AN EXACT SOLUTION OF NON-NEWTONIAN PERISTALTIC FLOW IN A TUBE : RABINOWITSCH FLUID MODEL

With the development of medical science, non-Newtonian fluids have taken on added significance with a variety of applications in real life. The flow of non-Newtonian fluids in tubes and pipes plays a vital role in daily life. The examples of such applications are medical instruments, human body, machines, etc. The study of such mechanisms with the Newtonian fluids has not been found satisfactory due to nonlinear behavior of stress strain relationship. In the present analysis, the study of peristaltic flow in a tube has been carried out taking into account the non-Newtonian fluid : Rabinowitsch fluid model. Considering the significant terms in Navier-Stokes equations, solutions have been derived for fluid flow in axial direction in terms of pressure gradient. Using the continuity of flow, and exact solution has been derived for fluid pressure at wall. To establish the applicability of the solution, results for pressure rise at wall, pressure gradient and streamlines have been presented graphically.


INTRODUCTION
In last few decades, the analysis of the peristaltic flows has played very important role in the development of medical science, and therefore received much attention of the bio-mechanics researchers, though, it has various applications engineering and industries too. Some important daily-life applications of peristaltic flow in a tube are sanitary fluid transport and transport of corrosive fluids, blood flow in vessels, blood pumps in heartlung machines, etc., and some examples of peristaltic flow in medical science can be found  Corresponding author Email: professorupsingh@gmail.com 2 in gall bladder, gastro-intestinal tract, female fallopian tube and medicine injector equipment. A more detail can be also found in well recognized articles by Mekheimer (2005), Mernone and Mazumdar (2002), Srinivas and Gayathri (2009), Misra and Pandey (1995), Li and Brasseur (1993), Vajravelu, Sreenadh and Babu (2006).
It has been found that the use of high molecular weight polymer solutions (viscosity index improvers) can give rise to a non-Newtonian fluid with minimized sensitivity of change in shearing strain rate (Spikes, 1994). However, the use of additives changes the stress-strain relationship of the fluid. To study the flow properties of such fluids, many traditional non-Newtonian models such as couple stress, power law, micropolar and Casson models are employed. Amongst these models, Rabinowitsch fluid model ) is an established model (Wada and Hayashi, 1971) to analyse the non-Newtonian behaviour of the fluid. The following stress-strain relation holds for Rabinowitsch fluid model for one dimensional fluid flow: where  is the initial viscosity and  is the non-linear factor responsible for the non-Newtonian effects of the fluid which will be referred to as coefficient of pseudoplasticity in this paper. This model can be applied to Newtonian lubricants for 0   , dilatant lubricants for 0   and pseudoplastic lubricants for 0   . The advantage of this model lies in the fact that the theoretical analysis for this model is verified with the experimental justification by Wada and Hayashi (1971). After Wada and Hayashi, many researchers used this model to theoretically analyse the performance characteristics of bearing performance with non-Newtonian lubricants (Bourging and Gay, 1984;Hashmimoto and Wada, 1986;Lin, 2001). Recently, this model was used by Singh et al. (2011a,b;2012a,b;2013) to investigate the performance of different types of hydrostatic, hydrodynamic and squeeze film bearing systems. Therefore, by the two reasonsfirst, Rabinowitsch fluid model fits a wide range of viscosity data (Wada and Hayashi, 1971), and second, none of the investigators have studied the flow characteristics of peristaltic flow in a tube with the Rabinowitsch fluid model, the present investigation is motivated.

ANALYSIS
The Schematic diagram of a peristaltic flow through a uniform tube is shown in  Figure 1) is defined as where, i r is the tube radius at inlet, a is the wave amplitude,  is the wavelength, c is propagation velocity and t is time.
The equations of motion of the flow in tube are and the related boundary conditions are where, u and w are velocity components in radial and axial directions, respectively;  is the density of fluid, ( , , ) ab a b r z   are components of stress and p is pressure.

Schematic diagram of peristaltic flow in a circular tube.
Taking the coordinate transformation z z ct  , w w c  together with the dimensionless parameters 2 , , , where, the terms of order  and higher have been dropped with the assumption that 1   , that is, the wavelength ()  is longer than the inlet radius () i r of the tube, which is a common situation in the creeping flow in medical science such as endoscopy.
Further, the boundary conditions (6-7) are transformed to Integrating the equation (13) By substituting equation (17) in equation (18) The exact solution of equation (19) is obtained as using Tartaglia's method for cubic The pressure rise can be calculated from equation (21)

CONCLUSION
The influence of non-Newtonian pseudoplastic and dilatant type fluids on the nature of peristaltic flow in a tube has been analyzed by means of Rabinowitsch fluid model. Analytical expressions for radial velocity, pressure gradient and pressure rise at walls have been obtained. Both the pseudoplastic as well as dilatant fluids showed significant influence on the flow behavior, however the influence is observed to be dependent on instantaneous flow rate.