Pressure-Driven Thermal Slip Flow in the Elliptical Channel with Radial Oscillatory Wall

This paper is aimed at presenting thermal slip flow driven by oscillatory pressure gradient in a deformable microchannel of elliptic cross-section. The fully developed flow of Newtonian fluid is considered, and Navier slip is applied on the boundary. The boundary value problem is formulated and applied to the coronary blood flow-heat transfer phenomenon during thermotherapy treatment. Its semianalytical solutions of velocity and temperature fields are carried out by the Ritz method. The effects of oscillatory wall and slip length on velocity and temperature fields of blood are investigated.


Introduction
Experimental and numerical studies related to heat transfer and flow characteristics of fluid in an oscillating body are of great interest during the last few decades due to a wide range of engineering and physiological applications, namely, fuel pump in the aircraft fuel system, electric centrifugal pumps, pulsatile blood flow through large arteries, and gas exchange in human lung during high-frequency oscillatory ventilation.
Oscillatory flows of fluids have attracted several researchers in the literature. Experimentally, numerous devices have been set up for studying such flow driven by either a pressure drop or a volumetric pump. Some devices include a Starling resistor fixed between two rigid tubes and placed inside a pressure chamber [1] and a low-speed recirculating water channel with a hydrogen-bubble-generating device [2]. These studies conducted in experimental models at different operating conditions have provided invaluable information on the main characteristics of oscillatory flows such as drag reduction effects, wall-shear stress reduction, transitional Reynolds number, velocity timescales, and pressure fields.
Due to limitation of experimental models, a number of numerical investigations have been carried out with an aim to better understanding the effects of wall oscillation on the channel flow of a fluid. In literature, the governing equations of the oscillatory flow of fluid in the duct with oscillating wall are the continuity equation and the Navier-Stokes equations subject to either no-slip or slip boundary condition. Various studies considered two-dimensional oscillating flow in the duct with fully oscillating wall [3][4][5][6][7][8][9], partially oscillating wall [10,11], and stretching sheet [12]. Three different directions of wall oscillation including the transverse (axial), vertical (radial), and both directions have been applied to the model.
For transverse oscillation of the wall, either fully or partially oscillatory wall has been investigated. Quadrio and Sibilla [3] applied a direct numerical simulation (DSN) to simulate turbulent flow in a pipe oscillating around the longitudinal axis. The effect of drag reduction was concerned due to the reference turbulent flows in a fixed pipe and a steadily rotating pipe. The maximum amount of drag reduction may be induced by the oscillating pipe. Xu and Huang [4] studied the impact of spanwise oscillating wall in the first two periods of oscillation on the turbulent plane channel flow at low Reynolds number using DSN. The periodic conditions in spanwise and streamwise directions were used, and no slip condition was imposed at the wall. Thomas et al. [5] applied a perturbation method to study the flow velocity and power requirements for fluid movement near oscillating wavy walls in transverse direction. It indicated that the more oscillating frequency of the wavy wall was, the faster the fluid oscillation and the more power requirements. Whittaker et al. [6] examined the energetic flow through a flexible tube in axial direction to determine the case of self-excited oscillations by investigating the effect of wall oscillation with varying a small amplitude and evaluating the energy budget.
In the case of partially transverse oscillation of the wall, Mateescu and Venditti [10] analysed the unsteady confined viscous flows with time variations of inflow velocities and partially transverse oscillation of the lower wall. The finite difference method was applied to the problem. They reported that the Reynolds number, oscillating frequency, and amplitude of the inflow velocity influenced on the formation of the flow separation regions. Yudhistira and Skote [11] presented a DSN to analyse the effect of a partially oscillating wall in the spanwise direction on the turbulent boundary layer. The flow characteristics including drag reduction and spatial development of drag reduction were related to the previous experiments.
The effects of wall oscillation in the radial direction have also been considered to analyse the drag reduction. Espin and Papageorgiou [7] investigated the stability of viscous pressure-driven flows in the channel with vertically oscillating walls and determined the numerical solutions for the Reynolds number ranges as an increase of wall oscillations amplitude. Shupti et al. [8] studied the two-dimensional pulsatile flow of blood through a stenosed artery which was assumed to be moving sinusoidally in the cross-section direction. The computational domain where a cosine-shaped aneurysm occurring after an appearance of a cosine-shaped stenosis causes the variation of the height of the domain. Blyth et al. [9] investigated the effect of wall amplitude and Reynolds number on an axisymmetric flow of a viscous incompressible fluid driven by a time-periodic wall motion of a cylindrical solid tube. By introducing a time-dependent stream function, the governing equations subject to the no-slip condition in the stream function vorticity form were solved in cylindrical polar coordinates. The results indicate that flow dynamics depend on the Reynolds number and the wall amplitude. A small or an intermediate value of the wall amplitude leads to quasiperiodic fields in time, and the flow field is time periodic at a large wall amplitude as an increase of Reynolds number. Adesanya and Makinde [13] investigated the effect of wall slip on twodimensional oscillatory flow of an incompressible viscous fluid through a channel of nonuniform wall temperature filled with a porous medium. In the last decade, heat transfer and fluid flow in the channel with oscillating wall have attracted too many researchers. Umuvathi et al. [14] determined the unsteady oscillatory flow and heat transfer in a horizontal channel filled with composite porous medium. Jha and Ajibade [15,16] presented a free convective oscillatory flow of an incompressible viscous fluid between two periodically heated infinite vertical parallel plates and carried out some interesting results on the free convective oscillatory flows with given the time-dependent boundary conditions. Abdul-Hakeem and Sathiyanathan [17] derived analytical solutions of velocity and temperature fields for two-dimensional oscillatory flow of an incompressible viscous fluid on free convective radiation through a highly porous medium bounded by an infinite vertical plate. Sobh [18] used perturbation technique to study the peristaltic slip flow of viscoelastic fluid with heat and mass transfer in a uniform tube. The model is considered as the chyme movement in the small intestine. The analytic solutions show that the velocity, pressure gradient, temperature profile, and concentration field depend on the combined effect of slip parameter, heat, and mass transfer.
Various problems have been carried out to study the effects of slip flow and heat transfer in the channel but excluding the oscillating wall. Wiwatanapataphee et al. [19] studied transient oscillating pressure-driven slip flow and heat transfer in a microchannel of elliptic cross-section using the Ritz method. The results show the effects of the slip length and the aspect ratio of the microchannel on the velocity and temperature distribution. Mohyud-Din et al. [20] studied the effects of velocity and temperature slip of nanofluid in diverging and converging channels using variation iteration and variation of parameter methods. However, a few attempts have studied the pulsatile flow in the channel with wall oscillation in the radial direction, and most studies have ignored the slip condition, the oscillatory nature of the pressure gradient, and the heat transfer in the flow channel.
Recently, there has been considerable interest in developing sound and faithful thermal slip flow models that describe the coronary blood flow-heat transfer phenomenon during thermotherapy treatment. In the cancer treatment process, temperatures within a range of 42°C to 45°C can kill tumour cells without affecting normal tissue. As thermal therapy can cause improper blood flow and blood clots, understanding the complex phenomena of blood flow and heat transfer in a human coronary artery with oscillatory wall is important for the most efficacious treatment. This paper presents the driven pressure fluid flow and heat transfer in an elliptical channel with the oscillatory wall to predict the behaviour of circulatory flows inside the coronary arteries. Effects of oscillation amplitude and slip length on the velocity and temperature fields are investigated. The results are compared with existing ones available in the literature.

Governing Equations
In this section, we propose a mathematical model to study the behaviour of slip flow and heat transfer in an elliptical channel with the oscillating wall. Based on the assumption of fully developed flow, the problem is thus governed by the following boundary value problem (BVP) consisting of the system of the Navier-Stokes equations (1) and energy equation (2) subject to the initial condition (3) and the boundary conditions (4) and (5).
Navier-Stokes equation: Journal of Applied Mathematics Energy equation: Initial condition: Boundary conditions: where u * ≡ u * ðt, x, y, zÞ and T * ≡ T * ðt, x, y, zÞ are, respectively, the velocity field and temperature field of fluid; p is the pressure; ρ, μ, k, and c p are, respectively, the density, viscosity, thermal conductivity, and specific heat of fluid; q p is the heat flux; β and u wall , represent, respectively, the slip length and tangential velocity of the wall; h ∞ denotes the coefficient of convective heat transfer; and T ∞ is the uniform external temperature. We also consider no swirling flow with pressure variation in the flow direction [19], i.e., where p is an average pressure. In this study, the pressure gradient is prescribed by equation (8) based on the wall deformation, i.e., where aðt, ω, γÞ and bðt, ω, γÞ are semimajor and minor axes of the channel cross-section which are functions of frequency ðωÞ, amplitude of oscillation ðγÞ, and time ðtÞ, and p 0 , a 0 , and b 0 are constants. Substituting equations (7) and (8) into the system of equations (1) and (2) yields the following BVP: subject to the initial condition (3), the boundary conditions (5), and In equation (9), u ≡ uðt, x, yÞ is the function of x, y, and t. To determine the wall velocity ðu wall Þ, we consider the sinusoidal movement of the channel in the cross-stream direction. Let the variations in equation (11) of the semimajor and semiminor axes define, respectively, by By parametric equation of an ellipse, we have, on its perimeter, where 0 ≤ ψ ≤ 2π. Thus, the wall velocity u wall can be determined by

Ritz Method
To apply the Ritz method for solving the above BVP (9), (10), (11), we firstly approximate the solution function of u and T * in the following forms [19]: with In equation (15), x and y varies over time and we assume that rates of change of the x and y coordinates with respect to t at the center ðx, yÞ = ð0, 0Þ are fixed, i.e., dx 0 /dt = dy 0 /dt = 0 and the rates of change of the x and y coordinates with respect to t on the boundary wall are defined by dx w /dt = − γω sin ωt cos ψ and dy w /dt = −γω sin ωt sin ψ. Then, the rates of change of the x and y coordinates with respect to t in the fluid domain can be expressed as 3 Journal of Applied Mathematics From equation (15), we obtain In equation (16), two parameters including α = α i and λ = λ i are fixed. We then obtain where dh/dt is the rate of change of thermal parameter. Substituting equations (15), (16), (17), (18), (19), (21), (22) into equations (9) and (10), we obtain subject to initial condition hð0, 0, z, 0Þ = 1 and two boundary conditions In equation (24), gðt, x, y, zÞ and f ðt, x, y, zÞ are defined by where δ = ρc p /k. To simplify the problem, we consider the flow at a crosssection z = z n , and t = t n at instant time. We then solve the BVP (23), (24), (26) for functions vðt n , x, yÞ and hðt n , x, y, z n Þ by applying the quadratic minimization such that where Bðw, wÞ and F are, respectively, bilinear and linear forms [21]. Hence, the BVP (23), (24), (26) is equivalent to the following system where We now approximate functions vðx, yÞ and hðx, yÞ as a linear combination of N known functions ϕ i ðx, yÞ and φ i ðx, yÞ as follows: with ϕ i x, y ð Þ= 1, x 2 , y 2 , x 4 , x 2 , y 2 , y 4 , x 2 y 4 , x 4 y 2 , y 6 , ⋯ È É , φ i x, y ð Þ= xy, x 2 , y 2 , x 4 , x 2 , y 2 , y 4 , x 2 y 4 , x 4 y 2 , y 6 , ⋯ È É : Substituting equation (32) into equation (29) and imposing ∂I v /∂c i = 0ði = 1, ⋯, NÞ yield the following linear system of equations: 4 Journal of Applied Mathematics with Similarly, substituting equation (33) into equation (30) and setting ∂I h /∂d i = 0ði = 1, ⋯, NÞ give with The problem is thus reduced to find the coefficients c j and d j by solving the following linear systems (35) and (37). Finally, the velocity and temperature fields are expressed by

Numerical Examples
To describe the effect of wall deformation on the slip flows and heat transfer in the microchannel, we choose the computational domain of a coronary artery with the elliptical crosssection having the lengths of major semiaxis aðt, ω, γÞ and minor semiaxis bðt, ω, γÞ as given in Table 1. The average lengths of major/minor semiaxes depend on the variation of the hydraulic diameter D h of the ellipse. In this study, D h between 0.108 cm and 0.231 cm corresponding to the typical size of the coronary artery [22] and oscillation frequency and period corresponding to human heartbeat pattern are chosen. However, owing to the lack of data, we could not find realistic values of oscillation amplitude γ and slip length β.
Heartbeat induces the oscillatory pressure gradient of blood and oscillating movement of the coronary arterial wall. To study oscillatory blood flow, blood is assumed to be an incompressible Newtonian fluid having the properties as shown in Table 2. Figure 1 represents variations of radial wall and pressure gradient obtained from the model with wall oscillation for oscillation amplitude γ = 0:03 cm and frequency ω = 5π/2. It is noted that the wall is expanding during dp/dz < 0, and the wall is contracting during dp/dz > 0. At the time in which the highest negative pressure gradient exists ðdp/dz = −2:4Þ, expansion of major/minor axes is the biggest, i.e., a = 0:13 cm and b = 0:105 cm. On the other hand, at the time in which the highest positive pressure gradient exists ðdp/dz = 2:4Þ, contraction of major/minor axes is the smallest, i.e., a = 0:07 cm and b = 0:045 cm.   Figure 2 illustrates the axial velocity in the first full wave cycle for a 0 = 0:1 cm, a 0 = 0:075 cm, oscillation amplitude γ = 0:3 cm, frequency ω = 5π/2, and a slip length β = 0:05. The velocity decreases during periods of wall expansion, t ∈ ½0, 0:2Þ, and wall contraction, t ∈ ½0:4, 0:6Þ, but increases during periods of wall expansion, t ∈ ð0:6, 0:8 and wall contraction, t ∈ ð0:2, 0:4. Effect of oscillatory wall motion and compliance on flow pattern have been observed in flow channel.
To examine the effect of oscillation wall and slip length on the temperature field, we plot the temperature profiles and its contour on the cross-section z = 5 cm at the 30 th wave cycle. In Figure 3, it is observed that temperature varies between 0.1 and 1.05°C at the 30 th wave cycle for the slip length of 0.05 and oscillation amplitude of 0.03 cm.
As the frequency increases, the wall responds by oscillating with increasing amplitude. To investigate the impact of oscillation amplitude, γ, on the velocity and temperature field, we choose the values of γ between 0.01 and 0.05 cm and select the investigated point at the center of the elliptical cross-section ðx, yÞ = ð0, 0Þ. The results indicate that during the expansion period as shown in Figure 4(a), there is a linear relationship between oscillation amplitude and velocity uð0, 0, tÞ as the increase of oscillation amplitude rises the velocity and there is an increase of the velocity during the contraction period but the fluid moves in the opposite direction as shown in Figure 4(b). It is noted that an increase of oscillation amplitude rises the highest level of flow pressure gradient and thus enhances fluid flow rate. The obtained results are in well agreement with others [7,14]. Figure 5 demonstrates the influence of slip length on the temperature field. Three values of the slip length β including 0.005, 0.05, and 0.07 are chosen for the investigation. The results show that the temperature increases as the slip length decreases. It is further to mention that the rise in the slip length causes the fluid flow to move faster. This induces the observed decrease in blood temperature. These results agree with the results obtained by [18]- [20].
Furthermore, we analyse the effect of oscillation amplitude and slip length on the average temperature field by   varying γ between 0.01 and 0.05 cm and β between 0.005 and 0.07 at three different wave cycles including the 30 th , 40 th , and 50 th cycles. It is found that the average temperature is inversely proportional to the oscillation amplitude and the slip length as shown in Figures 6(a) and 6(b). In addition, it is observed that the average temperature is higher in the higher wave cycle.

Conclusions
This paper studies the pressure-driven flow of fluid and heat transfer in an elliptical microchannel with radial oscillatory wall. Based on the assumption that there is no swirling flow, wall oscillation and pressure gradient are prescribed, and slip condition and heat convection are applied on the boundary; we proposed a mathematical model to investigate the effect of wall oscillation and slip length on the velocity and temperature field in the coronary artery. The results indicate that the slip length has a significant effect on blood flow and heat transfer during thermotherapy treatment. The wall oscillation also has a primary influence on the blood flow pattern and temperature distribution in the flow channel.

Data Availability
The data used to support the findings of this study are included within the supplementary material.   Figure 6: The effect of γ and β on the average temperature on the elliptic cross-section at three wave cycles. 8 Journal of Applied Mathematics