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Elliptical flows perturbed by shear waves

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Abstract

We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin–Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier–Stokes fluid in an orthogonal rheometer).

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Notes

  1. This time-dependent generalization of the motion in (1.2) is not a dynamically possible solution for simple fluids but it is an interesting class of possible flows when inertia terms are ignored.

  2. If in (1.9) we change the trigonometric functions \(\cos \) and \(\sin \) with the hyperbolic functions \(\cosh \) and \(\sinh \) respectively, we obtain a motion with hyperbolic streamlines [25].

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Acknowledgements

EP and GS have been partially supported by GNFM of INDAM. We would like to thank K. R. Rajagopal and L. Vergori for their comments on a previous draft of this work.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli Studi di Perugia, 06125, Perugia, Italy

    Edvige Pucci & Giuseppe Saccomandi

  2. School of Mathematics, Statistics and Applied Mathematics, NUI Galway, University Road, Galway, Ireland

    Giuseppe Saccomandi

Authors
  1. Edvige Pucci
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  2. Giuseppe Saccomandi
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Correspondence to Giuseppe Saccomandi.

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In expression of our esteem, gratitude and affection this paper is dedicated to Tommaso Ruggeri on the occasion of his 70th birthday.

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Pucci, E., Saccomandi, G. Elliptical flows perturbed by shear waves. Ricerche mat 67, 509–524 (2018). https://doi.org/10.1007/s11587-017-0325-0

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