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

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.

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 𝜌div𝐕=0,𝑑𝐕𝑑𝑡=−∇𝑝+div𝝉, (2.1) 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] 𝜏_𝑖𝑗+Γ𝜕𝜏_𝑖𝑗+𝜕𝑡𝑔^𝑘𝑘𝑔_𝑖𝑖𝑔_𝑗𝑗𝑣_𝑘𝜕𝜕𝑥^𝑘𝑔^𝑖𝑖𝑔^𝑗𝑗𝜏_𝑖𝑗−𝑔^𝑘𝑘𝑔_𝑗𝑗𝜏_𝑘𝑗𝜕𝜕𝑥_𝑘𝑔^𝑖𝑖𝑣_𝑖−𝑔^𝑘𝑘𝑔_𝑖𝑖𝜏_𝑖𝑘𝜕𝜕𝑥_𝑘𝑔^𝑗𝑗𝑣_𝑗=−𝜇𝛾̇_𝑖𝑗, (2.2) in which 𝜏_𝑖𝑗,𝑖,𝑗,𝑘=1,2,3 are the components of the extra stress tensor, 𝑔_𝑖𝑖 and 𝑔^𝑗𝑗 are respectively the diagonal components of covariant and contravariant metric tensor, 𝑣_𝑖 are the velocity components, 𝜇 is the fluid viscosity, Γ is relaxation time, and 𝛾̇_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 temperature 𝑇₀ and concentration 𝐶₀. In the fixed cylindrical coordinate system (𝑅,𝑍), the geometry of the problem, as can be seen in Figure 1, is ℎ𝑍,𝑡=𝑎+𝑏sin2𝜋𝜆𝑍−𝑐𝑡, (3.1) 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 𝑟=𝑅,𝑧=𝑍−𝑐𝑡,𝑢=𝑈,𝑤=𝑊−𝑐, (3.2) 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 𝜏₁₁+Γ𝑢𝜕𝜏₁₁𝜕𝑟+𝑤𝜕𝜏₁₁𝜕𝑧−2𝜏₁₁𝜕𝑢𝜕𝑟−2𝜏₁₃𝜕𝑢𝜕𝑧=−𝜇𝛾̇₁₁,𝜏₁₃+Γ𝑢𝜕𝜏₁₃𝜕𝑟+𝑤𝜕𝜏₁₃𝜕𝑧−𝜏₃₃𝜕𝑢𝜕𝑧−𝜏₁₁𝜕𝑤𝜕𝑟+𝑢𝑟𝜏₁₃=−𝜇𝛾̇₁₃,𝜏₂₂+Γ𝑢𝜕𝜏₂₂𝜕𝑟+𝑤𝜕𝜏₂₂𝜕𝑧−2𝑢𝑟𝜏₂₂=−𝜇𝛾̇₂₂,𝜏₃₃+Γ𝑢𝜕𝜏₃₃𝜕𝑟+𝑤𝜕𝜏₃₃𝜕𝑧−2𝜏₃₃𝜕𝑤𝜕𝑧−2𝜏₁₃𝜕𝑤𝜕𝑟=−𝜇𝛾̇₃₃, (3.3) where the components of the strain-rate tensor are given by 𝛾̇₁₁𝜕=2𝑢𝜕𝑟,𝛾̇₂₂=2𝑢𝑟,𝛾̇₃₃𝜕=2𝑤𝜕𝑧,𝛾̇₁₃=𝜕𝑢𝜕𝑧+𝜕𝑤𝜕𝑟. (3.4) Using the following non-dimensional variables and parameters: 𝑟=𝑟𝑎,𝑧=𝑧𝜆,𝑤=𝑤𝑐𝜆,𝑢=𝑢,𝛾̇𝑎𝑐_𝑖𝑗=𝑎𝑐𝛾̇_𝑖𝑗𝑎,𝑝=²𝑝𝑐𝑐𝜆𝜇,𝑡=𝑡𝜆𝑎,𝛿=𝜆,Re=𝜌𝑐𝑎𝜇,𝜏_𝑖𝑗=𝑎𝜏_𝑖𝑗𝑐𝜇,Wi=𝑐Γ𝑎,𝑇=𝑇𝑇₀,𝐶=𝐶𝐶₀,𝑃𝑟=𝜇𝑐_𝑝𝑘𝑐,𝐸=²𝑇₀𝑐_𝑝,Sr=𝜌𝐷_𝑚𝐾_𝑇𝑇₀𝜇𝑇_𝑚𝐶₀𝜇,Sc=𝜌𝐷_𝑚,ℎ=ℎ𝑎𝑏=1+𝑎sin2𝜋𝑧=1+𝜑sin2𝜋𝑧, (3.5) we obtain the non-dimensional continuity equation, momentum equations, constitutive equations, energy equation, and concentration equation as follows: 1𝑟𝜕()+𝜕𝑟𝑟𝑢𝜕𝑤𝜕𝑧=0,Re𝛿³𝑢𝜕𝑢𝜕𝑟+𝑤𝜕𝑢𝜕𝑧=−𝜕𝑝1𝜕𝑧−𝛿𝑟𝜕𝜕𝑟𝑟𝜏₁₁+𝛿𝜕𝜏₁₃−𝜏𝜕𝑧𝑟²²,Re𝛿𝑢𝜕𝑤𝜕𝑟+𝑤𝜕𝑤𝜕𝑧=−𝜕𝑝−1𝜕𝑧𝑟𝜕𝜕𝑟𝑟𝜏₁₃+𝛿𝜕𝜏₃₃,𝜏𝜕𝑧₁₁+𝛿Wi𝑢𝜕𝜏₁₁𝜕𝑟+𝑤𝜕𝜏₁₁𝜕𝑧−2𝜏₁₁𝜕𝑢𝜕𝑟−2𝛿𝜏₁₃𝜕𝑢𝜕𝑧=−2𝛿𝜕𝑢,𝜏𝜕𝑟₁₃+Wi𝛿𝑢𝜕𝜏₁₃𝜕𝑟+𝑤𝜕𝜏₁₃𝜕𝑧−𝛿𝜏₃₃𝜕𝑢+𝑢𝜕𝑧𝑟𝜏₁₃−𝜏₁₁𝜕𝑤𝛿𝜕𝑟=−²𝜕𝑢+𝜕𝑧𝜕𝑤,𝜏𝜕𝑟₂₂+𝛿Wi𝑢𝜕𝜏₂₂𝜕𝑟+𝑤𝜕𝜏₂₂−𝜕𝑧2𝑢𝑟𝜏₂₂𝑢=−2𝛿𝑟,𝜏₃₃+𝑊𝑖𝛿𝑢𝜕𝜏₃₃𝜕𝑟+𝑤𝜕𝜏₃₃𝜕𝑧−2𝜏₃₃𝜕𝑤𝜕𝑧−2𝜏₁₃𝜕𝑤𝜕𝑟=−2𝛿𝜕𝑤,𝑢𝜕𝑧𝛿Pr𝜕𝑇𝜕𝑟+𝑤𝜕𝑇=1𝜕𝑧𝑟𝜕𝑟𝜕𝑟𝜕𝑇𝜕𝑟+𝛿²𝜕²𝑇𝜕𝑧²−Br𝛿𝜏₁₁𝜕𝑢𝜕𝑟+𝜏₁₃𝜕𝑤𝜕𝑟+𝛿²𝜏₃₁𝜕𝑢𝜕𝑧+𝛿𝜏₃₃𝜕𝑤,𝑢𝜕𝑧𝛿Re𝜕𝐶𝜕𝑟+𝑤𝜕𝐶=1𝜕𝑧1Sc𝑟𝜕𝑟𝜕𝑟𝜕𝐶𝜕𝑟+𝛿²𝜕²𝐶𝜕𝑧²1+Sr𝑟𝜕𝑟𝜕𝑟𝜕𝑇𝜕𝑟+𝛿²𝜕²𝑇𝜕𝑧², (3.6) where 𝐶 is the concentration of the fluid, 𝑇 is the temperature, 𝑇_𝑚 is the temperature of the medium, 𝐷_𝑚 is the coefficient of mass diffusivity, 𝐾_𝑇 is the thermal diffusion ratio, 𝜇 is the viscosity, 𝑐_𝑝 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 𝑢=0,𝜕𝑤𝜕𝑟=0,𝜕𝑇𝜕𝑟=0,𝜕𝐶𝜕𝑟=0,at𝑟=0,𝑤=−1+𝐾_𝑛𝜏₁₃,𝑢=𝑑ℎ𝑑𝑧,𝑇=1,𝐶=1,at𝑟=ℎ, (3.7) where 𝐾_𝑛=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 parameter 𝛿 as follows: 𝑤=𝑤₀+𝛿𝑤₁+𝛿²𝑤₂+𝑂𝛿³,𝑢=𝑢₀+𝛿𝑢₁+𝛿²𝑢₂+𝑂𝛿³,𝑝=𝑝₀+𝛿𝑝₁+𝛿²𝑝₂+𝑂𝛿³,𝑓=𝑓₀+𝛿𝑓₁+𝛿²𝑓₂+𝑂𝛿³,𝜏_𝑖𝑗=𝜏⁽⁰⁾_𝑖𝑗+𝛿𝜏⁽¹⁾_𝑖𝑗+𝛿²𝜏⁽²⁾_𝑖𝑗+𝑂𝛿³,𝑖,𝑗=1,2,3,𝑇=𝑇₀+𝛿𝑇₁+𝛿²𝑇₂+𝑂𝛿³,𝐶=𝐶₀+𝛿𝐶₁+𝛿²𝐶₂+𝑂𝛿³, (4.1) where 𝑓=2∫^ℎ₀𝑟𝑤𝑑𝑟 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] 𝜑𝜃=𝑓+1+²2. (4.2) 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 1𝑟𝜕𝜕𝑟𝑟𝑢₀+𝜕𝑤₀𝜕𝑧=0,𝜕𝑝₀𝜕𝑟=0,𝜕𝑝₀=1𝜕𝑧𝑟𝜕𝑟𝜕𝑟𝜕𝑤₀,1𝜕𝑟𝑟𝜕𝑟𝜕𝑟𝜕𝑇₀𝜕𝑟=−Br𝜕𝑤₀𝜕𝑟²,10=1Sc𝑟𝜕𝑟𝜕𝑟𝜕𝐶₀1𝜕𝑟+Sr𝑟𝜕𝑟𝜕𝑟𝜕𝑇₀,𝜕𝑟 (4.3) with the boundary conditions 𝜕𝑤₀𝜕𝑟=0,𝜕𝑇₀𝜕𝑟=0,𝜕𝐶₀𝑤𝜕𝑟=0,at𝑟=0,₀=−1−𝐾_𝑛𝜕𝑤₀𝜕𝑟,𝑇₀=1,𝐶₀=1,at𝑟=ℎ(𝑧). (4.4)

The solution of (4.3), subject to the boundary conditions (4.4), is 𝑤₀(𝑟,𝑧)=𝑎₁𝑟²+𝑎₂,𝑢₀(𝑟,𝑧)=𝑎₃𝑟³+𝑎₄𝑟,𝑑𝑝₀8𝑓𝑑𝑧=−⁰+ℎ²ℎ⁴+4ℎ³𝐾_𝑛.𝑇₀()1𝑟,𝑧=−Br64𝑑𝑝₀𝑑𝑧²𝑟⁴−ℎ⁴𝐶+1,₀()1𝑟,𝑧=SrScBr64𝑑𝑝₀𝑑𝑧²𝑟⁴−ℎ⁴+1, (4.5) where 𝑎₁, 𝑎₂, 𝑎₃,  and 𝑎₄ are stated in the appendix.

4.2. First Order System

Consider 1𝑟𝜕𝜕𝑟𝑟𝑢₁+𝜕𝑤₁𝜕𝑧=0,𝜕𝑝₁𝑢𝜕𝑟=0,Re₀𝜕𝑤₀𝜕𝑟+𝑤₀𝜕𝑤₀𝜕𝑧=−𝜕𝑝₁−1𝜕𝑧𝑟𝜕𝜕𝑟𝑟𝜏⁽¹⁾₁₃𝜕+2Wi𝜕𝑧𝜕𝑤₀𝜕𝑟²,𝜏⁽¹⁾₁₃=−𝜕𝑤₁𝑢𝜕𝑟+Wi₀𝜕²𝑤₀𝜕𝑟²+𝑤₀𝜕²𝑤₀+𝑢𝜕𝑟𝜕𝑧𝑟⁰𝜕𝑤₀𝜕𝑟−2𝜕𝑢₀𝜕𝑟𝜕𝑤₀,1𝜕𝑟𝑟𝜕𝑟𝜕𝑟𝜕𝑇₁𝑢𝜕𝑟=RePr₀𝜕𝑇₀𝜕𝑟+𝑤₀𝜕𝑇₀𝜏𝜕𝑧+Br⁽¹⁾₁₃𝜕𝑤₀−𝜕𝑟𝜕𝑤₀𝜕𝑟𝜕𝑤₁𝜕𝑟−2Wi𝜕𝑤₀𝜕𝑟²𝜕𝑤₀,𝑢𝜕𝑧Re₀𝜕𝐶₀𝜕𝑟+𝑤₀𝜕𝐶₀=1𝜕𝑧1Sc𝑟𝜕𝑟𝜕𝑟𝜕𝐶₁1𝜕𝑟+Sr𝑟𝜕𝑟𝜕𝑟𝜕𝑇₁.𝜕𝑟 (4.6)(4.7)(4.8)(4.9)(4.10)(4.11)

The boundary conditions are 𝜕𝑤₁𝜕𝑟=0,𝜕𝑇₁𝜕𝑟=0,𝜕𝐶₁𝑤𝜕𝑟=0,at𝑟=0,₁=𝐾_𝑛𝜏⁽¹⁾₁₃,𝑇₁=0,𝐶₁=0at𝑟=ℎ(𝑧). (4.12)(4.13)

Substituting the zero order solution into (4.9), we obtain 𝜏⁽¹⁾₁₃ in the form 𝜏⁽¹⁾₁₃=−𝜕𝑤₁𝜕𝑟+𝑊𝑖2𝑎₁𝑎^′₁−8𝑎₁𝑎₃𝑟³+2𝑎₂𝑎^′₁𝑟. (4.14)

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 𝜕𝑤₁=1𝜕𝑟2𝑑𝑝₁𝑑𝑧𝑟+Wi−24𝑎₁𝑎₃+𝑎₁𝑎^′₁𝑟³+2𝑎₂𝑎^′₁𝑟+Re2𝑎₁𝑎₃+𝑎₁𝑎^′₁6𝑟⁵+2𝑎₁𝑎₄+𝑎₁𝑎^′₂+𝑎₂𝑎^′₁4𝑟³+𝑎₂𝑎^′₂2𝑟. (4.15)

The solution of (4.15), subject to the boundary condition (4.13) is given by 𝑤₁()=1𝑟,𝑧4𝑑𝑝₁𝑟𝑑𝑧²−ℎ²−2𝐾_𝑛ℎ+Re𝑎₅𝑟⁶+𝑎₆𝑟⁴+𝑎₇𝑟²+𝑎₈+Wi𝑎₉𝑟⁴+𝑎₁₀𝑟²+𝑎₁₁, (4.16) where 𝑎₅,…𝑎₁₁ are stated in the appendix.

The dimensionless mean flow rate in the wave frame 𝑓₁ is given by 𝑓₁=2^ℎ₀𝑟𝑤₁1𝑑𝑟=−8𝑑𝑝₁ℎ𝑑𝑧⁴𝑎+Re⁵4ℎ⁸+𝑎₆3ℎ⁶+𝑎₇2ℎ⁴+𝑎₈ℎ²𝑎+Wi⁹3ℎ⁶+𝑎₁₀2ℎ⁴+𝑎₁₁ℎ². (4.17)

On solving (4.17) for 𝑑𝑝₁/𝑑𝑧, one finds 𝑑𝑝₁𝑑𝑧=−8𝑓₁ℎ⁴+4𝐾_𝑛ℎ³−Reℎ⁴+4𝐾_𝑛ℎ³2𝑎₅ℎ⁸+83𝑎₆ℎ⁶+4𝑎₇ℎ⁴+8𝑎₈ℎ²−Wiℎ⁴+4𝐾_𝑛ℎ³83𝑎₉ℎ⁶+4𝑎₁₀ℎ⁴+8𝑎₁₁ℎ². (4.18)

Using zero order solution together with the first order solution of 𝑤₁(𝑟,𝑧), (4.16), into (4.10), (4.11) and applying the boundary conditions, we get 𝑇₁(𝑟,𝑧)=−Br32𝑑𝑝₀𝑑𝑧𝑑𝑝₁𝑟𝑑𝑧⁴−ℎ⁴−BrRe𝑎₁₂𝑟⁸−ℎ⁸+𝑎₁₃𝑟⁶−ℎ⁶+𝑎₁₄𝑟⁴−ℎ⁴+𝑎₁₅𝑟²−ℎ²−BrWi𝑎₁₆𝑟⁶−ℎ⁶+𝑎₁₇𝑟⁵−ℎ⁵+𝑎₁₈𝑟⁴−ℎ⁴+𝑎₁₉𝑟³−ℎ³,𝐶₁(𝑟,𝑧)=BrSrSc32𝑑𝑝₀𝑑𝑧𝑑𝑝₁𝑟𝑑𝑧⁴−ℎ⁴+BrSrScRe𝑎₂₀𝑟⁸−ℎ⁸+𝑎₂₁𝑟⁶−ℎ⁶+𝑎₂₂𝑟⁴−ℎ⁴+𝑎₂₃𝑟²−ℎ²+BrSrScWi𝑎₁₆𝑟⁶−ℎ⁶+𝑎₁₇𝑟⁵−ℎ⁵+𝑎₁₈𝑟⁴−ℎ⁴+𝑎₁₉𝑟³−ℎ³, (4.19)(4.20) where 𝑎₁₂,…𝑎₁₉ are defined in the appendix.

The results of our analysis can be expressed to first order by defining 𝑓=𝑓+𝛿𝑓₁, (4.21) then substituting into zero and first order solutions and neglecting all terms of higher than 𝑂(𝛿), we find 𝑤()=1𝑟,𝑧4𝑑𝑝𝑟𝑑𝑧²−ℎ²−2𝐾_𝑛ℎ−1+𝛿Re𝑏₅𝑟⁶+𝑏₆𝑟⁴+𝑏₇𝑟²+𝑏₈+𝛿Wi𝑏₉𝑟⁴+𝑏₁₀𝑟²+𝑏₁₁,𝑑𝑝𝑑𝑧=−8𝑓+ℎ²ℎ⁴+4𝐾_𝑛ℎ³−𝛿Reℎ⁴+4𝐾_𝑛ℎ³2𝑏₅ℎ⁸+83𝑏₆ℎ⁶+4𝑏₇ℎ⁴+8𝑏₈ℎ²−𝛿Wiℎ⁴+4𝐾_𝑛ℎ³83𝑏₉ℎ⁶+4𝑏₁₀ℎ⁴+8𝑏₁₁ℎ²,𝑇(𝑟,𝑧)=−Br64𝑑𝑝𝑑𝑧²𝑟⁴−ℎ⁴𝑏+1−𝛿BrRe₁₂𝑟⁸−ℎ⁸+𝑏₁₃𝑟⁶−ℎ⁶+𝑏₁₄𝑟⁴−ℎ⁴+𝑏₁₅𝑟²−ℎ²+Wi𝑏₁₆𝑟⁶−ℎ⁶+𝑏₁₇𝑟⁵−ℎ⁵+𝑏₁₈𝑟⁴−ℎ⁴+𝑏₁₉𝑟³−ℎ³,𝐶(𝑟,𝑧)=BrSrSc64𝑑𝑝𝑑𝑧²𝑟⁴−ℎ⁴𝑏+1+𝛿BrSrScRe₂₀𝑟⁸−ℎ⁸+𝑏₂₁𝑟⁶−ℎ⁶+𝑏₂₂𝑟⁴−ℎ⁴+𝑏₂₃𝑟²−ℎ²+Wi𝑏₁₆𝑟⁶−ℎ⁶+𝑏₁₇𝑟⁵−ℎ⁵+𝑏₁₈𝑟⁴−ℎ⁴+𝑏₁₉𝑟³−ℎ³, (4.22) 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]. 𝑎=1.25cm,𝜆=8.01cm,𝛿=𝑎/𝜆=0.156. (5.1)

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, Figures 2–23 are prepared using the MATHEMATICA package.

Figure 2 

Pressure gradient versus 𝑧 for 𝜙=0.6,  Wi=0,  Re=0,  𝛿=0,  𝜃=0.1.

Figure 3 

Pressure gradient versus 𝑧 for 𝜑=0.6,  Wi=0.04,  Re=10,  𝛿=0.02,  𝜃=0.1.

Figure 4 

Pressure gradient versus 𝑧 for 𝜑=0.6,  Wi=0.04,  Re=10,  𝜃=0.1,  𝐾𝑛=0.05.

Figure 5 

Pressure gradient versus 𝑧 for 𝜑=0.6,  Wi=0.04,  𝛿=0.01,  𝜃=0.1,  𝐾𝑛=0.05.

Figure 6 

Pressure gradient versus 𝑧 for 𝜑=0.6,  Re=10,  𝛿=0.02,  𝜃=0.1,  𝐾𝑛=0.05.

Figure 7 

Temperature distribution for 𝑧=0.2,  Br=1,  Pr=1,  𝛿=0,  Re=0,  Wi=0,  𝜃=0.5,  𝜙=0.2.

Figure 8 

Temperature distribution for 𝑧=0.2  ,Br=1,  Pr=1,  𝛿=0.02,  Re=10,  Wi=0.03,  𝜃=0.5,  𝜑=0.2.

Figure 9 

Temperature distribution for 𝑧=0.2,  Br=1,  Pr=1,  𝛿=0.156,  Re=10,  𝐾𝑛=0.02,  𝜃=0.5,  𝜑=0.4.

Figure 10 

Temperature distribution for 𝑧=0.2,  Br=1,  𝐾𝑛=0.02,  Pr=1,  Re=10,  Wi=0.04,  𝜃=0.5,  𝜑=0.4.

Figure 11 

Temperature distribution for 𝑧=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 for 𝑧=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 for 𝑧=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 for 𝑧=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 for 𝑧=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 for 𝑧=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 for 𝑧=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 for 𝑧=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 for 𝑧=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.

(a)

(b)

(c)

Figure 20 

Streamlines for Wi=0.03,Re=10,𝛿=0.02,𝜃=0.3,𝜑=0.4,𝐾𝑛=(0,0.05,0.1).

(a)

(b)

(c)

Figure 21 

Streamlines for Wi=0.03,  Re=10,  𝐾𝑛=0.05,  𝜃=0.3,  𝜑=0.3,  𝛿=(0,0.03,0.05).

(a)

(b)

(c)

Figure 22 

Streamlines for Wi=0.03,𝐾𝑛=0.05,𝛿=0.03,𝜃=0.3,𝜑=0.3,Re=(0,10,20).

(a)

(b)

(c)

Figure 23 

Streamlines for Re=1,𝛿=0.05,  𝜑=0.3,𝜃=0.3,𝐾𝑛=0.05,Wi=(0,0.04,0.08).

5.1. Pumping Characteristics

The effect of the slip parameter 𝐾_𝑛 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 𝐾_𝑛. 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.

Figures 4 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,𝐾_𝑛=0.05,Re=10,Wi=0.04,(𝛿=0,0.02,0.04) and 𝜃=0.1,  𝜑=0.6,𝐾_𝑛=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.

Figure 6 depicts the effect of the Weissenberg number Wi on the pressure gradient of the viscoelastic fluid at 𝜃=0.1,𝜑=0.6,𝐾_𝑛=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

Figures 7–12 are devoted to explain the effect of emerging parameters on the temperature distribution. The effect of slip parameter 𝐾_𝑛 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 Figure 9, we consider the variation of the temperature with 𝑟 for 𝑧=0.2,  Br=1,  Pr=1,  𝜃=0.5,𝜑=0.4,  𝐾_𝑛=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 Figures 10 and 11, respectively. One can observe that the temperature profiles increase with increasing both wave and Reynolds numbers.

In Figure 12, the temperature distribution is graphed versus 𝑟 for 𝑧=0.2,  Re=10,  Pr=1,  𝜃=0.5,𝜑=0.4,𝐾_𝑛=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

Figures 13–19 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 (𝐾_𝑛=0,0.02,0.04). It is observed that the concentration profiles are increasing with increasing slip parameter 𝐾_𝑛. This means that the concentration for slip flow is greater than for no-slip flow.

Figures 14, 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 Figures 17, 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 Figure 20. It is observed that the trapping is symmetric about the centre line and the volume of the trapped bolus decreases with increasing 𝐾_𝑛.

Figures 21 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, Figure 23 shows the graph of streamlines for 𝜃=0.3,𝜑=0.3,  𝛿=0.05,  Re=1,  𝐾_𝑛=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 parameter 𝐾_𝑛.(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 parameter 𝐾_𝑛 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.

Appendix

𝑎₁2=−2𝑓₀+ℎ²ℎ⁴+4ℎ³𝐾_𝑛,𝑎₂=−𝑎₁ℎ²+2ℎ𝐾_𝑛−1,𝑎₃𝑎=−^′14,𝑎₄𝑎=−^′22,𝑎₅=2𝑎₁𝑎₃+𝑎₁𝑎^′₁36,𝑎₆=2𝑎₁𝑎₄+𝑎₁𝑎^′₂+𝑎₂𝑎^′₁16,𝑎₇=𝑎₂𝑎^′₂4,𝑎₈=−ℎ⁶+6𝐾_𝑛ℎ⁵𝑎₅−ℎ⁴+4𝐾_𝑛ℎ³𝑎₆−ℎ²+2𝐾_𝑛ℎ𝑎₇,𝑎₉=−4𝑎₁𝑎₃+𝑎₁𝑎^′₁2,𝑎₁₀=𝑎₂𝑎^′₁,𝑎₁₁=−ℎ⁴+4𝐾_𝑛ℎ³𝑎₉−ℎ²+2𝐾_𝑛ℎ𝑎₁₀,𝑎₁₂=9𝑎₁𝑎₅32Pr+3𝑎₁𝑎₅8,𝑎₁₃=2𝑎²₁𝑎₄+𝑎₁𝑎₂𝑎^′₁72Pr+4𝑎₁𝑎₆9,𝑎₁₄=𝑎₁𝑎₇2−𝑎²₁𝑎^′₁ℎ⁴32Pr,𝑎₁₅𝑎=−¹𝑎₁₀8𝑎Pr,₁₆=4𝑎₁𝑎₉+2𝑎²₁𝑎^′₁9,𝑎₁₇=8𝑎₁𝑎₃−2𝑎₁𝑎^′₁25,𝑎₁₈=𝑎₁𝑎₁₀+𝑎²₁𝑎^′₂2,𝑎₁₉=−2𝑎₁₀9,𝑎₂₀=𝑎₁₂+9𝑎₁𝑎₅32Sc,𝑎₂₁=𝑎₁₃+2𝑎²₁𝑎₄+𝑎₁𝑎^′₁𝑎₂𝑎72Sc,₂₂=𝑎₁₄−𝑎²₁𝑎^′₁ℎ⁴32Sc,𝑎₂₃=𝑎₁₅−𝑎₁𝑎₁₀ℎ⁴8𝑏Sc,₁2=−2𝑓+ℎ²ℎ⁴+4ℎ³𝐾_𝑛,𝑏₂=−𝑏₁ℎ²+2ℎ𝐾_𝑛−1,𝑏₃𝑏=−^′14,𝑏₄𝑏=−^′22,𝑏₅=2𝑏₁𝑏₃+𝑏₁𝑏^′₁36,𝑏₆=2𝑏₁𝑏₄+𝑏₁𝑏^′₂+𝑏₂𝑏^′₁16,𝑏₇=𝑏₂𝑏^′₂4,𝑏₈=−ℎ⁶+6𝐾_𝑛ℎ⁵𝑏₅−ℎ⁴+4𝐾_𝑛ℎ³𝑏₆−ℎ²+2𝐾_𝑛ℎ𝑏₇,𝑏₉=−4𝑏₁𝑏₃+𝑏₁𝑏^′₁2,𝑏₁₀=𝑏₂𝑏^′₁,𝑏₁₁=−ℎ⁴+4𝐾_𝑛ℎ³𝑏₉−ℎ²+2𝐾_𝑛ℎ𝑏₁₀,𝑏₁₂=9𝑏₁𝑏₅32Pr+3𝑏₁𝑏₅8,𝑏₁₃=2𝑏²₁𝑏₄+𝑏₁𝑏₂𝑏^′₁72Pr+4𝑏₁𝑏₆9,𝑏₁₄=𝑏₁𝑏₇2−𝑏²₁𝑏^′₁ℎ⁴32Pr,𝑏₁₅𝑏=−¹𝑏₁₀8𝑏Pr,₁₆=4𝑏₁𝑏₉+2𝑏²₁𝑏^′₁9,𝑏₁₇=8𝑏₁𝑏₃−2𝑏₁𝑏^′₁25,𝑏₁₈=𝑏₁𝑏₁₀+𝑏²₁𝑏^′₂2,𝑏₁₉=−2𝑏₁₀9,𝑏₂₀=𝑏₁₂+9𝑏₁𝑏₅32Sc,𝑏₂₁=𝑏₁₃+2𝑏²₁𝑏₄+𝑏₁𝑏^′₁𝑏₂𝑏72Sc,₂₂=𝑏₁₄−𝑏²₁𝑏^′₁ℎ⁴32Sc,𝑏₂₃=𝑏₁₅−𝑏₁𝑏₁₀ℎ⁴8Sc. (A.1)

Acknowledgments

The author would like to thank Professor Dr. Engineer Rolf Radespiel, Head of the Institute of Fluid Mechanics, TU Braunschweig, for his invaluable discussions and constant support. Furthermore, the author thanks the DAAD for financial support to his research visit to Germany where this work was accomplished.