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Converging gravity currents of power-law fluid

Published online by Cambridge University Press:  05 May 2021

S. Longo Open the ORCID record for S. Longo [Opens in a new window] ,
L. Chiapponi Open the ORCID record for L. Chiapponi [Opens in a new window] ,
D. Petrolo Open the ORCID record for D. Petrolo [Opens in a new window] ,
A. Lenci Open the ORCID record for A. Lenci [Opens in a new window]  and
V. Di Federico Open the ORCID record for V. Di Federico [Opens in a new window]
S. Longo
Affiliation:
Department of Engineering and Architecture, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124Parma, Italy
L. Chiapponi*
Affiliation:
Department of Engineering and Architecture, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124Parma, Italy
D. Petrolo
Affiliation:
Department of Engineering and Architecture, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124Parma, Italy
A. Lenci
Affiliation:
Department of Civil, Chemical, Environmental, and Materials Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136Bologna, Italy
V. Di Federico
Affiliation:
Department of Civil, Chemical, Environmental, and Materials Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136Bologna, Italy
*
†Email address for correspondence: [email protected]
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Abstract

We study non-Newtonian effects associated with power-law rheology of behaviour index $n$ on the propagation of horizontal gravity currents. Two different set-ups are examined: (i) converging flow toward the origin in a channel of gap thickness $b(x) \propto x^k$ and $k<1$ and (ii) converging flow along $r$ toward the centre in a cylinder. The front of the current propagates in the negative $x$ or $r$ direction reaching the origin in a finite touch-down time $t_c$ during the pre-closure phase; in the post-closure phase, the current flows back in the positive direction and progressively levels out. Under the classical viscous-buoyancy balance, the current propagation is described by a differential problem amenable to a self-similar solution of the second-kind coupling space and reduced time $t_r=t_c-t$. The problem formulation in the phase plane leads to an autonomous system of differential equations which requires numerical integration and yields the shape of the current and its front as $\xi _f\propto t_r^{\delta _c}$, $\xi _f$ being the self-similar variable value at the front and $\delta _c$ being the critical eigenvalue. The latter is a function of fluid rheology $n$ and of channel geometry $k$ for the first set-up; it is a function only of $n$ for the second set-up. The dependency on $n$ is modest. The theoretical formulation is validated through experiments conducted during both pre- and post-closure phases and aimed at measuring the front position and the profile of the current. Experimental results are in fairly good agreement with theory and allow quantitative determination of the time interval of validity of the intermediate asymptotics regime, when self-similarity is achieved and when it is lost.

JFM classification

Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Hele-Shaw flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 918 , 10 July 2021 , A5
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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