Peristalsis is an essential mechanism of fluid transport in biological processes. Mathematical models on peristaltic transport with single fluid and their applications to the physiological fluid mechanics have been reported extensively in the literature (see Refs. [6, 13, 14, 26, 30]). The flow of physiological fluids in different organs of living body systems such as blood through micro-vessels, capillaries and veins; food through esophagus; chyme through small intestine; and the urine through ureter has a wall like structure, which perform the pumping is normally coated with a fluid of different properties from those fluids being pumped. In order to examine the influence of fluid coating on the transport, the single phase fluid analysis of peristaltic pumping has been extended to two phase fluid analysis.
At low shear rate, the rheology of human blood in micro-circulation with connecting vessels of diameter smaller than 500
The experimental investigations of Bugliarello and Sevilla [3], and Cokelet [4] reveal that the blood flow in small vessels is a two layered model in which the core layer is a region of suspension of all erythrocytes and the peripheral layer consisting of plasma. Srivastava and Srivastava [27, 28] reported the peristaltic flow of two layered fluid model in non-uniform tubes and they have interpreted this model to the ductus efferentes of reproductive tract and the blood flow in small vessels. Brasseur et al. [1] studied the influence of Newtonian peripheral layer on another Newtonian fluid of different viscosity. Brasseur [2] also modelled the mucous coating effects on the peristaltic esophageal food bolus transport with two-fluid model. Peristaltic flow of two immiscible viscous fluids in a circular tube has been addressed by Ramachandra Rao and Usha [23]. Also, Ramachandra Rao and Usha [32] extended their model to two layered Power-law fluids. Some investigators have engaged in the progress of peristaltic transport of two layered fluid models [15, 29 and 33].Very recently, Kavitha et al. [10] presented the peristaltic pumping of a Jeffrey fluid in contact with a Newtonian fluid in an inclined channel. All these authors have been reported the effects of interface shape and pressure rise with the time averaged flux.
The energy transfer in biological living systems can be identified in metabolic heat generation, blood perfusion, skin burning, hypothermia, fever, convectional heat exchange between blood and tissue. The variation of temperature strongly influences the non-Newtonian behavior of blood in various parts of the circulatory system. The results of heat transfer with peristaltic flow have many biomedical and bioengineering applications such as hemodialysis, oxygenation and hypothermia therapy. Considering multi-fluid flow situations and heat transfer, Umavathi et al. [31] studied the unsteady mixed convection flow of two immiscible viscous fluids through a channel characterized by one irregular wall and one flat wall. Farooq et al. [5] analyzed the two layered flow of third grade nano-fluids in a vertical channel with viscous dissipation effects. Very recently, Ponalagusamy and Selvi [22] addressed the combined effects of plasma layer thickness, heat transfer and magnetic field on the flow of blood through stenosed arteries.
In blood vessels, glycocalyx is a thin layer of glycoproteins and proteoglycans which covers the surface of endothelial cells. The increased endothelial cell layer permeability facilitates the accumulation of low density lipoproteins in the artery wall and accelerates atherosclerosis [12]. Investigations of peristaltic flow with porous walls provide insight into the disease involved in arteries and gastrointestinal tract. Vajravelu et al. [34] examined the influence of permeable wall on the peristaltic flow of a Casson fluid in contact with a Newtonian fluid in a circular pipe by considering the Saffman slip [25] in the absence of heat transfer. The flow of non-Newtonian fluids with slip effects and heat transfer has its wide range of applications in chemical and polymer processing. Nadeem and Akram [16] obtained the exact and numerical solutions for slip effects on the peristaltic transport of a Jeffrey fluid in an asymmetric channel under the influence of induced magnetic field. A few investigations of the effects of velocity slip and temperature jump conditions on the peristaltic flow of generalized Newtonian fluid models have been reported in [8, 18, 35].
In view of the above studies, an attempt is made in this paper to study the peristaltic transport of a two-layered fluid model consisting of a core region with a Jeffrey fluid and a peripheral layer with a Newtonian fluid. The effects of the velocity slip, temperature jump boundary conditions and the heat transfer in the channel are assessed. The results obtained throw light on the impact of peristalsis in the circulation of blood in small blood vessels and further may be useful in understanding the transport of physiological fluids in esophagus and gastrointestinal tract. Also, it is expected that the results obtained will not only provide useful information for industrial applications but also complement the earlier works.
The constitutive equations for an incompressible Jeffrey fluid are
where
We consider the peristaltic flow of incompressible and immiscible physiological fluids occupying core with a Jeffrey fluid of viscosity
The wall deformation due to infinite sinusoidal wave train of peristaltic wave is given by
where
The subsequent deformation of the interface separating the core and the peripheral layer is denoted by
The transformation from fixed to wave frame of references are given by
where
We introduce the following non-dimensional quantities;
Here superscript values (
Under the long wave length approximation (applicable in physiological flows), the Reynolds numberis small and hence the curvature and inertia terms are negligible. Thus the governing equations of two layered fluid flow reduce to (dropping bars)
Here
The velocity slip boundary condition Eq.(2.14) is used as in Saffman [25]. Further the velocity is continuous at the fluid interface. The temperature jump boundary condition is given in Eq.(2.18), where
Here the total flux
The non-dimensional average volume flow rate
Solving Eqs. (2.6)-(2.10) together with boundary conditions (2.11) -(2.20), we get
where
As
The axial pressure gradient is inferred from (2.6) or (2.7) as
Integrating Eq. (3.5) over one wavelength, we get the pressure rise (drop) over one cycle of wave as
where
The time average flux at zero pressure rise is denoted by
and the pressure rise required to produce zero average flow rate is denoted by Δ
The dimensionless frictional force at the wall across one wavelength is given by
where
The interface is also a stream line. For a given wave geometry and the time averaged flux
Since
The shape of interface obtained from Eq. (4.1) is a stream line in the wave frame. The unique interface
Equation (3.6) gives the expression for the pressure rise Δ
Equation (3.8) represents the maximum pressure difference Δ
Temperature field in core and peripheral layers are calculated from Eqs.(3.3) and (3.4) in terms of
The wave frame stream lines for core and peripheral regions are determined from Eqs. (3.1) and (3.2).Figs. 7–9 are drawn to observe the influences of
The present article is concerned with the analysis of peristalsis and heat transfer on the physiological flow of two-fluid model occupying the core region with a Jeffrey fluid and the peripheral region with a Newtonian fluid in a symmetric channel bounded by permeable walls.Analytical solutions for the stream function and the temperature field, interface, pressure rise, frictional force at the wall are determined. The obtained numerical results are presented through graphs and are discussed in detail. We have highlighted here some of the interesting observations.
The shape of interface for higher values of Jeffrey parameter
In the pumping region (Δ
For a given amplitude ratio, an increase in Jeffrey parameter
Temperature field decreases by increasing
An increase in the ratio of viscosity
The results of no slip conditions can be deduced from this analysis when
The size of the trapped bolus increases with an increase in the non-Newtonian Jeffrey number and decreases with an increase in the slip parameter.
It is observed that when