Application of Recursive Theory of Slow Viscoelastic Flow to the Hydrodynamics of Second-Order Fluid Flowing through a Uniformly Porous Circular Tube

Slow velocity fluid flow problems in small diameter channels have many important applications in science and industry. Many researchers have modeled the flow through renal tubule, hollow fiber dialyzer and flat plate dialyzer using Navier Stokes equations with suitable simplifying assumptions and boundary conditions. The aim of this article is to investigate the hydrodynamical aspects of steady, axisymmetric and slow flow of a general second-order Rivlin-Ericksen fluid in a porous-walled circular tube with constant wall permeability. The governing compatibility equation have been derived and solved analytically for the stream function by applying Langlois recursive approach for slow viscoelastic flows. Analytical expressions for velocity components, pressure, volume flow rate, fractional reabsorption, wall shear stress and stream function have been obtained correct to third order. The effects of wall Reynolds number and certain non-Newtonian parameters have been studied and presented graphically. The obtained analytical expressions are in agreement with the existing solutions in literature if non-Newtonian parameters approach to zero. The solutions obtained in this article may be considered as a generalization to the existing work. The results indicate that there is a significant dependence of the flow variables on the wall Reynolds number and non-Newtonian parameters.


Introduction
The problem of finding the dynamics of fluid flow through a small diameter cylindrical tube with porous walls is being of much interest among the researchers and scientists for last few decades because of its application in many physical and physiological processes. Such flows occur in hollow fiber dialyzer, flat plate dialyzer, renal tubule and in desalination processes with reverse osmosis. Under the conditions when walls are permeable or semipermeable, classical Poiseuille's law is not appropriate to predict the pressure and flow relationship and therefore a modified kind of Poiseuille's law will work.
A considerable contribution in the study of hydrodynamics of flow of Glomerular Filtrate in renal tubule was made by Macey [1]. He solved Stokes equations by assuming reabsorption as a linear function of longitudinal length of the tubule. Macey showed that if reabsorption is uniform then the solution resembles with the Poiseiulle's law. Kelman [2] showed that the bulk flow passing through a cross section of the tubule at any point is decreasing exponentially in major flow direction. Considering Kelman's findings, Macey [3] showed that under such conditions reabsorption rate can also be taken decaying exponentially. Another quantitative description of the fluid motion in a small diameter porous tube is given by Palat et al. [4] after Macey. They assumed that the fluid loss through are obtained in dimensionless formulation. In Section 9 the results are discussed via graphical representation. In Section 10 concluding remarks are given and main findings are listed.

Governing Equations
The basic equations governing the flow of incompressible second-order fluid neglecting thermal effects are the following: where ρ is constant density of the fluid, f is net body force per unit mass, V is velocity field and T is Cauchy stress tensor. The constitutive equation for second-order fluid is the following stress deformation relationship proposed by Rivlin-Ericksen [25]: where I is identity tensor, µ is dynamic viscosity, p is pressure, A 1 and A 2 are called Rivlin-Ericksen tensors and α 1 and α 2 are fluid parameters called normal stress modulii. A 1 and A 2 are defined as follows: where the operator D Dt is the material time derivative. Using (3) and (2) Dunn and Fosdick [33] have derived the following field equation for an incomporessible second-order fluid which is independent of coordinate system: where ∇ 2 denotes Laplacian operator and |A 2 1 | = trace(A 1 A T 1 ). In case of unsteady flows through tubes, cylindrical coordinates (r, θ, z) with velocity components V = (u r , u θ , u z ) respectively are chosen and due to axisymmetry u θ = 0 and ∂(.) ∂θ = 0. For the sake of simplicity it is further assumed that the body forces are also absent. Thus in case of axisymmetric flows velocity field is: V = u r (r, z, t), 0, u z (r, z, t) .
Using (7) continuity Equation (1) takes the form: The goal is to set the vector Equation (6) in component (r − z) form. Using above equations the following expressions are obtained: where ∇ 2 V is a vector function being Laplacian of a vector function whereas ∇ 2 u r and ∇ 2 u z are scalar functions being Laplacian of scalar functions. Using Equations (9)- (16), (6) is written in component form as: r-component: z-component: The following relations are obtained: Using the above results, Equations (17) and (18) take the following simplified form.

r-component:
z-component: where modified pressurep is defined as: Stream function ψ is defined for 2D and 3D incompressible (divergence-free) flows with axisymmetry. The velocity components can be expressed as the derivatives of scalar stream function as follows: Note that the continuity Equation (8) is identically satisfied and with the use of Equation (22) vorticity function Ω can be expressed as: where E 2 = ∂ 2 ∂r 2 − 1 r ∂ ∂r + ∂ 2 ∂z 2 and using Equation (23) following results are obtained: ∂Ω ∂r Using these results and Equation (22) in Equations (19) and (20), the following is obtained: r-component: z-component: In order to obtain the compatibility equation pressure, terms will be eliminated. Differentiate Equation (26) with respect to z and Equation (27) with respect to r and subtract the two equations to obtain: where the compatibility relation ∂z∂r is used, the reason (28) is named as the compatibility equation. For any functions f (r, z) and g(r, z) following notion have been adopted:

Langlois Recursive Approach
To find an exact analytical solution to the system of nonlinear Equations (19), (20) and (8) is almost impossible, therefore we seek for an approximate analytical solution. W. E. Langlois in 1963 proposed a method [31] known as "Recursive Approach" which is best to solve the system of equations governing the slow flow of steady state, incompressible Rivlin-Ericksen fluid analytically. He takes the flow field as a perturbation of the state of rest and following is set: and These assumptions lead to the linear dynamic equations and boundary conditions for each of the sets [u r i , u z i , p i ], i = 1, 2, 3, ..., so that [u r , u z , p] as given by (29)-(31) provides a solution to the equations of motion with appropriate boundary conditions for an arbitrary Rivlin-Ericksen fluid.
The equations corresponding to [u r 1 , u z 1 , p 1 ] are the same that govern the flow of a Newtonian fluid. The equations corresponding to [u r 2 , u z 2 , p 2 ] are similar except that they contain non-homogeneous terms involving [u r 1 , u z 1 ]. Similarly the equations governing [u r 3 , u z 3 , p 3 ] are similar but they contain non-homogeneous terms which involve lower order solutions [u r i , u z i ], i = 1, 2 and this continues recursively. Hence at each stage it is required to solve a linear system of equations involving solutions which are obtained of previous all stages.

Problem Description
Consider the steady, axisymmetric flow of a second-order incompressible fluid in a small diameter, circular, cylindrical, porous-walled tube. It is assumed that the tube is uniformly porous so that the radial velocity is to have constant value U 0 at the tube-wall, where is a small dimensionless parameter. Clearly the assumption of uniform porosity does not imply the constant rate of absorption but the developments made here will provide us with the useful insights and these will be helpful in formulating much improved theory. Keeping the problem geometry in consideration cylindrical coordinates (r, θ, z) are used with velocity components V = (u r , u θ , u z ) respectively. With the additional assumption of axisymmetry u θ = 0 and ∂(.) ∂θ = 0. Consider axis of the tube in z-direction. At any point of the flow field the velocity has the form (7) and following steady state compatibility, the equation is obtained from (28): where stream function ψ is defined using (22) and E 4 ( * ) = E 2 E 2 * )). The boundary condition in case of uniform porosity are given as: Using (22) above boundary conditions are expressed in terms of stream function ψ as follows: ∂ ∂r

Problem Solution
It is assumed that, in some sense best defined a posteriori, the flow is slow enough and seeks the solution ψ (r, z) of the highly nonlinear compatibility Equation (32) subject to boundary conditions (38) to (42). With the approach of Langlois [31] the flow field is taken as perturbation of a state of rest. Instead of using presumptions (29)-(31), the following is equivalently set: and where is a small dimensionless parameter. This leads to the compatibility equations and boundary conditions for ψ i (r, z) , i = 1, 2, 3 so that ψ (r, z) as given by (43) provides a solution to the compatibility Equation (32). The equation for ψ 1 corresponds exactly to the compatibility equation governing the flow of Newtonian fluid and is solved subject to the given non-homogeneous boundary conditions. The equation for ψ 2 contains the non-homogeneous term involving ψ 1 and the equation for ψ 3 contains the non-homogeneous terms involving both ψ 1 and ψ 2 and are solved subject to corresponding homogeneous boundary conditions. Using (43) in (32), the following is obtained: (45)

First Order System and the Solution
Equating terms involving i , i = 1 from Equation (45), the following homogeneous equation is obtained: and using (43) in Equations (38)-(42) the corresponding boundary conditions are obtained in terms of ψ 1 as below: ∂ ∂r 1 r The reverse solution is obtained for ψ 1 by assuming the form of the stream function a priori. Boundary conditions (47) to (51) suggest the following form: With this assumption, (46) reduces to: where (53) is true if F 1 and G 1 satisfy the differential equations: On substituting assumed form of stream function (52) the boundary conditions (47) to (51) reduce to: where ψ 1 (0, 0) = 0 is taken conventionally and therefore G 1 (0) = 0. The solution to the Equations (54) and (55) subject to conditions (56) to (58) is obtained as: On substituting these expressions in (52), the following is obtained: (61)

Second Order System and the Solution
Equating terms involving i , i = 2 from Equation (45) and Equations (38)-(42) the following non homogeneous partial differential equation governing the second order solution and the associated homogeneous boundary conditions are obtained. Non homogeneous terms in the equation contain the first order solution ψ 1 .
∂ ∂r 1 r Again a reverse solution is sought by assuming the stream function of the form ψ 2 (r, z) = z F 2 (r) + G 2 (r), a priori. On substituting this and obtained first order solution ψ 1 in (62) following is obtained: where H = ∂ 2 ∂r 2 − 1 r ∂ ∂r and H 2 ( * ) = H(H( * )). (68) is true if F 2 and G 2 satisfy the differential equations: On substituting the assumed form of the stream function for ψ 2 (r, z), boundary conditions (63) to (67) reduce to: The solution to Equations (69) and (70) subject to conditions (71) and (72) is obtained as: and hence: (75)

Third Order System and the Solution
Equating terms involving i , i = 3 from Equation (45) and Equations (38)-(42) the following non homogeneous partial differential equation governing the second order solution and the associated homogeneous boundary conditions are obtained. Non homogeneous terms in the equation contain both first and second order solutions ψ 1 and ψ 2 respectively. and ∂ ∂r 1 r Again a reverse solution is sought by assuming the stream function of the form ψ 3 (r, z) = z F 3 (r) + G 3 (r), a priori. On substituting this and the obtained first and second order solutions ψ 1 and ψ 2 in (76) following is obtained: On substituting the assumed form of the stream function for ψ 3 (r, z), boundary conditions (77) to (81) reduce to: The solution to the Equations (82) and (83) subject to conditions (84) and (85) is obtained as: and hence the third order solution is obtained as:

Expression for Stream Function Correct to Third Order
With the notations U * = U 0 and Q * = Q 0 and using (43), the third order approximate expression for the stream function ψ(r, z) is written as: (89)

First Order Velocity Terms
As ∂r , velocity components u r 1 and u z 1 are obtained as: It is worth mentioning that these first order velocity components match exactly those obtained by Macey [1] for the Newtonian fluid assuming uniform re-absorption rate, that is, by setting a 1 = 0. It is noted that (90) shows that first order radial velocity u r 1 vanishes at axis of the tube and begins to increase towards the wall, having the maximum value of 4 3 2 3 U 0 within the tube at r = 2 3 R, this property is directly evident from the continuity equation. It is also noted from (91) that first order longitudinal velocity w r 1 has same parabolic profile as in case of Poisseuille's law.

Second Order Velocity Terms
∂r , second order velocity components u r 2 and u z 2 are obtained as: (93)

Third Order Velocity Terms
∂r , third order velocity components u r 3 and u z 3 are obtained as: (95)

Expressions for Velocity Components Correct to Third Order
With the notations U * = U 0 and Q * = Q 0 and using the approach common with Langlois [31], the third order approximate expression for the velocity components u and w is written. By adding Equations (90), (92) and (94) following is obtained: and similarly by adding Equations (91), (93) and (95), the following is obtained: It has not escaped our notice that the last four terms in above expression of axial velocity vanish if the inertial terms in the momentum Equations (17) and (18) are neglected.

First Order Characteristic Pressure Terms and Pressure Drop
In Equations (26) and (27) with ∂( * ) ∂t = 0 and using (43) and (44) the equations governing first order pressurep 1 are obtained as follows: On substituting (61) it is found that: On integrating (100) with respect to r it is found that: where A(z) is arbitrary function of z. On differentiating (102) with respect to z, ∂p 1 ∂z = A (z) and in comparison with (101) we get: On substituting (103) into (102), first order pressure distribution is obtained as follows: and using (21) first order hydrostatic pressure is p 1 =p 1 . It is added here ,as commented earlier, that the set [u r 1 , u z 1 , p 1 ] is the solution to corresponding Newtonian flow. For this subsection our discussion is confined to the first order solutions (Newtonian case) only and define volume flow rate Q(z) and the mean flow Q(z) between points 0 and z as: Using (91) in these definitions it is found that: and hence We are seeking for the average pressure drop. Define the mean pressure taken over any cross section of the tube as: where A is area of cross section of the tube. On substituting (109) and initial condition Q(0) = Q 0 the mean pressure drop along the tube is obtained as: which is similar to the Hagen-Poiseulle's equation.

Second Order Characteristic Pressure Terms
Again in Equations (26) and (27) with ∂( * ) ∂t = 0 and using (43) and (44) the equations governing second order pressurep 2 are obtained as follows: On substituting (61) and (75) and then integration gives expression for second order pressure as: (114)

Third Order Characteristic Pressure Terms
Again in Equations (26) and (27) with ∂( * ) ∂t = 0 and using (43) and (44) the equations governing third order pressurep 3 are obtained as follows: On substituting (61), (75) and (88) in above, then integration gives expression for third order pressure as: It is worth to mention that all the terms contributing in third order characteristic pressure may vanish if the inertial terms in governing Equations (17) and (18) are neglected.

Characteristic Pressure Correct to Third Order
With the notations U * = U 0 and Q * = Q 0 , and using the approach common with Langlois [31], the third order approximate expression for modified pressurep is written. With the expressions (104), (114) and (117) in hand the expression for characteristic pressure correct to third order is obtained in Appendix A.

Various Important Expressions in Dimensionless Form
After doing a dimension analysis of the solution obtained in Section 4, in particular for Equations (96) and (97) the following dimensionless quantities are defined: where λ 1 is known dimensionless elastic parameter, λ 2 is known as dimensionless cross-viscosity parameter, W RE is wall porosity parameter which is named as wall Reynolds number and N RE is known as inlet flow Reynolds number.

Velocity Components
Using the above transformations in Equations (96) and (97) dimensionless velocity components are obtained as:

Volume Flow Rate
As volume flow rate is defined as: or in terms of dimensionless variables given in (118)-(120) and making use of Equation (122) this can be expressed as: In terms of the dimensionless parameters defined in (118)-(120), the volume flow rate can be further simplified to the following expression: At this point it is worth mentioning that this expression of volume flow rate is independent of non-Newtonian parameters λ 1 and λ 2 . In fact, if it is compared with the Newtonian case under same conditions [1,16], the same expression is obtained, which is surprising. Note that for no reverse flow Q (ζ) ≥ 0 at ζ = 1, this implies that the parameters W RE , N RE and δ should satisfy the following inequality if reverse flow is avoided near the exit of the tube.
Using Equation (124) axial velocity given in Equation (122) can be written in a more convenient form as:

Wall Shear Stress
As wall shear stress is defined as: or in terms of dimensionless variables given in (118)-(120) and making use of Equation (122) this can be expressed as:

Fractional Rabsorption
As fractional reabsorption in a tube of length L is defined as: or in terms of dimensionless variables given in (118)-(120) the following simple linear relationship between FR and wall Reynolds number W RE is obtained:

Leakage Flux
As leakage flux is defined as: where q = Lq/Q * is dimensionless leakage flux. The non-dimensionalization of the variables reduces the number of parameters and makes the graphical representation of the physical quantities more convenient. The axial velocity profile W for different values of wall Reynolds number at three cross sections, ζ = 0.1 (beginning), ζ = 0.5 (middle) and ζ = 0.9 (end) of the tube is shown in the Figures 1-3 respectively. It may be noted that the values of wall permeability parameter W RE have significant effect on the magnitude of the axial velocity in the middle and the end of the tube, however it has negligible influence in the beginning of the tube. It is found that the magnitude of axial velocity component decreases if the magnitude of the suction W RE increases. The reverse flow effect is observed when W RE assumes the threshold value W RE = 1.28 × 10 −5 , see Figure 3. Furthermore the variation in non-Newtonian parameters λ 1 and λ 2 does not show any significant change in the axial velocity profile.   It is interesting to note that the expression (125) of the dimensionless volume flow rate Q is independent of the elastic parameter λ 1 and cross-viscosity parameter λ 2 . It can be noted that this expression matches exactly with the special cases present in References [1,16]. Behavior of the volume flow rate with wall Reynolds number given in the Figure 4 which shows that volume flow rate decreases with increase of W RE . Wall shear stress τ also decreases in the major flow direction and the role of W RE is to diminish it further, see Figure 5.  The relationship between mean pressure drop in longitudinal direction and W RE is given in Figure 6. Pressure drop increases with the increase value of wall Reynolds number. It can also be observed the elastic parameter λ 1 does not bring significant variation in pressure drop, however small changes in cross-viscosity parameter λ 2 bring significant changes in mean pressure drop (Figure 7).
In the Figures 8-11 stream-lines are depicted for different values of W RE . Far away from the inlet of the tube there can be a point where flow rate becomes zero-stagnation point, beyond which the pressure starts to increase downstream and the fluid moves in −z direction. This phenomena is called reverse flow, which can be seen clearly in Figure 11. The relationship between wall permeability parameter W RE and FR is shown in Table 2.      Expression of radial velocity (121) is independent of ζ and it attains maximum somewhere within the tube as shown in Figure 12. Using Newton Raphson method it is found that U r (γ) is maximum at γ ≈ 0.8164914227 and this critical point is independent of the wall Reynolds number W RE . However in Figure 12 the radial velocity profile can be seen to have a direct relationship with wall Reynolds number.

Conclusions
The foremost objective of this article was to investigate the effects of wall porosity parameter W RE and non-Newtonian parameters λ 1 and λ 2 on the variables of slow flow of a second order Rivlin-Ericksen fluid in a small diameter permeable tube. Although earlier work done by Macey [1] and Narasimhan [16] might be suitable for such flow problems, current study presents a more general analysis and results obtained by Macey and Narasimhan can be considered as special cases of solutions obtained here. Moreover our solution can be used in mathematical modeling of hollow fiber dialyzer when coupled with convection diffusion equation, in bio-sciences as well as in industry.
Emphasizing on the effects of wall porosity parameter W RE , elastic parameter λ 1 and cross-viscosity parameter λ 2 following conclusions are drawn:

•
Elastic parameter λ 1 does not bring any significant change in any of the flow variables in case of slow flow with small amount of cross flow.

•
The magnitude of axial velocity component U z decreases if the magnitude of suction W RE increases. Reverse flow is observed when wall porosity parameter assumes value threshold value of W RE = 1.28 × 10 −5 . • Volume flow rate is found to be independent of both the elastic parameter λ 1 and the cross-viscosity parameter λ 2 . • Volume flow rate and wall shear stress decrease in major flow direction if W RE increases.
• Mean pressure drop in major flow direction increases with the increase of the value of wall Reynolds number W RE . It is also observed that elastic parameter λ 1 does not bring significant change in pressure drop, however pressure drop increases with increase of cross-viscosity parameter λ 2 .

•
Fractional reabsorption (FR) also increases with increase of W RE , this relationship is given in Table 2. Funding: This research received no external funding.

Acknowledgments:
The authors are thankful to Sukkur IBA University for providing excellent research facilities.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations and/or symbols are used in this manuscript: