Abstract
Motivated by a range of applications (biomedical, industrial, engineering, environmental) this contribution is focussed on a mathematical study of (a) constriction/distortion and (b) branching in a vessel or network of vessels containing fluid flow. The central interest addressed is in medium-to-high Reynolds numbers where asymptotic approaches and matching yield much insight. The main reasoning, order arguments and scaling factors within various parts of the vessels are presented. Theory and corresponding analysis are described for aspect (a) in symmetric and nonsymmetric cases and aspect (b) over short or long length scales with or without viscous–inviscid interactions, where attention is given to side-branching, large networks, viscous wall layers, flow reversal, eddies and upstream influence. Three-dimensional effects for (a) and (b) are also investigated. A final discussion includes suggestions of future project topics.
Highlights
Internal fluid dynamics associated with the flow through a rigid vessel or network of vessels is described theoretically in the present article which is concentrated in particular on two fundamental aspects, namely (a) constriction and (b) branching of the Communicated by S.K
The full approach is expressed more systematically in terms of asymptotic expansions for specific contexts in Sect. 3 onwards. It soon becomes clear for many two-dimensional steady configurations that at least three major length scales can apply readily to the internal flows of concern at medium to large Re values. This holds quite generally for distorted channel flows say with constrictions, dilations, roughnesses or corners we keep the application of branching flows firmly in mind
The results confirm in particular that most of the change in uW occurs within a short distance O(α) of the daughter mouth when α is small, e.g. 0.1 or less, the other distinguished length scale being O(1) ahead of and after the mouth. Over the latter global scale the full profile u0 has effect but the daughter acts as a sink-like disturbance at the wall, Fig. 5 i Symmetric branching flow: MW is the width of the mother-vessel, DW is that of the daughter. ii Scaled slip velocity induced at outer wall in three cases, versus x. iii Further cases, showing the induced scaled displacement, pressure and shear stress at the outer wall whereas over the O(α) local scale the details of the daughter mouth are apparent and only the incident shear flow λy drives the local flow
Summary
Internal fluid dynamics associated with the flow through a rigid vessel or network of vessels is described theoretically in the present article which is concentrated in particular on two fundamental aspects, namely (a) constriction and (b) branching of the Communicated by S.K. Possible practical limitations of the approach which can vary from case to case should be mentioned They centre mostly on turbulence, three-dimensionality, Fig. 1 Representative non-dimensional picture for internal fluid motion with a constriction or branching non-rigid wall effects, non-Newtonian effects, wall roughness and the use of asymptotics.
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