PERISTALTIC FLOW OF CASSON LIQUID IN AN INCLINED POROUS TUBE WITH CONVECTIVE BOUNDARY CONDITIONS AND VARIABLE LIQUID PROPERTIES

The primary objective of this paper is to examine the impact of variable viscosity and thermal conductivity on peristaltic transport of Casson liquid in a convectively heated inclined porous tube. The viscosity differs over the radial axis, and temperature dependent thermal conductivity is taken into account. The perturbation technique is utilized to solve the governing nonlinear equations under the assumption of long wavelength and small Reynolds number. The analytical solutions are obtained for velocity, streamlines, pressure rise, frictional force, and temperature when subjected to slip and convective boundary conditions. The impacts of related parameters on physiological quantities of interest are discussed and analyzed through graphs. It is seen that the variable viscosity has a noteworthy part in upgrading the velocity profiles. The investigation additionally demonstrates that the size of trapped bolus diminishes with an expansion in the velocity slip parameter.


INTRODUCTION
Peristaltic mechanism of Newtonian/non-Newtonian liquids assumes a critical part in understanding the physiological behavior of various biological fluids. Specifically, in the movement of blood flow in small arteries and the chyme transport in the gastrointestinal tract helps the biomedical engineers to design and construct dialysis and heart-lung machines. The mechanism of peristalsis can be found in various regular frameworks, for instance, bolus transport through the esophagus, the advancement of spermatozoa in the cervical channel of the male reproductive tract and the stream of urine through the ureter. The early literature on peristaltic stream was carried out by taking Newtonian fluid with different supposition and geometries (Shapiro et al. (1969)). But most of the liquids occurring in nature are non-Newtonian, where there exists a nonlinear relation between stress and strain. Mainly, a significant portion of the biological fluids like blood and chyme transport in gastrointestinal tracts fall under the non-Newtonian classifications. Thus, it is necessary to analyze the flow behavior of these liquids by considering the non-Newtonian fluid models. Keeping this in mind, Raju and Devanathan (1972) investigated the non-Newtonian behavior of peristaltic transport by using the power law model. Vajravelu et al. (2005) analyzed the peristaltic transport of Herschel-Bulkley fluid in a slanted tube and compared the results for Newtonian, Power-law and Bingham-Plastic models. Recently, Rajashekhar et al. (2018) investigated the peristaltic transport of two-layered † Corresponding author. Email: gudekote_m@rediffmail.com blood flow in an axisymmetric tube. The impacts of heat transfer on classical and biological fluids play a significant role in nuclear plants, bioengineering devices and have different applications in industries. The heat transfer in most of the systems can occur in three modes; convection, conduction, and radiation. Specifically, the convective way of heat transfer plays an essential role in understanding the heat exchange between many biological tissues. Additionally, the impact of porosity along with heat transfer has been of vital importance because of its application in understanding the different mechanism in lungs, gallbladder, movement of blood flow through narrow arteries, and so on. In the human body, a substantial part of the muscle is subjected to porous structures. These structures are necessary to supply the supplements to each cell, and their proper working fundamentally relies on the blood coursing through them. In such circumstances, the nearness of slip on the boundary because of the porosity of the wall has an essential influence in examining the flow of blood in arteries. Motivated by the applications of porosity along with the convective conditions at the walls, Akbar (2014) analyzed the peristaltic transport of a nanofluid flowing through porous channels with convective boundary conditions. Recently, several researchers have examined the heat transfer characteristics on classical and biological fluids in different geometries and configurations (Nadeem and Akbar (2009);Ramesh (2016); Hayat et al. (2016); Mebarek-Oudina and Bessaih (2016); Kumar and Abzal (2017); Wakif et al. (2018a,b,c); Sankad and Patil (2018)). Among the several non-Newtonian models (Hayat et al. (2018a,b,c); Vaidya et al. (2018)) the Casson model is more appropriate for understanding the complex rheological behavior of blood. Srivastava and Srivastava (1984) investigated the peristaltic transport by utilizing the Casson model for the flow of blood. Mernone et al. (2002) examined the two-layered peristaltic transport by employing the Casson model and inferred that the Casson model could be utilized for investigating the physiological behavior of blood in small arteries and urine flow through the ureter. Ramesh and Devakar (2015) examined the impact of slip velocity on peristaltic transport of Casson fluid. Recently, various researchers have utilized the Casson model to understand the complex behavior of many classical and biological fluids in different geometries and configuration (Khalid et al. (2015); Vajravelu et al. (2017); Manjunatha and Rajashekhar (2018); Khan et al. (2018); Zia et al. (2018)). The more significant part of the above examinations has investigated the Newtonian/non-Newtonian characteristics of the fluid flow by taking constant thermophysical properties of the fluid. These properties may change regarding temperature variation, especially the thermal conductivity. Therefore, it is essential to consider these properties to explore the peristaltic transport of biological fluids in various geometries and configurations. Thus, considering variable liquid properties will give rise to the broader understanding of classical and biological fluids (Hayat et al. (2014); Hussain et al. (2015); Prasad et al. (2017a,b)). Along these lines, the supposition of constant viscosity fails to clear up the peristaltic mechanism involved in blood flow, lymphatic vessels and bolus transport in the esophagus. In these organs, the viscosity of the fluid contrasts over the thickness of the channel/tube (Hayat and Ali (2008); Sinha et al. (2015); Bhatti and Zeeshan (2016)). Inspired by the examinations carried out by different researchers, the endeavor has been made to investigate the peristaltic transport of Casson liquid under the influence of variable viscosity, thermal conductivity and convective boundary conditions in an inclined porous tube. The governing nonlinear equations are solved by using perturbation technique subjected to slip and convective boundary conditions. The analytical solutions are obtained for velocity, streamlines, flow rate, pressure rise, frictional force, and temperature. Furthermore, the influences of appropriate parameters on physiological quantities of interest are analyzed and discussed graphically.

FORMULATION OF THE PROBLEM
Consider the peristaltic transport of incompressible viscous liquid in an inclined tube of radius a (Fig. 1). Casson model is used to represents the non-Newtonian behavior of the fluid. The flow happens through an axisymmetric porous tube and encourages the decision of fixed frame (R, Θ, Z) to study the problem. Let u and w be the radial and axial velocity components respectively. In the region between r = 0 and r = rp (plug flow region), we have τrz ≤ τ0 . The region between r = rp and r = H (core region), we have considered τrz ≥ τ0. The wall deformation due to a sinusoidal wave trains traveling with the wave speed c along the walls of a distensible tube is given by (Nadeem and Akbar (2009) where b is the amplitude, z is the axial coordinate, λ is the wavelength and t is the time.
The flow turns out to be steady in the wave frame (r, θ, z) moving with velocity c from the fixed frame (R, Θ, Z) given by (Nadeem and Akbar (2009) where ψ and Ψ are stream functions, p and P are pressures, in wave and fixed frames of references, respectively. The expressions for variable thermal conductivity and variable viscosity are given by (Hayat et al. (2014); Sinha et al. (2015)) where φ and α1 are the coefficients of variable thermal conductivity and variable viscosity respectively. To non-dimensionalize the mathematical formulation, we define the following non-dimensional quantities The non-dimensional equations of motion and energy in the wave frame of reference, moving with velocity c, under the lubrication approach is as follows (Nadeem and Akbar (2009)): where u and w are the radial and axial velocities, Re is the Reynolds number, θ is the temperature, δ is the wave number, Pr is the Prandtl number, Br is Brinkman number, r is the radial coordinate and τ is the shear stress. The constitutive equation for Casson's fluid in the non-dimensional form is given by (Casson (1959)) Incorporating Eq. (8) in Eqs. (5)- (7), and by using long wavelength and small Reynolds number approximation, we obtain The corresponding non-dimensional slip and convective boundary conditions are (Hayat et al. (2014)) ∂θ ∂r = 0, τrz is finite at r = 0 where α is the velocity slip parameter, Da is the porous parameter and Bi is the Biot number.

SOLUTION OF THE PROBLEM
The closed form solutions for the velocity in the core region is obtained by solving Eqs. (10) and (11) with the boundary conditions (13) and (14).
We obtain the velocity expression as where P = − ∂p ∂z and f = sinβ F 1 . Using the condition τ0 = P rp 2 at r = rp, the upper limit of plug flow region is obtained as rp = 2τ 0 P . Also, by using the condition τrz = τ h at r = h (Bird et al. (1976)), we obtain Hence, Using relation (17) in Eq. (15), we obtain the plug flow velocity as Integrating Eqs. (15) and (18) and using the conditions ψp = 0 at r = 0 and ψ = q 2 at r = h, the stream function for rp ≤ r ≤ h is given by and stream function for plug flow in the region 0 ≤ r ≤ rp can be written as The instantaneous volumetric flow rate in the wave frame is given by Thus, we have The dimensionless time-averaged flux (Q) across one wavelength is The non-dimensional expression for pressure rise (∆P ) across one wavelength is given as follows: The presence of nonlinear terms in heat transfer Eq. (12) makes it highly complex and hence it is difficult to get an exact solution. However, in the larger part of the practical problems, the value of φ is small which enables us to utilize a perturbation procedure to solve the nonlinear equation.

Perturbation Method
In order to obtain the perturbed solution about the temperature Eq. (12), the following procedure is employed. Expansion of θ leads to On solving zeroth and first order system, and neglecting the terms higher than O(φ), we obtain the solution for temperature.

RESULTS AND DISCUSSION
The Eq (25) is numerically integrated by using Weddle's rule in MAT-LAB. This section illustrates the role of physiological quantities such as axial velocity (w), pressure rise (∆P ), time averaged flow rate (Q) and trapping phenomenon (ψ) on the peristaltic transport of Casson fluid. On the other hand, the role of variable thermal conductivity (φ) and convective boundary conditions (Bi) on temperature (θ) play a significant role in understanding the heat transfer characteristics of biofluids. Hence the present section shows the effects of yield stress (τ ), porous parameter (Da), velocity slip parameter (α), angle of inclination (β), amplitude ratio ( ), thermal conductivity (φ), Biot number (Bi) and Brinkmann number (Br) on physiological quantities.

Velocity profile
The effects of pertinent parameters on velocity distribution in the presence and absence of variable viscosity are elucidated through Fig. 2. Velocity profiles exhibit a parabolic nature with a maximum velocity occurs at the center of the tube. Fig. 2a is sketched to show the variation of τ on velocity profile. Here velocity profile decreases for higher values of τ . Also, the peak velocity is attained when τ = 0. This is because of the yield stress parameter present in the model requires some amount of energy to begin the flow and hence an increase in the value of this parameter results in a decrease in the velocity. Figs 2b and 2c are drawn to show the effects of Da and α on velocity. Results indicate that the velocity is an increasing function of Da. However, the opposite behavior is noticed for higher values of α. Fig. 2d indicates that the velocity field enhances for an increase in the value of β. However, in all the cases, the velocity field enhance for a higher value of viscosity (α1 = 0.1). Thus, the presence of variable viscosity plays a vital role in enhancing the velocity field in an inclined porous tube.

Temperature profile
The effects of α, τ, Da, φ, Bi and Br on temperature are plotted and discussed in the Figs. 3 and 4. Fig. 3a shows the variation of α1 on temperature. From the figure, it is clear that the temperature diminishes for higher values of variable viscosity. This decay in temperature subject to a decrease in α1 is uniform throughout the tube. Fig. 3b reveals that the enhancement in the magnitude of temperature is because of the higher values of τ . Fig. 4a portrays the variation of Da on temperature. Here the decay in temperature is observed near the axis of the tube, and opposite behavior is noticed near the walls of the tube. Fig. 4b is graphed to illustrate the effect of φ on temperature. An increment in φ results in an increase in the temperature near the axis of the tube and the effect is negligible as we move towards the wall. This is because higher values of φ allow the fluid to dissipate or absorb the heat to its surroundings. Thus, when the temperature of the fluid is lower than the temperature of the boundary, an increment in the value of φ enhances the temperature near the axis of the tube. Fig. 4c depicts the variation of temperature due to the influence of Bi. An increase in the value of Bi results in the reduction of temperature. Fig. 4d shows the effect of Br on temperature.
Here an increment in Br enhances the temperature. Because Br occurs due to the viscous dissipation effects and it enhances the temperature.

Pumping characteristics
Pressure rise per wavelength is analysed in three different regions which are pumping region (∆P > 0,Q > 0), free pumping region (∆P = 0) and augmented region (∆P < 0,Q < 0). The quantitative analysis for the effects of pertinent parameters on pumping performance in the intervals forQ, when ∆P > 0 and ∆P < 0 are presented in Table 1.
We observe that an increase in the value of Da and β increases the length of the interval in the pumping region and opposite behavior is observed in the augmented region. This is mainly because an increase in the value of Da increases the porosity of the wall and thusQ decreases. Also, for a fixed value of Da and β, the length of the pumping region increases in the case of variable viscosity with that of constant viscosity. Further, the effects of τ, α, and decrease the length of the interval for ∆P > 0 and opposite behavior is observed when viscosity increases from 0 to 0.1. This information on τ helps in equalizing the pumping rate of Newtonian and Casson fluid for some value ofQ by adjusting the peristalsis velocity.

CONCLUSIONS
The present investigation analyzes the impact of variable liquid properties on the peristaltic mechanism of Casson liquid in an inclined porous tube. The examination finds its application in understanding the complex rheological behavior of biological fluids. Specifically, the flow of blood in narrow arteries and the movement of chyme in the gastrointestinal tract. The future investigation can be made by including the magnetic effect to the present model so that it can give a better understanding of the flow of blood when exposed to an external magnetic field. The main findings from the current model are listed below: • The axial velocity field is an increasing function of Da, , and β while it reduces for τ and α.
• The magnitude of temperature decreases for higher value of α1 and Bi, and it increases for an increase in the value of τ .
• Temperature of the fluid enhances for a larger value of φ.
• Pressure rise enhances for τ, α, β and , and it diminishes for Da.
• Variable viscosity plays a significant role in enhancing pressure rise.
• The size of trapped bolus increases for τ, Da and , and it reduces for α.