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Stability of plane Poiseuille and Couette flows of Navier–Stokes–Voigt fluid

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Abstract

The temporal stability of plane Poiseuille and Couette flows for a Navier–Stokes–Voigt type of viscoelastic fluid is examined. The primary unidirectional flow is between two infinite rigid parallel plates, which are either fixed or in relative motion. To investigate the instability of the basic flow, a numerical solution of the resulting eigenvalue problem is performed. Despite the base flow remains to be unaltered, the stability properties differ from those of a Newtonian fluid. In the case of plane Poiseuille flow, two values of the Reynolds number are found to be needed to specify the linear instability criteria owing to the existence of closed neutral stability curves and also the instability emerges only in a certain range of the Kelvin–Voigt parameter \(\Lambda\). To the contrary, instability occurs for all nonzero values of \(\Lambda\) in the case of plane Couette flow and a single critical value of the Reynolds number is adequate to discuss the stability/instability of fluid flow due to the parabolic nature of the neutral stability curves. The sensitivity of the Kelvin–Voigt parameter is clearly discerned on the stability of both types of flows. The variations of streamlines at the dominant mode of instability are analyzed in comprehending the underlying instability mechanism and shedding light on the secondary flow pattern.

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Acknowledgements

We are indebted to two anonymous referees for their valuable comments which have improved the manuscript.

Funding

Funding was provided by PES University (Grant No. PESUIRF/Math/2020/11).

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Authors and Affiliations

  1. Department of Mathematics, PES University, Bangalore, India

    B. M. Shankar

  2. Department of Mathematics, Bangalore University, Bangalore, India

    I. S. Shivakumara

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  1. B. M. Shankar
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Correspondence to B. M. Shankar.

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Shankar, B.M., Shivakumara, I.S. Stability of plane Poiseuille and Couette flows of Navier–Stokes–Voigt fluid. Acta Mech 234, 4589–4609 (2023). https://doi.org/10.1007/s00707-023-03624-0

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