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Article

Two-Dimensional Steady Boussinesq Convection: Existence, Computation and Scaling

Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, Dayton, OH 45433, USA
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editor: Mehrdad Massoudi
Fluids 2021, 6(12), 425; https://doi.org/10.3390/fluids6120425 (registering DOI)
Received: 28 September 2021 / Revised: 15 November 2021 / Accepted: 19 November 2021 / Published: 25 November 2021
(This article belongs to the Special Issue Convection in Fluid and Porous Media)
This research investigates laser-induced convection through a stream function-vorticity formulation. Specifically, this paper considers a solution to the steady Boussinesq Navier–Stokes equations in two dimensions with a slip boundary condition on a finite box. A fixed-point algorithm is introduced in stream function-vorticity variables, followed by a proof of the existence of steady solutions for small laser amplitudes. From this analysis, an asymptotic relationship is demonstrated between the nondimensional fluid parameters and least upper bounds for laser amplitudes that guarantee existence, which accords with numerical results implementing the algorithm in a finite difference scheme. The findings indicate that the upper bound for laser amplitude scales by O(Re2Pe1Ri1) when RePe, and by O(Re1Pe2Ri1) when PeRe. These results suggest that the existence of steady solutions is heavily dependent on the size of the Reynolds (Re) and Peclet (Pe) numbers, as noted in previous studies. The simulations of steady solutions indicate the presence of symmetric vortex rings, which agrees with experimental results described in the literature. From these results, relevant implications to thermal blooming in laser propagation simulations are discussed. View Full-Text
Keywords: convection; Navier–Stokes; steady state; existence convection; Navier–Stokes; steady state; existence
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MDPI and ACS Style

Lane, J.S.; Akers, B.F. Two-Dimensional Steady Boussinesq Convection: Existence, Computation and Scaling. Fluids 2021, 6, 425. https://doi.org/10.3390/fluids6120425

AMA Style

Lane JS, Akers BF. Two-Dimensional Steady Boussinesq Convection: Existence, Computation and Scaling. Fluids. 2021; 6(12):425. https://doi.org/10.3390/fluids6120425

Chicago/Turabian Style

Lane, Jeremiah S., and Benjamin F. Akers 2021. "Two-Dimensional Steady Boussinesq Convection: Existence, Computation and Scaling" Fluids 6, no. 12: 425. https://doi.org/10.3390/fluids6120425

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