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The Equation for the Stokesian Stream Function and Its Solutions

  • Günter BrennEmail author
Chapter
Part of the Mathematical Engineering book series (MATHENGIN)

Abstract

This chapter presents and discusses the equation for the Stokesian stream function. The equation emerges as the one non-zero component of the curl of the two-dimensional momentum equation with the velocity components given as spatial derivatives of the stream function. The stream function is defined such that its derivatives yield a solenoidal velocity field. The analyses of the flows discussed in Part I of this book are based on this function. In view of our search for analytical solutions, we are restricted to laminar two-dimensional flow in simple geometries. The equations of change therefore need no turbulence modelling, the concept of the Stokesian stream function can be applied for representing the flow velocity, and the boundary conditions are easy to formulate and implement analytically in the general solutions. The fluids are treated as incompressible and Newtonian or linear viscoelastic. The linear viscoelastic liquids exhibit a viscosity depending on frequency, but not on shear rate. Furthermore, we restrict this analysis to flows without heat and mass transfer, i.e. we solve the continuity and momentum equations and disregard the influence of viscous dissipation on the energy budget of the flow. We are therefore restricted to flow without viscous heating. Problems of heat and mass transfer are the subjects of Part II of this book.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover Publications, New York (1972)zbMATHGoogle Scholar
  2. 2.
    Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. Wiley, New York (1960)Google Scholar
  3. 3.
    Tomotika, S.: On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. A. 150, 322–337 (1935)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of Fluid Mechanics and Heat TransferGraz University of TechnologyGrazAustria

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