# Peristaltic Slip Flow of a Viscoelastic Fluid with Heat and Mass Transfer in a Tube

Authors: Ayman Mahmoud Sobh
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The paper discusses the combined effect of slip velocity and heat and mass transfer on peristaltic flow of a viscoelastic fluid in a uniform tube. This study has numerous applications. It serves as a model for the chyme movement in the small intestine, by considering the chyme as a viscoelastic fluid. The problem is formulated and analysed using perturbation expansion in terms of the wave number as a parameter. Analytic solutions for the axial velocity component, pressure gradient, temperature distribution, and fluid concentration are derived. Also, the effects of the emerging parameters on pressure gradient, temperature distribution, concentration profiles, and trapping phenomenon are illustrated graphically and discussed in detail.


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## Body

## 1. Introduction

Peristalsis is an important mechanism for mixing and transporting fluid which is generated by a progressive wave of contraction or expansion moving on the wall of the tube. It occurs widely in many biological and biomedical systems. In physiology, it plays an indispensable role in various situations. For examples, the transport of urine from kidney to the bladder, the movement of chyme in the gastrointestinal tract, transport of spermatozoa in the ducts efferentes of the male reproductive tracts, movement of ovum in the female fallopian tube, transport of lymph in the lymphatic vessels, vasomotion of small blood vessels such as arterioles, venules, and capillaries, and so on.The peristaltic flow of non-Newtonian fluids has gained considerable interest during the recent years because of its applications in industry and biology. In biology, it is well known that most physiological fluids behave like non-Newtonian fluids. Hence, the study of peristaltic transport of non-Newtonian fluids may help to get a better understanding for some biological systems. Now, several theoretical and numerical investigations have been carried out to understand the peristaltic mechanism in different situations. Some of the recent studies on peristaltic flow of non-Newtonian Fluids can be seen through references [1–10].Recently, investigations of heat and mass transfer in peristalsis have been considered by some researchers due to its applications in the biomedical sciences. Srinivas and Kothandapani [11] investigated the influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls. Eldabe et al. [12] studied the mixed convective heat and mass transfer in a non-Newtonian fluid at a peristaltic surface with temperature-dependent viscosity. The influence of radially varying MHD on the peristaltic flow in an annulus with heat and mass transfer has been studied by Nadeem and Akbar [13]. Moreover, Akbar et al. [14] investigated the effect of heat and mass transfer on the peristaltic flow of hyperbolic tangent fluid in an annulus. Moreover, Hayat et al. [15] investigated the peristaltic flow of pseudoplastic fluid under the effects of an induced magnetic field and heat and mass transfer in a channel. Furthermore, Hayat et al. [16] studied the heat and mass transfer effects on the peristaltic flow of Johnson-Segalman fluid in a curved channel with compliant walls.Problems that involve slip boundary conditions may be useful models for flows through pipes in which chemical reactions occur at the walls, flows with laminar film condensation, and certain two phase flows. Motivated by this, several studies were made to investigate the effect of slip velocity on peristaltic transport. Some of these studies have been done by Sobh [17], Hayat and Mehmood [18], Noreen et al. [19], Hayat et al. [20], and Saleem et al. [21].It is noticed from the available literature that no analysis has been made yet for the peristaltic flow of a viscoelastic fluid with heat and mass transfer in a tube in the presence of slip conditions on the tube wall. For this purpose, the peristaltic slip flow of an Oldroyd fluid, as a viscoelastic fluid, in a uniform tube is considered here in the presence of heat and mass transfer. This analysis can model movement of the chyme in the small intestine by considering chyme as an Oldroyd fluid. The flow analysis is developed in a wave frame of reference moving with the same velocity of the wave travelling down the tube wall. The perturbation technique is used to obtain an analytic solution for the governing equations in terms of the wave, Reynolds, and Weissenberg numbers. The derived solutions for pressure gradient, temperature field, and concentration profiles are plotted and analyzed in detail. The trapping phenomenon is also discussed.


## 2. Mathematical Modeling

The continuity and momentum equations for an incompressible fluid, in the absence of body forces, are given by(2.1)divV=0,ρdVdt=-∇p+divτ,
where ρ is the density of the fluid, V is the velocity vector, p is the pressure, τ is the extra stress tensor, and d/dt is the material time derivative. The constitutive equation for Oldroyd fluid is given by [22]
(2.2)τij+Γ[∂τij∂t+(gkkgiigjj)vk∂∂xk(giigjjτij)-gkkgjjτkj∂∂xk(giivi)-gkkgiiτik∂∂xk(gjjvj){∂τij∂t(gkkgiigjj)}]=-μγ˙ij,
in which τij,i,j,k=1,2,3 are the components of the extra stress tensor, gii and gjj are respectively the diagonal components of covariant and contravariant metric tensor, vi are the velocity components, μ is the fluid viscosity, Γ is relaxation time, and γ˙ij are the components of strain-rate tensor.


## 3. Formulation of the Problem

Consider the peristaltic flow of an incompressible Oldroyd fluid in an axisymmetric tube of a sinusoidal wave travelling down its wall. The wall of the tube is maintained at temperatureT-0 and concentration C-0. In the fixed cylindrical coordinate system (R-,Z-), the geometry of the problem, as can be seen in Figure 1, is
(3.1)h-(Z-,t-)=a+bsin[2πλ(Z--ct-)],
where Z- is the axis lies along the centreline of the tube, R- is the distance measured radially, a is the radius of the tube, b is the wave amplitude, λ is the wavelength, and c is the propagation velocity.Figure 1
Geometry of the problem.Let us introduce a wave frame(r-,z-) moving with velocity c away from the fixed frame (R-,Z-) by the transformation
(3.2)r-=R-,z-=Z--ct-,u-=U-,w-=W--c,
where (U-,W-), (u-,w-) are the velocity components in the fixed and wave frames, respectively. For the case of axisymmetric tube, the constitutive equations (2.2), in the wave frame, become
(3.3)τ-11+Γ(u-∂τ-11∂r-+w-∂τ-11∂z--2τ-11∂u-∂r--2τ-13∂u-∂z-)=-μγ˙-11,τ-13+Γ(u-∂τ-13∂r-+w-∂τ-13∂z--τ-33∂u-∂z--τ-11∂w-∂r-+u-r-τ-13)=-μγ˙-13,τ-22+Γ(u-∂τ-22∂r-+w-∂τ-22∂z--2u-r-τ-22)=-μγ˙-22,τ-33+Γ(u-∂τ-33∂r-+w-∂τ-33∂z--2τ-33∂w-∂z--2τ-13∂w-∂r-)=-μγ˙-33,
where the components of the strain-rate tensor are given by
(3.4)γ˙-11=2∂u-∂r-,γ˙-22=2u-r-,γ˙-33=2∂w-∂z-,γ˙-13=(∂u-∂z-+∂w-∂r-).
Using the following non-dimensional variables and parameters:
(3.5)r=r-a,z=z-λ,w=w-c,u=λu-ac,γ˙ij=acγ˙-ij,p=a2p-cλμ,t=ct-λ,δ=aλ,Re=ρcaμ,τij=aτ-ijcμ,Wi=cΓa,T=T-T-0,C=C-C-0,Pr=μcpk,E=c2T-0cp,Sr=ρDmKTT-0μTmC-0,Sc=μρDm,h=h-a=1+basin2πz=1+φsin2πz,
we obtain the non-dimensional continuity equation, momentum equations, constitutive equations, energy equation, and concentration equation as follows:
(3.6)1r∂∂r(ru)+∂w∂z=0,Reδ3(u∂u∂r+w∂u∂z)=-∂p∂z-δ(1r∂∂r(rτ11)+δ∂τ13∂z-τ22r),Reδ(u∂w∂r+w∂w∂z)=-∂p∂z-(1r∂∂r(rτ13)+δ∂τ33∂z),τ11+δWi[u∂τ11∂r+w∂τ11∂z-2τ11∂u∂r-2δτ13∂u∂z]=-2δ(∂u∂r),τ13+Wi[δ(u∂τ13∂r+w∂τ13∂z-δτ33∂u∂z+urτ13)-τ11∂w∂r]=-(δ2∂u∂z+∂w∂r),τ22+δWi[u∂τ22∂r+w∂τ22∂z-2urτ22]=-2δur,τ33+Wi[δ(u∂τ33∂r+w∂τ33∂z-2τ33∂w∂z)-2τ13∂w∂r]=-2δ(∂w∂z),δPr(u∂T∂r+w∂T∂z)=(1r∂∂r(r∂T∂r)+δ2∂2T∂z2)-Br(δτ11∂u∂r+τ13∂w∂r+δ2τ31∂u∂z+δτ33∂w∂z),δRe(u∂C∂r+w∂C∂z)=1Sc(1r∂∂r(r∂C∂r)+δ2∂2C∂z2)+Sr(1r∂∂r(r∂T∂r)+δ2∂2T∂z2),
where C is the concentration of the fluid, T is the temperature, Tm is the temperature of the medium, Dm is the coefficient of mass diffusivity, KT is the thermal diffusion ratio, μ is the viscosity, cp is the specific heat at constant volume, k is the thermal conductivity, δ is the dimensionless wave number assumed to be small, Re is the Reynolds number, Wi is the Weissenberg number, Pr is the Prandtl number, E is the Eckert number, Sr is the Soret number, Sc is the Schmidt number, and Br=E  Pr is the Brinkman number.The dimensionless boundary conditions are(3.7)u=0,∂w∂r=0,∂T∂r=0,∂C∂r=0,atr=0,w=-1+Knτ13,u=dhdz,T=1,C=1,atr=h,
where Kn=K-n/a, is the dimensionless slip parameter.


## 4. Perturbation Solution

We begin the construction of the solution by expanding the following quantities as power series in the small parameterδ as follows:
(4.1)w=w0+δw1+δ2w2+O(δ3),u=u0+δu1+δ2u2+O(δ3),p=p0+δp1+δ2p2+O(δ3),f=f0+δf1+δ2f2+O(δ3),τij=τij(0)+δτij(1)+δ2τij(2)+O(δ3),i,j=1,2,3,T=T0+δT1+δ2T2+O(δ3),C=C0+δC1+δ2C2+O(δ3),
where f=2∫0hrwdr is the dimensionless mean flow rate in the wave frame which is related with the mean flow rate in the fixed frame θ by the relation [4]
(4.2)θ=f+(1+φ22).
Substituting the expansions (4.1) into (3.6) and (3.7) and collecting terms of like powers of δ we obtain the following systems of coupled differential equations.


### 4.1. Zero Order System

Consider(4.3)1r∂∂r(ru0)+∂w0∂z=0,∂p0∂r=0,∂p0∂z=1r∂∂r(r∂w0∂r),1r∂∂r(r∂T0∂r)=-Br(∂w0∂r)2,0=1Sc(1r∂∂r(r∂C0∂r))+Sr(1r∂∂r(r∂T0∂r)),
with the boundary conditions
(4.4)∂w0∂r=0,∂T0∂r=0,∂C0∂r=0,atr=0,w0=-1-Kn(∂w0∂r),T0=1,C0=1,atr=h(z).The solution of (4.3), subject to the boundary conditions (4.4), is
(4.5)w0(r,z)=a1r2+a2,u0(r,z)=a3r3+a4r,dp0dz=-8(f0+h2)(h4+4h3Kn).T0(r,z)=-Br[164(dp0dz)2(r4-h4)]+1,C0(r,z)=SrScBr[164(dp0dz)2(r4-h4)]+1,
where a1, a2, a3,  and a4 are stated in the appendix.


### 4.2. First Order System

Consider(4.6)1r∂∂r(ru1)+∂w1∂z=0,(4.7)∂p1∂r=0,(4.8)Re(u0∂w0∂r+w0∂w0∂z)=-∂p1∂z-1r∂∂r(rτ13(1))+2Wi∂∂z[(∂w0∂r)2],(4.9)τ13(1)=-(∂w1∂r)+Wi[u0∂2w0∂r2+w0∂2w0∂r∂z+u0r∂w0∂r-2(∂u0∂r)(∂w0∂r)],(4.10)1r∂∂r(r∂T1∂r)=RePr(u0∂T0∂r+w0∂T0∂z)+Br[τ13(1)∂w0∂r-(∂w0∂r)(∂w1∂r)-2Wi(∂w0∂r)2(∂w0∂z)],(4.11)Re(u0∂C0∂r+w0∂C0∂z)=1Sc(1r∂∂r(r∂C1∂r))+Sr(1r∂∂r(r∂T1∂r)).The boundary conditions are(4.12)∂w1∂r=0,∂T1∂r=0,∂C1∂r=0,atr=0,(4.13)w1=Knτ13(1),T1=0,C1=0atr=h(z).Substituting the zero order solution into (4.9), we obtain τ13(1) in the form
(4.14)τ13(1)=-(∂w1∂r)+Wi[(2a1a1′-8a1a3)r3+(2a2a1′)r].Substituting (4.14) together with the zero order solution into (4.8) and integrating subject to the boundary condition (4.12), taking into account that ∂p1/∂r=0, we obtain a differential equation for w1(r,z) in the form
(4.15)∂w1∂r=12(dp1dz)r+Wi[-2(4a1a3+a1a1′)r3+2a2a1′r]+Re[(2a1a3+a1a1′)6r5+(2a1a4+a1a2′+a2a1′)4r3+a2a2′2r].The solution of (4.15), subject to the boundary condition (4.13) is given by
(4.16)w1(r,z)=14(dp1dz)(r2-h2-2Knh)+Re(a5r6+a6r4+a7r2+a8)+Wi(a9r4+a10r2+a11),
where a5,…a11 are stated in the appendix.The dimensionless mean flow rate in the wave framef1 is given by
(4.17)f1=2∫0hrw1dr=-18(dp1dz)h4+Re[a54h8+a63h6+a72h4+a8h2]+Wi[a93h6+a102h4+a11h2].On solving (4.17) for dp1/dz, one finds
(4.18)dp1dz=-8f1(h4+4Knh3)-Re(h4+4Knh3)(2a5h8+83a6h6+4a7h4+8a8h2)-Wi(h4+4Knh3)(83a9h6+4a10h4+8a11h2).Using zero order solution together with the first order solution ofw1(r,z), (4.16), into (4.10), (4.11)  and applying the boundary conditions, we get
(4.19)T1(r,z)=-Br32(dp0dz)(dp1dz)(r4-h4)-BrRe[a12(r8-h8)+a13(r6-h6)+a14(r4-h4)+a15(r2-h2)]-BrWi[a16(r6-h6)+a17(r5-h5)+a18(r4-h4)+a19(r3-h3)],(4.20)C1(r,z)=BrSrSc32(dp0dz)(dp1dz)(r4-h4)+BrSrScRe[a20(r8-h8)+a21(r6-h6)+a22(r4-h4)+a23(r2-h2)]+BrSrScWi[a16(r6-h6)+a17(r5-h5)+a18(r4-h4)+a19(r3-h3)],
where a12,…a19 are defined in the appendix.The results of our analysis can be expressed to first order by defining(4.21)f=f+δf1,
then substituting into zero and first order solutions and neglecting all terms of higher than O(δ), we find
(4.22)w(r,z)=14(dpdz)(r2-h2-2Knh)-1+δRe(b5r6+b6r4+b7r2+b8)+δWi(b9r4+b10r2+b11),dpdz=-(8f+h2)(h4+4Knh3)-δRe(h4+4Knh3)(2b5h8+83b6h6+4b7h4+8b8h2)-δWi(h4+4Knh3)(83b9h6+4b10h4+8b11h2),T(r,z)=-Br64(dpdz)2(r4-h4)+1-δBr[Re{b12(r8-h8)+b13(r6-h6)+b14(r4-h4)+b15(r2-h2)}+Wi{b16(r6-h6)+b17(r5-h5)+b18(r4-h4)+b19(r3-h3)}],C(r,z)=BrSrSc64(dpdz)2(r4-h4)+1+δBrSrSc[Re{b20(r8-h8)+b21(r6-h6)+b22(r4-h4)+b23(r2-h2)}+Wi{b16(r6-h6)+b17(r5-h5)+b18(r4-h4)+b19(r3-h3)}],
where b5,…b23 are defined in the appendix.


## 4.1. Zero Order System

Consider(4.3)1r∂∂r(ru0)+∂w0∂z=0,∂p0∂r=0,∂p0∂z=1r∂∂r(r∂w0∂r),1r∂∂r(r∂T0∂r)=-Br(∂w0∂r)2,0=1Sc(1r∂∂r(r∂C0∂r))+Sr(1r∂∂r(r∂T0∂r)),
with the boundary conditions
(4.4)∂w0∂r=0,∂T0∂r=0,∂C0∂r=0,atr=0,w0=-1-Kn(∂w0∂r),T0=1,C0=1,atr=h(z).The solution of (4.3), subject to the boundary conditions (4.4), is
(4.5)w0(r,z)=a1r2+a2,u0(r,z)=a3r3+a4r,dp0dz=-8(f0+h2)(h4+4h3Kn).T0(r,z)=-Br[164(dp0dz)2(r4-h4)]+1,C0(r,z)=SrScBr[164(dp0dz)2(r4-h4)]+1,
where a1, a2, a3,  and a4 are stated in the appendix.


## 4.2. First Order System

Consider(4.6)1r∂∂r(ru1)+∂w1∂z=0,(4.7)∂p1∂r=0,(4.8)Re(u0∂w0∂r+w0∂w0∂z)=-∂p1∂z-1r∂∂r(rτ13(1))+2Wi∂∂z[(∂w0∂r)2],(4.9)τ13(1)=-(∂w1∂r)+Wi[u0∂2w0∂r2+w0∂2w0∂r∂z+u0r∂w0∂r-2(∂u0∂r)(∂w0∂r)],(4.10)1r∂∂r(r∂T1∂r)=RePr(u0∂T0∂r+w0∂T0∂z)+Br[τ13(1)∂w0∂r-(∂w0∂r)(∂w1∂r)-2Wi(∂w0∂r)2(∂w0∂z)],(4.11)Re(u0∂C0∂r+w0∂C0∂z)=1Sc(1r∂∂r(r∂C1∂r))+Sr(1r∂∂r(r∂T1∂r)).The boundary conditions are(4.12)∂w1∂r=0,∂T1∂r=0,∂C1∂r=0,atr=0,(4.13)w1=Knτ13(1),T1=0,C1=0atr=h(z).Substituting the zero order solution into (4.9), we obtain τ13(1) in the form
(4.14)τ13(1)=-(∂w1∂r)+Wi[(2a1a1′-8a1a3)r3+(2a2a1′)r].Substituting (4.14) together with the zero order solution into (4.8) and integrating subject to the boundary condition (4.12), taking into account that ∂p1/∂r=0, we obtain a differential equation for w1(r,z) in the form
(4.15)∂w1∂r=12(dp1dz)r+Wi[-2(4a1a3+a1a1′)r3+2a2a1′r]+Re[(2a1a3+a1a1′)6r5+(2a1a4+a1a2′+a2a1′)4r3+a2a2′2r].The solution of (4.15), subject to the boundary condition (4.13) is given by
(4.16)w1(r,z)=14(dp1dz)(r2-h2-2Knh)+Re(a5r6+a6r4+a7r2+a8)+Wi(a9r4+a10r2+a11),
where a5,…a11 are stated in the appendix.The dimensionless mean flow rate in the wave framef1 is given by
(4.17)f1=2∫0hrw1dr=-18(dp1dz)h4+Re[a54h8+a63h6+a72h4+a8h2]+Wi[a93h6+a102h4+a11h2].On solving (4.17) for dp1/dz, one finds
(4.18)dp1dz=-8f1(h4+4Knh3)-Re(h4+4Knh3)(2a5h8+83a6h6+4a7h4+8a8h2)-Wi(h4+4Knh3)(83a9h6+4a10h4+8a11h2).Using zero order solution together with the first order solution ofw1(r,z), (4.16), into (4.10), (4.11)  and applying the boundary conditions, we get
(4.19)T1(r,z)=-Br32(dp0dz)(dp1dz)(r4-h4)-BrRe[a12(r8-h8)+a13(r6-h6)+a14(r4-h4)+a15(r2-h2)]-BrWi[a16(r6-h6)+a17(r5-h5)+a18(r4-h4)+a19(r3-h3)],(4.20)C1(r,z)=BrSrSc32(dp0dz)(dp1dz)(r4-h4)+BrSrScRe[a20(r8-h8)+a21(r6-h6)+a22(r4-h4)+a23(r2-h2)]+BrSrScWi[a16(r6-h6)+a17(r5-h5)+a18(r4-h4)+a19(r3-h3)],
where a12,…a19 are defined in the appendix.The results of our analysis can be expressed to first order by defining(4.21)f=f+δf1,
then substituting into zero and first order solutions and neglecting all terms of higher than O(δ), we find
(4.22)w(r,z)=14(dpdz)(r2-h2-2Knh)-1+δRe(b5r6+b6r4+b7r2+b8)+δWi(b9r4+b10r2+b11),dpdz=-(8f+h2)(h4+4Knh3)-δRe(h4+4Knh3)(2b5h8+83b6h6+4b7h4+8b8h2)-δWi(h4+4Knh3)(83b9h6+4b10h4+8b11h2),T(r,z)=-Br64(dpdz)2(r4-h4)+1-δBr[Re{b12(r8-h8)+b13(r6-h6)+b14(r4-h4)+b15(r2-h2)}+Wi{b16(r6-h6)+b17(r5-h5)+b18(r4-h4)+b19(r3-h3)}],C(r,z)=BrSrSc64(dpdz)2(r4-h4)+1+δBrSrSc[Re{b20(r8-h8)+b21(r6-h6)+b22(r4-h4)+b23(r2-h2)}+Wi{b16(r6-h6)+b17(r5-h5)+b18(r4-h4)+b19(r3-h3)}],
where b5,…b23 are defined in the appendix.


## 5. Discussion of Results

It is clear that our results allow calculation of the velocity, the pressure gradient, the temperature, and the concentration field without any restrictions on the Reynolds and Weissenberg numbers but we have used a small wave number. Moreover, we note that the approximation we have used (small wave number,δ<1) holds for our application as the values of various parameters for transporting the chyme in the small intestine are [23].
(5.1)a=1.25cm,λ=8.01cm,δ=a/λ=0.156.This agrees with the small wave number approximation. In order to have an estimate of the quantitative effects of the various parameters involved in the results of the present analysis, Figures2–23 are prepared using the MATHEMATICA package.Figure 2
Pressure gradient versusz for ϕ=0.6,  Wi=0,  Re=0,  δ=0,  θ=0.1.Figure 3
Pressure gradient versusz for φ=0.6,  Wi=0.04,  Re=10,  δ=0.02,  θ=0.1.Figure 4
Pressure gradient versusz for φ=0.6,  Wi=0.04,  Re=10,  θ=0.1,  Kn=0.05.Figure 5
Pressure gradient versusz for φ=0.6,  Wi=0.04,  δ=0.01,  θ=0.1,  Kn=0.05.Figure 6
Pressure gradient versusz for φ=0.6,  Re=10,  δ=0.02,  θ=0.1,  Kn=0.05.Figure 7
Temperature distribution forz=0.2,  Br=1,  Pr=1,  δ=0,  Re=0,  Wi=0,  θ=0.5,  ϕ=0.2.Figure 8
Temperature distribution forz=0.2  ,Br=1,  Pr=1,  δ=0.02,  Re=10,  Wi=0.03,  θ=0.5,  φ=0.2.Figure 9
Temperature distribution forz=0.2,  Br=1,  Pr=1,  δ=0.156,  Re=10,  Kn=0.02,  θ=0.5,  φ=0.4.Figure 10
Temperature distribution forz=0.2,  Br=1,  Kn=0.02,  Pr=1,  Re=10,  Wi=0.04,  θ=0.5,  φ=0.4.Figure 11
Temperature distribution forz=0.2,  Br=1,  Kn=0.02,  Pr=1,  δ=0.02,Wi=0.04, θ=0.5,φ=0.4.Figure 12
Temperature distribution forz=0.2,Re=10,  Kn=0.02,Pr=1,δ=0.02,Wi=0.04,θ=0.5,φ=0.4.Figure 13
Concentration profiles forz=0.2,Br=1,Pr=1,Sr=0.3,  Sc=0.3,δ=0.02,  Re=10,Wi=0.03,θ=0.5,  φ=0.4.Figure 14
Concentration profiles forz=0.2,  Br=1,Pr=1,Sr=0.3,Sc=0.3,δ=0.156,Re=20,  Kn=0.02,  θ=0.5,  φ=0.4.Figure 15
Concentration profiles forz=0.2,Br=1,Pr=1,Sr=0.3,Sc=0.3, Re=10,  Wi=0.04,  Kn=0.02,  θ=0.5,  φ=0.4.Figure 16
Concentration profiles forz=0.2,  Br=1,Pr=1,  Sr=0.3,  Sc=0.3,  δ=0.02,  Wi=0.04,  Kn=0.02,  θ=0.5,  φ=0.4.Figure 17
Concentration profiles forz=0.2,Re=10,Pr=1,Sr=0.3,Sc=0.3,  δ=0.02,Wi=0.04,Kn=0.02,θ=0.5,  φ=0.4.Figure 18
Concentration profiles forz=0.2,Re=10,  Pr=1,  Br=1,Sc=0.3,  δ=0.02,Wi=0.04,Kn=0.02,θ=0.5,  φ=0.4.Figure 19
Concentration profiles forz=0.2,Re=10,Pr=1,Br=1,Sr=0.3,  δ=0.02,Wi=0.04,Kn=0.02,θ=0.5,  φ=0.4.Streamlines forWi=0.03,Re=10,δ=0.02,θ=0.3,φ=0.4,Kn=(0,0.05,0.1).
(a)
(b)
(c)Streamlines forWi=0.03,  Re=10,  Kn=0.05,  θ=0.3,  φ=0.3,  δ=(0,0.03,0.05).
(a)
(b)
(c)Streamlines forWi=0.03,Kn=0.05,δ=0.03,θ=0.3,φ=0.3,Re=(0,10,20).
(a)
(b)
(c)Streamlines forRe=1,δ=0.05,  φ=0.3,θ=0.3,Kn=0.05,Wi=(0,0.04,0.08).
(a)
(b)
(c)


### 5.1. Pumping Characteristics

The effect of the slip parameterKn on the pressure gradient for both Newtonian and Oldroyd fluids is shown in Figures 2 and 3, respectively. It is evident that the pressure gradient decreases by increasing the slip parameter Kn. Furthermore, from the two figures it can be noticed that in the wider part of the tube z∈[0,0.3] and [0.6, 1], the pressure gradient is small. This means that the flow can easily pass without imposition of a large pressure gradient. On the other hand, in the narrow part of the tube z∈[0.3,0.6], a large pressure gradient is required to maintain the flow to pass it.Figures4 and 5 illustrate the effect of wave number and Reynolds number on the pressure gradient of the Oldroyd fluid at θ=0.1,φ=0.6,Kn=0.05,Re=10,Wi=0.04,(δ=0,0.02,0.04) and θ=0.1,  φ=0.6,Kn=0.05,Wi=0.04,δ=0.01,(Re=0,15,30), respectively. The figures reveal that the pressure gradient increases by increasing both wave number and Reynolds number.Figure6 depicts the effect of the Weissenberg number Wi on the pressure gradient of the viscoelastic fluid at θ=0.1,φ=0.6,Kn=0.05,  Re=10, and δ=0.02. We can conclude that an increase in the Weissenberg number decreases the pressure gradient.


### 5.2. Temperature Distribution

Figures7–12 are devoted to explain the effect of emerging parameters on the temperature distribution. The effect of slip parameter Kn on the temperature distribution T at z=0.2,Br=1,  Pr=1,θ=0.5,φ=0.2 is shown in Figures 7 and 8 for both Newtonian and Oldroyd fluid, respectively. As shown, the temperature decreases as the slip parameter increases. The two figures also reveal that the behaviour of the temperature profiles is the same for both Newtonian and Oldroyd fluids.In Figure9, we consider the variation of the temperature with r for z=0.2,  Br=1,  Pr=1,  θ=0.5,φ=0.4,  Kn=0.02,  δ=0.156,  Re=10, and (Wi=0,0.04,0.08). The figure shows that an increase of the Weissenberg number lowers the temperature.The effects of the wave and Reynolds numbers on temperature distribution are shown in Figures10 and 11, respectively. One can observe that the temperature profiles increase with increasing both wave and Reynolds numbers.In Figure12, the temperature distribution is graphed versus r for z=0.2,  Re=10,  Pr=1,  θ=0.5,φ=0.4,Kn=0.02,Wi=0.04, and (Br=0.5,0.7,1). We notice that the temperature profile increases with increasing Brinkman number Br.


### 5.3. Concentration Profiles

Figures13–19 illustrate the behaviour of the fluid concentration for different values of the physical parameters. Figure 3 depicts the concentration field with the variation of r for z=0.2,  Br=1,Pr=1,θ=0.5,  φ=0.4,Re=10,  δ=0.02,  Wi=0.03,Sr=0.3,Sc=0.3, and (Kn=0,0.02,0.04). It is observed that the concentration profiles are increasing with increasing slip parameter Kn. This means that the concentration for slip flow is greater than for no-slip flow.Figures14, 15, and 16 show the effects of the Weissenberg, the wave, and the Reynolds numbers on the concentration profiles. It is seen that the concentration profiles increase as the Weissenberg number increases while it decreases by increasing the wave and the Reynolds numbers.The effects of the Brinkman, Soret, and Schmidt numbers on concentration field are shown in Figures17, 18, and 19 for different values of other physical parameters. The figures reveal that the concentration field decreases with increasing Br, Sr, and Sc.


### 5.4. Streamlines and Trapping Phenomenon

The phenomenon of trapping is another interesting topic in peristaltic transport. The formulation of an internally circulating bolus of the fluid by closed streamline is called trapping. This trapped bolus is pulled ahead along with the peristaltic wave. The effect of slip parameter on trapping can be seen in Figure20. It is observed that the trapping is symmetric about the centre line and the volume of the trapped bolus decreases with increasing Kn.Figures21 and 22 illustrate the effects of the wave and Reynolds numbers on the streamline at fixed values of other parameters. It is evident that the volume of the trapped bolus increases by increasing δ and Re. Moreover, we notice from the two figures that when δ andRe increase, another trapped bolus arises.Finally, Figure23 shows the graph of streamlines for θ=0.3,φ=0.3,  δ=0.05,  Re=1,  Kn=0.05, and (Wi=0,0.04,0.08). As shown, there is no effect of the Weissenberg number on the behaviour of streamlines.


## 5.1. Pumping Characteristics

The effect of the slip parameterKn on the pressure gradient for both Newtonian and Oldroyd fluids is shown in Figures 2 and 3, respectively. It is evident that the pressure gradient decreases by increasing the slip parameter Kn. Furthermore, from the two figures it can be noticed that in the wider part of the tube z∈[0,0.3] and [0.6, 1], the pressure gradient is small. This means that the flow can easily pass without imposition of a large pressure gradient. On the other hand, in the narrow part of the tube z∈[0.3,0.6], a large pressure gradient is required to maintain the flow to pass it.Figures4 and 5 illustrate the effect of wave number and Reynolds number on the pressure gradient of the Oldroyd fluid at θ=0.1,φ=0.6,Kn=0.05,Re=10,Wi=0.04,(δ=0,0.02,0.04) and θ=0.1,  φ=0.6,Kn=0.05,Wi=0.04,δ=0.01,(Re=0,15,30), respectively. The figures reveal that the pressure gradient increases by increasing both wave number and Reynolds number.Figure6 depicts the effect of the Weissenberg number Wi on the pressure gradient of the viscoelastic fluid at θ=0.1,φ=0.6,Kn=0.05,  Re=10, and δ=0.02. We can conclude that an increase in the Weissenberg number decreases the pressure gradient.


## 5.2. Temperature Distribution

Figures7–12 are devoted to explain the effect of emerging parameters on the temperature distribution. The effect of slip parameter Kn on the temperature distribution T at z=0.2,Br=1,  Pr=1,θ=0.5,φ=0.2 is shown in Figures 7 and 8 for both Newtonian and Oldroyd fluid, respectively. As shown, the temperature decreases as the slip parameter increases. The two figures also reveal that the behaviour of the temperature profiles is the same for both Newtonian and Oldroyd fluids.In Figure9, we consider the variation of the temperature with r for z=0.2,  Br=1,  Pr=1,  θ=0.5,φ=0.4,  Kn=0.02,  δ=0.156,  Re=10, and (Wi=0,0.04,0.08). The figure shows that an increase of the Weissenberg number lowers the temperature.The effects of the wave and Reynolds numbers on temperature distribution are shown in Figures10 and 11, respectively. One can observe that the temperature profiles increase with increasing both wave and Reynolds numbers.In Figure12, the temperature distribution is graphed versus r for z=0.2,  Re=10,  Pr=1,  θ=0.5,φ=0.4,Kn=0.02,Wi=0.04, and (Br=0.5,0.7,1). We notice that the temperature profile increases with increasing Brinkman number Br.


## 5.3. Concentration Profiles

Figures13–19 illustrate the behaviour of the fluid concentration for different values of the physical parameters. Figure 3 depicts the concentration field with the variation of r for z=0.2,  Br=1,Pr=1,θ=0.5,  φ=0.4,Re=10,  δ=0.02,  Wi=0.03,Sr=0.3,Sc=0.3, and (Kn=0,0.02,0.04). It is observed that the concentration profiles are increasing with increasing slip parameter Kn. This means that the concentration for slip flow is greater than for no-slip flow.Figures14, 15, and 16 show the effects of the Weissenberg, the wave, and the Reynolds numbers on the concentration profiles. It is seen that the concentration profiles increase as the Weissenberg number increases while it decreases by increasing the wave and the Reynolds numbers.The effects of the Brinkman, Soret, and Schmidt numbers on concentration field are shown in Figures17, 18, and 19 for different values of other physical parameters. The figures reveal that the concentration field decreases with increasing Br, Sr, and Sc.


## 5.4. Streamlines and Trapping Phenomenon

The phenomenon of trapping is another interesting topic in peristaltic transport. The formulation of an internally circulating bolus of the fluid by closed streamline is called trapping. This trapped bolus is pulled ahead along with the peristaltic wave. The effect of slip parameter on trapping can be seen in Figure20. It is observed that the trapping is symmetric about the centre line and the volume of the trapped bolus decreases with increasing Kn.Figures21 and 22 illustrate the effects of the wave and Reynolds numbers on the streamline at fixed values of other parameters. It is evident that the volume of the trapped bolus increases by increasing δ and Re. Moreover, we notice from the two figures that when δ andRe increase, another trapped bolus arises.Finally, Figure23 shows the graph of streamlines for θ=0.3,φ=0.3,  δ=0.05,  Re=1,  Kn=0.05, and (Wi=0,0.04,0.08). As shown, there is no effect of the Weissenberg number on the behaviour of streamlines.


## 6. Conclusion

The study examines the combined effect of slip velocity and heat and mass transfer on peristaltic transport of a viscoelastic fluid (Oldroyd fluid) in uniform tube. The problem can be considered as an application to the movement of chyme in small intestine. Using perturbation technique, analytical solutions for velocity, pressure gradient, temperature, and concentration fields have been derived without any restrictions on Reynolds number and Weissenberg number. The main results are summarized as follows.(1)
The pressure gradient decreases by increasing the slip parameterKn.(2)
The pressure gradient increases with increasing the wave and Reynolds numbers while it decreases with increasing the Weissenberg number.(3)
The temperature profiles decrease as the slip parameterKn increases.(4)
The temperature profiles decrease by increasing the Weissenberg number.(5)
The temperature profiles increase with increasing the wave, the Reynolds, and the Brinkman numbers.(6)
The concentration profiles are increasing when the slip parameter and Weissenberg number increase while it is decreasing when the wave, the Reynolds, the Brinkman, the Eckert, and the Soret numbers are increasing.(7)
The trapped bolus decreases with increasing slip parameter and increases with increasing wave number and Reynolds number.



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*Source: 815747-2012-12-25.xml*