Low Reynolds number developing flows

Abstract

Equations are given that relate the entrance length to Reynolds number for pipe and channel geometries with a flat velocity profile as the initial condition. These equations are linear combinations of the creeping flow and boundary layer solutions. The former is obtained by minimization of the viscous dissipation using the finite element method. The equation for the pipe entrance is shown to be in good agreement with experimental data.The entrance region for laminar flow has received considerable attention in recent years. Not only is such study of importance for industrial and viscometric applications, but also as a simple developing flow it provides a useful vehicle for the evolution and evaluation of numerical solution procedures for nonlinear partial differential equations, for example, the Navier-Stokes equations. In consequence, the restrictions imposed on the solution by use of the Prandtl boundary-layer hypothesis have been overcome and, in principle, it is now possible to achieve a solution for all Reynolds numbers and boundary conditions.The early numerical solutions were obtained by Bodoia and Osterle ( 1 ) for parallel plates, and Hornbeck ( 2 ) for pipes, both using a flat velocity profile as an initial condition and applying the boundary-layer hypothesis to yield initial value problems. The velocity and pressure fields were then obtained by finite difference procedures ( 3 ) . These studies have been extended to the lower Reynolds number region where the problem is of the boundary value type, by Wang and Longwell ( 4 ) in the case of parallel plates and for pipes by Vrentas, Duda, and Bargeron ( 5 ) . In the former, solutions are presented at a Reynolds number of 300 using both the flat velocity profile (Figure 1) and the stream-tube (Figure 2) as the initial condition, whereas in the latter, the stream-tube condition was used exclusively at Reynolds numbers of 0, 1, 50, 150, and 250. The significant features of these solutions lie in the transport of vorticity upstream of the tube entrance when using the stream-tube condition (4, S), the way the entrance length varies with Reynolds number ( 5 ) , the finite entrance length at zero Reynolds number ( 5 ) , and the fact that the maxima are predicted in the velocity profiles at locations other than the center line ( 4 ) .The associated experimental investigations, all in pipes, due to Nikuradse ( 6 ) , Reshotko (7), and Pfenninger (8) are discussed by Atkinson, Kemblowski, and Smith (9), who have confirmed beyond reasonable doubt the high Reynolds number asymptote of the Navier-Stokes equations applied to entrance region problems.Experimentally, one of two methods is used in an attempt to produce an acceptable approximation to a flat velocity profile for use as an initial condition. These are the type of device developed by Atkinson, et al. (9) and the conical entrance section (7,8).In view of the computations of Vrentas, et al. ( 5 ) application of the latter method is questionable at low Reynolds numbers because of the upstream axial diffusion of vo...

Simultaneous measurements of the flow velocities in a microchannel by wide/evanescent field illuminations with particle/single molecules

“…3A) are shown in Fig. 5 34 Hence, the laminar flow was fully developed at the imaging point (#2 cm away from the entry reservoir). Considering W/Hy30, the velocity profile of Poiseuille flows in two dimensions is presented as 35 ( Fig.…”

Developing laminar flows with visco-elastic and non-Newtonian liquids

“…A t y p i c a l flow p a t t e r n in t h e e n t r a n c e r e g i o n is s h o w n in fig. 3 while for c o m p a r i s o n , (17). F r o m t h e s e t h e 9 9 % e n t r a n c e l e n g t h s w e r e o b t a i n e d a n d are c o r r e l a t e d w i t h t h e g e n e r a l i z e d p i p e Reynolds n u m b e r in fig.…”

Aggregation of blood components on a surface in a microfluidic environment

“…The corresponding length of the affected region was estimated practically by several studies. [21][22][23][24] For the flow condition in this model system, the length is approximately six times the diameter of the circular tube. This could increase platelet collision frequency at region A, thus becoming part of the reason that a higher density of adhering platelets appears at its downstream region ͑region G͒.…”