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arXiv2014

Strange Attractors Characterizing the Osmotic Instability

In the present paper a simple dynamical model for computing the osmotically driven fluid flow in a variety of complex, non equilibrium situations is derived from first principles. Using the Oberbeck-Boussinesq approximation, the basic equations describing the process of forward osmosis have been obtained. It has been shown that these equations are very similar to the ones used to model the free Rayleigh-Benard convection. The difference is that while in the case of thermal convection the volume expansion is driven by the coefficient of thermal expansion, the key role for the osmotic instability is played by the coefficient of isothermal compressibility. In addition, it has been shown that the osmotic process represents a propagation of standing waves with time-dependent amplitudes and phase velocity, which equals the current velocity of the solvent passing through the semi-permeable membrane. The evolution of the amplitudes of the osmotic waves is exactly following the dynamics of a strange attractor of Lorenz type with corresponding control parameters.

physics.flu-dyn↗
arXiv2014

Linear Eckman friction in the mechanism of the cyclon-anticyclon vortex asymmetry and in a new theory of rotating superfluid

Until now it was not a success to identify a universal triggering mechanism for the formation of the observed cyclon-anticyclon vortex asymmetry phenomenon or the corresponding breaking of chiral vortex symmetry in the rotating medium. In this paper the new linear universal instability mechanism of breaking chiral vortex symmetry due to the super threshold medium rotation frequence in the presence of linear Eckman friction is established. A new non stationary solution to the problem of the disc oscillations in fluid in the connection with experiments on the rotating superfluid helium-II has been found.

physics.flu-dyn↗
arXiv2014

The Optimal Swimming Sheet

Propulsion at microscopic scales is often achieved through propagating traveling waves along hair-like organelles called flagella. Taylor's two-dimensional swimming sheet model is frequently used to provide insight into problems of flagellar propulsion. We derive numerically the large-amplitude waveform of the two-dimensional swimming sheet that yields optimum hydrodynamic efficiency; the ratio of the squared swimming speed to the rate-of-working of the sheet against the fluid. Using the boundary element method, we show the optimal waveform is a front-back symmetric regularized cusp that is 25% more efficient than the optimal sine-wave. This optimal two-dimensional shape is smooth, qualitatively different from the kinked form of Lighthill's optimal three-dimensional flagellum, not predicted by small-amplitude theory, and different from the smooth circular-arc-like shape of active elastic filaments.

physics.flu-dyn↗
arXiv2014

Collapse of Point Vortex Dipoles in a Bounded Fluid Layer and the Hydrodynamic Mechanism of Mutual Attraction of Like-Charged Micro-Particles in the Colloid or Dusty Plasma Systems

The new exact weak solution of the equations for the ideal incompressible fluid dynamics in the finite layer inside two plates with solid boundaries is obtained. The solution meets a devised non-linear finite dimension Hamiltonian dynamic system for the coordinates and the Lamb impulses of N point vortex dipoles (PVD), i.e. extremely small solid spherical particles, moving with respect to the fluid. For N=2 the necessary condition for the collapse (or the converging in one point during the finite time) of two PVD is stated. On the base of the proposed theory, the new hydrodynamic mechanism of converging for two small spherical particles (of the same radius) is introduced and used for interpreting the observed paradoxical effects of attraction for micro particles with the same sign of electrical charge in the colloid and dusty plasma systems. The correspondence of the condition for collapse of two PVD with the experimentally observed data where convergence of two like-charged micro-spheres moving in the fluid layer is observed only when definite necessary restrictions on the fluid layer depth are imposed from above and below (J. C. Crocker, D. G. Grier, 1996) is stated. We also state the possibility of correspondence between the obtained condition of collapse for two PVD in the limit of unbounded fluid and known numerical results on the threshold of the impact parameter for the realization of resonance bond state (and corresponding stochastic dynamic regime) for two finite size vortex dipoles during their scattering in unbounded fluid (S. V. Manakov, L. N. Shchur, 1983; L. Tophoj, H. Aref, 2008). Here we show that in the latter process the size of dipoles is not so important as it was thought before.

physics.flu-dyn↗
arXiv2014

Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any $H^2$ initial data. An $N$-dimensional logarithmic Sobolev embedding inequality, which bounds the $L^\infty$ norm in terms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of the first order derivatives, and a system version of the classic Gronwall inequality are exploited to establish the required a priori $H^2$ estimates for the global regularity.

math.AP↗
arXiv2014

Hilfer fractional advection-diffusion equations with power-law initial condition; a Numerical study using variational iteration method

We propose a Hilfer advection-diffusion equation of order $0<α<1$ and type $0\leqβ\leq1$, and find the power series solution by using variational iteration method. Power series solutions are expressed in a form that is easy to implement numerically and in some particular cases, solutions are expressed in terms of Mittag-Leffler function. Absolute convergence of power series solutions is proved and the sensitivity of the solutions is discussed with respect to changes in the values of different parameters. For power law initial conditions it is shown that the Hilfer advection-diffusion PDE gives the same solutions as the Caputo and Riemann-Liouville advection-diffusion PDE. To leading order, the fractional solution compared to the non-fractional solution increases rapidly with $α$ for $α> 0.7$ at a given time $t$; but for $α<0.7$ this factor is weakly sensitive to $α$. We also show that the truncation errors, arising when using the partial sum as approximate solutions, decay exponentially fast with the number of terms $n$ used. We find that for $α< 0.7$ the number of terms needed is weakly sensitive to the accuracy level and to the fractional order, $n\approx 20$; but for $α>0.7$ the required number of terms increases rapidly with the accuracy level and also with the fractional order $α$.

physics.flu-dyn↗
arXiv2014

Convective turbulence in a Rayleigh-Benard system with and without rotation in the infinite Prandtl number limit

Convective turbulence in a Rayleigh Benard system has shown a marked reluctance to exhibit clear scaling in the energy or entropy spectrum. The recent numerical simulation of Pandey, Verma and Mishra has shown significantly better evidence of scaling in the infinite Prandtl number limit. This prompted us to look at this limit analytically. We find that the inevitable presence of sweeping helps give a very good understanding of the results of Pandey et. al. In the presence of rotation, the Rayleigh number dependence of the Nusselt number shows a strong increase in slope in recent experiments and simulations at finite Prandtl number. In the infinite Prandtl number case we find that this steepening does not occur for any rotation speed and our results satisfy the Doering-Constantin bound.

physics.flu-dyn↗
arXiv2014

Experimental evidence of slippage breakdown for a superhydrophobic surface in a microfluidic device

A full characterization of the water flow past a silicon superhydrophobic surface with longitudinal micro-grooves enclosed in a microfluidic device is presented. Fluorescence microscopy images of the flow seeded with fluorescent passive tracers were digitally processed to measure both the velocity field and the position and shape of the liquid-air interfaces at the superhydrophobic surface. The simultaneous access to the meniscus and velocity profiles allows us to put under a strict test the no-shear boundary condition at the liquid-air interface. Surprisingly, our measurements show that air pockets in the surface cavities can sustain non-zero interfacial shear stresses, thereby hampering the friction reduction capabilities of the surface. The effects of the meniscus position and shape as well as of the liquid-air interfacial friction on the surface performances are separately assessed and quantified.

physics.flu-dyn↗
arXiv2014

Mixing by microorganisms in stratified fluids

We examine the vertical mixing induced by the swimming of microorganisms at low Reynolds and Péclet numbers in a stably stratified ocean, and show that the global contribution of oceanic microswimmers to vertical mixing is negligible. We propose two approaches to estimating the mixing efficiency, $η$, or the ratio of the rate of potential energy creation to the total rate-of-working on the ocean by microswimmers. The first is based on scaling arguments and estimates $η$ in terms of the ratio between the typical organism size, $a$, and an intrinsic length scale for the stratified flow, $\ell = \left ( νκ/ N^2 \right )^{1/4}$, where $ν$ is the kinematic viscosity, $κ$ the diffusivity, and $N$ the buoyancy frequency. In particular, for small organisms in the relevant oceanic limit, $a / \ell \ll 1$, we predict the scaling $η\sim (a / \ell)^3$. The second estimate of $η$ is formed by solving the full coupled flow-stratification problem by modeling the swimmer as a regularized force dipole, and computing the efficiency numerically. Our computational results, which are examined for all ratios $a/\ell$, validate the scaling arguments in the limit $a / \ell \ll 1$ and further predict $η\approx 1.2 \left ( a / \ell \right )^3$ for vertical swimming and $η\approx 0.15 \left ( a / \ell \right )^3$ for horizontal swimming. These results, relevant for any stratified fluid rich in biological activity, imply that the mixing efficiency of swimming microorganisms in the ocean is at very most 8\% and is likely smaller by at least two orders of magnitude.

physics.flu-dyn↗
arXiv2014

Energy spectrum of ensemble of weakly nonlinear gravity-capillary waves on a fluid surface

In this Letter we regard nonlinear gravity-capillary waves with parameter of nonlinearity being $\varepsilon \sim 0.1 ÷0.25$. For this nonlinearity time scale separation does not occur and kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in \emph{Kartashova, \emph{EPL} \textbf{97} (2012), 30004.} We compute for the first time an analytical expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function depending on the ratio of surface tension to the gravity acceleration. It is shown that its two limits - pure capillary and pure gravity waves on a fluid surface - coincide with the previously obtained results. We also discuss relations of the model of D-cascade with a few known models used in the theory of nonlinear waves such as Zakharov's equation, resonance of the modes with nonlinear Stokes corrected frequencies and Benjamin-Feir index. These connections are crucial in the understanding and forecasting specifics of the energy transport in a variety of multi-component wave dynamics, from oceanography to optics, from plasma physics to acoustics.

physics.flu-dyn↗
arXiv2014

What can go wrong when applying wave turbulence theory

Many new models of wave turbulence -- frozen, mesoscopic, laminated, decaying, sand-pile, etc. -- have been developed in the last decade aiming to solve problems seemingly not solvable in the framework of the existing wave turbulence theory (WTT). In this Letter we show that very often the reason of these discrepancies is that some necessary conditions of the WTT are not satisfied: initial energy distribution is not according to the assumptions of the theory; nonlinearity is not small enough; duration of an experiment is not sufficient to observe kinetic time scale; etc. Two alternative models are briefly presented which can be used to interpret experimental data, both giving predictions at the dynamical time scale: a) a dynamical energy cascade, for systems with narrow initial excitation and weak and moderate nonlinearity, and b) an effective evolution equation, for systems with distributed initial state and small nonlinearity.

physics.flu-dyn↗
arXiv2014

Saturation of a large scale instability and non linear stuctures in a rotating stratified flow

In this letter, we present non linear structures resulting from the saturation of a large scale instability in rotating stratified fluids with small scale forced turbulence which we found in \cite{[1]}. These structures are of helical kinks type. The different solutions that we plotted correspond to solutions linking two different stationnary points for different values of the Rayleigh and the Taylor numbers.

physics.flu-dyn↗
arXiv2014

The interplay of curvature and vortices in flow on curved surfaces

Incompressible fluids on curved surfaces are considered with respect to the interplay between topology, geometry and fluid properties using a surface vorticity-stream function formulation, which is solved using parametric finite elements. Motivated by designed examples for superfluids, we consider the influence of a geometric potential on vortices for fluids with finite viscosity and show numerical examples in which a change in the geometry is used to manipulate the flow field.

physics.flu-dyn↗
arXiv2014

Hydrodynamic model of Fukushima-Daiichi NPP Industrial site flooding

While the Fukushima-Daiichi was designed and constructed the maximal tsunami height estimate was about 3 m based on analysis of statistical data including Chile earthquake in 1960. The NPP project industrial site height was 10 m. The further deterministic estimates TPCO-JSCE confirmed the impossibility of the industrial site flooding by a tsunami and therefore confirmed ecological safety of the NPP. However, as a result of beyond design earthquake of 11 March 2011 the tsunami height at the shore near the Fukushima-Daiichi NPP reached 15 m. This led to flooding and severe emergencies having catastrophic environmental consequences. This paper proposes hydrodynamic model of tsunami emerging and traveling based on conservative assumptions. The possibility of a tsunami wave reaching 15 m height at the Fukushima-Daiichi NPP shore was confirmed for deduced hydrodynamic resistance coefficient of 1.8. According to the model developed a possibility of flooding is determined not only by the industrial site height, magnitude and distance to the earthquake epicenter, but also by duration of seismic impacts, power dissipation conditions, epicenter size and other conditions.

physics.flu-dyn↗
arXiv2014

Transpose-free Fast Fourier Transform for Turbulence Simulation

Pseudo-spectral method is one of the most accurate techniques for simulating turbulent flows. Fast Fourier transform (FFT) is an integral part of this method. In this paper, we present a new procedure to compute FFT in which we save operations during interprocess communications by avoiding transpose of the array. As a result, our transpose-free FFT is 15\% to 20\% faster than FFTW.

cs.MS↗
arXiv2014

Identify the diapycnical eddy diffusivities by salt fingers and turbulence with vertical microstructure measurements

Diapycnical eddy diffusivities are formulated from physical relations according to a simple fact that the different formulas are identical for the same parameter. It is found that the dispassion ratio Γis a crucial parameter. When it is above a critical value (about 0.2), the flow is salt finger type; otherwise, it is turbulence. All the density ratio $R_ρ=1/(1-Γ^2)$, eddy flux ratio $γ=1/(1+Γ)$, and eddy diffusivity ratio $R_k=1-Γ$ are simply dependent on the dispassion ratio gamma Γ. We apply these relations to the measurement data in the western tropical Atlantic Ocean. The effective diapycnal diffusivity is $0.96*10^{-4} m^2/s$ that agrees quite well with the observation from 0.8 to $0.9*10^{-4} m^2/s$ by tracer.

physics.ao-ph↗
arXiv2014

Internal bores and gravity currents in a two-fluid system

In this paper, a unified theory of internal bores and gravity currents is presented within the framework of the one-dimensional two-layer shallow-water equations. The equations represent four basic physical laws: the theory is developed on the basis of these laws. Though the first three of the four basic laws are apparent, the forth basic law has been uncertain. This paper shows first that this forth basic law can be deduced from the law which is called in this paper the conservation law of circulation. It is then demonstrated that, within the framework of the equations, an internal bore is represented by a shock satisfying the shock conditions that follow from the four basic laws. A gravity current can also be treated within the framework of the equations if the front conditions, i.e. the boundary conditions to be imposed at the front of the current, are known. Basically, the front conditions for a gravity current also follow from the four basic laws. When the gravity current is advancing along a no-slip boundary, however, it is necessary to take into account the influence of the thin boundary layer formed on the boundary; this paper describes how this influence can be evaluated. It is verified that the theory can satisfactorily explain the behaviour of internal bores advancing into two stationary layers of fluid. The theory also provides a formula for the rate of advance of a gravity current along a no-slip lower boundary; this formula proves to be consistent with some empirical formulae. In addition, some well-known theoretical formulae on gravity currents turn out to be obtainable on the basis of the theory.

physics.flu-dyn↗