Wolfgang Schöpf

Boundary layer development on a semi-infinite suddenly heated vertical plate

The flow resulting from suddenly heating a semi-infinite, vertical wall immersed in a stationary fluid has been described in the following way: at any fixed position on the plate, the flow is initially described as one-dimensional and unsteady, as though the plate is doubly infinite; at some later time, which depends on the position, a transition occurs in the flow, known as the leading-edge effect (LEE), and the flow becomes two-dimensional and steady. The transition is characterized by the presence of oscillatory behaviour in the flow parameters, and moves with a speed greater than the maximum fluid velocities present in the boundary layer. A stability analysis of the one-dimensional boundary layer flow performed by Armfield & Patterson (1992) showed that the arrival times of the LEE determined by numerical experiment were predicted well by the speed of the fastest travelling waves arising from a perturbation of the initial one-dimensional flow. In this paper, we describe an experimental investigation of the transient behaviour of the boundary layer on a suddenly heated semi-infinite plate for a range of Rayleigh and Prandtl numbers. The experimental results confirm that the arrival times of the LEE at specific locations along the plate, relatively close to the leading edge, are predicted well by the Armfield & Patterson theory. Further, the periods of the oscillations observed following the LEE are consistent with the period of the maximally amplified waves calculated from the stability result. The experiments also confirm the presence of an alternative mechanism for the transition from one-dimensional to two-dimensional flow, which occurs in advance of the arrival of the LEE at positions further from the leading edge.

Multicritical Behaviour in Binary Fluid Convection

The nonlinear coefficients of the amplitude equations for the stationary pattern, the travelling waves, and the standing waves are calculated for the first time rigorously for binary fluid mixtures with realistic boundary conditions. We find that the travelling-wave pattern bifurcates subcritically except in the immediate vicinity of the codimension-2 point and has a larger amplitude than standing waves. The tricritical points for the stationary pattern and the travelling waves lie always at negative values for the separation ratio Ψ with the codimension-2 point between them. In the limit of small Lewis numbers both tricritical points and the codimension-2 point converge to the interesting, highly degenerate point Ψ = 0.

Convection onset for a binary mixture in a porous medium and in a narrow cell: a comparison

The hydrodynamic equations describing the fluid motion in a narrow cell and in a porous medium become identical in the limit of infinite height-to-width ratio (Hele-Shaw limit) in the first and zero permeability in the second case. The properties of the convection onset are, however, indistinguishable from an experimental point of view away from these limits. For realistic (rigid, impermeable) boundary conditions the critical Rayleigh number, the critical wavenumber and in the case of a Hopf bifurcation additionally the critical frequency are derived in both cases for a binary fluid mixture

Thermal convection in a thermosensitive colloidal suspension

Thermal convection is investigated experimentally in a microgel suspension that consists of core-shell colloids, which change their size with temperature. The swelling and shrinking of the particles strongly modify their volume fraction in the carrier fluid and therefore the viscosity of the suspension. In this experiment, thermal convection in a Hele–Shaw-like apparatus is monitored using the shadowgraph technique. When compared to a normal fluid, the threshold temperature difference is reduced dramatically, which is interpreted as a manifestation of the Soret effect, i.e. the temperature gradient applied to the suspension induces an unstable gradient of the colloid concentration. The wavelength in the nonlinear regime is very different from the one observed in water. Furthermore, transient oscillations of the patterns are detected in the nonlinear regime and are investigated as a function of the applied temperature difference.

Amplification of thermal noise via convective instability in binary-fluid mixtures

In the convectively unstable regime of a binary-fluid mixture in a quasi-one-dimensional ramped convection channel an erratic spatio-temporal behaviour of very small amplitudes is observed. These patterns are interpreted as the amplification of intrinsic noise. Their measured temporal correlation functions are compared with the theoretical ones obtained from a Ginzburg-Landau equation containing a noise term. The measured intensity confirms the assumption that the fluctuations are due to thermal noise.

The influence of thermal noise on the onset of travelling-wave convection in binary fluid mixtures : an experimental investigation

The paper presents a study on the convective instability occurring in binary fluid mixtures for a negative separation ratio which is utilized for amplifying intrinsic thermal fluctuations. Experiments are performed in a 1D convection channel which prevents the reflection of traveling waves from the sidewalls. The temperature variations are quantified using the shadowgraph method. The temporal correlation function is calculated from measured light intensity caused by the temperature variations and are compared with a theoretical expression obtained from a Ginzburg-Landau equation containing a noise term. It is assumed that thermal noise is the reason for these fluctuating convecting rolls.

Exploring cogging free magnetic gears

The coupling of two rotating spherical magnets is investigated experimentally, with particular emphasis on those motions where the driven magnet follows the driving one with a uniform angular speed, which is a feature of the so called cogging free couplings. The experiment makes use of standard equipment and digital image processing. The theory for these couplings is based on fundamental dipole-dipole interactions with analytically accessible solutions. Technical applications of this kind of coupling are foreseeable particularly for small machines, an advantage which also comes handy for classroom demonstrations of this feature of the fundamental concept of dipole-dipole coupling.

Experimental Study of Binary Fluid Convection in a Quasi 1-Dimensional Cell without Reflections from the Sidewalls

With properly chosen parameters, convection in binary fluid mixtures sets in via a Hopf bifurcation leading to travelling waves. In this case one has to distinguish between a convectively unstable and an absolutely unstable situation. We present an observation in a cell where the fluid becomes unstable at the absolute instability point, not at the convective one as in most other experiments. This is done by preventing the sides of the cell from reflecting the travelling wave.

The Degenerate Amplitude Equation Near the Codimension-2 Point in Binary Fluid Convection

Hydrodynamical systems like Rayleigh-Benard convection play a major role in the investigation of self-organization in nonequilibrium systems and in various bifurcation problems. In binary fluid mixtures one can either have a transition from the heat conducting state to the convective state via a stationary bifurcation or via a Hopf bifurcation, depending on the separation ratio Ψ.