Theoretical Study of Heat Transfer on Peristaltic Transport of Non- Newtonian Fluid Flowing in a Channel: Rabinowitsch Fluid Model

The present investigation is concerned with the problem of heat transfer and peristaltic flow of non-Newtonian fluid using Rabinowitsch fluid model through a channel under long wavelength and low Reynolds number approximation. Expressions for velocity, pressure gradient, pressure rise, friction force and temperature have been obtained. The effect of different parameters on velocity, pressure gradient, pressure rise, streamlines, friction force and temperature have been discussed through graphs. KeywordsPeristalsis, Uniform channel, Heat transfer, Rabinowitsch fluid model.


Introduction
Peristalsis is an innate property of several biological systems and it occurs in the movement of urine from the kidney to bladder, vasomotion of small blood vessels, ovum transport in the fallopian tube and movement of chyme in the gastro-intestinal tract. It is an important transport mechanism not only in physiological processes but has a wide range of applications in engineering and industry. The principle of peristalsis has been used to develop pumps having physiological and industrial applications.
Probably the first pioneering investigation was done by Latham (1966) in which a theoretical attempt has been made to understand the peristaltic action. Fung and Yih, (1968), Shapiro et al. (1969), Yin and Fung (1969), Gupta and Seshadri (1976) and Machireddy and Kattamreddy (2016) have made a significant contribution to understand the peristaltic flow of Newtonian fluid. Theory of non-Newtonian fluid has attracted researchers during past few decades. To understand the flow properties of physiological fluids, Raju and Devanathan (1972) considered a non-Newtonian model (Power law fluid) of peristaltic motion in an axisymmetric tube caused due to the proliferation of sinusoidal wave on the walls. Considering the arbitrary wave shape, Kaimal (1978) studied the peristaltic motion of a particle-fluid mixture in an axisymmetrical tube at low Reynolds number. Misra and Pandey (2002) studied the peristaltic transport of blood in small blood vessels. In this study, the core layer is casson fluid and the outer layer is an incompressible Newtonian viscous fluid. Vajravelu et al. (2005) presented peristaltic pumping of incompressible fluid with Herschel-Bulkley fluid model in a channel. The analytical solution for velocity, pressure gradient and stream function were determined. Pandey and Chaube (2011) investigated wall properties during the action of peristaltic motion of couple stress fluid and disclosed the result that means velocity reduces by raising couple stress parameter and increases with increasing wall tension. Reddy et al. (2011) investigated the peristaltic motion of Carreau fluids through an inclined channel in presence of magnetic field. The perturbation method is used to analyze the flow. Akbar and Butt (2015) studied heat transfer under the action of the peristaltic flow of viscous fluid with Herschel-Bulkley fluid model while considering that fluid flowing through a non-uniform inclined channel. Study of peristaltic flow of non-Newtonian fluid with power law fluid model through the channel of varying width was done by Chaube et al. (2015). Consequences of changes in fluid behavior index, slip parameter and angle between the walls on velocity profile, pressure gradient and trapping phenomenon were discussed.
The study of heat transfer in association with peristalsis is essential because it plays a significant role in physiology. For instance, the thermodynamic features of blood are very useful in oxygenation and hemodialysis. Radhakrishnamacharya and Murty (1993) studied the heat transfer in the connection to peristaltic transport in a channel of varying width and found a perturbed solution for temperature and heat transfer coefficient. The problem of peristaltic pumping and heat transfer through an asymmetric channel was considered by Srinivas and Kothandapani (2008). Sinha et al. (2015) studied heat transfer analysis MHD flow through the asymmetric channel with variable viscosity. Bhatt et al. (2017) recently studied peristaltic transport and heat transfer through non-uniform geometry. Walls are assumed to be permeable and found that temperature decreases with increase in Darcy number. The study of heat and mass transfer in connection to the peristaltic transport of hyperbolic tangent fluid through a channel of varying width in presence of magnetic field was done by Sarvana et al. (2016).
Rabinowitsch fluid model is a well-established model for studying the non-Newtonian nature of the fluid. The shearing stress and shearing strain for the Rabinowitsch fluid model is connected by the relation as given below: where the nonlinear term  is called coefficient of pseudoplasticity and the non-Newtonian nature of fluid depends on this parameter,  is the viscosity of the fluid, U is velocity, X and Y are axial and transverse coordinate respectively. This model exhibits dilatant, Newtonian and pseudoplastic fluids nature for 0, Wada and Hayashi (1971) experimentally analyzed this model to justify theoretical results. Singh et al. ( , 2012 and Singh, (2013) have utilized Rabinowitsch fluid model to investigate the performance of different types of hydrostatic, hydrodynamic and squeeze film bearing systems. Over the most recent couple of years, some researchers have explored peristaltic transport for the Rabinowitsch fluid model. Singh and Singh (2014) investigated peristaltic flow in a tube using Rabinowitsch fluid model. Akbar and Nadeem (2014) studied Rabinowitsch fluid model in peristalsis. Maraj and International Journal of Mathematical, Engineering and Management Sciences Vol. x, No. x, xxx-xxx, 20xx ISSN: 2455-7749 3 Nadeem (2015) studied peristaltic flow in a curved channel for Rabinowitsch fluid. In the present paper, heat transfer and peristaltic transport of Rabinowitsch fluid flowing through a channel have been investigated.
The present investigation is motivated because none of the investigators above have studied the effect of heat transfer on flow characteristics of peristaltic flow in a uniform channel with the Rabinowitsch fluid model. The present work is helpful to enhance the knowledge of physiological fluid of pseudoplastic nature.

Analysis
Consider the flow of a non-Newtonian fluid complying Rabinowitsch fluid model through a channel of uniform thickness. Sinusoidal wave proliferates on the wall of the channel and moving with speed c . Taking   , XY as a rectangular coordinate in a fixed frame, the geometry of peristaltic flow is shown in Fig. 1. The geometry of wall surface is given as where a is half channel width, b is the amplitude of the wave, ' t is time,  is the wavelength.

Continuity equation
The transformation between fixed frame and wave frame is given by where ', ', ', ' u v x y are axial velocity, transverse velocity, axial coordinate and transverse coordinate respectively in wave frame.
Introducing non-dimensional parameters as follows Using Eq. (7) On solving Eq. (11) and Eq. (13) The corresponding stream function (cf. Appendix A) is obtained as: y y h dp dp y y h y dx dx The volume flow rate in fixed frame   ' Q and in wave frame   ' q is given by Using Eq. (7), Eq. (19) and Eq. (20) Average flow This can be reduced in dimensionless form as The pressure rise and friction force are given by

Results and Discussion
In the present section of this paper, the attempts have been made to analyze the effects of various parameters on pressure gradient, fluid velocity, pressure rise, friction force and fluid temperature in case of Newtonian, dilatant and pseudoplastic fluids. Looking at the need to scrutinize the qualitative effects of different parameters, the graphs have been drawn using MATHEMATICA. In order to perform numerical computation, following values of the parameters have been used: Amplitude ratio: 01  .
The variation of pressure rise against the flow rate is shown in Fig. 2  . The pressure rise linearly connected to flow rate for the case of Newtonian fluids which agrees with the result derived by Shapiro et al. (1969) whereas the same is non-linear in the pseudoplastic and dilatant fluid. It is unambiguous from Fig. 2 that when the parameter of pseudopalsticity is large then the fluid will be thinner and small pressure rise is observed as compared to Newtonian and dilatant fluids in peristaltic pumping domain. The pressure rise for Newtonian fluid increases by increasing amplitude ratio in the region of peristaltic pumping whereas shows opposite behavior in the copumping region. The result of pressure rise for Newtonian fluid at agrees with the established result of Vajravelu et al. (2005) for yield stress 0 and flow behavior index 1. Pressure rise decreases more rapidly for dilatant nature of the fluid in the peristaltic pumping region. Maximum pressure rise is observed for dilatant fluid at zero flow rate.   . Fig. 4 explains that pressure rise increases indefinitely for Newtonian and dilatant nature of the fluid at a constant flow rate but for pseudoplastic fluid, firstly pressure rise increases and then decreases indefinitely. On increasing flow rate pressure rise decreases at zero amplitude ratio (at no pumping). . It is apparent that the effect of flow rate on friction force is in contrast to the effect of flow rate on pressure rise. Variation of velocity for various values of flow rate and amplitude ratio has been presented in Fig. 7(a-b). For Newtonian fluid, the magnitude of velocity is noticed to be highest in the middle of the channel. The magnitude of velocity decreases for pseudoplastic fluid and increases for dilatant nature of the fluid with an increase in flow rate. Opposite behavior is seen at the wall of the channel. A similar behavior is observed with an increasing value in amplitude ratio.
. It is also clear that there is minor change noticed in the temperature close to the middle of the channel but a fast change in temperature is observed in the vicinity the wall of the channel. Maximum temperature is obtained for dilatant fluid and minimum temperature is obtained for pseudoplastic fluid at the middle of the channel.  . It is evident from Fig. 11 that temperature increases with increase in amplitude ratio. Maximum temperature is obtained in the middle of the channel and decreases near the wall of the channel for all types of fluid.
The temperature profile is plotted against y as illustrated in Fig. 12  . It is clear from Fig. 12 that temperature decreases continuously by an increase in flow rate for pseudeoplastic, dilatant and Newtonian fluid. Fig. 13(a-b) shows the variation of heat transfer coefficient along the axial direction for different value of 0.1, . It is clear from Fig. 13(ab) that the magnitude of the heat transfer coefficient increases with increase in amplitude ratio irrespective of the fluids nature. The maximum and minimum magnitude of the heat transfer coefficient is observed for dilatant and pseudeoplastic fluid respectively. This analysis is expected to be useful in physiology as well as in industry.  Fig. 14(a-c) shows the effect of coefficient of pseudoplasticity on the formation of bolus for 0.8 Q  , 0.6   . It is clear that the size of bolus decreases when nature of the fluid changes gradually from pseudoplastic (thinner fluid) to dilatant. More number of the boluses is formed for Newtonian fluid as compared to pseudoplastic and dilatant fluid. Fig. 15(a-b) reveals the interesting fact that with an increase in amplitude ratio, the size of the bolus decreases and gets shifted towards the wall of the channel.

Conclusions
The following conclusion can be summarized:  Pressure rise for pseudopalstic fluid is small as compared to Newtonian and dilatant fluid in the peristaltic pumping region.  Friction force reflects the reverse trend to that of pressure rise.
International Journal of Mathematical, Engineering and Management Sciences Vol. x, No. x, xxx-xxx, 20xx ISSN: 2455-7749 20  It is noticed that as the nature of fluid changes from Newtonian to pseudoplastic, the pressure rise decreases, whereas while changing from Newtonian to dilatant nature, the pressure rise increases irrespective of the value of flow rate.  It is clearly seen that the magnitude of velocity is maximum for Newtonian fluid at the middle of the channel and decreases with pseudoplasticity but increases with dilatant nature of the fluid.
 It is analyzed that on moving from the inlet   0 x  to downstream   0 x  , temperature decreases for all types of fluid.  With increasing amplitude ratio, temperature increases for all types of fluid.  It is seen that bolus size decreases when nature of the fluid shifted from pseudoplastic (thinner fluid) to dilatant. More number of the boluses is formed for Newtonian fluid as compared to pseudoplastic and dilatant fluid.  The consequence of increasing amplitude ratio comes in the form of decrease in the size of the trapped bolus.