The equation of the channel wall is given by

Introducing a wave frame （x， y）moving with velocity c away from fixed frame （X， Y）by the transformation

The constitutive equation for a Carreau fluid is

where τis the extra stress tensor，η∞is the infinite shear rate viscosity，η0is the zero shear rate viscosity， Γis the time constant，n is Power-law index andn is defined as

whereΠis the second invariant of strain-rate tensor. Note that the above model reduces to Newtonian model for n =1or Γ=0. The equations of motion for a channel flow in a wave frame of reference are：

whereρis the density，T is the temperature，kis the thermal conductivity，Cρis the specific heat at constant pressure，g is the acceleration due to gravity andαis the coefficient of thermal expansion of the fluid. The appropriate boundary conditions in a moving frame are：

where β1is the slip coefficient having dimension of length. Consider the following non-dimensional variables：

and the dimensionless parameters as follows：

Reynolds number Re=cap/η0， wave number δ=a/λ， Prandtl number Pr=（cpη0）/k， Eckert number Ec=c2/［c（T-T）］， Brickhman numberp10Br=PrEc ， Grashof number Gr=［gα a3（ T-1）ρ2］/， Weissenberg number We=cΓ/a， Biot number Bi=γa/ k， Slip parameter β=β1/λ.

By using Eq.（10） （after dropping the bars） and the dimensionless parameters， we have：

where

Using the long wavelength approximation in Eqs.（12-19） and consider the terms free ofδonly， it follows that：

The corresponding non-dimensional boundaryconditions are：

2. Solution procedures

Introducing the dimensionless stream function ψ（x， y）such that

We find that Eq.（20） is satisfied identically. The compatibility equation， which governs the flow in terms of the stream， function ψ（x， y）after eliminating the pressure gradient from Eqs.（21） and （22）， is

and the energy Eq.（23） will be in the form

The corresponding non-dimensional boundary conditions are：

where

qis the non-dimensional flow rate in the wave frame and the relation between the time-mean flowsand q in the fixed and wave frames is

3. The HAM solution of the problem

For HAM solutions of the governing Eqs.（26） and （27）， we choose the initial approximations of ψ andθ（satisfy the boundary conditions） as follows：

and the auxiliary linear operators are L（ψ）=d4ψ/1dy4and L（θ）=d2θ/dy2. These auxiliary linear ope-2rators satisfy：

where c1，c2，c3，c4，c5，c6are constants. Introducing a non-zero auxiliary parameter?， we develop the zeroth-order deformation problems as follow：

with the boundary conditions

where the nonlinear operators，L1［ψ（y； p）］and L2［θ（y； p）］are defined as：

whenp increases from 0 to 1，ψ（y； p）and θ（y； p） vary from ψ0（y）and θ0（y）to ψ（y）and θ（y）respectively. Using Taylor's theorem ψ（y； p）and θ（y； p）can be expanded in power series of pas follows：

where

a non-zero auxiliary parameter?is chosen in such a way that the series （40） and （41） are convergent at p =1. Suppose that the auxiliary parameter?is selected such that the series （40） and （41） are convergent at p =1. Then we have：

Differentiating the zeroth-order deformation Eqs.（35） and （36），m times with respect topand then dividing them by m！and finally setting p=0， we have the following mth-order deformation problem：

where

are recurrence formulae， in which

with the boundary conditions

We use MATHEMATICA software to obtain the solution of these equations. The first deformations of the coupled solutions may be presented as follow：

where

The higher-order solutions of ψmand θmare too long to list here.

To determine the auxiliary parameter?the so called ?-curves and optimization method are used. In the optimization method， the optimal convergence control parameters are fixed by demanding minimum of the square residual error integrated in the whole region. Let F（ ?）denote the square residual error of the governing Eqs.（26）-（27） and express as

whereψ，θare mentioned in Eqs.（43） and （44）. The optimal value of?is given by solving a nonlinear algebraic equation

The pressure rise Δpfor a channel of lengthL， in non-dimensional form， is given by

The integral in Eq.（55）， not integrable in closed form and is evaluated numerically using a digital computer.

Fig.2?-curve for the stream function at 5th order approximation for different values of the Weissenberg numberWe

Fig.3?-curve for the stream function at 5th order approximation for different values of the Power-law index n

Fig.4?-curve for the temperature at 5th order approximation for different values of the Weissenberg numberWe

Fig.5?-curve for the temperature at 5th order approximation for different values of the Power-law indexn

Table 1 The optimal values of?at 5th order approximation for the stream function at the fixed values of x =0.2，α=0.4，β=0.01，Bi =0.01，Br =0.5，Gr =0.5，Q=2

4. Convergence of the solution

It is noticed that the solutions （51） and （52） contain the auxiliary parameter? . As pointed out by Liao［16］， the convergence region and rate of approximations given by the HAM are strongly dependent upon?. For fixed values of the parameters x =0.2， α=0.4，β=0.01，Bi =0.01，Br =0.5，Gr =0.5， Q =2and n=0.398and with two different values of Weissenberg numberWe （namely，We =0and We =0.2） and x =0.2，α=0.4，β=0.01，Bi =0.01， Br =0.5，Gr =0.5，Q =2and We=0.2with two different values of power-law indexn （namely，n= 0.398 and n =1） the range for admissible values of? for the stream function is -1.3≤?≤-0.7and for temperature is -1.25≤?≤-0.8（see Figs.2-5）. Also， the optimal values of?for different parameters are givened in Tables 1 and 2.

Table 2 The optimal values of ?at 5th order approximation for the temperature at the fixed values of x= 0.2，α=0.4，β=0.01，Bi =0.01，Br =0.5，Gr =0.5，Q=2

5. Numerical results and discussion

This section is divided into three subsections. In the first subsection， the effects of various parameters on the pumping characteristics are investigated. The heat characteristics are discussed in the second subsection. The trapping phenomenon is illustrated in the last subsection.

Fig.6 The velocity distributionu ， across the channel with different values of βandGrat x =0.2，We =0.3，n =0.398，Bi =0.1，Q =2，Br =1and φ=0.4

5.1 Distribution of velocity

For different values of Grashof numberGr ， slip parameterβ， Weissenberg numberWe ， Biot number Bi ， and the Power-law indexn， Figs.6-9 present the distribution of axial velocity. Figure 6 shows the effect of Grashof numberGr and slip parameterβonthe velocity through the channel with other parameters fixed. It is clear that the velocity profile distributes symmetrically about the center of the channel when Gr =0and β=0， because there is no natural convection and no slip velocity on the walls of the channel. However， with the values ofGrandβelevating， we notice that the velocity is less or large than -1 on the walls （slip condition） and the velocity decreases from near the wall y=hto the center of the channel， but it increases in the other half for increasingGr . Figures 7 and 8 reveal that the magnitude of the axial velocity is large in a Newtonian fluid （We =0or n=1） compared with a non-Newtonian fluid. However， the forward flow region is predominant here since the time averaged flow rate is positive. The effect of Biot numberBi on the axial velocity is shown in Fig.9. It is evident that near the wall y=hthe magnitude of axial velocity is enhanced by increasing the Biot numberBi ， because the convection process offers much buoyancy force which leads to the increase in the magnitude of axial velocity. But in the other half of the channel the convection process offers little resistance to the flow therefore the magnitude of axial velocity diminishes.

Fig.7 The velocity distribution u， across the channel with different values of We at x =0.2，Gr =0，β=0.01，n =0.398，Bi =0.4，Q =2，Br =1and φ=0.4

Fig.8 The velocity distributionu ， across the channel with different values ofn at x =0.2，Gr =0，β=0，We =0.4，Bi =0.3，Q =2，Br =1and φ=0.4

Fig.9 The velocity distribution u， across the channel with different values of Bi at x =0.2，Gr =1，β=0，n =0.398，We =0.4，Q =2，Br =1and φ=0.4

Fig.10 Variation of pressure rise Δpover the length versus Q with different values of We at Gr =1，β=0.01，n =0.398，Bi =0.01，Br =1and φ=0.4

Fig.11 Variation of pressure rise over the length versusQ with different values of n at Gr =2，β=0.01，We =0.3，Bi =0.01，Br =1and φ=0.4

5.2 Pumping characteristics

In this subsection， we aim to study the influence of the apparent parameters on the different pumping regions. To discuss this phenomenon we prepared Figs.10-12. For our study of the peristaltic transport of a non-Newtonian fluid， the relationship between the pressure rise and the flow rate is found to be nonlinear. Moreover， for a Newtonian fluid （We =0orn=1） the flow rate averaged over one wave varies linearly with the pressure rise as shown in Figs.10 and 11. Plots in Figs.10 and 11 indicate that the peristaltic pumping region （Δp＞0 and Q＞0） enhances in a New tonian fluid （We =0 or n=1） and become little in a non-New tonian fluid， that is because the shear thickening appears in the non-New tonian fluid. The effect of Grashof number Gr and the Biot number Bi on the pressure rise Δp is illustrated in Fig.12. It is clear that the peristaltic pumping region increases by increasing the Grashof number Gr while it decreases by increasing the Biot number Bi . Moreover， the rate of the reduce in the peristaltic pumping by the effect of the Biot number Bi in the case of the low values of Grashof number Gr is small， compared w ith the large values of Grashof number Gr.

Fig.12 Variation of pressure rise Δp over the length versus Q w ith different values of Gr and Bi at n =0.398，β=0.01，We =0.2，Br =1 and φ=0.4

Fig.13 Temperature distribution θversus y for different values of We at x =0.2，Gr =1，β=0.01，n= 0.398，Br =0.2，Br =1，Q =2 and φ=0.4

5.3 Heat characteristics

A ll figures representing the temperature are plotted at the cross section of the channel （i.e，x=0.2）. To study the behavior of emerging parameters in the temperature distribution some figures （Figs.13-15）have been displayed. It may be observed from Figs.13 and 14 that at a fixed cross section of the channel the temperature distribution is higher in the case of the New tonian fluid （We =0 or n=1） compared w ith a non-New tonian fluid. This behavior is related to the thermal properties of the non-New tonian fluids. The effects of Grashof number Gr and the Biot number Bi on the temperature distribution are shown in Fig.15. It is clear that the temperature distribution increases through the channel by increasing the Biot number Bi while it increases near the wall y=h by increasing the Grashof number Gr but away from this wall the temperature decreases.

Fig.14 Tem perature distribution θversus y for different values of We at x =0.2，Gr =1，β=0.01，n= 0.398，Bi =0.2，Br =1，Q =2 and φ=0.4

Fig.15 Tem perature distribution θversus y for different values of Bi and Gr at x =0.2，β=0.01，We= 0.2，n =0.398，Br =1，Q =2 and φ=0.4

5.4 Stream lines and fluid trapping

It is well known that， one of the significant features of peristaltic transport is the phenomenon of trapping. It occurs when stream lines on the central line are split to enclose a bolus of fluid particles circulating along closed stream lines in the wave frame of reference. The trapped bolus moves w ith a speed equal to that of the wave. Figure 16 is an illustration of the stream lines for different values of Weissenberg number（We）.Two different areas of trapped bolus appearing abo ut the ce nter， but they are different in number and size.As Weissenberg number（We），increases （non-Newtonian fluid） some sort of rigidity appears and the number of bolus decreases. Streamlines for different Grashof numberGr and Biot numberBi are depicted in Figs.17 and 18. These figures indicate that occurrence and number of trapping is strongly influenced by the value of the Grashof number Gr and Biot numberBi.

Fig.16 Streamlines for different values ofWe The other parameters chosen are φ=0.4，Q =2，Br =1，Bi =0.3，β=0.01，Gr =1and n =0.398where y∈［-h， h］

Fig.17 Streamlines for different values of GrThe other parameters chosen are φ=0.4，Q =2，Br =1，Bi =0.3，β=0.01，We =0.2 and n =0.398 where y∈［-h， h］

Fig.18 Streamlines for different values ofBi the other parameters chosen are φ=0.4，Q =2，Br =1，Gr =1，β=0.01，We =0.2and n =0.398where y∈［-h，h］

6. Concluding remarks

The present paper deals with the peristaltic motion of a non-Newtonian fluid （namely a Carreau fluid）through a vertical channel. Thus the present investigation bears the potential of significant application in biomedical engineering and technology. The system of governing equations is reduced to a system of nonlinear PDE by using the long wavelength approximation. A homotopy analysis method （HAM） is used to obtain the solutions for velocity and temperature fields. The convergence region and the optimal values of the auxiliary parameter are discussed explicitly. The present study reveals that the velocity profiles distributes symmetrically about the center of the channel when there is no natural convection and also no slip velocity on the walls of the channel. The region of peristaltic flow advances， if the value of the Grashof number is raised. Moreover， the present study shows that in anon-Newtonian fluid the peristaltic pumping has small effect compared with Newtonian fluid. The region of retrograde flow （the upper left-hand quadrant denotes the region of retrograde pumping （or backward pumping） where Q＜0and Δp＞0） depreciate at a faster rate， if the values of（We）are raised.

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* Biography： ABD ELMABOUD Y. （1976-）， Male， Ph. D.，Associate Professor