Exact Similarity Solutions of Coupled Nonsteady Navier--Stokes and Energy Equations in Liquids

Exact similarity solutions are obtained for coupled Navier--Stokes and energy equations that govern the time-dependent motion of a gravitation-free viscous liquid with variable density in one- and three-dimensional (1D and 3D) spaces. The 1D case deals with propagation and diffusion of an initial algebraic density hump C/x that is subjected to a constant flow rate at the origin. We demonstrate that the initial temperature distribution is convected without being diffused at very long times. We also demonstrate that two different propagation velocities exist for short and long times. The 3D case addresses the implosion of an insulated closed system with an initial radially symmetric algebraic density hump C/r3 . We demonstrate that if viscous dissipation and liquid compressibility terms are neglected in the energy equation, very strong shock-like pressure distributions may occur that may lead to a "black hole" within a finite time.A comprehensive analysis is also carried out for density fields with an initi...