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Continuum Mechanics and Thermodynamics

, Volume 9, Issue 2, pp 73–82 | Cite as

The \(\vec\omega-\vec{D}\) fluid: general theory with special emphasis on stationary two dimensional flows

  • P.O. Brunn
  • E. Ryssel
Original Article

In this paper a theory is presented in which the extra stress tensor \(\vec\tau\) is allowed to depend not only on the rate of strain tensor \(\vec{D}\) but also on the relative vorticity \(\vec\omega\) of the fluid, i.e. on the vorticity relative to the rate of rotation of the principal straining directions. This theory has its origin in an expansion of \(\vec\tau\) in terms of kinematic tensors in the limit of stationarity in a material sense (constant stretch history flows). For two dimensional flows of an incompressible fluid three tensors suffice to completely specify \(\vec\tau\). The three material functions which appear can depend only on two invariants, namely the second invariant of \(\vec{D}\) and on \(\omega^2\). Using the predictions of an Oldroyd 8 constant fluid in a homogeneous planar flow of constant stretch history, the three material functions are studied in detail. For the special case of a quasi-Newtonian fluid shear thinning and extension thickening can directly be accounted for in the “viscosity” function.

Key words: rheology, invariants, rotation of the principal straining directions, shear thinning and extension thickening, two dimensional flows 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • P.O. Brunn
    • 1
  • E. Ryssel
    • 1
  1. 1. Lehrstuhl für Strömungsmechanik, Universität Erlangen-Nürnberg, Erlangen, GermanyDE

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