61 research outputs found

    Nonlinear long-wave interfacial stability of two-layer gas-liquid flow

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    The flow of two immiscible viscous fluids in a thin inclined channel is considered, in either a cocurrent or countercurrent regime. Following the air-water case, which is found in a variety of engineering systems, we allow the upper fluid to be either compressible or incompressible. The disparity of the length scales and the density and viscosity ratios of the two fluids is exploited through a lubrication approximation of the conservation of mass and the Navier-Stokes equations. As a result of this long-wave theory, a coupled nonlinear system of partial differential equations is obtained that describes the evolution of the interfacial thickness and the leading-order pressure. This system includes the effects of viscosity stratification, inertia, shear, and capillarity, and reduces to the single-phase falling film Benney equation for sufficiently thin liquid films and constant gas density. The case of two incompressible fluids is investigated first. Since the experimental conditions for this effective system are unclear, we consider several ways to drive the flow: either by fixing the volumetric flow rate of the gas phase or by fixing the total pressure drop over a downstream length of the channel, or by fixing liquid flow rate and gas pressure drop. The forcing with prescribed pressure drop results in a single evolution equation whose dynamics depends nonlocally on the interfacial shape. From weakly nonlinear analysis in this case, we obtain the modified Kuramoto-Sivashinsky equation with an additional integral term, influencing the speed of propagation but not the shape of the interfacial wave. For the strongly nonlinear case, admissible criteria for Lax shocks, undercompressive shocks and rarefaction waves are investigated. Through a numerical verification we find that these criteria do not depend significantly on the inertial effects within the more dense layer. The choice of the local/nonlocal boundary conditions appears to play a role in the transient growth of undercompressive shocks, and may relate to the phenomena observed near the onset of flooding. We then perform a linear stability analysis when the gas phase is compressible. The base-state profile for the density is spatially dependent when a pressure drop over the length of the channel is prescribed. The case when zero pressure drop is prescribed is amenable to a normal-mode analysis. When the liquid film thickness is sufficiently thin, the stability matches that of the single-phase falling film case with the exception that the compressible quiescient gas is stabilizing. When the liquid film thickness is sufficently thick, the density mode within the thin gas layer is destabilizing. In the general case, over a finite domain, a general stability diagram of film thickness and pressure drop is found. For sufficently large countercurrent pressure drops, the interfacial mode becomes unstable, with the location of the largest deformation found near the liquid inlet

    Simulation of thin film flows with a moving mesh mixed finite element method

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    We present an efficient mixed finite element method to solve the fourth-order thin film flow equations using moving mesh refinement. The moving mesh strategy is based on harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To achieve a high quality mesh, we adopt an adaptive monitor function and smooth it based on a diffusive mechanism. A variety of numerical tests are performed to demonstrate the accuracy and efficiency of the method. The moving mesh refinement accurately resolves the overshoot and downshoot structures and reduces the computational cost in comparison to numerical simulations using a fixed mesh.Comment: 18 pages, 10 figure

    Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

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    We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.Comment: Revision from v2; 57 pages, 19 figure

    Stability of Traveling Waves in Thin Liquid Films Driven by Gravity and Surfactant

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    A thin layer of fluid flowing down a solid planar surface has a free surface height described by a nonlinear PDE derived via the lubrication approximation from the Navier Stokes equations. For thin films, surface tension plays an important role both in providing a significant driving force and in smoothing the free surface. Surfactant molecules on the free surface tend to reduce surface tension, setting up gradients that modify the shape of the free surface. In earlier work [12, 13J a traveling wave was found in which the free surface undergoes three sharp transitions, or internal layers, and the surfactant is distributed over a bounded region. This triple-step traveling wave satisfies a system of PDE, a hyperbolic conservation law for the free surface height, and a degenerate parabolic equation describing the surfactant distribution. As such, the traveling wave is overcornpressive. An examination of the linearized equations indicates the direction and growth rates of one-dimensional waves generated by small perturbations in various parts of the wave. Numerical simulations of the nonlinear equations offer further evidence of stability to one-dimensional perturbations

    A theory for undercompressive shocks in tears of wine

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    We revisit the tears of wine problem for thin films in water-ethanol mixtures and present a new model for the climbing dynamics. The new formulation includes a Marangoni stress balanced by both the normal and tangential components of gravity as well as surface tension which lead to distinctly different behavior. The prior literature did not address the wine tears but rather the behavior of the film at earlier stages and the behavior of the meniscus. In the lubrication limit we obtain an equation that is already well-known for rising films in the presence of thermal gradients. Such models can exhibit non-classical shocks that are undercompressive. We present basic theory that allows one to identify the signature of an undercompressive (UC) wave. We observe both compressive and undercompressive waves in new experiments and we argue that, in the case of a pre-coated glass, the famous "wine tears" emerge from a reverse undercompressive shock originating at the meniscus
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