BioMed Research International

Volume 2017 (2017), Article ID 5284816, 12 pages

https://doi.org/10.1155/2017/5284816

## Optimal Branching Structure of Fluidic Networks with Permeable Walls

^{1}Department of Mechanical Engineering, Federal University of Rio Grande do Sul (UFRGS), Porto Alegre, RS, Brazil^{2}Mechanical Engineering Graduate Program, University of Vale do Rio dos Sinos (UNISINOS), São Leopoldo, RS, Brazil^{3}Department of Physics, School of Science and Technology, University of Evora, Evora, Portugal

Correspondence should be addressed to Antonio F. Miguel

Received 1 January 2017; Revised 16 March 2017; Accepted 29 March 2017; Published 21 May 2017

Academic Editor: Kazunori Uemura

Copyright © 2017 Vinicius R. Pepe et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Biological and engineering studies of Hess-Murray’s law are focused on assemblies of tubes with impermeable walls. Blood vessels and airways have permeable walls to allow the exchange of fluid and other dissolved substances with tissues. Should Hess-Murray’s law hold for bifurcating systems in which the walls of the vessels are permeable to fluid? This paper investigates the fluid flow in a porous-walled T-shaped assembly of vessels. Fluid flow in this branching flow structure is studied numerically to predict the configuration that provides greater access to the flow. Our findings indicate, among other results, that an asymmetric flow (i.e., breaking the symmetry of the flow distribution) may occur in this symmetrical dichotomous system. To derive expressions for the optimum branching sizes, the hydraulic resistance of the branched system is computed. Here we show the T-shaped assembly of vessels is only conforming to Hess-Murray’s law optimum as long as they have impervious walls. Findings also indicate that the optimum relationship between the sizes of parent and daughter tubes depends on the wall permeability of the assembled tubes. Our results agree with analytical results obtained from a variety of sources and provide new insights into the dynamics within the assembly of vessels.

#### 1. Introduction

In nature, the function of many flow systems is to deliver a fluid flow from a finite-size volume to one point (and vice versa) [1]. Tree-shaped networks provide the solution to house and facilitate fluid flow. They play a vital role in the organization and operation of tracheobronchial system, blood vessels, river basins, and so forth. These networks branch by dichotomy with a regular reduction of their length and diameter, and they have been found to have fractal properties [2–5]. The tip of each tube bifurcates to form two daughter tubes, along the length of the network. Repetition of these bifurcations generates the stereotyped, hierarchically organized branched architecture of the tree [6]. Therefore, bifurcation is the building block of trees and deserves to be analyzed.

The sizes of the tubes in the bifurcation are important factors in determining the efficiency of physiological processes. Despite the remarkable variety and complexity of natural tree-shaped flow networks, a relationship exists between parent and daughter tube sizes [1, 6–8]. The studies of Hess [9] and Murray [10] show that, in the cardiovascular system, when a parent vessel branches into daughter vessel, the cube of the parent vessel’s diameter equals the sum of the cubes of the daughter vessels’ diameters. This reduction of diameter of daughter vessels is essential for a proper functioning of the cardiovascular system and is usually termed as Hess-Murray law. This law predicts the relationship between the diameters of parent and daughter tubes for internal flows obeying laminar conditions. Hess-Murray’s law has been most often applied to symmetry and asymmetric branching. In both cases it is assumed that the pressure drop over the daughter tubes is similar which is valid for a symmetry bifurcation but is only appropriated for a very low degree of branching asymmetry [11]. For a symmetric bifurcation, the ratio of daughter and parent tubes’ diameters can be written more simply as equal to a homothetic factor of 2^{-1/3}.

Hess-Murray law can be derived based on minimization of the energy required to synthesize, maintain, and pump blood (principle of minimum work) [10], minimization of flow resistance under the constraints posed by the space (constructal law) [1, 12], minimization of volume under constant pressure drop and flow rate [13], applying to a constant shear stress in all tubes [14], and so forth.

Using the constructal law, Bejan et al. [12] later derived an equation predicting the lengths of branching tubes by minimizing the overall flow resistance over a finite-size space. For laminar flow, they also found that the cube of the length of a parent tube should be equal to the sum of the cubes of the lengths of the daughter tubes. This means that in a symmetric bifurcation, for example, successive tubes are also homothetic with a size ratio of 2^{-1/3}. Uylings [15] and Bejan et al. [12] derived equations predicting the sizes of branching tubes whose internal flows obey turbulent conditions. Revellin et al. [16] and Miguel [17] also presented extensions of Hess-Murray’s law for non-Newtonian fluids that exhibit shear-thinning and shear-thickening behavior. Miguel [18] analyzes the optimal arrangement of vessels when there is dependence of apparent viscosity of blood on vessel diameter and hematocrit (Fåhræus-Lindqvist effect). Following the Haynes marginal zone theory, he obtain comprehensive expressions for the branching sizes of parent and daughter vessels providing easier flow access.

Although targeting the cardiovascular tree [9], experimental data have shown Hess-Murray’s law holds well in medium sized blood vessels [17], in the respiratory tract airways of warm-blooded vertebrates such as humans and dogs [10], and in the tubes for fluid transport in plants [19]. This means the vascular and bronchial trees not only in warm-blooded vertebrates but also in tree flow systems of animals and plants have reached similar design solutions.

From the considerable amount of literature comparing Hess-Murray’s law to physiological studies, there are also cases where vessels deviate from the optimum perspective [20–22]. Studies show deviations from the Hess-Murray law for proximal bifurcations of aorta and some pulmonary veins [20, 21]. The diameter of acinar airways (respiratory region of lungs) seems to fall less steeply than that of conducting airways that obey Hess-Murray law [22, 23]. Some authors point out that these deviations might be due to structural constraints. They argue that since deviations are small, the penalty of deviating from Hess-Murray’s law is also small. Thus, structural constraints are likely to influence their design [21]. Miguel [23] derived analytical expressions for the optimum way to connect parent and daughter vessels together having permeable walls. This work has brought out the idea for the optimum branching sizes of vessels influenced by the wall permeability. Although important, the analytical approach presented in this paper is based on several assumptions [23] and is lacking 3D details that can enrich the understanding of the influence of the permeable boundary [24].

Generally, in the body, fluid flow is laminar and there are evidences that turbulent flows may even pose health risk [25, 26]. Notice also that tubes that are part of the circulatory and respiratory systems may be able to transport fluids and solutes across the walls. For example, a network of capillaries (vascular tree) surrounds the respiratory tree (acinus zone) and brings blood into close proximity with air within the alveolus [27]. The exchange of oxygen and carbon dioxide is accomplished through semipermeable walls of both alveoli and capillaries.

This paper seeks to answer the following question: Are small deviations from the Hess-Murray law observed in permeable vessels tolerated because of small punitive increase of resistance penalty, or is there optimum way to connect parent and daughter vessels together to achieve the higher performance? Here, we present a numerical study devised to investigate the influence of wall permeability on the optimum geometrical relationship governing the ratio of sizes of the tubes in a branching network. The study focuses specifically on T bifurcation transporting fluid under steady laminar flow of incompressible fluids. Analytical expression for the optimum daughter-parent sizes ratio is presented and compared with Hess-Murray law.

#### 2. Methods

##### 2.1. Geometrical Configuration of Branching Tubes in a T-Junction

Figure 1 illustrates the symmetrical T-shaped assembly of cylindrical tubes. Parent tube bifurcates and its size may change by a certain factor. The ratio of successive diameters and lengths (daughters to parent vessels) can be written aswhere and are scale factors or homothety ratios, is the diameter, is the length, and the subscripts 1 and 2 mean parent and daughter tubes. According to Hess-Murray law, for laminar flow, = = 2^{-1/3} but in this paper branching tubes with homothety ratios between 0.1 and 1.0 are also studied. In order to compare the performance of each assembly of tubes to fluid flow, the following constraints are considered:The constraints represented by these equations physically define that both the total volume occupied by the tubes and the total space occupied by the planar assembly of tubes are fixed, respectively.