Fields of research
Current research: Surface instabilities of magnetic fluids
  Thermomagnetic convection in magnetic fluids
     
Flow dynamics and stability of the density oscillator
Pattern formation in water-sand mixtures
Electrohydrodynamic convection in liquid crystals and multiplicative noise


Surface instabilities of magnetic fluids


Magnetic fluids are stable colloidal suspensions of ferromagnetic nanoparticles (typically magnetite or cobalt) dispersed in a carrier liquid (typically oil or water). The nanoparticles are coated with a layer of chemically adsorbed surfactants to avoid agglomeration. Magnetic fluids are



The behaviour of MF is characterized by the complex interaction of their hydrodynamic and magnetic properties with external forces. Magnetic fluids show many fascinating effects, as the labyrinthine instability or the Rosensweig instability (for a review see [1]). The latter instability occurs when a layer of MF with a free surface is subjected to a uniform and vertically oriented magnetic field. Above a certain threshold of the magnetic field that surface becomes unstable, giving rise to a regular arrangment of pattern of peaks, typically hexagons or squares.
[1] R. E. Rosensweig, Ferrohydrodynamics, (Dover Publications, Mineola, 1997).


Particular interests

Linear aspects of the Rosensweig instability
  • wave number with maximal growth rate for any combination of layer thickness and fluid viscosity
  • scaling behaviour of the maximal growth rate
  • frequency and propagation velocity of the decaying patterns
  • growth rate of surface ondulations
Nonlinear aspects of the Rosensweig instability
  • amplitude equation for a finite layer with a free surface
  • pattern selection and stability
Faraday instability on magnetic fluids
  • Faraday instability on magnetic fluids in a horizontal magnetic field
  • Faraday instability on magnetic fluids with chains in a horizontal magnetic field
Publications

Thermomagnetic convection in magnetic fluids


To describe the appearance of the thermomagnetic convection a horizontal layer of magnetic fluid between two rigid boundary plates is considered. The lower one is cooled, the upper one is heated. Additionally an external magnetic field Hext oriented vertically is applied. Due to the temperature gradient ∇T in the fluid layer, the magnetization of the magnetic fluid shows a gradient ∇M pointing in opposite direction as ∇T. The gradient in the magnetization causes now another gradient, that of the internal magnetic field, ∇Hint, which is antiparallel to ∇M. If a fluid element with the magnetization M from the bottom is moved adiabatically to the top of the layer, where the magnetization M-ΔM prevails, a resulting force

Fres=M ∇Hint - (M-ΔM) ∇Hint = + ΔM ∇Hint

is generated. This force has the same positive z-direction as the adiabatic movement. Thus initial disturbances will be enhanced and can lead to the onset of thermomagnetic convection if the stabilizing effects of heat conduction and internal friction can be overcome. This type of heat and mass transfer was first theoretically analyzed by Finlayson [1] and later on experimentally studied by Schwab [2-3] for the very first time.
[1] B. A. Finlayson, J. Fluid Mech. 40, 753 (1970).
[2] L. Schwab and K. Stierstadt, J. Magn. Magn. Mat. 65, 315 (1987).
[3] L. Schwab, J. Magn. Magn. Mat. 85, 199 (1990).


Particular interests

Thermomagnetic problems in magnetic fluids
  • Kelvin force in a layer of magnetic fluid
  • thermal convection in a two-dimensional cylindrical geometry
  • thermodiffusion and the influence of magnetic fields on the Soret coefficient
  • thermomagnetic convection in spatially modulated magnetic fields
Publications

Flow dynamics and stability of the density oscillator


The research in fluid dynamics and hydrodynamic instabilities often fruitfully combines classical concepts with novel approaches from nonlinear dynamics. The research is also enhancing our understanding of phenomena that are typically studied in fields such as oceanography, meteorology, or geophysics. While this applied research mainly focuses on the dynamics in infinite systems, certain hydrodynamic instabilities are strongly related to the presence of spatial constraints. A fascinating example is the density oscillator that was discovered by Martin in 1970 [1]. It combines the unstable stratification of two fluids known from Rayleigh-Taylor experiments with the spatio-temporal constraints of the ordinary Poiseuille flow. Due to its experimental simplicity and high degree of reproducibility it also appears to be an excellent model system for the study of self-induced relaxation oscillations [2].

The density oscillator consists of two containers separating the fluids (picture on the left). The inner and outer containers hold the dense and light fluids, respectively. The small inner container has a vertical capillary tube attached to its bottom that allows the dense fluid to flow into the large container and vice versa. Typical fluids are aqueous NaCl solutions and water. Surprisingly, one finds an oscillatory flow through the capillary giving rise to beautiful jets of ascending water and descending salt water. The jets and their oscillatory change of flow direction can be easily followed with the naked eye due to differences in refraction. The picture on the right (taken by Oliver Steinbock) shows the end of the capillary. The jet of salt water (dark) starts to shrink after leaving the orifice of the cappilary. This effect happens few seconds before the flow reversal sets in.



Our study presents quantitative measurements of the height evolution h(t), the salt water level in the small container. Critical heights and flow velocities characterizing the state of flow reversal are presented. The experimental results are compared with calculations that address the key problem of how to derive the conditions for flow reversal from the unstable pressure equilibria. The calculations follow an approach that has been successfully used for a particular example of heat transfer [3]. There is an excellent agreement between the theoretical und experimental values for the difference between the lower unstable equilibrium level and the critical level where the downflow actually reverses.
[1] S. Martin, Geophys. Fluid Dyn. 1, 143 (1970).
[2] P. S. Landa, Nonlinear Oscillations and Waves in Dynamical Systems, (Kluwer Academic Publishers, Dordrecht, 1996)
[3] T. Mahmood and J. H. Merkin, Mass and Heat Transfer 24, 257 (1989).
Publication

Pattern formation in water-sand mixtures


Hydrodynamic approaches to granular material are few and are associated with restrictions. Nevertheless, there are striking phenomenological similarities in the observed patterns for pure granular materials and pure fluids. Experiments with an inclined chute [1,2] or a vertically vibrated container [3,4] show the most noteable analogy. It is clear that granular media are different from fluids but under certain conditions these differences are not prevailing.

Here, experiments are performed with sand dispersed in water. The occuring shear rates, the mean particle diameters, and the viscosity of water result in a Bagnold number of about 1 [5]. This motivates the idea to consider the water-sand mixture as fluid-like. In the experiments we observe that the initial flat water-sand interface evolves into a finger-like pattern. To model this behavior we test a continuum approach which is based on a well-known hydrodynamic instability, the Rayleight-Taylor instability.

We choose a two-fluid system as a model. Carrying out a linear stability analysis for the interface between the two fluids we calculate the growth rates from the dispersion relation for a finite-size cell. The theoretical results agree with the essence in the experimental findings when assuming a relative viscosity of the water-sand mixture approximately 100 times higher than that of pure water, in agreement with other experimental findings [6]

The picture shows typical images of the sand-water cell at certain stages of the sedimentation. 20 ms after the series of snapshots is started, the initial flat sand layer is modulated at small scales (a). These disturbances are enhanced and give rise to sand fingers as seen in (c). At later stages the fingers evolve to a mushroom-like pattern ((d) and (e)).

[1] O. Pouliquen, J. Delour, S.B. Savage, Nature 386, 816 (1997); O. Pouliquen, S.B. Savage, Fingering instability in granular chute flows,preprint (1996).
[2] H.E. Huppert, Nature 300, 427 (1982).
[3] F. Melo, P. Umbanhowar, H.L. Swinney, Phys. Rev. Lett. 72, 172 (1994).
[4] S. Fauve, K. Kumar, C. Laroche, D. Beysens, Y. Garrabos, Phys. Rev. Lett. 68, 3160 (1992).
[5] M. Schröter, Diploma thesis, University of Magdeburg, 1997.
[6] L. Arnaud, C. Boutin, in Theoretical and Applied Rheology, edited by P. Moldenaers, R. Keunings (Elsevier, Amsterdam, 1992), p. 640.

Publications

Electrohydrodynamic convection in liquid crystals and multiplicative noise

Electroconvection (EC) in a thin layer of nematic liquid crystal (NLC) has become the prime example for pattern formation in anisotropic systmes. NLC molecules have an orientational order relative to each other, but no positional order. The direction parallel to the average alignment is noted by the director ñ. If ñ is oriented uniformly along a particular axis in the plane of the fluid, the case is known as planar alignment (a). For EC the NLC is sandwiched between two conducting plates with an ac electric field of amplitude V and frequency f applied across. There is a critical value Vc of V for which a transition from the spatially uniform state to a convecting state occurs (b).

Figure by courtesy of Thomas John.


Varying V > Vc and f, a great variety of spatiotemporal patterns has been observed, including time-dependent rolls, traveling waves, defect chaos, chaos at onset, and localized states. Some of the structures can be seen here and for an introduction into the subject the reader is refered to [1].

The influence of stochastic modulation of parameters in spatially extended systems is a subject of recent interest [2]. Especially well investigated are electrohydrodynamic instabilities in nematic liquid crystals where the convection is driven by an external (spatially homogeneous) time dependent stochastic electric field. The electric field is the superposition of a 'slow' (harmonically modulated or constant) deterministic component and a 'fast' stochastic component. Slow and fast refer to the characteristic times of the liquid crystal describing the relaxation of space charge and director in absence of external electric fields.

In experiments [3] it was found that the superposition of a fast stochastic field increases the threshold for the deterministic field (i.e. stabilizes the nematic phase) up to a certain critical value of the stochastic field. Beyond this value the nematic phase is unstable which leads to a discontinuous behaviour of the threshold curve as a function of the stochastic field.

Theoretically, this phenomenon found --at least qualitatively-- an explanation [4] by considering the stability of moments (the stochastic averages of space charge and director). This theory explained qualitatively (i) the discontinuous behaviour of the threshold at a critical strength of the noise, (ii) the change from discontinuous to continuous behaviour of the threshold with increasing characteristic time of the noise, and (iii) the change from a stabilizing to a destabilizing effect of the noise if its correlation time becomes comparable to the characteristic times of the system.

The quantitative agreement was, however, not satisfactory. It remained an open question whether the quantitative discrepancies were the result of the approximate treatment of the nemato-electrohydrodynamic equations, of poor knowledge of material parameters, of the choice of the stochastic stability criterion, or depending on other reasons. Since we would restrict ourself to a linear theory to extend the benefits of a previous developed method [4] the options for a better qualitative and quantitative agreement between theory and experiment are limited:
(i) To test the new patterns of oblique rolls and rhombic cells in the context of stochastic excitation.
(ii) The flexoeffect may be included in the calculations of the thresholds.
(iii) To test different stochastic stability criteria. The criteria are based, for example, on the stability of (first or higher) moments, on the bifurcation of the most probable value, or on the concept of sample stability describing the stability of one stochastic trajectory (a sample).

[1] P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, Oxford, 1993); L. Kramer and W. Pesch, Annu. Rev. Fluid Mech. 27, 515 (1995); Pattern Formation in Liquid Crystals, edited by L. Kramer and A. Buka (Springer, New York, 1996).
[2] Noise in nonlinear dynamical systems, Vols. 1-3, edited by F. Moss and P. V. E. McClintock (Cambridge University Press, Cambridge, 1989).
[3] S. Kai, T. Kai, M. Takata, and K. Hirakawa, J. Phys. Soc. Jpn. 47, 1379 (1979); H. R. Brand, S. Kai, and S. Wakabayashi, Phys. Rev. Lett. 54, 555 (1985); S. Kai, H. Fukunaga, H. R. Brand, J. Stat. Phys 54, 1133 (1989).
[4] U. Behn and R. Müller, Phys. Lett. 113A, 85 (1985); R. Müller and U. Behn, Z. Phys. B 69, 185 (1987); R. Müller and U. Behn, Z. Phys. B 78, 229 (1990).
Publications

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