Abstract
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.
1. Introduction
The fluid flow problem in permeable tubule has significance in various physiological and industrial processes. The complete presentation of such flows can be found in mass transfer and filtration processes, such as reverse osmosis desalination and blood flow through a nonnatural kidney; in the human body, the process of lymphatic flow through lymphatic vessels network and the purification of blood occur in the nephron’s renal tubules. In direction to research the pressure fields 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 fluid reabsorption.
In renal tubule, the theoretic flow of study was first investigated by Macey [1, 2]. They supposed a crawling movement of a viscous fluid over a constricted porous tube. They foretold that an exponentially decomposing flow rate occurs along the tube. In addition, the work for the flow through porous walls’ duct for small Reynolds number is conferred by Kozinski [3]. The effects of the variable cross-section tube for the flow through tube were analyzed by Radhakrishnacharya et al. [4]. The exact closed form solution for a viscous flow over a permeable tubule was presented by Marshall et al. [5]. They have deserted the inertial terms by assuming a creeping flow situation. Palatt et al. [6] resolved the viscous flow through a permeable tube by imposing the supposition that fluid loss across the wall of tubule is a linear function of the pressure gradient through wall. Effects of variable wall permeability on the creeping flow of a viscous fluid through a tubule were examined by Chaturani and Ranganatha [7]. Siddiqui et al. [8] studied the effects of an external applied MHD on the theoretic model of the flow for renal tubule. Recently, Sajid et al. [9] investigated the Ellis fluid flow in renal tubule.
The literature survey designates that most of the theoretic studies of flow in renal tubules are examined for viscous fluids. 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. These 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. The 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 , 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. The hybrid nanofluid contains gold and tantalum nanoparticles with thermal effects of radiation. However, there have been some developments in the realm of non-Newtonian fluids in recent years [23–26].
The 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. The mathematical formulation is investigated in Section 2. The 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.
2. Mathematical Formulation
We assume the axial symmetric, incompressible creeping flow of Carreau fluid in a long thin permeable tubule of length and radius whose axis extend in the -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 , the equations that control the flow are as follows.
Continuity equation for incompressible fluid [19] is
Momentum equation isin which
From equations (1)–(3), we obtain
The heat transfer equation of Carreau fluid isunder boundary conditions,where and are the axial and radial components of velocity, respectively, is the intertubular hydrostatic pressure, is the specific heat at constant pressure, is the hydrodynamic permeability coefficient of the tubule wall, , is hydrostatic pressure, the osmotic pressure is osmotic pressure, and 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:
Along with boundary conditions,where , and . The dimensionless number is the permeability coefficient of the tubule wall.
3. Solution of the Problem
Let be the magnitude of the outward redial velocity at the wall and be the mean axial velocity; at any cross-section perpendicular to the direction, then, in view of the physiological data [6], we have and . This allows ignoring terms of order and higher order. Consequently, equations (10) and (11) can be written as
Using regular perturbation method to find the value of , and , solve equation (15) by using conditions and for finding the values of and , and we obtain
Therefore, the total axial velocity is
Substitute (18) in equation (10) and integrate from to :
Therefore, the total redial velocity is
Solving equation (16) by using conditions and for finding the values of and , we obtain
Therefore, the total heat transfer is
Solve equation (10) for and using the boundary condition :where .
Solve equations (25) and (26) parallel and use conditions and , and we have
From equations (26) and (27), we have
Thus, total axial velocity, redial velocity, and temperature become
3.1. Average Pressure
The difference between mean pressure drop and the inlet of the tubule at some point is defined as
3.2. Wall Shear Stress
The dimensionless total wall shear stress is obtained as
3.3. Flow Rate
The total nondimensional volume flow rate is
3.4. Heat Transfer Rate
The total heat transfer rate is
4. Result and Discussion
4.1. Velocity Profile
This 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. The case corresponds the Poisseuille flow for a nonporous tube. Equations (28) and (33), respectively, can be written aswhere . To investigate the volume flow rate in response to various values of , 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. There are important regions of interest for different . For , the flow rate monotonically increases from at and as . For , and flow rate also present the same properties. For , the flow rate also decreases from at and approaches to as . When , the flow rate monotonically decreases from at and as . For the case , monotonically decreases from at . Negative flow rate is obtained after that point; it decreases monotonically and as ; this predicts the reverse flow phenomena that may not be admissible in many physically situations. Figure 2 is plotted for with , for different values of which present the same properties. Values of the axial velocity and radial velocity on the length of the tube can be deliberated from equations (22) and (23), respectively, for various values of , and . Figures 3 and 4 presented the variation of with for different values of and , respectively, by keeping other parameters fixed. From figures, we noted that the axial velocity reduces by increasing the value of permeability parameter and . The variation of radial velocity is presented in Figures 5 and 6 with for different values of and , respectively, by keeping other parameters fixed. It is observed that the velocity increases around the axis of tubule by increments in and . For the fixed value of and , the radial velocity in the interval is increased and reduced in the interval , where is the root of the equation. Mean pressure drop is presented in Figures 7 and 8 for various values of and , respectively. It is observed that mean pressure drop reduces by increasing , which increases by raising the value of . Figures 9 and 10 represent the deviations of wall shear stress with for various values of and , respectively. It is clear from figure that increasing the value of , the wall shear stress reduces and increases by increasing the value of . The variation of fractional reabsorption with is presented in Figures 11 and 12 for different values of and , respectively. Fractional reabsorption increases by increasing the value of and decreases by increasing .
4.2. Temperature
The effect of on temperature at various locations is depicted in Figure 13. The effects of on temperature are greatest in the middle and then decrease to zero at the surface. The temperature of the tube’s entry area is founded to be higher than that of the tube’s middle and exit regions. The temperature at each axial position rises as 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. The effects of Brinkman number on temperature at various locations are depicted in Figure 14. The figure shows that the temperature increases as increases. The effects of Weissenberg number against temperature is plotted in Figure 15. The Weissenberg number and temperature have a direct relationship. The 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 in the axial direction as a function of . There is a maximum at the entrance region for the same inlet velocity that decreases in the axial direction. In addition, boosts in the existing field.
5. 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)The flow rate and pressure can be controlled by increasing the value of (iii)The radial velocity , stress tensor , and fractional reabsorption increases by increasing the permeability coefficient ; also, the radial velocity increases by increasing the value of , where stress tensor and fractional reabsorption decrease by increasing (iv)The axial velocity decreases by increasing the permeability coefficient and , while the mean pressure drops decrease by increasing and increase by increasing (v)The temperature field increases at the tube’s centerline as the permeability coefficient decreases(vi)The 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 increasesNomenclature
| : | Viscosity at zero shear rate |
| μ∞: | Viscosity at infinite shear rate |
| : | Relaxation time |
| n: | Power index |
| m: | Ratio of vcosity at infinite and zero shear rate |
| Lp: | Hydrodynamics permeability coefficient |
| pe: | Hydrostatic pressure |
| : | |
| pm: | Difference between hydrosatic pressure and osmotic pressure |
| K: | Permeability coefficient |
| a: | Radius of tubule |
| L: | Length of tubule |
| Um: | Mean axial velocity |
| : | Outward radial velocity |
| δ: | Ratio between radius and length of tubule |
| We2: | Weissenberg number |
| : | Velocity components |
| : | Cylinderical coordinates |
| ∆p: | Pressure drop |
| : | Wall shear stress |
| Br: | Brinkman number |
| cp: | Specific heat at constant pressure. |
Data Availability
All data used to support the findings of the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.