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- Title
Analytical solutions to the pressureless Navier–Stokes equations with density-dependent viscosity coefficients.
- Authors
Dong, Jianwei; Xue, Hongxia; Zhang, Qiao
- Abstract
In this paper, we construct a class of spherically symmetric and self-similar analytical solutions to the pressureless Navier–Stokes equations with density-dependent viscosity coefficients satisfying h (ρ) = ρ , g (ρ) = (− 1) ρ for all > 0. Under the continuous density free boundary conditions imposed on the free surface, we investigate the large-time behavior of the solutions according to various > 1 and 0 < < 1. When the time grows up, such solutions exhibit interesting information: Case (i) If the free surface initially moves inward, then the free surface infinitely approaches to the symmetry center and the fluid density blows up at the symmetry center, or the free surface tends to an equilibrium state; Case (ii) If the free surface initially moves outward, then the free surface infinitely expands outward and the fluid density decays and tends to zero almost everywhere away from the symmetry center, or the free surface tends to an equilibrium state. We also study the large-time behavior of the solutions for = 1 without any boundary conditions.
- Subjects
NAVIER-Stokes equations; ANALYTICAL solutions; FREE surfaces; VISCOSITY; BLOWING up (Algebraic geometry)
- Publication
Communications in Contemporary Mathematics, 2024, Vol 26, Issue 5, p1
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199723500220