Mathematical Modeling of Carreau Fluid Flow and Heat Transfer Characteristics in the Renal Tubule

This study looks into the steady heat transfer issue of a flow of an incompressible Carreau fluid. Carreau fluid exhibits the shear thinning and thickening characteristics at low, moderate, and high shear rates. At the tubule wall, the fluid absorption is used as a function of pressure gradient and wall permeability through the tubule wall. Supposing the tubule radius considerably small in comparison to its length, the governing equations are considerably simplified. Significant quantities of interest are computed analytically by using perturbation, and the influence of emergent parameters is discussed through graphical results. The comparisons of results with existing data are set up to be good agreement.


Introduction
e uid ow problem in permeable tubule has signi cance in various physiological and industrial processes. e complete presentation of such ows can be found in mass transfer and ltration processes, such as reverse osmosis desalination and blood ow through a nonnatural kidney; in the human body, the process of lymphatic ow through lymphatic vessels network and the puri cation of blood occur in the nephron's renal tubules. In direction to research the pressure elds and velocity in such circumstances, since there occurs a normal competent of velocity, the rule of Poisuille's law cannot be applied directly on the tubule wall due to uid reabsorption.
In renal tubule, the theoretic ow of study was rst investigated by Macey [1,2]. ey supposed a crawling movement of a viscous uid over a constricted porous tube. ey foretold that an exponentially decomposing ow rate occurs along the tube. In addition, the work for the ow through porous walls' duct for small Reynolds number is conferred by Kozinski [3]. e e ects of the variable crosssection tube for the ow through tube were analyzed by Radhakrishnacharya et al. [4]. e exact closed form solution for a viscous ow over a permeable tubule was presented by Marshall et al. [5]. ey have deserted the inertial terms by assuming a creeping ow situation. Palatt et al. [6] resolved the viscous ow through a permeable tube by imposing the supposition that uid loss across the wall of tubule is a linear function of the pressure gradient through wall. E ects of variable wall permeability on the creeping ow of a viscous uid through a tubule were examined by Chaturani and Ranganatha [7]. Siddiqui et al. [8] studied the e ects of an external applied MHD on the theoretic model of the ow for renal tubule. Recently, Sajid et al. [9] investigated the Ellis uid ow in renal tubule. e literature survey designates that most of the theoretic studies of ow in renal tubules are examined for viscous uids. It is now a well-known fact that many physiological and industrial fluids do not obey Newton's law of viscosity. On the basis of investigational studies, many relationships of the apparent viscosity are anticipated in the literature. ese fluids are generally explained as generalized Newtonian fluids, and in such fluids, the fluid responses to an applied shear stress at an instant do not depend on the response at some previous instant. e generalized fluid models are widely applied to discuss the physiological flows such as peristaltic flows and blood flow. Ali et al. [10] used Newtonian fluid to examine peristaltic waves in a curved channel. In curved channel, the effect of MHD on peristaltic flow of non-Newtonian fluid was investigated by Hayat et al. [11]. Hina et al. [12] demonstrated peristaltic pseudoplastic fluid flow in a curved with complaint wall. Abbassi et al. [13] discussed in curved channel peristaltic transport of Eyring-Powell fluid. Narla et al. [14] studied the peristaltic motion of viscoelastic fluid in a curved channel using the fractional second-grade model. Ali et al. [15] addressed heat transfer study of peristaltic flow in a curved channel by third-grade fluid. Canic and Kim [16] investigated quasilinear properties mathematically in the hyperbolic blood flow model over complaint axisymmetric vessel. Zaman et al. [17] looked at how unsteady and non-Newtonian rheology affected blood flow in a stuck time-variant stenotic artery. Unsteady and non-Newtonian blood flow over a tapered coinciding stenosed catheterized vessel was discussed by Ali et al. [18]. Zaman et al. [19] investigated unsteady blood flow over a vessel numerically by using the Sisko fluid model.
Keeping this fact in mind, we have revisited the hydrodynamical model of renal tubule using a Carreau fluid model. Carreau fluid is the general form of viscous fluid and power law fluid. At n � 1, it acts like a viscous fluid, while on strong shear rates, it behaves like a power law fluid. Akbar and Nadeem [20] looked into the Carreau fluid model for blood flow over a stenosis in a tapered artery. Salahuddin [21] used a Keller box and a shooting tool to study the Carreau fluid model for stretching a cylinder. Under the impacts of an induced and applied magnetic field, Bhatti and Abdelsalam [22] investigated the peristaltically driven motion of Carreau fluid in a symmetric channel. e hybrid nanofluid contains gold (Au) and tantalum (Ta) nanoparticles with thermal effects of radiation. However, there have been some developments in the realm of non-Newtonian fluids in recent years [23][24][25][26].
e main goal of this study looks into the steady heat transfer issue of a flow of an incompressible Carreau fluid. We have utilized the constitutive equations of a Carreau fluid model to discuss the characteristics of renal tubule flow. e mathematical formulation is investigated in Section 2. e analytical solutions are presented for the velocity distribution, temperature, pressure, shear stress, mean pressure drop, fractional reabsorption, and heat transfer rate and are investigated in Section 3. Section 4 is denoted for the analysis of obtained results in Section 3. On the basis of presented results, some conclusions are compiled in Section 5.

Mathematical Formulation
We assume the axial symmetric, incompressible creeping flow of Carreau fluid in a long thin permeable tubule of length L and radius a whose axis extend in the z-axis. We supposed that hydrostatic pressure is the sole driving force for fluid association and that osmotic pressure and hydrostatic pressure outside the tubule are both constant along the length of the tubule.
For velocity V � (v, 0, u), the equations that control the flow are as follows.

Momentum equation is
in which From equations (1)-(3), we obtain 2 Journal of Mathematics e heat transfer equation of Carreau fluid is under boundary conditions, where u and v are the axial and radial components of velocity, respectively, p is the intertubular hydrostatic pressure, c p is the specific heat at constant pressure, L p is the hydrodynamic permeability coefficient of the tubule wall, p m � p e − π e , p e is hydrostatic pressure, the osmotic pressure π e is osmotic pressure, and Q 0 is the constant flow rate at the inlet of the tubule. In order to converted the problem in nondimensional form, we define the following parameter: Using the above parameters, equations (4)-(6) are transformed into the following nondimensional form: Journal of Mathematics Along with boundary conditions, where 4 , and Q(z) � Q(z)/Q 0 . e dimensionless number K is the permeability coefficient of the tubule wall.

Solution of the Problem
Let V w be the magnitude of the outward redial velocity at the wall and U m be the mean axial velocity; at any cross-section perpendicular to the z direction, then, in view of the physiological data [6], we have δ ≪ 1 and V w ≪ U m . is allows ignoring terms of order δ, δ 2 and higher order. Consequently, equations (10) and (11) can be written as Using regular perturbation method to find the value of p(z), u(r, z), and v(r, z), solve equation (15) by using conditions zu/zr(0, z) � 0 and u(1, z) � 0 for finding the values of u 0 (r, z) and u 1 (r, z), and we obtain erefore, the total axial velocity is Substitute (18) in equation (10) and integrate from r � 0 to r � r: erefore, the total redial velocity is Solving equation (16) by using conditions zθ/zz(0, z) � 0 and θ(1, z) � 1 for finding the values of θ 0 (r, z) and θ 1 (r, z), we obtain erefore, the total heat transfer is Solve equation (10) for p 0 (z) and p 1 (z) using the boundary condition v(1, z) � Kp(1, z): where ξ 2 � 16K. Solve equations (25) and (26) parallel and use conditions p(r, 0) � P 0 and 2 1 0 ru(r, 0)dr � 1, and we have From equations (26) and (27), we have us, total axial velocity, redial velocity, and temperature become 3.1. Average Pressure. e difference between mean pressure drop and the inlet of the tubule at some point z is defined as

Wall Shear Stress.
e dimensionless total wall shear stress is obtained as Journal of Mathematics

Flow
Rate. e total nondimensional volume flow rate is

Heat Transfer
Rate. e total heat transfer rate is

Velocity Profile.
is section shows how the Carreau fluid material parameters affects the pressure gradient, pressure distribution, flow rate, velocity profile, wall shear stress, fractional reabsorption, and mean pressure drop. e case n � 1 corresponds the Poisseuille flow for a nonporous tube. Equations (28) and (33), respectively, can be written as where β � P 0 ξ/8. To investigate the volume flow rate Q(z) in response to various values of β, Table 1 shows the physiological data for the rat proximal convoluted tubule and Table 2 shows the physiological data for normal hydropenic rats. Figure 1 is plotted for Carreau fluids and Newtonian fluid. ere are important regions of interest for different β. For −∞ < β < 0, the flow rate monotonically increases from Q � 1 at z � 0 and Q ⟶ + ∞ as z ⟶ ∞. For β � 0, Q � coshξz and flow rate also present the same properties. For 0 < β < 1, the flow rate also decreases from Q � 1 at z � 0 and Q approaches to +∞ as z ⟶ ∞. When β � 1, the flow rate monotonically decreases from Q � 1 at z � 0 and Q ⟶ 0 as z ⟶ ∞. For the case 1 < β < ∞, Q monotonically decreases from Q � 0 at z � (1/2ξ)ln(β + 1/β − 1). Negative flow rate is obtained after that point; it decreases monotonically and Q ⟶ − ∞ as z ⟶ ∞; this predicts the reverse flow phenomena that may not be admissible in many physically situations. Figure 2 is plotted for p(z)/P 0 with z, for different values of β which present the same properties. Values of the axial velocity u and radial velocity v on the length of the tube can be deliberated from equations (22) and (23), respectively, for various values of P 0 , K, and z. Figures 3 and  4 presented the variation of u with r for different values of K and We 2 , respectively, by keeping other parameters fixed.
From figures, we noted that the axial velocity reduces by increasing the value of permeability parameter K and We 2 . e variation of radial velocity v is presented in Figures 5  and 6 with r for different values of K and We 2 , respectively, by keeping other parameters fixed. It is observed that the velocity increases around the axis of tubule by increments in K and We 2 . For the fixed value of K and We 2 , the radial velocity in the interval r ∈ (0, r m ) is increased and reduced in the interval r ∈ (r m , 1), where r m is the root of the equation. Mean pressure drop is presented in Figures 7 and 8 for various values of K and We 2 , respectively. It is observed that mean pressure drop reduces by increasing K, which increases by raising the value of We 2 . Figures 9 and 10          represent the deviations of wall shear stress τ ω with z for various values of K and We 2 , respectively. It is clear from figure that increasing the value of K, the wall shear stress reduces and increases by increasing the value of We 2 . e variation of fractional reabsorption FR with P 0 is presented in Figures 11 and 12 for different values of K and We 2 , respectively. Fractional reabsorption increases by increasing the value of K and decreases by increasing We 2 .

Temperature.
e effect of K on temperature θ at various locations is depicted in Figure 13. e effects of K on temperature are greatest in the middle and then decrease to zero at the surface. e temperature θ of the tube's entry area is founded to be higher than that of the tube's middle and exit regions. e temperature at each axial position rises as K rises. It is self-evident that the presence of porous material generates greater fluid flow restriction, causing the fluid to decelerate. As a result, as the permeability parameter increases, the barrier to fluid motion increases, and hence, velocity fall that causes temperature rises. e effects of Brinkman number B r on temperature θ at various locations are depicted in Figure 14. e figure shows that the temperature increases as B r increases. e effects of Weissenberg number We 2 against temperature θ is plotted in Figure 15. e Weissenberg number and temperature have a direct relationship. e temperature profile increases as the Weissenberg number increases. As the Weissenberg number increases, the thermal boundary layer shrinks, causing the temperature profile to increase. Figure 16 depicts the variance of Nusselt number N u � _ θ /θ in the axial direction as a function of K. ere is a maximum N u at the entrance region for the same inlet velocity that decreases in the axial direction. In addition, K boosts N u in the existing field.

Conclusion
(i) In this study, we examined the flow of the non-Newtonian Carreau fluid in insignificant diameter of permeable tubule with an application to the renal tubule (ii) e flow rate and pressure can be controlled by increasing the value of β (iii) e radial velocity v, stress tensor τ w , and fractional reabsorption FR increases by increasing the permeability coefficient K; also, the radial velocity increases by increasing the value of We 2 , where stress tensor and fractional reabsorption decrease by increasing We 2 (iv) e axial velocity u decreases by increasing the permeability coefficient K and We 2 , while the mean pressure drops decrease by increasing K and increase by increasing We 2 (v) e temperature field increases at the tube's centerline as the permeability coefficient decreases (vi) e Nusselt number at the tube's wall decreases as it travels down the tube, peaking at the tube's exit region as the permeability coefficient increases