Characterizing Networks Admitting k Arc-disjoint Branching Flows

An s-branching flow f in a network N = (D,c) (where c is the capacity function) is a flow that reaches every vertex in V(D) \ {s} from s while loosing exactly one unit of flow in each vertex other than s. In other words, the difference between the flow entering a vertex v and a flow leaving a vertex v is one whenever v is different from s. It is known that the hardness of the problem of finding k arc-disjoint s-branching flows in network N is linked to the capacity c of the arcs in N: the problem is solvable in polynomial time if every arc has capacity n - l, for fixed l, and NP-complete in most other cases, with very few cases open. We further investigate a conjecture by Costa et al. from 2019 that aims to characterize networks admitting k arc-disjoint s-branching flows, generalizing a classical result by Edmonds that provides such characterization when all arcs have capacity n-1. We show that, in general, the conjecture is false. However, on the positive side, it holds for digraphs formed by out-branchings together with parallel arcs.

NONSYMMETRIC BRANCHING OF FLUID FLOWS IN 3D VESSELS

Nonsymmetric branching flow through a three-dimensional (3D) vessel is considered at medium-to-high flow rates. The branching is from one mother vessel to two or more daughter vessels downstream, with laminar steady or unsteady conditions assumed. The inherent 3D nonsymmetry is due to the branching shapes themselves, or the differences in the end pressures in the daughter vessels, or the incident velocity profiles in the mother. Computations based on lattice-Boltzmann methodology are described first. A subsequent analysis focuses on small 3D disturbances and increased Reynolds numbers. This reduces the 3D problem to a two-dimensional one at the outer wall in all pressure-driven cases. As well as having broader implications for feeding into a network of vessels, the findings enable predictions of how much swirling motion in the cross-plane is generated in a daughter vessel downstream of a 3D branch junction, and the significant alterations provoked locally in the shear stresses and pressures at the walls. Nonuniform incident wall-shear and unsteady effects are examined. A universal asymptotic form is found for the flux change into each daughter vessel in a 3D branching of arbitrary cross-section with a thin divider.

Optimal Branching Structure of Fluidic Networks with Permeable Walls

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.

Peculiarities of Parameter Definition of Products Routing in Branching Flow-Spatial Technological Systems on the Basis of Multidimensional Algebra of Groups

: In this work there were developed fundamentals in parameter definition of products routing in flow-spatial technological systems which have branching flows of products. In work there have been established initial conditions for definition of parameters of products routing.In work there were considered peculiarities of operation of flow-spatial technological systems with branching flows of products. There were developed basics of definition of parameters of products routing on the basis of multidimensional algebra of groups.

Edge Effects on the Flow Characteristics in a 90deg Tee Junction

Measurements of pressure drop were carried out for the flow of a Newtonian fluid in 90deg tee junctions with sharp and round corners. Rounding the corners reduced the energy losses by between 10 and 20%, depending on the flow rate ratio, due to the reduction in the branching flow loss coefficient, whereas the straight flow basically remained unaffected. The corresponding detailed measurements of mean and turbulent velocities for a Reynolds number of 31,000 and flowrate ratio of 50% showed that rounding the corner lead to an increase in turbulence in the branch pipe. The increased turbulence diffused momentum more efficiently thus reducing the length of the recirculation by 25% with its width and strength also decreasing in magnitude. The overall effect of the increased dissipation due to turbulence and reduced dissipation due to mean flow irreversibilities in the recirculation was a decrease in the corresponding loss coefficient.

Thermal Characteristics of Microscale Fractal-Like Branching Channels

Heat transfer through a fractal-like branching flow network is investigated using a three-dimensional computational fluid dynamics approach. Results are used for the purpose of assessing the validity of, and providing insight for improving, assumptions imposed in a previously developed one-dimensional model for predicting wall temperature distributions through fractal-like flow networks. As currently modeled, the one-dimensional code fairly well predicts the general wall temperature trend simulated by the three-dimensional model; hence, demonstrating its suitability as a tool for design of fractal-like flow networks. Due to the asymmetry in the branching flow network, wall temperature distributions for the proposed branching flow network are found to vary with flow path and between the various walls forming the channel network. Three-dimensional temperature distributions along the various walls in the branching channel network are compared to those along a straight channel. Surface temperature distributions on a heat sink with a branching flow network and a heat sink with a series of straight, parallel channels are also analyzed and compared. For the same observed maximum surface temperature on these two heat sinks, a lower temperature variation is noted for the fractal-like heat sink.