Variational methods of solving dynamic problems for fluid-containing bodies. Nonlinear problems of the dynamics of elastic shells partialiy filled with a liquid. Observation of standing waves of large amplitude in harmonic Faraday resonance. Faraday resonance in water waves at nearly critical depths.

Three-dimensional arbitrary Lagrangian-Eulerian numerical prediction method for non-linear free surface oscillation. Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries. Experimental study of surface waves with Faraday resonance excitation. Single-mode Faraday waves in small cylinders. Study on the seismic response of a reactor vessel of a pool type LMFBR including fluid-structure interaction. Instabilities caused by oscillating accelerations normal to a viscous fluid-fluid interface.

Containment of a silicone fluid free surface in reduced gravity. Violent liquid sloshing in vertically excited cylindrical containers.

## Liquid Interfacial Systems: Oscillations and Instability - CRC Press Book

Parametric and autoparametric vibrations of an elevated water tower, part I: Non-linear parametric response. Drop formation mechanism in a vertically vibrated liquid column. Design of a thermocapillary flow experiment in reduced gravity. Surface waves in closed basins under parametric and internal resonances. Parametric excitation of surface waves.

Axisymmetric liquid sloshing under low-gravity conditions. Nonlinear Faraday resonance. Pattern Competition Leads to Chaos. Non-linear parametric liquid sloshing under wide band random excitation. Surface disintegration and bubble formation in vertically vibrated liquid column. Stability of vibrations of a cylindrical shell filled with a liquid under conditions of nonlinear resonances.

Nonlinear liquid motion in a longitudinally excited container with elastic bottom. Linear oscillations of viscous liquids in tanks. Fluid behavior in a longitudinally excited, cylindrical task of arbitrary sector-annular cross section. The onset of turbulence: Convection, surface waves, and oscillators. The motion of a fluid in a cylindrical container with a free surface following vertical impact.

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Survey of dynamic analysis methods for flight control design. A survey of dynamic analysis methods for airframe-flight control design.

Mechanical model for longitudinally excited sloshing motion. Optimal satellite orientation and acquisition problem. As one might expect, various stresses occur along the length of any tip-growing cell because of the varying properties of cell wall from one region to another when exposed to the turgor pressure [28]. The distribution of the wall stresses as well as deformation of the particular wall layers can in principle be calculated by solving equilibrium equations of elasticity theory.

The equilibrium equation may be derived both for materials deformed elastically deformation vanishes when force equals zero or non-elastically plastic deformation survives, even when the acting force is removed.

## Liquid Interfacial Systems: Oscillations and Instability

In this article we concentrate on elastic properties, because the oscillatory growth of pollen tubes is our main concern [7] , [9] , [8]. As an aside, note that whereas the hydrodynamic model as it is used by Zonia [7] proposes gradual increase in turgor until a threshold when rupture of individual links between cell wall polymers occurs, Winship and coworkers [3] state that turgor is essentially stable, but an exocytosis—induced relaxation of the wall causes expansion.

They postuale that variations in cell wall mechanical properties cause the oscillations and that variations in turgor if there are any are a passive consequence due to cell wall relaxation. Several possible mechanisms have been proposed to account for the behaviour of growing pollen tubes in quantitative terms [31] , [13] , [32]. A model for calcium dependent oscillatory growth in pollen tubes has been put forward by Kroeger et al.

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More recently, Fayant et al. Other models have included a chemically mediated mechanism of mechanical expansion of the pollen tube cell wall in which deposition causes a turnover of cell wall cross-links, thereby facilitating mechanical deformation [34]. The possible role of wall ageing in the self-regulation of tip-growth was considered in [35] ; while a model based on plasma membrane flow and cyclosis regulation in growing pollen tubes was discussed in [36].

Campas and Mahadevan treat the irreversible expansion of the cell wall during growth as an example of the extension of an inhomogeneous viscous fluid shell under the action of turgor pressure, fed by a material source in the neighborhood of the growing tip [37]. Lastly, a realistic osmotic model of the growing pollen tube has been proposed recently by Hill et al. However, none of the models produced oscillations on mechanical basis. Indeed, the model presented in [36] actually does not describe the cell wall, and hence, can not predict oscillations.

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The authors assume that the growth oscillations are given to understand the phase angle differences between growth velocity and the regulating mechanisms. But yet, by definition, oscillation is the repetitive variation, typically in time, of some measure about a central value often a point of equilibrium , and pollen tubes do not exhibit such form of oscillations. In fact, what we observe in pollen tubes is a periodical elongation without shrinking phase , and can be a result of transitions between two or more different states, which is another definition of oscillation, we adopt in this paper.

Our approach does not invalidate previous studies. We explore the relationship between turgor pressure and nontrivial cell geometry in terms of mutually exclusive instability models [39] based on cylindrical and spherical geometries, both of which are present in rapidly extending pollen tubes. We base our physical model on the parametrized description of a tip-growing cell that allows the manipulation of cell size, cell geometries, cell wall thickness, and local mechanical properties.

However, the mechanical load contrary to op. An important feature of pollen tubes elongation is the observation that the growth rate oscillates and that additionally, many of the underlying processes also oscillate with the same fundamental period, but usually with different phase e.

The possible roles of these oscillating ion gradients and fluxes in the control of pollen tube growth [40] is beyond the scope of this paper and will not be discussed here. We share the view expressed in [34] , [41] that the wall extension is primarily a biophysical mechanical process, and although ultimately dependent on enzymatic activity, under conditions where the enzymatic background can be subtracted out the biophysical process still proceedes normally.

The origin of the oscillation is still unclear, though all hypotheses agree in that, the cell wall mechanics are essential to the oscillation [36]. Any new model should deliver testable and quantitative predictions that can be validated by experimental data. In case of pollen tubes, it is necessary to present predictions that go beyond stress values which are inherently difficult to measure. The presented model satisfies this requirement by offering, among other things, an experimentally testifiable power law which relates the turgor pressure and the oscillation angular frequency.

In order to construct a stress-mechanical model of observed pollen tube growth oscillations we begin with an analysis of the mechanical properties of a tip growing cell. In particular we focus on the circular perimeter located at the junction of hemispherical dome and the cylindrical shank. This is based on the observation that cell wall assembly by exocytosis occurs mainly in an annular region around the apical pole of the cell [42] , [6] and that the concomittant turgor driven deformation of the cell wall causes characteristic strain in the hemisphere shaped apex of the cell [18] , [34].

The dynamic properties of such a complex growing system should be self-consistent meaning that the turgor pressure and the wall mechanical properties are conjugate magnitudes that usually form coupled equations, which have to be solved iteratively. Nevertheless, in the first approximation the following heuristic solution can be proposed. Assuming an intrinsic and constant turgor pressure [38] , and a much lower external pressure, presumably atmospheric pressure producing an effective pressure within the wall material , the equilibrium equation for the displacement vector , which is the shortest distance from the initial to the final position of a moving point, takes the form [43] 1 where is the Poisson coefficient.

For our purposes Young's modulus and Poisson coefficient are assumed to be picewise constant functions, so they remain constant on the interface Fig. Note, that by acting divergence operator on both sides of Eq. In particular it can be solved exactly for both spherical and cylindrical symmetries given the fact that both symmetries are necessarily present Fig. It follows that the symmetries of both subdomains must be used in any description of pollen tube shape and dynamical properties.

By assuming cylindrical symmetry for the displacement vector field , which is obtained under the assumption that the total length of the cylinder part remains constant the axial elongation of the more rigid shank upon application of a constant internal pressure we assume as negligible , and hence in first approximation we accept in instead of , which would lead to unavoidable numerical solution of Eq. For displacement in the spherical tip the introduction of spherical coordinates with the origin in the center of a sphere allows us to define the displacement field as a function of the radius :.

Therefore and Eq. Hence for displacement spherical symmetry yields 4 The upper index in Eqs 3 and 4 has been substituted to differentiate solutions for cylindrical c and spherical s geometries.

With these assumptions a realistic cell geometry for an elongating pollen tube can be described in terms of a cylinder of radius capped by a prolate half-spheroid with short radius and a long radius [18] ; Fig. Our working model for pollen tube growth thus consists of a thin-walled hollow cylinder ending by a hemispherical shell immersed in an external pool of pressure and filled with a cell sap with turgor pressure we equate both radii , for simplicity. The inner radius of the cylinder and the sphere is , while the outer radius is Fig.

## Liquid Interfacial Systems Oscillations And Instability

Another simplifying assumption is that we deal with weak elastic interactions at the interface , because of ongoing wall building processes resulting in the fact that the deflection field may differ slightly on both sides of the interface. From the mechanical point of view the two domains may be treated as being weakly coupled. This is consistent with the view that new deposition is associated with wall-loosening whereby load—bearing cross—links are broken, while simultaneously creating new load—free cross—links; thereby effecting a fail-safe scenario for mechanical expansion [34].