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Register a new userExact solutions for nonlinear development of Kelvin-Helmholtz instability for counterflow of superfluid and normal components of Helium II
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A relative motion of the normal and superfluid components of Helium II results in Kelvin-Helmholtz instability (KHI) at their common free surface. We found the exact solutions for the nonlinear stage of the development of that instability. Contrary to the usual KHI of the interface between two fluids, the dynamics of Helium II free surface allows decoupling of the governing equations with their reduction to the Laplace growth equation which has the infinite number of exact solutions including the formation of sharp cusps at free surface in a finite time.
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The nonlinear dynamics of charged-surface instability development was investigated for liquid helium far above the critical point. It is found that, if the surface charge completely screens the field above the surface, the equations of three-dimensional (3D) potential motion of a fluid are reduced to the well-known equations describing the 3D Laplacian growth process. The integrability of these equations in 2D geometry allows the analytic description of the free-surface evolution up to the formation of cuspidal singularities at the surface.
We analyze nonlinear dynamics of the Kelvin-Helmholtz quantum instability of the He-II free surface, which evolves during counterpropagation of the normal and superfluid components of liquid helium. It is shown that in the vicinity of the linear stability threshold, the evolution of the boundary is described by the $|phi|^4$ Klein-Gordon equation for the complex amplitude of the excited wave with cubic nonlinearity. It is important that for any ratio of the densities of the helium component, the nonlinearity plays a destabilizing role, accelerating the linear instability evolution of the boundary. The conditions for explosive growth of perturbations of the free surface are formulated using the integral inequality approach. Analogy between the Kelvin-Helmholtz quantum instability and electrohydrodynamic instability of the free surface of liquid helium charged by electrons is considered.
A wide class of exact solutions is obtained for the problem of finding the equilibrium configurations of charged jets of a conducting liquid; these configurations correspond to the finite-amplitude azimuthal deformations of the surface of a round jet. A critical value of the linear electric charge density is determined, for which the jet surface becomes self-intersecting, and the jet splits into two. It exceeds the density value required for the excitation of the linear azimuthal instability of the round jet. Hence, there exists a range of linear charge density values, where our solutions may be stable with respect to small azimuthal perturbations.
In this paper, the coupled Rayleigh-Taylor-Kelvin-Helmholtz instability(RTI, KHI and RTKHI, respectively) system is investigated using a multiple-relaxation-time discrete Boltzmann model. Both the morphological boundary length and thermodynamic nonequilibrium (TNE) strength are introduced to probe the complex configurations and kinetic processes. In the simulations, RTI always plays a major role in the later stage, while the main mechanism in the early stage depends on the comparison of buoyancy and shear strength. It is found that, both the total boundary length $L$ of the condensed temperature field and the mean heat flux strength $D_{3,1}$ can be used to measure the ratio of buoyancy to shear strength, and to quantitatively judge the main mechanism in the early stage of the RTKHI system. Specifically, when KHI (RTI) dominates, $L^{KHI} > L^{RTI}$ ($L^{KHI} < L^{RTI}$), $D_{3,1}^{KHI} > D_{3,1}^{RTI}$ ($D_{3,1}^{KHI} < D_{3,1}^{RTI}$); when KHI and RTI are balanced, $L^{KHI} = L^{RTI}$, $D_{3,1}^{KHI} = D_{3,1}^{RTI}$. A second sets of findings are as below: For the case where the KHI dominates at earlier time and the RTI dominates at later time, the evolution process can be roughly divided into two stages. Before the transition point of the two stages, $L^{RTKHI}$ initially increases exponentially, and then increases linearly. Hence, the ending point of linear increasing $L^{RTKHI}$ can work as a geometric criterion for discriminating the two stages. The TNE quantity, heat flux strength $D_{3,1}^{RTKHI}$, shows similar behavior. Therefore, the ending point of linear increasing $D_{3,1}^{RTKHI}$ can work as a physical criterion for discriminating the two stages.
We report experimental results for the Kelvin-Helmholtz instability between two immiscible fluids in parallel flow in a Hele-Shaw cell. We focus our interest on the influence of the gap size between the walls on the instability characteristics. Experimental results show that the instability threshold, the critical wavelength, the phase velocity and the spatial growth rate depend on this gap size. These results are compared to both the previous two-dimensional analysis of Gondret and Rabaud (1997) and the three-dimensional analysis recently derived by Plourabou=E9 and Hinch (2001), showing that the agreement is still not complete especially when gap size increases.
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