We studied dendritic side-branching mechanism in the experiment of anisotropic viscous fingering. We measured the time dependence of growth speed of side-branch and the envelop of side-branches. We found that the speed of side-branch gets to be faster than one of the stem and the growth exponent of the speed changes at a certain time. The envelope of side-branches is represented as Y ~ X^1.47.
An understanding of the underlying mechanism of side--branching is paramount in controlling and/or therapeutically treating mammalian organs, such as lungs, kidneys, and glands. Motivated by an activator-inhibitor-substrate approach that is conjectur
ed to dominate the initiation of side--branching in pulmonary vascular pattern, I demonstrate a distinct transverse front instability in which new fingers grow out of an oscillatory breakup dynamics at the front line, without any typical length scale. These two features are attributed to unstable peak solutions in 1D that subcritically emanate from the Turing bifurcation and that exhibit repulsive interactions. The results are based on a bifurcation analysis and numerical simulations, and provide a potential strategy toward developing a framework of side--branching also of other biological systems, such as plant roots and cellular protrusions.
We consider the problem of viscous fingering in the presence of quenched disorder that is both weak and short-range correlated. The two point correlation function of the harmonic measure is calculated perturbatively, and is used in order to calculate
the correction the the box-counting fractal dimension. We show that the disorder increases the fractal dimension, and that its effect decreases logarithmically with the size of the fractal.
The growth dynamics of an air finger injected in a visco-elastic gel (a PVA/borax aqueous solution) is studied in a linear Hele-Shaw cell. Besides the standard Saffmann-Taylor instability, we observe - with increasing finger velocities - the existenc
e of two new regimes: (a) a stick-slip regime for which the finger tip velocity oscillates between 2 different values, producing local pinching of the finger at regular intervals, (b) a ``tadpole regime where a fracture-type propagation is observed. A scaling argument is proposed to interpret the dependence of the stick-slip frequency with the measured rheological properties of the gel.
In this paper the travelling wave solutions in the adiabatic model with two-step chain branching reaction mechanism are investigated both numerically and analytically in the limit of equal diffusivity of reactant, radicals and heat. The properties of
these solutions and their stability are investigated in detail. The behaviour of combustion waves are demonstrated to have similarities with the properties of nonadiabatic one-step combustion waves in that there is a residual amount of fuel left behind the travelling waves and the solutions can exhibit extinction. The difference between the nonadiabatic one-step and adiabatic two-step models is found in the behaviour of the combustion waves near the extinction condition. It is shown that the flame velocity drops down to zero and a standing combustion wave is formed as the extinction condition is reached. Prospects of further work are also discussed.
We study solitons in the two-dimensional defocusing nonlinear Schroedinger equation with the spatio-temporal modulation of the external potential. The spatial modulation is due to a square lattice; the resulting macroscopic diffraction is rotationall
y symmetric in the long-wavelength limit but becomes anisotropic for shorter wavelengths. Anisotropic solitons -- solitons with the square (x,y)-geometry -- are obtained both in the original nonlinear Schroedinger model and in its averaged amplitude equation.