The breakup of a jet of a viscous fluid with viscosity $μ_{1}$ immersed into another viscous fluid with viscosity $μ_{2}$ is considered in the limit when the viscosity ratio $λ=μ_{1}/μ_{2}$ is close to zero. We show that, in this limit, a transition from ordinary continuous selfsimilarity to discrete selfsimilarity takes place as $λ$decreases. The result being that instead of a single point breakup, the rupture of the inner jet occurs through the appearance of an infinite sequence of filaments of decreasing size that will eventually produce infinite sequences of bubbles of the inner fluid inside the outer fluid. The transition can be understood as the result of a Hopf bifurcation in the system of equations modelling the physical problem.
In the present paper, we examine the viscous flow evolution in a square cavity. Coupled with the stream function, the initial-boundary value problem of the vorticity is numerically solved by a method of iteration. The only boundary condition is the wall velocity which in turn defines the Dirichlet value for the stream function. We assert that the corner singularity in the cavity flow is in fact a theoretical artefact. By adopting suitably regulated lid velocity, excellent comparison with experiments is found because of the converged palinstrophy field. Our calculations demonstrate that asymmet…
We consider the problem of oscillation damping in air of a thermally actuated microlever as it is gradually approached towards an infinite wall in parallel geometry. As the gap is decreased from 20 nm down to 400 nm, we observe the increasing damping of the lever Brownian motion in the fluid laminar regime. This manifests itself as a linear decrease with distance of the lever quality factor accompanied by a dramatic softening of its resonance, and eventually leads to the freezing of the CL oscillation. We are able to quantitatively explain this behavior by analytically solving the Navier-Stoke…
We consider an axisymmetric, freely suspended fluid drop with surface tension, whose viscosity is so large that both inertia and forcing by an external fluid can be ignored. We show that whatever be the initial condition, pinchoff can never occur.
We study the collapse of an axisymmetric liquid filament both analytically and by means of a numerical model. The liquid filament, also known as ligament, may either collapse stably into a single droplet or break up into multiple droplets. The dynamics of the filament are governed by the viscosity and the aspect ratio, and the initial perturbations of its surface. We find that the instability of long viscous filaments can be completely explained by the Rayleigh-Plateau instability, whereas a low viscous filament can also break up due to end pinching. We analytically derive the transition betwe…
The experimental observation of D. Duft, T. Achtzehn, R. Muller, B. A. Huber, and T. Leisner, Nature 421, 128 (2003) on the sequential progression of the instability of a charged liquid drop points at the formation of a jet followed by the emission of progeny droplets as a crucial pathway of the Rayleigh fission process. In spite of considerable theoretical progress, a quantitative understanding of this breakup pathway through mathematical models has largely remained inconclusive. This limitation has mainly been due to the fact that the generally applied electrostatic boundary condition of the…
We report the theoretical and experimental demonstration of pattern formation in acoustics. The system is an acoustic resonator containing a viscous fluid. When the system is driven by an external periodic force, the ultrasonic field inside the cavity experiences different pattern-forming instabilities leading to the emergence of periodic structures. The system is also shown to possess bistable regimes, in which localized states of the ultrasonic field develop. The thermal nonlinearity in the viscous fluid, together with the far-from-equilibrium conditions, are is the responsible of the observ…
In this paper we compute the linear stability of similarity solutions of the breakup of viscous liquid threads, in which the viscosity and inertia of the liquid are in balance with the surface tension. The stability of the similarity solution is determined using numerical continuation to find the dominant eigenvalues. Stability of the first two solutions (those with largest minimum radius) is considered. We find that the first similarity solution, which is the one seen in experiments and simulations, is linearly stable with a complex nontrivial eigenvalue, which could explain the phenomenon of…
In this article, three dimensional (3D) lid-driven cubic cavity flows have been studied numerically for various values of Reynolds number ($Re$). The numerical solution of the Navier-Stokes equations modeling incompressible viscous fluid flow in a cubic cavity is obtained via a methodology combining a first order accurate operator-splitting, $L^2$-projection Stokes solver, a wave-like equation treatment of the advection and finite element methods. The numerical results obtained for Re$=$400, 1000, and 3200 show a good agreement with available numerical and experimental results in literature. S…
On the basis of the two-fluid hydrodynamics, an analogue of the famous Rayleigh-Plesse equation for the dynamics of a spherical bubble in superfluid helium is obtained. The mass flow velocity $v$ and the velocity of the normal component $v_{n}$ were chosen as independent variables. Due to the two-fluid nature of HeII, the cross terms in the evolution equation for the boundary position $\ R(t)$ appeared, which were absent in classical Rayleigh-Plesset equation in ordinary fluids. One of them renormilizes the coefficient in front of $(dR/dt)^{2}$. Another additional term formally coinciding with…
The evolution of suspension drops sedimenting under gravity in a viscous fluid close to a vertical wall was studied experimentally and numerically with the use of the point-force model, in the Stokes flow regime. The fluid inside and outside the drop was identical. The initial distribution of the suspended solid heavy particles was uniform inside a spherical volume. In the experiments and in the simulations, the suspension drops evolved qualitatively in the same way as in an unbounded fluid. However, it was observed, both experimentally and numerically that, on the average, the destabilization…
Helped by the careful analysis of their experimental data, Worthington (1897) described roughly the mechanism underlying the formation of high-speed jets ejected after the impact of an axisymmetric solid on a liquid-air interface. In this work we combine detailed boundary-integral simulations with analytical modeling to describe the formation and break-up of such Worthington jets in two common physical systems: the impact of a circular disc on a liquid surface and the release of air bubbles from an underwater nozzle. We first show that the jet base dynamics can be predicted for both systems us…
Two common concepts of nonreciprocity in sound propagation are based on nonlinear effects [1, 2] and on local circulation of fluid [3, 4]. They originate from two known methods of breaking a time reversal symmetry, that is necessary for observation of nonreciprocal effects. Both concepts require additional devices to be installed with their own power sources. Recently it was demonstrated that acoustical losses may serve as a source of T-symmetry violation, thus leading to nonreciprocity in reflection of sound from gradient-index metasurface [5]. Here, we explore viscosity of fluid as a natural…
Computations of the breakup of a liquid bridge are used to establish the limits of applicability of similarity solutions derived for different breakup regimes. These regimes are based on particular viscous-inertial balances, that is different limits of the Ohnesorge number $Oh$. To accurately establish the transitions between regimes, the minimum bridge radius is resolved through four orders of magnitude using a purpose-built multiscale finite element method. This allows us to construct a quantitative phase diagram for the breakup phenomenon which includes the appearance of a recently discover…
We analyze both theoretically and experimentally the breakup of a pendant water droplet loaded with Sodium Dodecyl Sulfate (SDS). The free surface minimum radius measured in the experiments is compared with that obtained from a numerical solution of the full Navier-Stokes equations for different values of the shear and dilatational surface viscosities. This comparison shows the small but measurable effect of the surface viscous stresses on the system dynamics for sufficiently small spatiotemporal distances from the breakup point, and allows to establish upper bounds for the values of the shear…
After impact onto a smooth dry surface, a drop of viscous liquid initially spreads in the form of a thick lamella. If the drop splashes, it first emits a thin fluid sheet that can ultimately break up into droplets causing the splash. Ambient gas is crucial for creating this thin sheet. The time for sheet ejection, $t_{ejt}$, depends on impact velocity, liquid viscosity, gas pressure and molecular weight. A central air bubble is trapped below the drop at pressures even below that necessary for this sheet formation. In addition, air bubbles are entrained underneath the spreading lamella when the…
In the minutes immediately preceeding the rupture of a soap bubble, distinctive and repeatable patterns can be observed. These quasi-stable transient structures are associated with the instabilities of the complex Marangoni flows on the curved thin film in the presence of a surfactant solution. Here, we report a generalised Cahn-Hilliard-Swift-Hohenberg model derived using asymptotic theory which describes the quasi-elastic wrinkling pattern formation and the consequent coarsening dynamics in a curved surfactant-laden thin film. By testing the theory against experiments on soap bubbles, we fin…
We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-Hückel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results…
In this fluid dynamics video prepared for the APS-DFD Gallery of Fluid Motion we study the collision of two identical viscous jets at different jet speeds and collision eccentricities. The dynamics of the jet motion are slowed down by orders of magnitude using a synchronized strobe effect coupled with precise timing control of the perturbation frequency imposed on one of the jets. Our results show that different shapes and morphologies appear as we change the collision eccentricity. At low jet speeds ($We_j = 45$) viscous threads and filaments are formed as the jets begin to impinge on each ot…
Electro-coalescence is the application of an electric field onto coalescing fluid bodies. The following fluid dynamics videos show a droplet coalescing into a fluid bath while embedded into a viscous medium and subject to a very high electric field. The concentration of electric stresses at the apex of the droplet cause it to break apart. The droplet is glycerol and the viscous medium is silicone oil.
When a viscous fluid, like oil or syrup, streams from a small orifice and falls freely under gravity, it forms a long slender thread, which can be maintained in a stable, stationary state with lengths up to several meters. We shall discuss the shape of such liquid threads and their surprising stability. It turns out that the strong advection of the falling fluid can almost outrun the Rayleigh-Plateau instability. Even for a very viscous fluid like sirup or silicone oil, the asymptotic shape and stability is independent of viscosity and small perturbations grow with time as $\exp({{\rm C} t^{1/…
A transition to unsteadiness of a flow inside a cubic diagonally lid-driven cavity with no-slip boundaries is numerically investigated by a series of direct numerical simulations (DNS) performed on 100^3 and 200^3 stretched grids. It is found that the observed oscillatory instability is setting in via a subcritical symmetry-breaking Hopf bifurcation. The instability evolves on two vortices in a coupled manner. Critical values of Reynolds number Recr=2320 and non-dimensional angular oscillating frequency omegacr=0.249 for transition from steady to oscillatory flow are accurately estimated. Char…
In this paper, we derive a viscous generalization of the Dysthe (1979) system from the weakly viscous generalization of the Euler equations introduced by Dias, Dyachenko, and Zakharov (2008). This "viscous Dysthe" system models the evolution of a weakly viscous, nearly monochromatic wave train on deep water. It contains a term which provides a mechanism for frequency downshifting in the absence of wind and wave breaking. The equation does not preserve the spectral mean. Numerical simulations demonstrate that the spectral mean typically decreases and that the spectral peak decreases for certain…
The compressible Neural Particle Method for Simulating Compressible Viscous Fluid Flows
Space and Astronautical Science, The Graduate University for Advanced Studies, SOKENDAI（空间与航天科学，高级研究大学，SOKENDAI）；Department of Aeronautics and Astronautics, Graduate School of Engineering, University of Tokyo（航空与航天系，工程研究生院，东京大学）
Electronic fluids bring into hydrodynamics a new setting: equipotential flow sources embedded inside the fluid. Here we show that nonlocal relation between current and electric field due to momentum-conserving inter-particle collisions leads to a total or partial field expulsion from such flows. That results in freely flowing currents in the bulk and boundary jump in electric potential at current-injecting electrodes. We derive the appropriate boundary conditions, analyze current distribution in free flows, discuss how the field expulsion depends upon geometry of the electrode, and link the ph…
It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid coincides with the equations of motion of a charged {\it compressible} fluid moving due to a self-consistent electromagnetic field. Transition to the Lagrangian description in a new hydrodynamics is equivalent for the original Euler equations to the mixed Lagrangian-Eulerian description - the vortex line representation (VLR). Due to compressibility of a "new" fluid the collapse of vortex lines can happen as the result of breaking (or overturning) of vortex lines. It is found that the Navier-Stokes equation in the vor…
A physical model of a three-dimensional flow of a viscous bubbly fluid in an intermediate regime between bubble formation and breakage is presented. The model is based on mechanics and thermodynamics of a single bubble coupled to the dynamics of a viscous fluid as a whole, and takes into account multiple physical effects, including gravity, viscosity, and surface tension. Dimensionless versions of the resulting nonlinear model are obtained, and values of dimensionless parameters are estimated for typical magma flows in horizontal subaerial lava fields and vertical volcanic conduits. Exact solu…
It is well known that the viscosity of a homogeneous electron liquid diverges in the limits of zero frequency and zero temperature. A nanojunction breaks translational invariance and necessarily cuts off this divergence. However, the estimate of the ensuing viscosity is far from trivial. Here, we propose an approach based on a Kramers-Krönig dispersion relation, which connects the zero-frequency viscosity, $η(0)$, to the high-frequency shear modulus, $μ_{\infty}$, of the electron liquid via $η(0) =μ_{\infty} τ$, with $τ$ the junction-specific momentum relaxation time. By making use of a simple…
