Session R19: Instability General II

Liquid-Bridge Breaking Limits

Wet adhesion by liquid bridges in large arrays shows promise for use in lightweight, controllable on-demand devices. Applications include grab/release of wafer substrates, transport of micron-sized tiles for use in 3D printing and micro-dosing of personalized pharmaceutical drugs. By wetting and spreading, a drop can form a bridge and thereby ``grab'' a nearby solid substrate. By volume decrease or extension, the bridge can break. The breaking limit corresponds to bridge instability which can be predicted, knowing the static mechanical response of the bridge. Mechanical behaviors include force-volume (FV), pressure-volume (pV) and force-length (FL) responses. Instability crucially depends on the mode of failure -- failure under constant-force or constant length are typical cases. We study single bridge equilibria for their breaking limits. FV diagrams for the pin-pin equal and pin-pin unequal radii boundary conditions for different bridge heights are measured in the laboratory. The FL response in the case of pin-pin equal radii is also measured. Results are compared to predictions of static theory. Static results are then used to compare to dynamical sequences where volume is driven quasistatically by syringe or an electro-osmotic pump. As the breaking limit is approached, the shape deformation accelerates leading to non-equilibrium shapes not captured by the static analysis.   

Scenarios on a rotating two-fluid interface with a density contrast - the morphology and the transitions

We study experimentally an oil-water interface maintained in a cylindrical container with its upper boundary rotating at constant speeds. The interface exhibits intriguing morphology as the rotation speeds up, making transitions from a smooth hump that presumably compensates the centrifugal-force induced pressure dip, to more fascinating geometries such as a spinning flap top (a plateau) or a mound with distinct spatial steps. The available scenarios can be controlled by varying the depths of two fluids as well. Increasing the rotation rates also tends to induce wavy patterns that break the axial symmetry of those base shapes, while a violent collapse of the oil-water interface occurs at sufficiently fast rotations. We attempt to give partial explanations for the different scenarios.   

Role of intrinsic flame instability in the excitation of combustion chamber instability

While considerable progress was made on understanding the various modes of flame instability at the fundamental level, and substantial empirical information and phenomenological descriptions was also accumulated on combustion instability within combustion chambers such as those of rocket engines, few attempts were made to explore the possible macro-scale excitation of the latter through the micro-scale manifestation of the former. Here we present an initial attempt towards identifying such a possibility and the associated coupling mechanisms. We shall incorporate the flame parameters into the classical theories of liquid-propellant rocket engines, and then implement the rocket dynamics into the analyses of premixed and diffusion flame segments. The analyses are conducted for the various instability modes, including the diffusional-thermal, Darrieus-Landau, and Rayleigh-Taylor (body-force) instabilities for premixed flames, and the Kelvin-Helmholtz and body-force instabilities for diffusion flames. The role of chamber-generated sound on stabilizing the inherent flame instabilities and triggering the parametric instability is also considered.   

Experiments on the Mode Selection in Faraday Instability

The resonance of a fluid interface with an oscillating acceleration field is studied for liquid-liquid systems in small aspect ratio cylindrical and rectangular containers. The resonant phenomenon, known as Faraday waves, is more typically studied in large aspect ratio systems at high frequencies where multiple modes are excited simultaneously and the associated nonlinear interactions give way to a variety of patterns. In this work the low excitation frequency allows for the selection of individual cell modes and their dynamics are considered. The onset threshold for instability is compared to the predictions of this model. It is seen for the cell modes that frequency bands are well predicted by the model and the amplitudes are very close, with deviation attributable to interfacial pinning and sidewall stresses. Supercritical and subcritical bifurctions are observed, along with other nonlinear phenomena such as co-dimension2 points and wave breaking.   

Spatially localized patterns in 2D and 3D doubly diffusive convection

Doubly diffusive convection, that is, convection driven by a combination of concentration and temperature gradients, is known to display a wealth of dynamical behavior whose properties depend on the gradients. In the present work, we first investigate spatially localized states in two-dimensional horizontal thermosolutal convection with no-slip boundary conditions at top and bottom and vertical gradients of temperature and concentration. Numerical continuation demonstrates the formation of stationary convectons in the form of 1-pulse and 2-pulse states of both odd and even parity while time integration reveals the presence of stable time dependent spatially localized states. We next turn to large scale three-dimensional vertical enclosures placed in horizontal thermal and solutal gradients. Different types of spatially localized states are computed and the results related to the presence of homoclinic snaking.   

Modulation instability of space-periodic oscillatory patterns

Pattern selection and stability of regular (periodic in space) regimes is a classical problem with a number of applications in fluid dynamics. For steady bifurcations both competition of perfect periodic patterns and their stability with respect to slow modulations in space (e.g. Eckhaus or zigzag instabilities) are well studied. In contrast, in the case of Hopf bifurcation, usually only selection of patterns that possess a certain symmetry was analyzed (Silber \& Knobloch, Nonlinearity, 1991; Roberts et al, Contemp. Math, 1986), whereas the set of Ginzburg-Landau equations was studied only in the one-dimensional case (rolls). Dealing with a wide class of problems, where the longwave oscillatory instability takes place, we consider a stability of regular oscillatory patterns that belong to either square or hexagonal lattices with respect to spatial modulations. By means of the multiple scale expansion, we derive instability criteria valid near the stability threshold. Useful classification of possible perturbations of a regular structure is introduced. As an example, the theory is applied to Marangoni convection in a layer of a binary mixture with the Soret effect. Domains of stability of space-periodic patterns are obtained.   

Break of the symmetry in a two-lid driven cavity

The lid driven cavity has applications in crystal growth as well as in the coating industry. We study the problem of a driven cavity with two parallel walls moving at the same speed and in the same direction. Time marching calculations using Chebyshev-spectral method were done with different aspect ratios. As the Reynolds number increases, the onset of the instability is characterized by the break of the symmetry which is described and explained. The critical Reynolds number depends on the aspect ratio. This dependence is explained.   

Transition to turbulence in a quasi-2D Kolmogorov flow

We describe a combined experimental and numerical study of quasi-2D flows to search for unstable, exact solutions to Navier-Stokes known as Exact Coherent Structures (ECS), which may provide a foundation for a simplified dynamical description of turbulence. We focus on a system that closely approximates Kolmogorov flow by inducing shear in a thin fluid layer using electromagnetic forces. PIV is used to obtain time series of velocity fields from images of the visualized flows in the lab; time series of velocity fields are calculated numerically for flows with forcing similar to that in the experiments. Discrepancies arising from differences in lateral boundary conditions between experiments (no slip) and simulations (periodic) are addressed in two separate ways: (1) experimentally studying a large system to approximate the effects of periodic boundary conditions and (2) adding padding regions in the simulations to mimic finite system size. We describe in detail the sequences of bifurcation leading to turbulence in both experiments and simulations.