A variety of extended dissipative systems, when driven externally far from thermodynamic equilibrium, show spatial structures at macroscopic scale. These patterns can be maintained in the non-equilibrium state by controlling the external parameter(s) and are known as dissipative structures. When one of the control parameter(s) is raised above a finite value, usually called the critical value, the system jumps from one state to another. In many cases, the motionless isotropic state changes to anisotropic state varying in space and/or time.

 

 

Patterns due to parametrically forced surface waves in fluids

 

Faraday experiment consists of a thin layer of viscous fluid subjected to sinusoidal vibration in vertical direction. This is done by taking small amount of a fluid in a container with flat base, which is then vibrated in vertical direction. In the vibrating (non-inertial) frame of reference, the effective gravity may be considered as time periodic parameter. The forcing amplitude a and the forcing frequency ω are two parameters, which can be fixed externally. For small values of a the fluid remains static with respect to the container. As soon as the value of a is raised above certain critical value , which strongly depends on the viscosity of the fluid, the free and flat surface of the fluid becomes unstable. This leads to excitation of standing waves on the free surface.

It is the result of the instability of the free surface rather than surface generated in fluids by a wave maker. Faraday did his original experiment with low viscosity fluids and always observed square patterns and predicted these are the only patterns likely in such experiments. Faraday also measured the frequency of these surface waves. The time period of the surface waves was found to be exactly double of the period of sinusoidal vibration imposed. This is similar to the parametric excitation a pendulum whose point of suspension is vibrated sinusoidally in the vertical direction. The parametrically excited standing waves at the free surface are also known as Faraday waves.

 

 

Depending upon the dissipation in the system, however, there may be one, two or three sets of standing waves with different phases as soon as a is raised just above . The top view of the resulting patterns may be stripes, squares, rhombs, competing equilateral triangles and hexagons, alternating small and big hexagons.

 

Movie 1 shows standing waves consisting of alternating small and big hexagons in water-glycerol mixture.

 

Movie 2 shows chaotic competition between squares and hexagons. The chaotic state is very close to the onset.

 

Granular materials in a vertically vibrating container

 

It is well known since the pioneering work of Faraday that fine granular material or powder on a thin vibrating plate forms interesting patterns. When a thin metallic plate is vibrated, nodes and antinodes are formed in the vibrating plate. The powder lying on the plate moves away from the locations of anti-nodes to the locations of nodes. These patterns are known as Chladni patterns. The patterns depend on the shape of the plates. If the plate is thick, then the whole plate oscillates uniformly. There are no distribution of nodes and anti-nodes. In such cases fine granular material or powder moves to form a heap, if the vibration amplitude is raised sufficiently to make grains move. Movie 3 shows the formation heap as the only stable state at the onset. Initially many small heaps are formed and they slowly interact and finally form a single conical heap.

 

One always wonders how the grains move in the stable heap just above the onset. Faraday suggested that the granular materials falling from the heap are sucked into the central part of the heap due to partial vacuum there and then move upward to reach the top of the heap. Movie 4 shows a simple experiment that studies interaction of two heaps of the same granular material but of different colors. As the two heaps of granular material come close to each other, falling particles of the small (black) heap are sucked into the bigger (white) one.The black particles are sucked into the white heap making a triangular part of the surface grey. The vibration is continued for sufficiently long time and then the amplitude of the vibration is suddenly brought to zero. Carefully cutting the heap horizontally with a thick card shows that the black particles have penetrated only into a thin layer of the material below the grey triangular area. This suggests that the particles in the central part of the heap do not move at the onset of heap formation. The black particles falling from the small heap penetrate only a thin layer of the white heap and then move upward.While in upward motion, many of the black particles come to the surface and keep falling to the bottom.The particles move in thin conical layers at the onset of heap formation.The motion of the grains is not like that in bulk convective motion at least close to the onset of heap formation.

 

Movie 5 shows the secondary instability as the vibration amplitude a is raised further. The continuous azimuthal symmetry of the free surface is broken. Further increase in the vibration amplitude allows more particles in the heap leaving the contact with the the vibrating plate and staying in flight for some time. This leads to trapping of more air near the base and more fluidization of the powder. Consequently the single big heap collapses into many small heaps, which interact with each other when they come in the close vicinity of each other. Movie 6 shows the interaction of many heaps for a much above the its critical value for the heap formation.

 

Movie 7 shows spatio-temporal behaviour with further increase of forcing amplitude a. The fluidization increases and fluctuation in the shape and the size of the heaps increases. In this movie, the value of a is slowly decreased. This leads to formation of bigger heaps and ultimately to a single heap. Movie 8 shows the coexistence of complete fluidization in one part of the granular material and partial fluidization in another part. This is a state of granular turbulence.

References

 

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