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Self-similar dynamics, often described by partial differential equations (PDEs) without noise terms, have been widely observed in nature and have attracted considerable attention from researchers in various fields, including physics, fluid mechanics, applied mathematics, and chemical engineering. The concept of universality class and the renormalization group (RG) theory, emerged from critical phenomena, which also have been widely observed in nature, are pillars of modern physics, with guiding the recent development of physics. While self-similarity appearing in fluid dynamics reminds physicists of critical phenomena, deep connections between them have not been explored. Here, we focus on a variation of a daily phenomenon, a drop falling from a faucet, and demonstrate self-similar dynamics for the shape of liquid-air interface in confined space with a scaling law for a characteristic length scale, which leads to a clear correspondence between the present self-similar dynamics and critical phenomena by identifying critical exponents, which define a universality class. We not only show that the observed master curve describing the self-similar shape is a solution to the Navier-Stokes equations, but also develop a renormalization group (RG) theory for the governing PDE to show the fixed point of RG corresponds to the solution, which becomes stable with time due to a scaling crossover. Our results will impact on our understanding of critical, scaling, and singular behaviors widely observed in nature.

Recently, simple scaling laws concerning the mechanical response and mechanical transition of Kirigami have been revealed through agreement between theory and experiment for kirigami made of paper [M. Isobe and K. Okumura, Sci. Rep. 2016]. Here, we provide experimental data obtained from kirigami made of soft elastic sheets to demonstrate good agreement with previous theories, although a number of assumptions in the theory are violated and the elastic modulus is three orders of magnitude smaller in the present case. This remarkable universality in the mechanics of Kirigami, which could be useful for applications, is reported with physical insights based on previous theories.

Critical phenomena, in which various physical quantities exhibit power laws, have been widely observed in nature and played a central role in physics. They generated the concept of universality class, which origin is elucidated by the renormalization group (RG) theory and which has guided the recent development of physics. Accordingly, identifying a rich variety of universality class is a major issue in modern physics. Here, we report that a daily phenomenon, similar to a drop falling from a faucet, possesses a strikingly close analogy with critical phenomena, with this version remarkably revealing the existence of an uncountably infinite number of universality classes. The key for our present findings is the confinement of a system into a thin cell, in which we observe the dynamics of air-liquid interface formed by entrained air into viscous liquid by a solid disk. The entrained air eventually detaches from the disk with or without breakup involving change in topology. While the time development of the interface shape is self-similar with characteristic length scales exhibiting power laws as in the case of non-confined previous studies, we here found that the observed self-similar shape corresponds to a solution to the governing equations, which we show a stable fixed point of the RG flow by combining a RG analysis and the dynamical system description (DSD) developed for singular dynamics in applied mathematics. We thus demonstrate a striking analogy with critical phenomena, showing critical exponents that define a universality class depend on "continuous numbers" characterizing the confinement. This is in contrast with the conventional critical phenomena, in which a class is defined by "discrete numbers." Our results open a new avenue for our understanding of critical phenomena, RG, and DSD, which impacts on the study of singular dynamics widely observed in nature.

We fill a viscous liquid in a vertically stood cell of millimeter thickness, called the Hele-Shaw cell, and insert a disk in the liquid whose thickness is smaller than the cell thickness. The disk starts falling in the liquid due to gravity opposed by viscous friction. We focus on the case in which lubricating films formed in the gap between the cell surface and the disk surface are thinner than the disk thickness. As a result, we find an apparent scaling regime for the falling velocity of a disk, in which the thickness of the lubricating film characterizes the dynamics. We further show that the apparent scaling regime is explained simply as a result of competition of two scaling regimes, elucidating the physics of the viscous friction. The present study is thus relevant to fundamental issues and applications in various fields in which small-scale physics in the flow at low Reynolds numbers is essential, such as microfluidics, bioconvection, and active matter. The simple scenario for explaining an apparent scaling law demonstrated in the present study would be useful in diverse fields, considering that the generality and strength of scaling analysis in science and that simple arguments usually lead to a few different scaling laws for a given problem., Comment: 13 pages, 3 figures

Rising motion of an obstacle in a vibrated granular medium is a classic problem of granular segregation, and called the Brazil nut (BN) effect. The controlling vibration parameters of the effect has been a long-standing problem. A simple possibility that the BN effect can be characterized solely by vibration velocity has recently been pointed out. The issue has become controversial before a long history of research, with only a few systems have provided for the simple possibility. Here, we investigate the rising motion of an obstacle in a vertically positioned one-layer granular bed under continuous vibrations. We find the rising motion is composed of two distinct regimes, and the first and second regimes are both governed, in terms of vibration parameters, solely by the vibration velocity. We further demonstrate simple scaling laws well describe the two regimes. Our results support the emergent possibility on the controlling parameters of the BN effect and suggests that this feature would be universal. We propose two possible mechanisms of convection and arch effect for the two distinct regimes and demonstrate these mechanism explain the scaling laws followed by our experimental data., Comment: 13 pages, 5 figures

Capillary invasion of a liquid into an empty tube, which is called capillary rise when the tube axis is in the vertical direction, is one of the fundamental phenomena representing capillary effects. Usually, the tube is actually filled with another pre-existing fluid, air, whose viscosity and inertia can be practically neglected. In this study, we considered the effect of the pre-existing fluid, when its viscosity is non-negligible, in a horizontally geometry. This geometry is free from gravity and thus simpler than the geometry of capillary rise. We observed the dynamics when a capillary tube that is submerged horizontally in a liquid gets in contact with a second liquid. An appropriate combination of liquids allowed us to observe that the second liquid replaces the first without any prewetting process, thanks to a careful cleaning of capillary tubes. Furthermore, we experimentally observed three distinct viscous dynamics: (i) the conventional slowing-down dynamics, (ii) an unusual accelerating dynamics, and (iii) another unusual dynamics, which is linear in time. We developed a theory in viscous regimes, which accounts well for the observations through a unified expression describing the three distinct dynamics. We also demonstrated a thorough experimental confirmation on the initial velocity of the replacement. We further focused on the replacement time, the time required for the invading fluid to replaces completely the pre-existing fluid in the horizontal geometry, which is again well explained by the theory., Comment: 15 pages, 5 figures

Recently, kirigami's high extensibility has been understood as a transition in the force-elongation curve. In this paper, we consider a model, which modifies our previous model, to show a striking analogy between the present theory and Landau theory of continuous thermodynamic transitions, if we regard a rotation angle and elongation of kirigami as the order parameter and the inverse temperature, respectively. The present study opens a new avenue in physics, pointing out the importance of the distinction between discontinuity and continuity of the high-extensibility transition in an elementary kirigami structure, and showing that the mechanical response of kirigami can be understood using the tools of statistical physics, which have been proved to be useful in many fields of physics., Comment: 20 pages, 5 figures, submitted to Phys Rev Research

Velocity jumps observed for crack propagation under a static boundary condition have been used as a controlling factor in developing tough rubbers. However, the static test requires many samples to detect the velocity jump. On the contrary, crack propagation performed under a dynamic boundary condition is timesaving and cost-effective in that it requires only a single sample to monitor the jump. In addition, recent experiments show that velocity jump occurs only in the dynamic test for certain materials, for which the velocity jump is hidden in the static test because of the effect of stress relaxation. Although the dynamic test is promising because of these advantages, the interrelation between the dynamic test and the more established static test has not been explored in the literature. Here, by using two simulation models, we elucidate this interrelation and clarify a universal condition for obtaining the same results from the two tests, which will be useful for designing the dynamic test., Comment: 21 pages, 8 figures, 1st revised version submitted to Polymer

Recently, it has been found experimentally that hydrated nacre exhibits a nonlinear mechanical response. While mechanical nonlinearity has been shown to be important in other biological structures, such as spider webs, the implications of mechanical nonlinearity in nacre have not been explored. Here, we show that the nonlinear mechanical response of nacre can be reproduced by an analytical model, which reflects a nacre-like layered structure, consisting of linear-elastic hard sheets glued together by weakly nonlinear-elastic soft sheets. We develop scaling analysis on this analytical model, and perform numerical simulations using a lattice model, which is a discrete counterpart of the analytical model. Unexpectedly, we find the weak nonlinearity in the soft component significantly contributes to enhancing toughness by redistributing the stress at a crack tip over a wider area. Beyond demonstrating a mechanism that explains the unusual properties of biological nacre, this study points to a general design principle for constructing tough composites using weak nonlinearity, and is useful as a guiding principle to develop artificial layered structures mimicking nacre., Comment: 20 pages, 5 figures, submitted to Phys. Rev. Mater. (1st revised version)

It has long been known for elastomers that the velocity of crack propagation jumps as a function of strain. On the other hand, such a jump has not been reported in the literature for polymers which do not exhibit a rubbery plateau in the storage-modulus plot. Here, we report observation of jumps in crack propagation for semi-crystalline polymer sheets without the rubbery plateau, as a result of pulling the sheets at a constant speed. We discuss the advantages of this crack-propagation test under constant-speed stretching and provide physical interpretation of the velocity jump observed for non-elastomer sheets on the basis of a recently proposed theory for the velocity jump in crack propagation., Comment: 14 pages, 4 figures,1st revised version submitted to Polymer (Elsevier)

We study the dependence of the fracture surface energy on the pulling velocity for nano-porous polypropylene (PP) sheets to find two components: the static and dynamic ones. We show that these terms can be interpreted respectively as plastoelastic and viscoelastic components, as has been shown for soft polyethylene (PE) foams in a previous work. Considering significant differences in the pore size, volume fraction and Young's modulus of the present PP and previous PE sheets, the present results suggest a universal physical mechanism for fracture of porous polymer sheets. The simple physical interpretation emerging from the mechanism could be useful for developing tough polymers. Equivalence of Griffith's energy balance in fracture mechanics to a stress criterion is also discussed and demonstrated using the present experimental data., Comment: 12 pages, 4 figures; submitted to Polymer J. (2nd revision)

In hydrodynamic topological transitions, one mass of fluid breaks into two or two merge into one. For example, in the honey-drop formation when honey dripping from a spoon, honey is extended to separate into two as the liquid neck bridging them thins down to micron scales. At the moment when topology changes due to the breakup, physical observables such as surface curvature locally diverges. Such singular dynamics have widely attracted physicists, revealing universality in their self-similar dynamics, which share much in common with critical phenomena in thermodynamics. Many experimental examples have been found, which include electric spout and vibration-induced jet eruption. However, only a few cases have been physically understood on the basis of equations that govern the singular dynamics and even in such a case the physical understanding is mathematically complicated inevitably involving delicate numerical calculations. Here, we study breakup of air film entrained by a solid disk into viscous liquid in a confined space, which leads to formation, thinning and breakup of the neck of air. As a result, we unexpectedly find that equations governing the neck dynamics can be solved analytically by virtue of two remarkable experimental features: only a single length scale linearly dependent on time remains near the singularity and universal scaling functions describing singular neck shape and velocity field are both analytic. The present solvable case would be essential for our better understanding of the singular dynamics and will help unveil the physics of unresolved examples intimately related to daily-life phenomena and diverse practical applications., Comment: 18 pages, 4 figures, accepted for publication in Phys Rev Fluids

We experimentally investigate the bursting dynamics of confined liquid film suspended in air and find a viscous dynamics distinctly different from the non-confined counterpart, due to lack of circular symmetry in the shape of expanding hole: the novel confined-viscous bursting proceeds at a constant speed and a rim formed at the bursting tip does not grow. We find a confined-viscous to confined-inertial crossover, as well as a nonconfined-inertial to confined-inertial crossover, at which bursting speed does not change although the circular symmetry in the hole shape breaks dynamically., Comment: 12 pages, 4 figures

In this review article, we discuss recent studies on drops and bubbles in Hele-Shaw cells, focusing on how scaling laws exhibit crossovers from the three-dimensional counterparts and focusing on topics in which viscosity plays an important role. By virtue of progresses in analytical theory and high-speed imaging, dynamics of drops and bubbles have actively been studied with the aid of scaling arguments. However, compared with three dimensional problems, studies on the corresponding problems in Hele-Shaw cells are still limited. This review demonstrates that the effect of confinement in the Hele-Shaw cell introduces new physics allowing different scaling regimes to appear. For this purpose, we discuss various examples that are potentially important for industrial applications handling drops and bubbles in confined spaces by showing agreement between experiments and scaling theories. As a result, this review provides a collection of problems in hydrodynamics that may be analytically solved or that may be worth studying numerically in the near future., Comment: 34 pages, 16 figures, accepted for publication in Adv. Colloid Interface Sci

We investigate crack propagation in a simple two-dimensional visco-elastic model and find a scaling regime in the relation between the propagation velocity and energy release rate or fracture energy, together with lower and upper bounds of the scaling regime. On the basis of our result, the existence of the lower and upper bounds is expected to be universal or model-independent: the present simple simulation model provides generic insight into the physics of crack propagation, and the model will be a first step towards the development of a more refined coarse-grained model. Relatively abrupt changes of velocity are predicted near the lower and upper bounds for the scaling regime and the positions of the bounds could be good markers for the development of tough polymers, for which we provide simple views that could be useful as guiding principles for toughening polymer-based materials., Comment: 17 pages, 5 figures, Final version (proof correction reflected)

Meandering instability is familiar to everyone through river meandering or small rivulets of rain flowing down a windshield. However, its physical understanding is still premature, although it could inspire researchers in various fields, such as nonlinear science, fluid mechanics and geophysics, to resolve their long-standing problems. Here, we perform a small-scale experiment in which air flow is created in a thin granular bed to successfully find a meandering regime, together with other remarkable fluidized regimes, such as a turbulent regime. We discover that phase diagrams of the flow regimes for different types of grains can be universally presented as functions of the flow rate and the granular-bed thickness when the two quantities are properly renormalized. We further reveal that the meandering shapes are self-similar as was shown for meandering rivers. The experimental findings are explained by theory, with elucidating the physics. The theory is based on force balance, a minimum-dissipation principle, and a linear-instability analysis of a continuum equation that takes into account the fluid-solid duality, i.e., the existence of fluidized and solidified regions of grains along the meandering path. The present results provide fruitful links to related issues in various fields, including fluidized bed reactors in industry., Comment: 13 pages, 4 figures

Needs to impart appropriate elasticity and high toughness to viscoelastic polymer materials are ubiquitous in industries such as concerning automobiles and medical devices. One of the major problems to overcome for toughening is catastrophic failure linked to a velocity jump, i.e., a sharp transition in the velocity of crack propagation occurred in a narrow range of the applied load. However, its physical origin has remained an enigma despite previous studies over 35 years. Here, we propose an exactly solvable model that exhibits the velocity jump incorporating linear viscoelasticity with a cutoff length for a continuum description. With the exact solution, we elucidate the physical origin of the velocity jump: it emerges from a dynamic glass transition in the vicinity of the propagating crack tip. We further quantify the velocity jump together with slow- and fast-velocity regimes of crack propagation, which would stimulate the development of tough polymer materials., Comment: 9+14 pages, 6+4 figures

We study both experimentally and theoretically the descending motion due to gravity of a fluid drop surrounded by another immiscible fluid in a confined space between two parallel plates, i.e., in the Hele-Shaw cell. As a result, we show a new scaling regime of a nonlinear drag friction in viscous liquid that replaces the well-known Stokes' drag friction through a clear collapse of experimental data thanks to the scaling law. In the novel regime, the dissipation in the liquid thin film formed between the drop and cell walls governs the dynamics. The crossover of this scaling regime to another scaling regime in which the dissipation inside the droplet is dominant is clearly demonstrated and a phase diagram separating these scaling regimes is presented., Comment: 11 pages, 3 figures

The hourglass is one of the apparatuses familiar to everyone, but reveals intriguing behaviors peculiar to granular materials, and many issues are remained to be explored. In this study, we examined the dynamics of falling sand in a special form of hourglass, i.e., a wedge-shaped hopper, when a suspended granular layer is stabilized to a certain degree. As a result, we found remarkably different dynamic regimes of bubbling and cavity. In the bubbling regime, bubbles of nearly equal size are created in the sand at a regular time interval. In the cavity regime, a cavity grows as sand beads fall before a sudden collapse of the cavity. Bubbling found here is quite visible to a level never discussed in the physics literature and the cavity regime is a novel phase, which is neither continuous, intermittent nor completely blocked phase. We elucidate the physical conditions necessary for the bubbling and cavity regimes and develop simple theories for the regimes to successfully explain the observed phenomena by considering the stability of a suspended granular layer and clogging of granular flow at the outlet of the hopper. The bubbling and cavity regimes could be useful for mixing a fluid with granular materials., Comment: 11 pages, 5 figures; SI movies (CC BY 4.0; http://creativecommons.org/licenses/by/4.0/) are available at http://www.phys.ocha.ac.jp/okumuralab/h16/listOfPubs.htm

We study, experimentally and theoretically, the mechanical response of sheet materials on which line cracks or cuts are arranged in a simple pattern. Such sheet materials, often called kirigami (the Japanese words, kiri and gami, stand for cut and paper, respectively), demonstrate a unique mechanical response promising for various engineering applications such as stretchable batteries: kirigami sheets possess a mechanical regime in which sheets are highly stretchable and very soft compared with the original sheets without line cracks, by virtue of out-of-plane deformation. However, this regime starts after a transition from an initial stiff regime governed by in-plane deformation. In other words, the softness of the kirigami structure emerges as a result of a transition from the two-dimensional to three-dimensional deformation, i.e., from stretching to bending. We clarify the physical origins of the transition and mechanical regimes, which are revealed to be governed by simple scaling laws. The results could be useful for controlling and designing the mechanical response of sheet materials including cell sheets for medical regeneration and relevant to the development of materials with tunable stiffness and mechanical force sensors., Comment: 7 pages, 4 figures; A SI movie (CC BY 4.0; http://creativecommons.org/licenses/by/4.0/) is available at http://www.phys.ocha.ac.jp/okumuralab/h16/listOfPubs.htm

Single crystal nematic elastomers undergo a transition from a strongly ordered phase N to an "isotropic" phase I. We show that: (a) samples produced under tension by the Finkelmann procedure are intrinsically anisotropic and should show a small (temperature dependent) birefringence in the high temperature I phase. (b) for the I->Ntransition via cooling there is a spinodal limit but for the N->I transition via heating there is no soft mode at the standard spinodal temperature. (c) the N->I transition is reminiscent of a martensitic transformation: nucleation of the I phase should occur in the form of platelets, making a well defined angle with the director., Comment: 7 pages, 3 figures (To appear in Europhys. Lett.)