Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T14:24:06.519Z Has data issue: false hasContentIssue false
  • English
  • Français

The viscous flow through symmetric collapsible channels

Part of: Biological fluid mechanics

Published online by Cambridge University Press:  26 February 2010

A. P. Rothmayer
A. P. Rothmayer
Affiliation:
Department of Aerospace Engineering, Iowa State University, Ames, Iowa, 50011, U.S.A.
Get access

Extract

A high Reynolds number theory is developed for a viscous fluid flowing through an elastic channel. Unlike the flow through rigid symmetric channels, the viscous flow through a symmetric elastic channel is found to admit free-interaction solutions, due solely to the interaction of the boundary layer with the elastic channel wall. The assumption of symmetry is found to be general providing that the streamwise extent of the channel collapse dilation is larger than O(K17) and the channel is allowed to deviate only slightly from a straight channel. These free-interactions are believed to be the viscous initiation of a sudden collapse or dilation of the channel, commonly observed in experiment. The collapse of the channel is found to occur over a wide range of possible streamwise length scales from O(l) to O(K). For a rigid channel which is coated with a thin elastic solid, the equations are found to reduce to the hypersonic strong interaction problem of triple-deck theory. The hypersonic triple-deck is known to admit both compressive and expansive free-interactions. The expansive free-interaction is found to correspond to a sudden collapse of the channel and an acceleration of the flow within the core of the channel. A cha nnel that is backed by a stagnant constant pressure fluid is also examined. For this problem, the pressure is proportional to the negativeof the fourth derivative of the channel wall displacement. This structure is also found to admit compressive and or expansive free-interactions, depending on whether the internal pressure within the channel is less than or greater than the constant pressure external to the channel. Terminal forms are developed for the expansive free-interaction and compared with numerical calculations.

MSC classification

Secondary: 76Z05: Physiological flows
Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 153 - 181
Copyright
Copyright University College London 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bertram, C. D., 1987. The Effect of Wall Thickness, Axial Strain and Endpoint Proximity on the Pressure-Area Relation of Collapsible Tubes. J. Biomeck, 20, 863876.Google Scholar
Brown, S. N.Stewartson, K. and Williams, P. G. 1974. On Expansive Free-Interactions in Boundary Layers. Proc. Roy. Soc. Edinburgh, 74A(21), 271283.CrossRefGoogle Scholar
Brown, S. N., Stewartson, K. and Williams, P. G., 1975. Hypersonic Self-Induced Separation. Physics of Fluids, 18 (6), 633639.Google Scholar
Cancelli, C. and Pedley, T. J., 1985. A Separated-Flow Model for Collapsible-Tube Oscillations. J. Fluid Mech, 157, 375404.CrossRefGoogle Scholar
Fung, Y. C., 1965. Foundations of Solid Mechanics (Prentice-Hall).Google Scholar
Gajjar, J. and Smith, F. T., 1983. On Hypersonic Self-Induced Separation, Hydraulic Jumps and Boundary Layers with Algebraic Growth. Mathematika, 30, 7793.CrossRefGoogle Scholar
Jensen, O. E. and Pedley, T. J., 1989. The Existence of Steady Flow in a Collapsed Tube. To appear.CrossRefGoogle Scholar
Moreno, A. H., Katz, A. I., Gold, L. D. and Reddy, R. V. 1970. Mechanics of Distension of Dog Veins and Other Very Thin-Walled Tubular Structures. Circulation Research, 27, 10691080.Google Scholar
Pedley, T. J., 1980. The Fluid Mechanics of Large Blood Vessels (Cambridge University Press).Google Scholar
Reyhner, T. A. and Flugge-Lotz, I., 1968. The Interaction of a Shock Wave with a Laminar Boundary Layer. Int. J. Nonl. Mech, 3, 173199.CrossRefGoogle Scholar
Rothmayer, A. P. and Hiemcke, C., 1988. The Stability of a Blasius Boundary Layer Flowing Over an Elastic Solid. Unpublished.Google Scholar
Rothmayer, A. P., 1989. The Computation of Separated Flows in Symmetric Channels. Proceedings of the 4th Conference on Numerical and Physical Aspects of Aerodynamic Flows.Google Scholar
Smith, F. T., 1976a. Flow Through Constricted or Dilated Pipes and Channels: Part 1. Q. Jl. Meek appl. Math., 29 (3), 344364.Google Scholar
Smith, F. T., 1976b. Flow Through Constricted or Dilated Pipes and Channels: Part 2. Q. Jl. Meek appl. Math., 29 (3), 365376.Google Scholar
Smith, F. T., 1977. Upstream Interactions in Channel Flows. J. Fluid Meek, 79 (4), 631655.CrossRefGoogle Scholar
Stewartson, K. and Williams, P. G., 1969. Self-Induced Separation. Proc. Roy. Soc., A312, 181206.Google Scholar
Stewartson, K. 1974. Multistructured Boundary Layers on Flat Plates and Related Bodies. Adv. appl. Meek, 14, 145239.Google Scholar
Tutty, O. R. 1984. High Reynolds Number Viscous Flow in Collapsible Tubes. J. Fluid Mech, 146, 451469.Google Scholar