Theoretical Studies On The Instability Of Non-ideal Fluids

Fluid mechanics is an important branch of classical mechanics. It has a long history and applied extensively in atmosphere, astrophysics, physical geography, the space exploration, and many other fields. Therefore, intensive studies on the fluid mechanics will be very important for probing into nature and improving daily life and so forth.The instability of hydrodynamics flows is a fundamental issue in fluid mechanics, and one of common phenomena in nature. Small disturbances in a multifluid system produce buoyancy and shear-driven instabilities at an interface between distinct fluids. The Rayleigh-Taylor and Richtmyer-Meshkov instabilities are the most important interfacial instabilities. The Rayleigh-Taylor instability occurs at a fluid interface where the density gradient and the acceleration are oppositely directed; The Richtmyer-Meshkov instability takes place when shock waves pass through an interface. They exist extensively in many fields ranging from astrophysics to magnetic or inertial confinement fusion and plasmas and play an important role.Since the early of the last century, the Rayleigh-Taylor and Richtmyer-Meshkov instabilities have aroused much attention. It is found that, at the early stage of instability evolution (i.e., the linear stage), the amplitude of disturbance at the interface increases exponentially with time. At the late time, when the amplitude of disturbance becomes comparable to its wavelength, the nonlinear structure occurs at the interface. In such a stage, there clearly appear bubbles of light fluid and spikes of heavy fluid, each penetrating into the opposite fluid. At the last, turbulent mixing takes place and the fluids flow in turbulence.As for dynamic evolution of the bubble, there are still debates in theory. There are two models for describing the bubble evolution in history. One is the Layzer model proposed by Layzer in 1955, and the other is the point-source model proposed by Zufiria in 1988. Both of the two models were originally proposed in the case of vacuum bubble, and then generalized to the case of arbitrary Atwood numbers by Goncharov and Sohn. The comparison between the two models has been made by Sohn, and the obtained solutions for the velocity and curvature of the bubble based on the point-source model Zufiria proposed were found to be better agreement with numerical and experimental results than that based on the Layzer model.Due to the complexity of instability, most of theoretical studies are devoted to incompressible and inviscid fluids, and the studies based on non-ideal fluids are rare. However, in real fluids, compression and viscosity exist extensively. In this article, we will investigate analytically the effects of head loss and viscosity on the evolution of the bubble in the Rayleigh-Taylor and Richtmyer-Meshkov instabilities based on the Zufiria's point-source model.First of all, we will investigate systematically the effects of head loss on the growths of the bubble for Rayleigh-Taylor and Richtmyer-Meshkov instabilities based on the Zufiria's point-source model. The late time solutions for the amplitude, velocity and curvature of the bubble were obtained analytically. We will find that the head loss depresses the amplitude and the velocity of the bubble but enhances the curvature of the bubble. Furthermore, we will find that the obtained amplitude of the bubble is in better agreement with recent experimental data, compared with that of previous theoretical results based on ideal fluids. More than that, we will find that for both of the instabilities, the loss coefficient approaches to a limit:For Rayleigh-Taylor instability, the loss coefficient approaches to 0.5, independently of the Atwood number; for Richtmyer-Meshkov instability, the loss coefficient approaches to 0.625-0.875 for the Atwood number equals to 0-1.Secondly, we will investigate systematically the effects of viscosity on the velocity and curvature of the bubble at the late time for Rayleigh-Taylor instability based on the Zufiria's point-source model. The Reynolds number dependence of the Froude number of the bubble will be derived analytically. We will find that the viscosity decreases the velocity but does not affect the curvature of the bubble, consistent with that based on the Layzer model. Our results are found to be better agreement with experimental data than that based on the Layzer model.The presented results in this article are helpful for studying the instability of non-ideal fluids (real fluids), the boundary layer of fluids as well as the turbulence.