Fluid Flow and Solute Transfer in a Tube with Variable Wall Permeability

1 Introduction

The formation of urine involves filtration and reabsorption processes in the flow through nephrons which are permeable to water and solutes of small molecules. The glomerulus is the place of urine formation, which filter fluid and solute compounds from blood. Blood enters the glomerulus by way of the afferent arteriole and leaves through the efferent arteriole. Glomerular filtration is a bulk process where water and dissolved components can move freely between the vascular space and Bowman’s capsule. The glomerular filtrate is formed due to the Starling’s forces in the glomerular capillaries. Under these conditions, there is a net movement of fluid from the glomerular capillaries into Bowman’s capsule [1], [2], [3].

Many researchers have developed theoretical and mathematical models to study the fluid flow in renal tubules and ultrafiltration process in the glomerular capillaries. These models which are related to the fluid flow through the permeable boundary of the tubules have wide range of applications in understanding the mechanics of physiological fluid flows [4], [5], [6], [7].

The ultrafiltration process and movement of the solute particles across the glomerular capillaries have been analysed by Brenner et al. [8], [9]. Deen et al. [10], [11] presented the concept of concentration distribution in a capillary using the relation between the transmembrane fluid flux and the differences of hydrostatic pressure with osmotic pressure. They also employed the relation between the concentration of proteins and osmotic pressure by considering the pressure gradient is constant at the capillary wall. A mathematical model for the glomerular ultrafiltration process by taking pressure gradient as a function of axial distance of the capillary was developed by Marshal and Trowbridge [12] and Papenfuss and Gross [13].

The above studies have not taken the movement of the solute particles across the membrane of the capillary. But in reality, the movement is significant in the physiological situation as mentioned by Guyton [1]. Papenfuss and Gross [14] and Salathe [15] have analysed the capillary tissue fluid exchange and the transport of solute across the wall, assuming uniform concentration at each cross-section. By neglecting osmotic pressure, Ross [16] investigated the mass transport in fluid flow through a permeable tube. Tyagi and Abbass [17] studied the distribution of solute concentration in a blood passage in a hallow-fibre artificial kidney.

The transcapillary movement of water across the glomerular capillary is governed by the Starling’s forces that control the fluid movement across the capillary wall. Chaturani and Ranganatha [18] mentioned that ultrafiltration occurs because of both hydrostatic and osmotic pressures and the movement of solutes across the permeable membrane by diffusion and convection under the assumption of constant wall permeability. It may be noted that the assumption of constant permeability of glomerular capillary is of mathematical simplicity. There can be a significant changes in the filtration process of individual glomerulus in different regions of the kidney, where some nephrons are having larger volume compared with the other nephrons. The ultrafiltration process is dependent on the hydraulic conductivity and capillary surface area and this surface area may vary in the individual glomerular capillaries (Pollak et al. [19]). Thus, it can be concluded that permeability is not constant in a glomerular capillary.

Motivated by the article [19], an attempt has been made to describe the solute transfer and fluid flow through a tube by considering variable wall permeability. We explored the model, analysis, and solution procedure in detail, in the following Sections.

Figure 1:

Geometrical representation of glomerular capillary.

2 Mathematical Model

We considered the axisymmetric transport of fluid in a permeable tube of radius R and the length L, as represented in Figure 1. The u(r, z) and v(r, z) are velocity components along axial (z) and radial (r) directions, respectively. Assuming the low Reynolds number flow (of order 10−3), the governing equations for incompressible Newtonian fluid flow of constant viscosity and the equation for solute transfer are,

where P and μ are the pressure and dynamic viscosity coefficient of the fluid, respectively and c and D are the concentration and diffusivity of the solute, respectively.

Now, we considered the boundary conditions for the above system of equations. At the inlet cross-section, solute enters with uniformly distributed concentration and the fluid flow has a constant hydrostatic pressure and a constant flux. Mathematically, they are represented as:

Fluid flow and solute concentration distribution are axisymmetric. Hence, we have

Fluid flow through the permeable tube wall is governed by Starling’s hypothesis [18], [19],

At the permeable wall, we shall assume the no-slip condition for axial velocity, given as,

Solute mass flux at the permeable wall is given as,

where Φ={c;VR>0cT;VR<0, ΔP=P−PT, ΔPa=Pa−PT, Δπ=π−πT, (P,PT) – the hydrostatic pressure inside and outside the tube wall and (π,πT) – the osmotic pressure inside and outside the tube wall (P_T and π_T are taken as constant). P_a is hydrostatic pressure at z = 0 (entrance) and Q₀ is volume flow rate at z = 0. σ is the reflection coefficient [20], c₀ is the solute concentration at z = 0, c_T is the concentration outside the tube, h is the permeability of solute at the wall and T_R is transmittance coefficient [21]. The k(z) is the arbitrary function of axial distance z, denoting variable wall permeability.

The osmotic pressure of solute is usually a polynomial function of its concentration, as in [13],

The solute mass flux J_S across the permeable tube wall [20] is given as,

The total solute clearance J_C is given as,

To analyze the fluid flow, the following dimensionless quantities (denoted by ^)

are introduced, in (1)–(4). The dimensionless equations are (after dropping ^)

Here, Rp=RΔPa/(μU0) is a nondimensional parameter and Pe=U0R/D is the Peclet number, which is a measure of the mass transfer by convection compared due to diffusion.

The dimensionless boundary conditions are

The non-dimensional form of J_S of (13) and J_C of (14) are given as,

where, Sh = hR/D is Sherwood number and ε=kaμ/R is the ultrafiltration parameter.

3 Analysis

We solved (15) and (16), for u and v, using the conditions (22) and (24) and we got the axial and radial velocities as,

The second order ordinary differential equation governing the hydrostatic pressure (ΔP) is determined by the (29) and (23), is given by,

The boundary conditions (20) and (21) will take the form,

where

Figure 2:

Grid detail for Crank–Nicolson scheme around node (i, j).

Figure 3:

Flow sheet representing the iterative method used to solve the flow parameters.

The dimensionless solute concentration (18),

with corresponding initial/boundary conditions,

Figure 4:

The hydrostatic pressure (ΔP) distributions for different values of U₀.

Figure 5:

The hydrostatic pressure (ΔP) distribution for different values of T_R and h.

The solutions of the (30) and (34), in closed form, are difficult to obtain due to the coupled nature of hydrostatic pressure and solute concentration. So, we need to solve (30) and (34) using boundary conditions (31), (32) and (35)–(37), respectively.

Figure 6:

Effect of variable permeability on axial distribution of hydrostatic pressure (ΔP) for different values of D and ΔPa.

4 Numerical Solution of Transport Equation

Now, we solved (34) for solute concentration (c), which is coupled and dependent on u, v, ΔP and Δπ as given in (28), (29), (30) and (33), respectively. A solution for c in terms of an analytical expression not possible in this case involving u and v, explicitly. Hence, we obtained numerical solution of c, satisfying the initial/boundary conditions (19)–(25) using an iterative procedure, outlined below.

Equation (34) is discretized by finite difference method of Crank–Nicolson scheme. Let the tube inlet and outlet be denoted by index i = 0 and i = N, respectively. Similarly, the axis of the tube and the boundary are represented by the index j = 0 and j = M. Δz and Δr are the increments in z and r directions, respectively (Refer Fig. 2). The finite difference scheme corresponding to (34) at a general grid point with node index (i, j), is written as,

where Aj=λ(2r−Δr+rPeΔrvi+1,j),Bj=−2r(2λ+Pe(ui+1,j+ui,j)),Ej=λ(2r+Δr−rPeΔrvi+1,j) and Rj=−λ(2r−Δr+rPeΔrvi,j)ci,j−1+2r(2λ−Pe(ui+1,j+ui,j))ci,j−λ(2r+Δr−rPeΔrvi,j)ci,j+1 with λ=Δz/(Δr)2.

Figure 7:

The osmotic pressure (Δπ) distribution for different values of U₀.

Figure 8:

The osmotic pressure (Δπ) distribution for different values of T_R and h.

Figure 9:

Effect of variable permeability on osmotic pressure (Δπ) for different values of D and ΔPa.

These coefficients A_j, B_j, E_j, and R_j are valid for interior nodes. For the nodes at the boundary and axis of the tube, we simplify (38) using the discretised form of conditions (36) and (37) and the details are given below:

Figure 10:

Effect of variable permeability on axial velocity u with r, at z = 15, z = 30 and z = 50.

Figure 11:

Effect of variable permeability on radial velocity (v) with r, at z = 15, z = 30 and z = 50.

Figure 12:

Concentration profiles (c) with r at different cross-sections z = 15, z = 25 and z = 50 the tube for different values of T_R and h.

Figure 13:

Effect of variable permeability on concentration profiles c with r.

At r = 0, we have,

where B0=−2λ−Pe(ui+1,0+ui,0),E0=2λ; and R0=(2λ−Pe(ui+1,0+ui,0))ci,0−2λci,1.

At r = 1, we have,

Equations (38), (39), and (40) will generate a system of linear, simultaneous equations for unknown ci,j for each node (i,j). The coefficients of the matrix can be represented in a tridiagonal form and is solved by well known Thomas algorithm to get c(r,z) at discretised grid points.

The c, ΔP and Δπ values are known, at i = 0 level. To get the c values at i = 1 level, we applied the following iterative procedure. For this, we assumed an approximate value of c, say c1,M0, at the boundary and it needs to be corrected based on discretised scheme of (38) along with condition of (37). The corrected c value (say c1,M1 at the boundary) is obtained by three point backward difference formula corresponding to the condition (37). The iterative process is continued to get the correct value of c at the boundary, at the level i = 1, with the tolerance error value of 10−6. With this c value, i = 2 level values of c at the boundary is calculated in a similar way and this procedure was repeated until i=N level. The various steps of the algorithm of the iterative process is shown in flow chart as in Figure 3.

5 Results and Discussion

In this analysis, we considered the tube wall with variable permeability (k(z)) for 0⩽z⩽1, which is more significant in physiological situations [1], [19], [22]. So, we took a linear function form [23],

where θ is the permeability parameter.

The computational results are obtained for a given set of parameters relevant to the physiological situation [11], [13]. The flow quantities such as osmotic and hydrostatic pressures, velocity, solute concentration and solute clearance have been computed from analytical and numerical solutions based on these parameter values.

5.4 Concentration at the Wall (cw(z))

Figure 14 illustrates the axial distribution concentration at wall, for different values of T_R and h. It is clearly seen that the effects of transmittance coefficient and solute wall permeability are significant on the wall concentration. That is, the c_w decreases as T_R and h increases. It may be noted that the concentration at the wall at any axial position increases with the increase in variable permeability θ (See Fig. 15).

Figure 14:

Distribution of wall concentration profiles c_w for different values of T_R and h.

Figure 15:

Effect of variable permeability on the axial distribution of wall concentration profiles c_w.

Figure 16:

Axial distribution of concentration at the wall c_w for (a) Δπ=0 and (b) Δπ≠0 for different values of ε.

Figure 16 represents the concentration at the wall c_w with axial distance, for the two cases Δπ=0 (Fig. 16a) and Δπ≠0 (Fig. 16b) for different values of ultrafiltration parameter ε. It may be noted that Δπ is a function of z as given in (12). Δπ=0 refers the case of the absence of osmotic pressure across the permeable boundary, which can be enforced explicitly in the (30).

It is observed from both the Figure 16a and b that the values of concentration at the wall are close to each other when ε=10−8. As ε increases, the concentration at the wall (c_w) increases in both cases of Δπ. The numerical values of c_w are comparatively large in the case of Δπ=0 than in the case of Δπ≠0, as noted in [16]. We mentioned that the osmotic pressure and ultrafiltraion have greater impact than the hydrostatic pressure, in this analysis.

Table 1:

Total solute clearance (J_C) for TR=0.2, U0=0.07cm/sec, D=1×10−6cm²/sec and ΔPa=40×103dyn/cm²

ε=7.98×10−7ε=1.98×10−6
θ	Sh = 0.005	Sh = 0.035	Sh = 0.005	Sh = 0.035
0.0	0.0214991	0.0349016	0.0442532	0.0586395
0.05	0.0412248	0.0554341	0.0735118	0.0894347
0.1	0.0556031	0.0705242	0.0805	0.0973311
0.2	0.059538	0.075044	0.0880607	0.10602

6 Conclusions

In this article, we have described the idea of fluid flow and solute transfer in a straight tube with wall permeability as a linear function of axial distance. Numerical solutions have been obtained for the governing coupled fluid flow and solute concentration equations. The effects of the variable permeability on hydrostatic pressure, osmotic pressure, velocity, concentration profiles, and solute clearance have been obtained.

1. The effect of variable permeable wall on hydrostatic pressure, shows that as θ increases the ΔP curves decreases linearly. The pressure drop variations show that in the variable permeable case, different parametric values are greater than the corresponding constant permeability case. Also the pressure drop rapidly decreases in a variable permeable tube for increase in θ or the axial distance z, in comparison to the constant permeable tube.

2. The osmotic pressure curves increases nonlinearly with axial distance.

3. The axial and radial velocity values at the boundary are increasing along the axial direction, as θ increases.

4. Concentration profiles have been observed and are affected by T_R and h. Solute concentration rises as the wall variable permeability increases. The wall concentration decreases with the increase in T_R and h and increases with θ.

5. The total solute clearance increases with the values of θ, h, and ε. It may be noted that the present analysis studies the solute transfer and fluid flow in a straight tube. But in physiological situations, the capillaries are not uniform. So, this model can be made realistic by extending the above study to non-uniform tube.