Peristaltic Creeping Flow of Power Law Physiological Fluids through a Nonuniform Channel with Slip Effect

A mathematical study on creeping flow of non-Newtonian fluids (power law model) through a nonuniform peristaltic channel, in which amplitude is varying across axial displacement, is presented, with slip effects included. The governing equations are simplified by employing the long wavelength and low Reynolds number approximations. The expressions for axial velocity, stream function, pressure gradient, and pressure difference are obtained. Computational and numerical results for velocity profile, pressure gradient, and trapping under the effects of slip parameter, fluid behavior index, angle between the walls, and wave number are discussed with the help of Mathematica graphs. The present model is applicable to study the behavior of intestinal flow (chyme movement from small intestine to large intestine). It is also relevant to simulations of biomimetic pumps conveying hazardous materials, polymers, and so forth.


Introduction
The transportation of physiological fluids due to continuous wavelike muscle contraction and relaxation of physiological vessels such as the oesophagus, stomach, intestines, ureter and blood vessels (arteries, veins, capillaries, etc.), and other hollow tubes is known as peristalsis [1]. Peristalsis is used in many diverse applications in the human body. These include urodynamic conveyance from the kidneys to the bladder [2], swallowing of food through the esophagus, the movement of chyme in gastrointestinal tract, intrauterine fluid motion, and the flow of spermatozoa in the ductus efferentes of the male reproductive tract. Further applications include the movement of ovum in the female fallopian tube, transport of lymph in the lymphatic vessels, and the vasomotion of small blood vessels such arterioles, venules, and capillaries. These are all internal peristaltic mechanisms. In biolocomotion, earthworms also use peristalsis as an external motion achieving very efficient "geonautical" mobility, aided by the secretion of lubricating mucus. This also serves to subject the soil to continuous biological "pistons" forcing air through burrowed tunnels, promoting aeration and soil mixing, and encouraging mineralization of nutrients and their uptake by vegetation [3]. Roller and finger pumps also operate on this principle and furthermore modern microand nanorobots are exploiting peristaltic mechanisms [4].
Kwang et al. [22] studied the peristaltic transport of a Newtonian fluid through a 2D microchannel where the slip effect is present. Ali et al. [23] investigated slip effects on the peristaltic transport of variable viscosity magnetic fluid. Hayat et al. [24] studied slip effect on the peristaltic motion of a third-order rheological fluid in an asymmetric channel. Ebaid [25] analyzed effects of magnetic field and wall slip conditions on the peristaltic transport of a Newtonian fluid in an asymmetric channel. Recently Tripathi et al. [26] studied slip effects in fractional viscoelastic Oldroyd gastric flows using a homotopy method, showing that pressure is decreased with increasing slip.
It has been pointed out by Charm and Kurland [27,28] that the flow behavior of blood in vessels of small diameter (0.02 cm) and at low shear rates (<20 s −1 ) can be represented by a power law fluid. Also, it is found that physiological organs are generally nonuniform ducts [29,30]. Remaining cognizant of these facts, in this paper we investigate peristaltic transport of power law fluid in a nonuniform channel under a slip boundary condition. The effects of slip parameter, fluid behavior index, angle between the walls, and wave number on pumping characteristics and trapping phenomenon are investigated numerically and depicted graphically.

Mathematical Formulation
We consider the peristaltic flow of power law fluid in a nonuniform channel under a hydrodynamic slip boundary condition (see Figure 1). Let the motion of the walls of the channel be governed by a sinusoidal nonuniform wave which is mathematically modelled as where ℎ, , , , , are transverse vibration of the wall, half width of the channel, amplitude, wavelength, axial displacement, and angle between walls of channels, respectively. The sinusoidal nature of peristaltic waves is established in numerous clinical studies and we refer readers to the standard monograph Keener and Sneyd [31].
The governing equations of the motion of power law fluids (see, e.g., [6] for two-dimensional channel flow) are given by where , , are the shear stress components and , , V, , are the fluid density, axial velocity, transverse velocity, transverse coordinate, and pressure, respectively.
We introduce the following dimensionless parameters: where , , , are the wave velocity, wave number, amplitude ratio, and viscosity, respectively, and is the fluid behavior index (i.e., < 1 is pseudoplastic and > 1 is the dilatant fluid and = 1 is the Newtonian fluid). Using the above nondimensional variables and taking into account long wavelength and low Reynolds number approximation, after dropping the primes, the governing equations for flow of a power law fluid reduce to Applied Bionics and Biomechanics 3 where sign is a Signum function and is defined as The nondimensional wall equation in the wave frame is Boundary conditions in the wave frame of reference are specified thus: where (= / ) is the dimensionless slip parameter and is the dimensional slip parameter.

Analytical Solutions
Integrating (5) with respect to and using condition (9) we get Again integrating (11) with respect to and using condition (10) we get The stream function is defined, based on Cauchy-Riemann equations, as = , Using (12) and (13) we get The nondimensional volumetric flow rate in the wave frame is defined as Figure 1: Geometry of nonuniform peristaltic channel.
The pressure gradient is obtained from (15) as follows: Integrating (16) with respect to , the pressure difference across the axial line is

Numerical Results and Interpretation
In this section, numerical calculations executed on Mathematica software are presented via graphs, that is, We systematically study the effects of slip parameter ( ), rheological fluid power index ( ), angle between the walls ( ), and wave number ( ) on the velocity profile, pressure gradient, and the trapping phenomenon. The slip boundary condition defined in (10) is a Navier modification of the conventional no-slip condition. In certain physiological fluids, a partial nonadherence of the fluid to a solid boundary is observed. This constitutes momentum or velocity slip. This has been observed over four decades ago in celebrated clinical physiological testing studies with Weissenberg rheogoniometry for both blood and intestinal liquids [32,33]. As such, to provide a more realistic appraisal of actual peristaltic transport, a slip condition is advisable. the core region also and generates a consistent acceleration in the flow. In the absence of the momentum slip effect ( = 0) the magnitude is suppressed. The implication is that, with a slip effect, the axial flow distribution receives a nontrivial modification (acceleration) which is generally ignored in the majority of peristaltic flow models, and this can influence the efficiency of the peristaltic pumping. It may further be noted that with heat and species diffusion present (not studied in the current analysis) thermal jump (slip) and solutal slip (mass slip) at the deformable boundaries can also be incorporated and this is being considered by the authors for future investigations.  In all these graphs the channel is diverging ( > 0). The axial velocity is generally enhanced in magnitude with greater wall slip effect (Figure 3(a)), and the amplitudes are progressively increased with progressive distance from the apex of the channel. With greater geometric divergence of the channel, the peristaltic wave is allowed to grow considerably and axial flow is substantially accelerated with increasing slip. Conversely with greater power law index, owing to an elevation in biofluid viscosity, the momentum in the propulsion is opposed and the axial velocity is depleted, as observed in Figure 3(b). Dilatant ( > 0) biofluids clearly propel slower than pseudoplastic ( < 0) biofluids. Figure 3(c) reveals that as the channel apex angle (inclination) is increased, the axial velocity along the pumping direction is markedly accelerated again. Naturally with an expanding frontier to propel into, the waves grow and the biofluid accelerates. Finally in Figure 3(d), we find that, with greater wave number, the axial velocity magnitudes are enhanced for > 0 whereas they are decreased for < 0.  Figure 4(b). It is found that the pressure gradient increases with fluid behavior index. Pressure gradient for pseudoplastic biofluid is a minimum and it is maximum for dilatant fluid. Figure 4(c) shows the effect of inclination between the peristaltic walls on pressure gradient for various values of ( = /3, /4, /6) at fixed values = 0.5, = 1, = 0.1, = 1, = 1. It is found that the pressure gradient is maximum with small inclination and minimum with large inclination. Figure 4 index ( ), angle between the walls ( ), and wave number ( ) is that the trapped bolus increases in size as , , and increase. However, the size of the trapped bolus decreases in size as increases. Evidently slip exerts a nontrivial influence on pressure gradient, velocity, and bolus magnitude and growth.

Conclusions
In this study the two-dimensional peristaltic flow of a power law physiological fluid with the effect of slip condition through a nonuniform channel has been investigated. On the basis of computational and numerical results, the main findings of the present study are as follows: (i) Axial velocity across the transverse displacement is parabolic in nature and shifted in a downward direction increasing and and the converse behavior with and .
(ii) Pressure gradient across the axial length increases with slip parameter, fluid behavior index, and wave number and decreases with increasing inclination between walls.
(iii) The size of trapped bolus increases with , , and increase and decreases with .
The present study has ignored curvature effects of the physiological vessel which are important in clinical applications and also biomimetic pumps employed in chemical engineering. These introduce a Coriolis effect and can lead to secondary vortex effects. They have been studied by other authors [34] and will be addressed imminently.
Applied Bionics and Biomechanics  S t r e a mf u n c t i o n : Wavelength : Angle between walls of channels , , : Shear stress components : Fluid density.