Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-12T18:07:34.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Conjugate-flow theory for heterogeneous compressible fluids, with application to non-uniform suspensions of gas bubbles in liquids

Published online by Cambridge University Press:  29 March 2006

T. Brooke Benjamin
T. Brooke Benjamin
Affiliation:
Fluid Mechanics Research Institute, University of Essex
Get access Rights & Permissions [Opens in a new window]

Abstract

Conjugate flows have been defined generally as flows uniform in the direction of streaming that separately satisfy the relevant hydrodynamical equations, so allowing a transition from one flow to its conjugate to be consistent with mass and energy conservation. In previous studies of various examples, certain general principles have been found to apply to conjugate flows: in particular, one in a pair of such flows is subcritical (subsonic) and the other supercritical (supersonic), the former having greater flow force (i.e. momentum flux plus pressure force). In this paper these principles are confirmed in another field of application, for which the theory of conjugate flows takes a novel course.

The theoretical model defined in § 2 consists of a straight duct of arbitrary cross-section filled with a perfect fluid whose constitutive properties vary with cross-sectional position, and whose primary, prescribed flow is axial with a velocity distribution that may be non-uniform. In § 3 the possibility of a conjugate flow in the same duct is investigated, and its principal properties relative to those of the primary flow are deduced from certain simple inequalities between integrals over the cross-section. A Lagrangian description of the conjugate flow is essential, but the properties in question are established without the necessity of determining this flow explicitly. At the end of § 3, a modification of the model is discussed accounting for dissipative, flow-force conserving transitions (shocks). The application of the theory to flows of non-uniform suspensions of gas bubbles is considered in § 4.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 54 , Issue 3 , 8 August 1972 , pp. 545 - 563
Copyright
© 1972 Cambridge University Press

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

Ackeret, J. 1930 Experimental and theoretical investigations on cavitation in water Techn. Mech. u. Thermodyn. 1, 1.Google Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon J. Fluid Mech. 14, 593.Google Scholar
Benjamin, T. B. 1965 Significance of the vortex breakdown phenomenon. Trans. Am. Soc. Mech. Engrs, J. Basic Engng. 87, 518 & 1091.Google Scholar
Benjamin, T. B. 1967 Some developments in the theory of vortex breakdown J. Fluid Mech. 28, 65.Google Scholar
Benjamin, T. B. 1971 A unified theory of conjugate flows. Phil. Trans. Roy. Soc. A 269, 587.Google Scholar
Campbell, I. J. & Pitcher, A. S. 1958 Shock waves in a liquid containing gas bubbles. Proc. Roy. Soc. A 243, 534.Google Scholar
Carstensen, E. L. & Foldy, L. L. 1947 Propagation of sound through a liquid containing bubbles J. Acous. Soc. Am. 19, 481.Google Scholar
Hadry, G. H., Littlewood, J. E. & Polya, G. 1952 Inequalities, 2nd edn. Cambridge University Press.
Hsieh, D.-Y. & Plesset, M. S. 1961 On the propagation of sound in a liquid containing bubbles Phys. Fluids, 4, 970.Google Scholar
Mallock, A. 1910 The damping of sound by frothy liquids. Proc. Roy. Soc. A 84, 391.Google Scholar
Meyer, E. & Skudzrijk, E. 1953 Über die akustischen Eigenschaften von Gasblasenshcleiern in Wasser Akust. Beih. 3, 434.Google Scholar
Prandtl, L. 1952 The Essentials of Fluid Dynamics. London: Blackie.
van Wijngaarden, L. 1968 On the equations of motion for mixtures of liquid and gas bubbles J. Fluid Mech. 33, 465.Google Scholar