Peristaltic Flow with Heat Transfer on Sisko Fluid in a Ciliated Arteries

The flow of blood through arteries is an important physiological problem. In the present investigation, we carried out to study the peristaltic of non-Newtonian incompressible blood flow with heat transfer through ciliated arteries. The blood flow is characterized by the generalized Sisko model. The nonlinear partial differential equations of the problem are simplified by using an approximation of long wavelength and low Reynolds number. The differential equations are solved analytically by using the perturbation method . We find that Sisko fluid parameter and the power index effects the behavior of the velocity where the velocity increase in the arteries then decreases near the wall, but the Sisko parameter give opposite behavior where the velocity decrease then increases near the wall of arteries. The velocity increase in arteries with the increase of cilia length and elliptic path. The temperature profile increases then decreases near the wall of arteries with the increase of power index, Sisko fluid parameter and Grashof number, while the temperature decrease then increase near the wall with increase of Sisko parameter. The effect of increase in the cilia length give an increase of the temperature. The pressure gradient increases with the increase of power index and elliptic path, while the pressure gradient decrease with an increase of elliptic path, Sisko parameter. The pressure gradient increases and decreases in a different interval with the increase the cilia length. Our results are illustrated through a set of Figures.


Introduction
In fluid mechanics, Cilia are small hair-like structures, those projects from the free surface of certain cells and play important role in many psychological processes such as locomotion, alimentation, sensory perception [1], respiration, reproduction, and development [2] in a wide range of eukaryotes including human. There are two types of cilia motile and non-motile cilia, motile cilia are important for clearance of mucosa (airways), transport of oocytes (fallopian tubes) and circulation of cerebrospinal fluid (brain) [3] and are found in groups. In the adult human body, epithelial cells with motile cilia are highly rich in airways, reproductive tracts, and specific brain regions [4]. Whereas, primary cilia or (non-motile cilia) are usually found only one at a time on cell. Examples of non-motile cilia can be found in human sensory organs such as the eye and the nose. Some various studies about cilia transport have been achieved by [5][6][7][8][9][10] Discuss the effects of viscous Nanofluid due to ciliary motion in an annulus. [11] Have studied a metachronal wave analysis for non-Newtonian fluid inside a symmetrical channel with ciliated walls. Cilia walls influence on peristaltically induced motion of magneto-fluid through a porous medium at moderate Reynolds number has studied by [12]. Ciliary bands are used to produce feeding currents that draw the food particles toward the mouth of the organism for feeding [13,14]. Cilia facilitate organism swimming [14,15]. Oscillatory wavy walled (peristalsis) is a form of fluid transport induced by a progressive wave of area contraction and expansion of the length of a distensible tube containing fluid. Peristalsis plays important phenomena in the transport of biofluid it is well known to physiologists to be one of the major mechanisms for fluid transport in many biological systems. Peristaltic is characterized by swap reduction and leisure. Which pushes foodstuff in the course of the digestive area towards its let go at the anus Peristaltic flow occurs widely in the functioning of the ureter, food mixing and chime movement in the intestine, movement of ovum in the fallopian tube, the transport of the spermatozoa in the cervical canal, transport of cilia and circulation of blood in small blood vessels. Also, peristalsis involves in many industrial and biomedical applications like sanitary fluid transport, blood pumps in the heart-lung machine and transport of corrosive fluid where the contact fluid with the machinery parts is prohibited the problem of the peristaltic transport has attracted the attention of many investigators. The idea of peristaltic transport in a mathematical point of view was first coined by Latham [16]. Peristaltic transport has been studied under various conditions by using different assumptions like long wavelength or small amplitude ratio. Srivastava and Saxena [17] studied a two-fluid model of non-Newtonian blood flow induced by peristaltic waves. Sucharitha et al. [18] described the peristaltic flow of non-Newtonian fluids in an asymmetric channel with a porous medium. The peristaltic flow of micropolar fluid in an asymmetric channel with permeable walls is discussed by Sreenadh et al. [19,20] Discuss the effect of slip and heat transfer on the peristaltic transport of Jeffery fluid in a vertical asymmetric porous channel. [21] Created MHD peristaltic transportation of conducting blood flow with a porous medium through an inclined coaxial vertical channel. [22] Have studied peristaltic transport of a couple stress fluids some applications to hemodynamics. The peristaltic flow of a Sisko fluid over a convectively heated surface with viscous dissipation is investigated by [23].
Mathematical modeling of Sisko fluid flow through a stenosed artery discussed by [24,25] Study the boundary layer equations and lie group analysis of a Sisko fluid. [26] Investigated the influences of the peristaltic flow of Sisko fluid in a uniform inclined tube. [27] Discussed the peristaltic Sisko nanofluid in an asymmetric channel. The focus of the study of the peristaltic flow of Sisko fluid with heat transfer on ciliated arteries is to obtaining details analytical solutions using the perturbation method. The flow is considered to be obeying the constitutive equation of Sisko's model fluid, this describes the proposed solution methodologies for the governing systems of non-linear partial differential equations, this presented the analytical results and discussion, finally the paper is concluded with a discussion of the results.

Formulation of the Problem
Consider the peristaltic flow of an incompressible Sisko fluid in ciliated arteries with heat transfer. The geometry of the problem is shown in Figure 1. The fluid under the action of a ciliary that generates a metachronal wave. The flow propagating with constant speed c along the walls of the arteries whose inner surface is ciliated. We are considering the Where a is the radius of the tube, ε is the cilia length, λ is the wavelength, c is the wave velocity of the metachronal wave and t is the time. The cilia tips moving in elliptical path which can be represented mathematically in the form.
Where 0 Z is a reference position of the cilia and α is a measure of the eccentricity of the elliptic path. The equations for conservation of mass, momentum and heat transfer can be written as Where ρ is the density, t represents the time, p c is the specific heat at constant volume, k is the thermal conductivity, P is the pressure, T is the temperature of fluid and S is the extra stress tensor of Sisko fluid is given by ( ) Where 1 A is the rate of deformation tensor, 1 1 , a b and n are the physical constants of Sisko fluid model, the Rivlin-Erickson tensor is defined as follows: The corresponding boundary conditions are: Let us introduce the following dimensionless: Making use of equations (1) and (11), the governing equations (4) to (7) take the form 0, u u w r r z With the boundary conditions ( ) By using the long-wavelength approximation 1 δ 〈〈 and small Reynolds number 1 e R 〈〈 , we neglecting the terms containing δ , e R and higher order, then equations (13) -(15) take the form.

Solution of the Problem
The exact solution of nonlinear equations (19) and (20) given by using perturbation method Substituting equation (21) in equations (19)

Results and Discussion
With a view to studied the mathematical model of  Figure 4(a) illustrate that the pressure gradient increases with the increase in the power index n . It is noticed that the pressure gradient decreases with increase of elliptic path α , Sisko fluid parameters s b , and s a respectively, observed that in Figures 4(b) to 4(d). Figure  4(e) depicts that the pressure gradient increases and decreases in different interval with increase in the cilia length ε .

Conclusion
In the present study, we have developed the peristaltic flow like Sisko fluid and heat transfer on ciliated arteries. The problem is simplified by using approximation of the long wavelength and low Reynolds number. Solutions have been obtained by using perturbation method. The results are illustrated analytically and graphically through a set of figures. It is observed that the velocity profile ( , ) w r z has increases in the interval 0