Conventional viscous fingering flow in radial Hele-Shaw cells employs a constant injection rate, resulting in the emergence of branched interfacial shapes. The search for mechanisms to prevent the development of these bifurcated morphologies is relevant to a number of areas in science and technology. A challenging problem is how best to choose the pumping rate in order to restrain the growth of interfacial amplitudes. We use an analytical variational scheme to look for the precise functional form of such an optimal flow rate. We find it increases linearly with time in a specific manner so that interface disturbances are minimized. Experiments and nonlinear numerical simulations support the effectiveness of this particularly simple, but nontrivial, pattern controlling process.
In quantum theory we refer to the probability of finding a particle between positions x and x + dx at the instant t, although we have no capacity of predicting exactly when the detection occurs. In this work, first we present an extended non-relativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between x and x + dx during [t,t + dt]. Then, we find a Schrödinger-like equation for a "mirror" wave function φ(t, x) associated with the probability of measuring the system between t and t + dt, given that detection occurs at x. In this framework, it is shown that energy measurements of a stationary state display a non-zero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to ad hoc assumptions, is a natural, built-in part of the formalism presented here. In Schrödinger quantum mechanics (QM) there is a clear asymmetry between time and space. Time is a continuous parameter that can be chosen with arbitrary precision and used to label the solution of the wave equation. In contrast, the position of a particle is seen as an operator, and therefore its value under a measurement is inherently probabilistic. It is common to hear that this asymmetry is due to the non-relativistic character of the Schrödinger equation (SE). Although partially correct, this argument is largely insufficient to justify all the disparity between space and time in the formalism of QM.A clear illustration is as follows. In a position measurement, ψ(x, t) = x|ψ(t) gives the probability amplitude of finding the particle within [x, x + dx], given that the time of detection is t. Would it not be equally reasonable, even in the non-relativistic domain, to ask about the probability of measuring the particle between x and x + dx, and t and t + dt? In this broader scenario, inquiring about the state of a particle at a given time t (as we often do), should make as much sense as asking about the state of that particle in a given position x (which we never do). In addition, if symmetry is to hold at this level, then there should exist a "mirror" wave function φ(t, x) = t|φ(x) , where x is a continuous parameter and t is the eigenvalue of an observable. If the location of particle becomes a physical reality only when a measurement is made, then it is a tenable position to expect that time should emerge in the same way. To earnestly consider these issues is the main goal of this manuscript.Time has been addressed in different contexts in QM . Common to several of these works is the attempt to remain within the borders of the standard theory. However, the solution to the arrival-time problem is considered by several authors to lay outside the framework of QM. It concerns the arrival of a particle in a spatially localized apparatus, where a time operator may be defined so that the relation [T ,Ĥ] = i is satisfied, and * corresponding author: eduardodias@df.ufpe.br † parisio@df.ufpe.br ...
It is well known that the constant injection rate flow in radial Hele-Shaw cells leads to the formation of highly branched patterns, where finger tip-splitting events are plentiful. Different kinds of patterns arise in the lifting Hele-Shaw flow problem, where the cell's gap width grows linearly with time. In this case, the morphology of the emerging structures is characterized by the strong competition among inward moving fingers. By employing a mode-coupling theory we find that both finger tip-splitting and finger competition can be restrained by properly adjusting the injection rate and the time-dependent gap width, respectively. Our theoretical model approaches the problem analytically and is capable of capturing these important controlling mechanisms already at weakly nonlinear stages of the dynamics.
The Mullins-Sekerka and the electric breakdown instabilities are well known to lead to the spontaneous formation of a variety of complex spatial structures, among them dendritic crystal shapes, and treelike electric discharge patterns. Controlling such systems by suppressing predominantly excited solutions offers the opportunity to manipulate and stabilize these patterns in a defined way for a wide range of technological applications. In this work, we employ a variational approach which enables one to systematically search for the ideal conditions under which the patterns grow, but where interfacial deformations are efficiently minimized. The effectiveness of our variational control method is illustrated via linear stability calculations on both two-dimensional and three-dimensional contour-dynamics models for crystal growth and electric discharge phenomena.
The injection of a fluid into another of larger viscosity in a Hele-Shaw cell usually results in the formation of highly branched patterns. Despite the richness of these structures, in many practical situations such convoluted shapes are quite undesirable. In this Brief Report, we propose an efficient and easily reproducible way to restrain these instabilities based on a simple piecewise-constant pumping protocol. It results in a reduction in the size of the viscous fingers by one order of magnitude.
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