Instability and dripping of electrified liquid films flowing down inverted substrates

We consider the gravity-driven flow of a perfect dielectric, viscous, thin liquid film, wetting a flat substrate inclined at a nonzero angle to the horizontal. The dynamics of the thin film is influenced by an electric field which is set up parallel to the substrate surface—this nonlocal physical mechanism has a linearly stabilizing effect on the interfacial dynamics. Our particular interest is in fluid films that are hanging from the underside of the substrate; these films may drip depending on physical parameters, and we investigate whether a sufficiently strong electric field can suppress such nonlinear phenomena. For a non-electrified flow, it was observed by Brun et al. [Phys. Fluids 27, 084107 (2015)] that the thresholds of linear absolute instability and dripping are reasonably close. In the present study, we incorporate an electric field and analyze the absolute and convective instabilities of a hierarchy of reduced-order models to predict the dripping limit in parameter space. The spatial stability results for the reduced-order models are verified by performing an impulse-response analysis with direct numerical simulations (DNS) of the Navier–Stokes equations coupled to the appropriate electrical equations. Guided by the results of the linear theory, we perform DNS on extended domains with inflow and outflow conditions (mimicking an experimental setup) to investigate the dripping limit for both non-electrified and electrified liquid films. For the latter, we find that the absolute instability threshold provides an order-of-magnitude estimate for the electric-field strength required to suppress dripping; the linear theory may thus be used to determine the feasibility of dripping suppression given a set of geometrical, fluid, and electrical parameters

On the dc Magnetization, Spontaneous Vortex State and Specific Heat in the superconducting state of the weakly ferromagnetic superconductor RuSr2_{2}GdCu2_{2}O8_{8}

Magnetic-field changes $<$ 0.2 Oe over the scan length in magnetometers that necessitate sample movement are enough to create artifacts in the dc magnetization measurements of the weakly ferromagnetic superconductor RuSr$_{2}$GdCu$_{2}$O$_{8}$ (Ru1212) below the superconducting transition temperature $T_{c} \approx$ 30 K. The observed features depend on the specific magnetic-field profile in the sample chamber and this explains the variety of reported behaviors for this compound below $T_{c}$. An experimental procedure that combines improvement of the magnetic-field homogeneity with very small scan lengths and leads to artifact-free measurements similar to those on a stationary sample has been developed. This procedure was used to measure the mass magnetization of Ru1212 as a function of the applied magnetic field H (-20 Oe $\le$ H $\le$ 20 Oe) at $T < T_{c}$ and discuss, in conjunction with resistance and ac susceptibility measurements, the possibility of a spontaneous vortex state (SVS) for this compound. Although the existence of a SVS can not be excluded, an alternative interpretation of the results based on the granular nature of the investigated sample is also possible. Specific-heat measurements of Sr$_{2}$GdRuO$_{6}$ (Sr2116), the precursor for the preparation of Ru1212 and thus a possible impurity phase, show that it is unlikely that Sr2116 is responsible for the specific-heat features observed for Ru1212 at $T_{c}$.Comment: 17 pages, 6 figure

Linear instability of supersonic plane wakes

In this paper we present a theoretical and numerical study of the growth of linear disturbances in the high-Reynolds-number and laminar compressible wake behind a flat plate which is aligned with a uniform stream. No ad hoc assumptions are made as to the nature of the undisturbed flow (in contrast to previous investigations) but instead the theory is developed rationally by use of proper wake-profiles which satisfy the steady equations of motion. The initial growth of near wake perturbation is governed by the compressible Rayleigh equation which is studied analytically for long- and short-waves. These solutions emphasize the asymptotic structures involved and provide a rational basis for a nonlinear development. The evolution of arbitrary wavelength perturbations is addressed numerically and spatial stability solutions are presented that account for the relative importance of the different physical mechanisms present, such as three-dimensionality, increasing Mach numbers enough (subsonic) Mach numbers, there exists a region of absolute instability very close to the trailing-edge with the majority of the wake being convectively unstable. At higher Mach numbers (but still not large-hypersonic) the absolute instability region seems to disappear and the maximum available growth-rates decrease considerably. Three-dimensional perturbations provide the highest spatial growth-rates

Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

The nonlinear stability of immiscible two–fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni ef- fects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatiotemporal evolution of the interface and its local surfactant concentration. The system is nonlocal and arises by appropri- ately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled P ́eclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two– and three–dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three–dimensional distur- bances can be more unstable than two–dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evo- lution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two–dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stabil- ity via Hopf bifurcations to time–periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics become more complex and include time–periodic, quasi–periodic as well as chaotic fluctuations. It is also found that one–dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three–dimensional flows with interfacial profiles that are two–dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three–dimensional disturbances but not two–dimensional ones. These are found to have a one–dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form

Noise induced state transitions, intermittency and universality in the noisy Kuramoto-Sivashinsky equation

We analyze the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation in a regime close to the instability onset. We show that when the noise is highly degenerate, in the sense that it acts only on the first stable mode, the solution of the KS equation undergoes several transitions between different states, including a critical on-off intermittent state that is eventually stabilized as the noise strength is increased. Such noise-induced transitions can be completely characterized through critical exponents, obtaining that both the KS and the noisy Burgers equation belong to the same universality class. The results of our numerical investigations are explained rigorously using multiscale techniques.Comment: 4 pages, 4 figure

Ice formation within a thin film flowing over a flat plate

We present a model for ice formation in a thin, viscous liquid film driven by a Blasius boundary layer after heating is switched off along part of the flat plate. The flow is assumed to initially be in the Nelson et al. (J. Fluid Mech., vol. 284, 1995, pp. 159–169) steady-state configuration with a constant flux of liquid supplied at the tip of the plate, so that the film thickness grows lik

Capturing nonlinear dynamics of two-fluid Couette flows with asymptotic models

The nonlinear stability of two-fluid Couette flows is studied using a novel evolution equation whose dynamics are validated by direct numerical simulations (DNS). The evolution equation incorporates inertial effects at arbitrary Reynolds numbers through a non-local term arising from the coupling between the two fluid regions, and is valid when one of the layers is thin. The equation predicts asymmetric solutions and exhibits bistability, features that are essential observations in the experiments of Barthelet et al. (1995). Related low-inertia models have been used in qualitative predictions rather than the direct comparisons carried out here, and ad hoc modifications appear to be necessary in order to predict asymmetry and bistability. Comparisons between model solutions and DNS show excellent agreement at Reynolds numbers of O(10³) found in the experiments. Direct comparisons are also made with the available experimental results of Barthelet et al. (1995) when the thin layer occupies 1/5 of the channel height. Pointwise comparisons of the travelling wave shapes are carried out and once again the agreement is very good