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The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood. [ABSTRACT FROM AUTHOR]

A coupled system of three nonlinear evolution equations is derived for a pair of collinear wave packets over finite depth fluid. The wave packets are narrow banded having different carrier wave frequencies. The evolution equations are employed to perform stability analysis of a pair of collinear wave trains. It is observed that the region of instability as well as the growth rate of instability for counter-propagating waves is greater than that for co-propagating waves when everything else remains same. The region of instability increases with the increase in the depth of the medium. It is also found that the growth rate of instability of a wave train increases with the increase in wave-steepness of the second wave train. [ABSTRACT FROM AUTHOR]

We analyze the dynamics of small two-dimensional disturbances in stable plane-parallel inviscid shear flows under linear theory. Using a velocity profile V x = U (y) with an inflection point but stable according to Fjørtoft's theorem, we illustrate that the continuum spectrum of van Kampen modes, possessing real phase velocities c = ω / k , aggregates into Landau damping solutions or "quasi-modes," which exhibit exponential decay. It was found that the real part of the complex phase velocity c L (k) of these solutions may lie outside the allowable range for van Kampen modes, suggesting a non-resonant damping mechanism for these quasi-modes. This conclusion was reached by solving the eigenvalue problem and observing the evolution of initial perturbations, calculated by directly solving the evolutionary equation for vorticity as well as by decomposing the initial disturbance into van Kampen modes. Landau damping of the total vorticity across the channel emerges as an intermediate stage before transitioning to power-law damping. [ABSTRACT FROM AUTHOR]

In arbitrary water depths, the influence of uniform flow, which includes transverse and longitudinal flows, on the generation of three-dimensional (3D) freak waves is examined. A modified Davey–Stewartson equation is derived using potential flow theory and the multiscale method. This equation describes the evolution of 3D freak wave amplitude under the influence of uniform flow. The relationship between two-dimensional (2D) modulational instability (MI) and the generation of 3D freak waves, as represented by the modified 3D Peregrine Breather solution, is explored. The characteristics of 2D MI depend on the orientation of the longitudinal and transverse perturbations. In shallow waters, the generation of freak waves by MI is challenging due to the minimal orientation difference, and longitudinal flows hardly affect the occurrence of MI. Variations in relative water depth can contribute to forming shallow-water freak waves. In finite-depth waters, oblique modulation leads to MI, whereas in deep and infinite-depth waters, longitudinal modulation gains significance. In environments of finite-depth, deep, and infinite-depth waters, the divergence (convergence) effect of longitudinal favorable (adverse) currents reduces (increases) the MI growth rate and suppresses (facilitates) freak wave generation. [ABSTRACT FROM AUTHOR]

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., "An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model," J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works. [ABSTRACT FROM AUTHOR]

A long-standing question in the Reynolds-averaged modeling of turbulent flow is how to predict accurately flows with temporal or spatial unsteadiness. In this paper, we present a "return to equilibrium" approach to modeling the unsteady Reynolds stress anisotropy bij in terms of differential transport equations, the solutions to which are constrained by algebraic closures for bij in flows at equilibrium and by the corresponding rapid-distortion-theory solutions for flows far from equilibrium. When coupled with scale equations for the turbulent kinetic energy and the dissipation rate, these anisotropy evolution equations comprise a complete closure scheme for which no additional model coefficients are required. Evaluations of this closure scheme, in which two different existing equilibrium algebraic models for bij were employed, were carried out for homogeneous turbulence in flows with a step imposition of shear, with continuously oscillatory shear at different frequencies and with a prescribed time-dependent plane-strain rate. Good agreement was achieved between model predictions and experimental/simulation target data in all test cases, with either Girimaji's bij model or a structure-based bij model used as the equilibrium algebraic closure. [ABSTRACT FROM AUTHOR]

We propose a novel method for efficient production of microbubbles based on a tapered capillary with an interiorly attached filament. When gas–liquid displacement driven by an input pressure occurs in the capillary, the gas cone ruptures close to the orifice of the capillary. The generated microbubbles can be pushed out of the capillary and collected by a liquid tank when the pressure is appropriately selected. A liquid column is employed in the straight part of the capillary, which can sustain the liquid film near the capillary orifice and hence the bubble generation by transporting liquid along the filament. Within the working pressure range, increasing the input air pressure leads to a decrease in the microbubble diameter. The minimum diameter of the microbubbles is approximately equal to the orifice diameter of the tapered capillary. In our experiments, microbubbles with a minimum diameter of 1.56 μm can be realized. Theoretically, we derive a one-dimensional unsteady lubrication equation describing the evolution of the gas–liquid interface in a tapered tube. The bubble pinch-off is justified by the numerical solution of the lubrication equation. In particular, the predicted bubble diameters are in agreement with the experimental measurements. [ABSTRACT FROM AUTHOR]

In this paper, the spatial discontinuous Galerkin (DG) approximation coupled with the temporal spectral deferred correction (SDC) evolution for the Stokes equations is adopted to construct the higher-order discretization method. First, the artificial compressibility strategy method is used to convert the Stokes equations into the Cauchy–Kovalevskaja type equations. Second, the corresponding equations can be rewritten as a first-order system by introducing the new variable equal to the gradient of the velocity. Then, the DG and the SDC methods are properly combined to construct the expected higher-order method. Theoretically, the stability analysis of the second-order fully discrete method is proved. The numerical experiments are given to verify the effectiveness of the presented methods. [ABSTRACT FROM AUTHOR]

The current work examines three (1 + 1)-dimensional Mikhailov–Novikov–Wang (MNW) equations. The Painlevé criteria are employed for testing the integrability of the evolution equations. Using the simplified Hirota's approach, multiple soliton solutions for the family of the MNW equation are derived. Significant physical solutions, such as shock waves, periodic solutions, and many others, are also obtained for each equation under consideration. The current investigation provides insights into the integrability features of these evolution equations. The obtained outcomes will contribute to comprehending and studying many enigmatic phenomena that consistently manifest in nature and various nonlinear media, including optical fiber, fluid mechanics, and plasma physics. [ABSTRACT FROM AUTHOR]

The evaluation and consideration of the mean flow in wave evolution equations are necessary for the accurate prediction of fluid particle trajectories under wave groups, with relevant implications in several domains, from the transport of pollutants in the ocean to the estimation of energy and momentum exchanges between the waves at small scales and the ocean circulation at large scale. We derive an expression of the mean flow at a finite water depth, which, in contrast to other approximations in the literature, accurately accords with the deep-water limit at third order in steepness and is equivalent to second-order formulations in intermediate water. We also provide envelope evolution equations at fourth order in steepness for the propagation of unidirectional wave groups either in time or space that include the respective mean flow term. The latter, in particular, is required for accurately modeling experiments in water wave flumes in arbitrary depths. [ABSTRACT FROM AUTHOR]

Properties of dust-ion acoustic waves (DIAWs) in an unmagnetized dusty plasma consisting of mobile ions, superthermal electrons, and negatively charged dust clouds with charge fluctuation are investigated. Burgers' equation is derived from the fluid equation of the plasmas by considering the reductive perturbation technique. The Lax pair of the evolution equation is obtained, and the characteristics of the DIAWs are investigated by employing the Darboux transformation. Some new waves (one-soliton, two-soliton-shocks, etc.) other than usual shocks are observed as the Darboux transformation is used through the Lax pair. It is also shown that all the physical plasma parameters have a significant effect on the structure of the nonlinear waves. [ABSTRACT FROM AUTHOR]

The aim of this paper is on the propagations of barotropic–baroclinic coherent structures based on the two-layer quasi-geostrophic model (2LQG) through a Fourier spectrum compliant approach. First, by introducing the barotropic and baroclinic stream functions starting from the 2LQG model, a new coupled Gardner-type evolution equations, representing the interaction processes between the barotropic flow and baroclinic one, are obtained by combining the multi-scale method and the perturbation expansion method. Second, based on the obtained coupled model equations, the physical mechanisms of the nonlinear barotropic–baroclinic interaction are analyzed qualitatively. Within the range of parameters chosen in this paper, quantitative results show that the basic flow, the β effect, and the bottom topography are necessary factors to excite the nonlinear Rossby isolated waves. The results also declare that the dipole-like blockings are readily excited in the flow field and move slowly eastward in both barotropic and baroclinic flow fields. [ABSTRACT FROM AUTHOR]

The impact of thixotropy on the settling behavior of a solid sphere is investigated utilizing a finite element-computational fluid dynamics simulation. Flow behavior is evaluated by coupling the Navier–Stokes equations with the dynamic evolution of an initially heterogeneous fluid's microstructure. Studying the structure dynamics around the settling sphere allows us to identify a variety of irregular and linear settling regimes. Settling regimes are varied by the degree of structuring, the degree of associated heterogeneity, the local morphology of the heterogeneous microstructure, and the stress induced by the sphere. In addition, the settling velocity profile of the relatively light spheres temporarily fluctuates in a case where the settling time of the sphere is long enough to capture the local heterogeneity. Ultimately, we compare the results of the simulation of dropping spheres with those of the numerical simulation of different rheological tests. This illustrates that the competition between kernels of orthokinetic and perikinetic build-up and shear-induced break-down of the microstructure indeed allows an understanding of the connection between the fluids' flow curve and the settling behaviors. Furthermore, settling regimes are characterized based on the rates of build-up and break-down of the microstructure. Moreover, the loss of fore-aft symmetry is observed in the flow field around the settling sphere as a result of a viscosity gradient behind and ahead of the sphere. [ABSTRACT FROM AUTHOR]

An energy approach to the problem of theoretical description of axisymmetric and quasi-one-dimensional oil product spills is presented. Within the framework of these models, approximate equations describing the evolution of the size of oil spots over time were derived. Solution of these equations describing the characteristics of spills of both a limited area (for machine oil stains) and unlimited spills of crude oil is obtained. A number of laboratory experiments have been carried out to study the dynamics of spreading of a spot of reference engine oil and crude oil for axisymmetric and quasi-one-dimensional flows. The presented theoretical and experimental results are in good agreement for both spreading regimes. [ABSTRACT FROM AUTHOR]

The purpose of this study is to examine the properties of the dust-acoustic solitary waves in a complex magnetoplasma made up of negatively charged moving dust grains in the lower ionospheric region and inertialess electrons and ions obeying Maxwell and kappa distributions, respectively. In this context, the reductive perturbation technique is carried out to obtain the Zakharov–Kuznetsov (ZK) equation within the given framework. The obtained evolution equation, i.e., ZK equation is transformed to a planar dynamical system for studying the qualitative behavior of the solitary waves. The impact of important physical parameters, such as the dust number density, ion concentration, ion temperature, superthermality, and the background magnetic field, on the profile of the nonlinear structures is numerically investigated. The findings may be utilized to comprehend the low-frequency irregularities that are detected in the lower ionosphere. [ABSTRACT FROM AUTHOR]

The age of the nearest particle pair is introduced as the difference between the current time and the most recent time when the nearest particle pair was formed. The evolution equation for the age-included nearest pair distribution function is derived. With the assumption of random destruction of the nearest particle pairs, the evolution equation predicts the exponential probability distribution of the ages of the nearest particle pairs. Particle-resolved numerical simulations with moving particles are performed to verify this prediction. The equation is then used to derive the evolution equation for the particle–fluid–particle (PFP) stress, which is known to be related to hyperbolicity of the two-fluid equations. It is found that the relaxation time of the age probability distribution is also the relaxation time for the PFP stress. Guided by the closure terms in the PFP stress evolution equation, we study kinematics of the nearest particle pairs in the particle-resolved simulations for flows caused by sedimentation of the particles with initially isotropic and homogeneous particle distributions. At the steady states, the particle Reynolds numbers are around 20. Anisotropy and inhomogeneity of particle distributions are seen to develop in these flows. The mean distances to the nearest particles and evolution of the distribution of the Voronoi cell volumes are studied. We also found the PFP stress is closely related to the changes in these inter-particle scale quantities. [ABSTRACT FROM AUTHOR]

We have studied the effect of randomness on the stability of interfacial gravity waves on deep water in the presence of uniform wind flow. The fourth-order nonlinear evolution equations for two Stokes wavetrains are used here to obtain the wave-kinetic equations for narrow-band of gravity wavetrains. Employing these kinetic equations we have then made the stability analysis of two initially homogeneous Lorentz form of wave spectra subject to infinitesimal perturbations. The effect of randomness is observed to reduce the growth rate and the extent of the instability region. The key result of the present analysis is that the fourth-order terms in the evolution equations significantly modify the modulational instability properties and produce a decrease on the growth rate. It is found that the growth rate of instability increases due to the effect of wind velocity. [ABSTRACT FROM AUTHOR]

In this paper, an optimized discrete unified gas kinetic scheme (DUGKS) is presented for both continuum and rarefied flows. The present scheme can be considered as a new version of the DUGKS. At first, we follow the original DUGKS to obtain the evolution equation by finite volume method. Then, we propose a new method to evaluate the flux. Different from the original DUGKS, the flux is evaluated by the distribution function at the node instead of the interface center. This makes the present scheme easier to implement and more efficient than the original DUGKS. To validate the present scheme, several numerical tests are performed, including the doubly periodic shear layers, the canonical two-dimension, and three-dimension Taylor–Green vortex flows, as well as the pressure-driven Couette flow and micro-Couette flow. Numerical results demonstrate that the present scheme preserves almost the same accuracy as compared with the original DUGKS, while it exhibits a lower numerical dissipation, and the computational efficiency and numerical stability can be significantly improved. [ABSTRACT FROM AUTHOR]

A temperature evolution equation for compressible ferrofluids is derived using basic laws of thermodynamics of moving electromagnetic media. Along with the compressibility, the change in kinetic energy is also included in the laws of thermodynamics to make the equation suitable for studying convective instabilities and fully developed turbulence in compressible ferrofluids as is shown by an order of magnitude analysis. The derived equation is found to be consistent with the total energy conservation when the forcing and dissipative effects are neglected. One can indeed recover the previously derived temperature evolution equations under suitable limits. In the hydrodynamic limit, the equation is reduced to the temperature evolution of a neutral compressible fluid and can be potentially important for studying astrophysical turbulence. [ABSTRACT FROM AUTHOR]

In this paper, the log-conformation representation method (LCR) is applied in an orthogonal curvilinear coordinate system to study the Giesekus fluid flow in a curved duct. Derivations for evolution equations of LCR in this curvilinear coordinate system are presented. Secondary flow patterns and oscillation solutions are computed by using the collocation spectral method. The influence of a wide range of Dean number, Weissenberg number, and dimensionless mobility parameter α on fluid behaviors is studied. A six-cell secondary flow pattern is found under very low Dean number and relatively high Weissenberg number and α. Moreover, both Weissenberg number and α are able to facilitate the development of the secondary flow. In addition, simulations under critical Reynolds number for oscillation imply that Giesekus fluid flow with W e ≥ 0.1 is not able to retain a four-cell secondary flow pattern in a steady state, which is different from Newtonian fluids. [ABSTRACT FROM AUTHOR]

Proper characterization of the transient interaction between shock front motion and chemical energy release is the key for further understanding of unsteady detonation dynamics. We have shown that the gradients could be the adequate intermediate quantities to reflect the transient interaction between shock front motion and chemical energy release through the Lagrangian particle analysis of the simulated direct detonation initiation (DDI) process in H2–O2–Ar mixtures. Specifically, the "shock change equations" are verified to describe the direct relation between the shock front motion and the gradients immediately behind the shock. Moreover, given the time derivatives of shock speed, the gradient evolution in the induction zone can be reproduced by the gradient evolution equations that are deduced from the Euler equations, no matter if the shock front undergoes rapid deceleration or acceleration. While in the reaction zone where the heat release is significant, it is demonstrated that the evolutions of velocity, pressure, and their gradients can be described by the Zel'dovich–von Neumann–Döring model with a constant wave speed that is below the Chapman–Jouguet speed, no matter if the shock front undergoes rapid deceleration or acceleration. These two distinct controlling mechanisms are verified in both planar and spherical DDI processes, showing their general applicability. This work suggests a new perspective, in terms of gradient evolution, for further understanding the unsteady detonation dynamics. [ABSTRACT FROM AUTHOR]

We compare initial value and eigenvalue problems for two-dimensional perturbations of the inviscid shear flow in a channel. Singular solutions, known in plasma physics as van Kampen (vK) modes, are constructed. They form a complete set of eigenfunctions for decomposition of any initial perturbation for stable wavy perturbations. A pair of discrete modes appears to ensure completeness in the unstable case. Expansion coefficients for eigenmodes are found, and equivalence of temporal evolution obtained with the help of the evolutionary equation for vorticity and expansion over eigenmodes is presented. This alternative description of the evolution using vK modes is analogous to ones found earlier in plasma and in stellar dynamics. In particular, for stable wavy perturbations, an initial state decays first exponentially due to Landau damping, then algebraically. It has been established (numerically and analytically) that the final decay law is t − 1 . Also, we numerically demonstrate that Landau-damped perturbations are not true eigenmodes, but rather a superposition of vK-modes with a real frequency, which does not retain its shape over time. However, solution on contours in the complex plane may exhibit properties of a true eigenmode, that is, decay without changing its spatial form. Energy redistribution between perturbation and the flow, in stable and unstable regimes, is analyzed. [ABSTRACT FROM AUTHOR]

Since its emergence, the pseudopotential lattice Boltzmann (LB) method has been regarded as a straightforward and practical approach for simulating single-component multiphase flows. However, its original form always results in a thermodynamic inconsistency, which, thus, impedes its further application. Several strategies for modifying the force term have been proposed to eliminate this limitation. In this study, four typical and widely used improved schemes—Li's single-relaxation-time (SRT) scheme [Li et al., "Forcing scheme in pseudopotential lattice Boltzmann model for multiphase flows," Phys. Rev. E 86, 016709 (2012)] and multiple-relaxation-times (MRT) scheme [Li et al., "Lattice Boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model," Phys. Rev. E 87, 053301 (2013)], Kupershtokh's SRT scheme [Kupershtokh et al., "On equations of state in a lattice Boltzmann method," Comput. Math. Appl. 58, 965 (2009)], and Huang's MRT scheme [Huang and Wu, "Third-order analysis of pseudopotential lattice Boltzmann model for multiphase flow," J. Comput. Phys. 327, 121 (2016)]—are systematically analyzed and intuitively compared after an extension of the MRT framework. The theoretical and numerical results both indicate that the former three schemes are specific forms of the last one, which thus help further understand the improvements of these pseudopotential LB models for achieving thermodynamic consistency. In addition, we modified the calculation of the additional source term in the LB evolution equation. Numerical results for stationary and moving droplets confirm the higher accuracy. [ABSTRACT FROM AUTHOR]

The complex flow modeling based on machine learning is becoming a promising way to describe multiphase fluid systems. This work demonstrates how a physics-informed neural network promotes the combination of traditional governing equations and advanced interface evolution equations without intricate algorithms. We develop physics-informed neural networks for the phase-field method (PF-PINNs) in two-dimensional immiscible incompressible two-phase flow. The Cahn–Hillard equation and Navier–Stokes equations are encoded directly into the residuals of a fully connected neural network. Compared with the traditional interface-capturing method, the phase-field model has a firm physical basis because it is based on the Ginzburg–Landau theory and conserves mass and energy. It also performs well in two-phase flow at the large density ratio. However, the high-order differential nonlinear term of the Cahn–Hilliard equation poses a great challenge for obtaining numerical solutions. Thus, in this work, we adopt neural networks to tackle the challenge by solving high-order derivate terms and capture the interface adaptively. To enhance the accuracy and efficiency of PF-PINNs, we use the time-marching strategy and the forced constraint of the density and viscosity. The PF-PINNs are tested by two cases for presenting the interface-capturing ability of PINNs and evaluating the accuracy of PF-PINNs at the large density ratio (up to 1000). The shape of the interface in both cases coincides well with the reference results, and the dynamic behavior of the second case is precisely captured. We also quantify the variations in the center of mass and increasing velocity over time for validation purposes. The results show that PF-PINNs exploit the automatic differentiation without sacrificing the high accuracy of the phase-field method. [ABSTRACT FROM AUTHOR]

We study the fingering instability in a droplet simultaneously induced to spread by a surfactant and temperature. The use of the lubrication approximation yields coupled evolution equations for the film thickness, surfactant concentration, and temperature. A direct numerical simulation is performed, and a stability analysis based on the disturbance energy is conducted. Four cases are considered for the substrate temperature field: a nonheated substrate, an isothermally heated substrate, a nonisothermally heated substrate, and a thick substrate. The results show that fluids always tend to "flee" from hotter areas and surfactant-enriched areas, and that the flow stability is greatly influenced by this effect. The uneven distribution of the velocity field caused by surface tension is the fundamental reason for the formation of fingerlike patterns. The contributions of the capillary effect, the solutocapillary effect, and the thermocapillary effect as driving forces are quantified in terms of their locations and relative strength during spreading. The solutocapillary and thermocapillary effects exert a destabilizing effect on the spreading. On a nonisothermally heated substrate, a stronger thermocapillary effect strengthens the unevenness of the surfactant, leading to the most unstable flow. Finally, a variable viscosity model is considered and the flow stability is examined. The results show that on a nonisothermally heated substrate, the unevenness of the surfactant and temperature distribution is strengthened due to better fluidity in hotter areas, leading to a more unstable flow. On an isothermally heated substrate, the overall liquidity increases the spreading velocity but does not affect the stability. [ABSTRACT FROM AUTHOR]

We investigate the linear evolution of the Richtmyer–Meshkov instability (RMI) in the framework of an ideal two-fluid plasma model. The two-fluid plasma equations of motion are separated into a base state and a set of linearized equations governing the evolution of the perturbations. Different coupling regimes between the charged species are distinguished based on a non-dimensional Debye length parameter d D , 0 . When d D , 0 is large, the coupling between ions and electrons is sufficiently small that the induced Lorentz force is very weak and the two species evolve as two separate fluids. When d D , 0 is small, the coupling is strong and the induced Lorentz force is strong enough that the difference between state of ions and electrons is rapidly decreased by the force. As a consequence, the ions and electrons are tightly coupled and evolve like one fluid. The temporal dynamics is divided into two phases: an early phase wherein electron precursor waves are prevalent and a post-ion shock-interface interaction phase wherein the RMI manifests itself. We also examine the effect of an initially applied magnetic field in the streamwise direction characterized by the non-dimensional parameter β0. For a short duration after the ion shock-interface interaction, the growth rate is similar for different initial magnetic field strengths. Time progresses the suppression of the instability because the magnetic field is observed. The growth rate shows oscillations with a frequency that is related to the ion or electron cyclotron frequency. The instability is suppressed due to the oscillation of vorticity on the interface caused by the perturbed Lorentz force. [ABSTRACT FROM AUTHOR]

The effects of the Coriolis force in inhomogeneous rotating turbulence are studied in the paper. Linear analyses and numerical simulations both reveal that energy is transported to the slowly rotating fields, and the energy distribution is proportional to Ω 3 − 2 (x 3). The scale energy is almost spatially self-similar, and the inverse cascade is reduced by inhomogeneous rotation. The corresponding evolution equation of the scale energy, i.e., the generalized Kolmogorov equation, is calculated to study the scale transport process in the presence of inhomogeneity. The equation is reduced to twice the energy transport equation at sufficiently large scales, which is verified by numerical results. In addition, the results reveal the dominant role of the corresponding pressure of the Coriolis force in the spatial energy transport. An extra turbulent convention effect in r-space solely in slowly rotating fields is also recognized. It can be associated with the small-scale structures with strong negative vorticity, whose formation mechanism is similar to rotating condensates. Finally, by vortex dynamic analyses, we find that the corresponding pressure of the Coriolis force transports energy by vorticity tube shrinking and thickening. The effects of the Coriolis force can be divided into two components: one is related to the gradient of rotation, and the other is associated with the strength of rotation. [ABSTRACT FROM AUTHOR]

In this work, a nonlinear phenomenological model for neutrally buoyant force-free active suspension of nematodes is proposed and tested. Just a few limited studies were found linked with nonlinear viscoelastic response of the active suspension investigated in this paper. The stress is decomposed through Fourier transform into elastic and viscous stress contributions. The stress response at large strain deviates drastically from the harmonic forcing in a nonlinear regime. In this case, the standard linear viscoelastic moduli cannot describe the nonlinear response of the fluid. Lissajous–Bowditch loops are used as rheological fingerprints to examine the behavior of nonlinear response of the investigated active fluid. The results show time-strain separable nonlinearity, therefore providing a new physically meaningful interpretation. When self-propelled particles interact with each other (i.e., a collective effect), they produce stresses that result in dynamic self-organization at spatial and temporal scales much larger than those of single particles. Complex rheological behavior in active matter depends on the interplay between the external forcing and the circulating flow induced by active agents. The active matter examined in this work is based on the nematode Caenorhabditis elegans motion, whose shape is defined by a dynamic balance between elastic, hydrodynamic, and muscular forces. The orientational instabilities of the active suspension of C. elegans observed in recent experiments carried out by the authors are considered in the present theoretical study. A new time evolution equation for the active stress tensor is proposed in terms of an Oldroyd–Maxwell upper convected material derivative for a dilute active suspension in the absence of thermal or active fluctuations. On the other hand, the Gordon–Schowalter material derivative is used in order to modify the model for the case of non-diluted suspensions. The constitutive equations are nondimensional, and the results are addressed on both linear (small amplitude oscillatory shear) and nonlinear (large amplitude oscillatory shear) regimes. We show results of the viscoelastic moduli as a function of strain in the linear region and in the nonlinear region. The associated Lissajous loop curves illustrating the nonlinear response and the transitions of elastic to viscous behavior of the material at high strain are also presented. The dissipated energy over oscillation cycle is associated with the area enclosed by the closed Lissajous loops curves. Lissajous–Bowditch loops are also computed for the first normal stress differences using our theoretical model, and the results are compared with experimental work that was previously published by the authors. [ABSTRACT FROM AUTHOR]

In line with recent ongoing efforts to collect crucial information about the mechanisms of virus diffusion and put them in relation to the effective complexity of the several natural or artificial environments where human beings leave and operate, the present study deals with the dispersion of evaporating saliva droplets in the cabin of an interregional train. A relevant physical model is constructed taking into account the state of the art in terms of existing paradigms and their ability to represent some fundamental aspects related to the evolution in time of a cloud of multi-disperse droplets. Conveniently, such a theoretical framework is turned into a computational one that relies on low Mach-number asymptotics and can therefore take advantage of the typical benefits (relatively low computational cost) associated with pressure-based methods. Numerical simulations are used to predict the flow established in the cabin as a result of the ventilation systems and related settings dictated by considerations on passenger comfort. The solution of two-way coupled Lagrangian evolution equations is used to capture the associated dynamics of the dispersed phase and predict its transport in conjunction with the peculiar topology of the considered flow and morphology of solid surfaces, which bound it (including the human beings). Typical physiological processes such as talking or coughing are considered. An analysis on the impact of the multiplicity of droplet sources is also conducted, thereby providing some indications in terms of potential risks for the cabin occupants. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid lattice-Boltzmann finite-difference method is developed for the simulation of ternary fluids near immersed solid objects of general shapes. The flow equations are solved by the lattice-Boltzmann method and the coupled Cahn–Hilliard equations for interface evolutions are solved by the finite-difference method. A special implementation of the wetting boundary condition on a surface of general shapes immersed inside the domain was extended for ternary fluids within the phase-field framework with no need to use complicated interpolations. Several two and three dimensional problems with three immiscible fluids were studied by using the proposed method and the results agree well with analytical predictions and/or previous numerical and experimental studies. In particular, the inclusion of properly chosen free energy to handle total spreading enabled us to numerically reproduce the encapsulation of a small droplet by another bigger one of different component on a round fiber. The proposed method is expected to be useful to investigate a variety of multiphase problems involving ternary fluids and surfaces with different configurations, including the challenging total spreading regime. [ABSTRACT FROM AUTHOR]

We study a granular gas of viscoelastic particles (kinetic energy loss upon collision is a function of the particles' relative velocities at impact) subject to a stochastic thermostat. We show that the system displays anomalous cooling and heating rates during thermal relaxation processes, this causing the emergence of thermal memory. In particular, a significant Mpemba effect is present, i.e., an initially hotter/cooler granular gas can cool down/heat up faster than an in comparison cooler/hotter granular gas. Moreover, a Kovacs effect is also observed, i.e., a nonmonotonic relaxation of the granular temperature—if the gas undergoes certain sudden temperature changes before fixing its value. Our results show that both memory effects have distinct features, very different and eventually opposed to those reported in theory for granular fluids under simpler collisional models. We study our system via three independent methods: approximate solution of the kinetic equation time evolution and computer simulations (both molecular dynamics simulations and direct simulation Monte Carlo method), finding good agreement between them. [ABSTRACT FROM AUTHOR]

In this paper, a lattice Boltzmann model for the coupled Allen–Cahn–Navier–Stokes equations in three dimensions is presented. Two equations are solved: one for the fluid velocity and one for the order parameter. Both are written within the general multiple-relaxation-time framework, where all the equilibrium and forcing terms are described by using the full set of Hermite polynomials. The resultant practical implementation is compact. The gradient of the order parameter can be computed by the non-local finite differences or the local central moments. The latter suffers from grid-scale oscillations. The very good accuracy properties are demonstrated against nine well-consolidated benchmark tests. Specifically, two groups of tests are tackled. In the former, the velocity field is superimposed. Hence, only the equation for the evolution of the order parameter is solved. These numerical experiments demonstrate the ability of the proposed scheme to capture the correct evolution of the interface. In the latter, two immiscible fluids are considered and the two equations are solved. Simulations of the vertical penetration of a wedge-shaped body, two- and three-dimensional Rayleigh–Taylor instability prove that two-fluids systems can be successfully simulated by our approach. [ABSTRACT FROM AUTHOR]

In Ref. [1], there was an error in the second integral of Eq. (14) due to an erroneous application of Eq. (A3) in Appendix A of that paper. Correction of the above error requires consideration of a more general expression for the gradient of the chemical potential HT ht (obtained from a general equilibrium thermodynamics reference, such as Refs. Note that only when this error is corrected, the (correct) equation for the time evolution for the entropy density, Eq. (29) of Ref. [1], can be derived. [Extracted from the article]

In this paper, we explore the instability of large-scale riverbed patterns, performing a linear stability analysis of a zero-pressure gradient free-surface flow in a wide straight channel with an erodible bed. The local depth-averaged turbulence state is governed by two key parameters: namely, the turbulent kinetic energy (TKE) and its dissipation rate. A depth-averaged flow model coupled with the transport equations of the TKE and its dissipation rate and the particle transport model are developed to examine the formation of large-scale patterns. Both the modes of particle transport as bedload and suspended load are considered herein, allowing for the extension of the conventional theories to cover from hydraulically smooth to transitional flow regimes. The classical Exner equation of the bed evolution is modified in the presence of suspended particles, whose concentration is coupled with the steady-state advection–diffusion equation. Applying a standard linearization technique, the periodic perturbations in both streamwise and spanwise directions are imposed on the bed to find the dispersion relationship. The stability maps for the growth rate of large-scale patterns are obtained as a function of streamwise and spanwise wavenumbers and of key parameters associated with the flow and particles. [ABSTRACT FROM AUTHOR]

In this work, we use molecular dynamics (MD) simulations coupled with continuum-based theoretical analysis to study the coalescence dynamics of two equal-sized nanobubbles (NBs). We first derive a governing equation for the evolution of the capillary bridge radius between two coalescing NBs from the axisymmetric Navier–Stokes equation. To verify the prediction from the governing equation, we carry out MD simulations of the coalescence of two NBs in a Lennard-Jones fluid system and directly measure the bridge radius, rb, as a function of time, t. By varying the bubble diameter, we change the NB Ohnesorge number from 0.46 to 0.33. In all cases, we find the theoretical prediction overestimates the expansion speed of the capillary bridge at early time of NB coalescence. However, once we take into account the curvature-dependent surface tension and restrict the minimum principal radius at the capillary bridge to the size of the atom in the model liquid, the theoretical prediction agrees with the MD data very well in both early time and later time of the coalescence process. From the theoretical model, we find neither liquid viscous force nor liquid inertial force dominates at later time of coalescence of the model NBs. In this case, the MD simulation results show rb(t) ∝ t0.76 ± 0.04 with the scaling exponent considerably higher than that in the scaling law rb(t) ∝ t0.5 for the viscous and inertial dominated regimes. The diameter ratio of fully merged NB to that of the original NB is about 2 , which is different from 2 3 for the coalescence of millibubbles and microbubbles. [ABSTRACT FROM AUTHOR]

In this work, a system of non-linear partial differential equations, which describes one-dimensional motion of an inviscid, self-gravitating, and spherically symmetric van der Waals gas cloud, is considered. By using the method based on the kinematics of shock waves, the evolution equation for spherical shock wave in an interstellar van der Waals gas cloud is derived. By applying the truncation approximation procedure, an infinite system of transport equations, which governs the shock propagation, is derived to study the kinematics of shock waves for the one-dimensional motion. The first, second, and third order transport equations, which describe the shock strength and the induced discontinuity behind it, are used to analyze the decay and growth behavior of the shock waves in a non-ideal gas. The results are obtained for the exponent obtained from the first, second, and third order approximations and compared with the results obtained by Whitham's characteristic rule (Chester–Chisnell–Whitham approximation). In addition, the effects of the parameters of non-idealness and cooling–heating function on the evolutionary behavior of shocks are discussed and shown graphically. [ABSTRACT FROM AUTHOR]

This study describes the evolutionary mechanisms of nonlinear barotropic–baroclinic interactions, especially, on the excitations, propagations, and decreases of nonlinear coherent structures. Starting from the classical two-layer quasi-geostrophic potential vorticity conservation model equations, the barotropic and baroclinic model equations are derived from the classical work of Pedlosky and Thomson [J. Fluid Mech. 490, 189–215 (2003)]. By considering the effects of bottom topography and beta-plane approximation, the coupled nonlinear Korteweg–de Vries model equations for the evolutions of barotropic and baroclinic coherent structures are obtained by using the methods of multiple scales and perturbation expansions, respectively. Solitary wave solutions are given according to the method of elliptic function expansions, and the physical mechanisms for the evolutions of the nonlinear barotropic–baroclinic interactive coherent structures are analyzed based on the obtained solitary wave solutions. It will be potentially useful for further theoretical investigations on atmospheric blocking phenomena or wave–flow interactions. [ABSTRACT FROM AUTHOR]

Stability analysis is performed for a gravity-driven thin liquid film flowing down a locally heated porous substrate. Using the lubrication approximation, the governing equations are simplified to derive the evolution equation for the free surface of the liquid film. The Beaver-Joseph condition is employed at the interface of the porous layer and the liquid film. The base profiles are mainly influenced by parameters that appear due to non-uniform heating. Linear stability analysis is performed and reported that both thermocapillary and rivulet instabilities are enhanced with increasing values of the Marangoni number, Biot number, and Beavers–Joseph coefficient and decreasing values of the Darcy number. Dependence of critical Darcy number on the porous layer thickness and the Beavers–Joseph coefficient is presented. It is also shown that the full Darcy model can be replaced with an approximated slip model. The growth rate from nonlinear computations is consistent with the linear stability analysis. [ABSTRACT FROM AUTHOR]

A plane wake or jet supports sinuous and varicose instability modes. The nonlinear interaction between them following their linear development was described previously by Leib and Goldstein ["Nonlinear interaction between the sinuous and varicose instability modes in a plane wake," Phys. Fluids A 1, 513–521 (1989)] using the strongly nonlinear non-equilibrium critical-layer approach in the case of the Bickley jet for which the frequencies of the sinuous and varicose modes have an integer ratio of 2. This paper develops the theory for general profiles where the frequencies of the sinuous and varicose modes are non-commensurable. The disturbance is quasi-periodic in time and space and must be expressed as a function of two phase variables. Using matched asymptotic expansions simultaneously with the multi-scale method, we derived a set of coupled evolution equations governing the development of the amplitudes and critical-layer vorticities of these modes. The evolution system is solved for the base-flow profiles mimicking those in experiments. The sinuous mode suppresses the varicose mode but also causes the latter to saturate in a highly oscillatory manner. The varicose mode inhibits the sinuous mode initially. However, in the later stage, it lends the sinuous mode a significantly higher saturating amplitude. For a wide range of initial modal compositions and Reynolds numbers, the ratio of the varicose mode amplitude to that of the sinuous mode eventually tends to an almost constant value in the range of 0.4–0.6, in line with the experimental measurement. Due to the self and mutual interactions, the vorticities roll up to form vortices, which are non-symmetric in the transverse direction and quasi-periodic in the streamwise direction as well as in time. With such an increased complexity, the vortices resemble those observed in experiments. The nonlinear interactions of the sinuous and varicose modes in the critical layer generate all harmonics in the main layer, as a result of which the perturbation is non-periodic and may even appear "random-like." [ABSTRACT FROM AUTHOR]

The flow of viscous fluid injected from a point source into the space between two horizontal plates initially filled with a second fluid of lesser density and different viscosity is studied theoretically and numerically. The volume of the dense input fluid increases with time in proportion to tα. When the fluid has spread far from the source, lubrication theory is used to derive the governing equations for the axisymmetric evolution of the interface between the fluids. The flow is driven by the combination of pressure gradients associated with buoyancy and pressure gradients associated with the input flux. The governing equation is integrated numerically, and we identify that with a constant input flux, the flow is self-similar at all times with the radius growing in proportion to t1/2. In the regimes of injection-dominated and gravity-dominated currents, we obtain asymptotic approximations for the interface shape, which are found to agree well with the numerical computations. For a decreasing input flux (0 < α < 1), at short times, the flow is controlled by injection; the current fills the depth of the channel spreading with radius r ∼ tα/2. At long times, buoyancy dominates and the current becomes unconfined with the radius growing in proportion to t(3α+1)/8. The sequence of regimes is reversed in the case of an increasing input flux (α > 1) with buoyancy dominating initially while the pressure associated with the injection dominates at late times. Finally, we consider the release of a fixed volume of fluid (α = 0). The current slumps under gravity and transitions from confined to unconfined, and we obtain asymptotic predictions for the interface shape in both regimes. [ABSTRACT FROM AUTHOR]

We revisit and derive the shock change equations relating the dynamics of a shock wave with the partial derivatives describing the motion of a reactive fluid with the general equation of state in a stream-tube with arbitrary area variation. We specialize these to a perfect gas in which we obtain all shock change equations in closed form. These are further simplified for strong shocks. We discuss the general usefulness of these equations in problems of reactive compressible flow and in the development of intrinsic evolution equations for the shock, such as the approximations made by Whitham and Sharma. [ABSTRACT FROM AUTHOR]

The nonlinear dynamics of the interface between both perfect and leaky dielectric liquid films, interposed between two parallel electrodes, are investigated under the effect of mixed-frequency electric fields. A coupled system of evolution equations is derived in dimensionless form, by employing the long-wave approximation. The linear stability analysis is implemented in accordance with the characteristics of each specific case, namely, the constant (DC) and the altering (AC) fields. In particular, the response of the system to the multi-mode AC electrical field is analyzed. Assisted by the conclusions of the theoretical investigation, the initial-boundary-value problem associated with the coupled system of evolution equations is solved numerically for several parameter sets. The system behavior is studied by monitoring the evolution process and by examining the steady/quasi-steady pillar formations in the nonlinear regime. The possibility to generate interface profiles of diverse topological forms, to manipulate their features, and to control the time dependent progress and the film rupture by imposing different combinations of frequencies and/or amplitudes of the corresponding mode is confirmed. [ABSTRACT FROM AUTHOR]

A filament of liquid is usually unstable and breaks up into small droplets, while a filament of polymer solution is known to be quite stable against such instability, and they form a stable configuration of a filament connecting two spherical droplets. If the droplets are fixed in space, the liquid flows from the filament region to the droplet region to reduce the surface energy and the filament gets thinner. If the whole liquid is placed in another viscous fluid, the droplets approach each other and the filament can get thicker. Here, we study the dynamics of such a system. We derive time evolution equations for the radius and the length of the filament taking into account the fluid flux from the filament to the droplets and the motion of the droplets. We will show that (a) if the centers of the droplets are fixed, the filament thins following the classical prediction of Entov and Hinch and (b) if the droplets are mobile (subject to the Stokes drag in the viscous medium), the thinning of the filament is suppressed and, under certain conditions, the filament thickens. This theory explains the phenomena observed by Yang and Xu ["Coalescence of two viscoelastic droplets connected by a string," Phys. Fluids 20, 043101 (2008)] in a four-roller mill device. [ABSTRACT FROM AUTHOR]

Performing accurate large eddy simulations in compressible, turbulent magnetohydrodynamics (MHDs) is more challenging than in nonmagnetized fluids due to the complex interplay between kinetic, magnetic, and internal energy at different scales. Here, we extend the subgrid-scale gradient model, so far used in the momentum and induction equations, to also account for the unresolved scales in the energy evolution equation of a compressible ideal MHD fluid with a generic equation of state. We assess the model by considering box simulations of the turbulence triggered across a shear layer by the Kelvin-Helmholtz instability, testing cases where the small-scale dynamics cannot be fully captured by the resolution considered, such that the efficiency of the simulated dynamo effect depends on the resolution employed. This lack of numerical convergence is actually a currently common issue in several astrophysical problems, where the integral and fastest-growing-instability scales are too far apart to be fully covered numerically. We perform a priori and a posteriori tests of the extended gradient model. In the former, we find that, for many different initial conditions and resolutions, the gradient model outperforms other commonly used models in terms of correlation with the residuals coming from the filtering of a high-resolution run. In the second test, we show how a low-resolution run with the gradient model is able to quantitatively reproduce the evolution of the magnetic energy (the integrated value and the spectral distribution) coming from higher-resolution runs. This extension is the first step toward the implementation in relativistic MHDs. [ABSTRACT FROM AUTHOR]

Mathematical derivation and numerical verification of a wave transformation model in the frequency domain are discussed. The model is a fully dispersive nonlinear wave model and is derived based on the boundary value problem. Transforming the problem in the frequency domain and using multiple scale analysis in space and perturbation theory, the model is expanded up to second order in wave steepness. This fully dispersive nonlinear wave model is a set of evolution equations which explicitly contains quadratic near-resonant interactions. The comparison between the presented model, the existing fully dispersive model, and a nearshore model with different sets of laboratory and field data shows that the presented model provides significant improvements particularly at higher frequency. [ABSTRACT FROM AUTHOR]

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow—an observation attributed to the thermostress convection effects at the microscale. The dynamics of the overall thermostress convection process is governed by the Boltzmann equation—an integrodifferential equation describing the evolution of the molecular distribution function in six-dimensional phase space—which models dilute gas behavior at the molecular level to accurately describe a wide range of flow phenomena. Approaches for solving the full Boltzmann equation with general intermolecular interactions rely on two perspectives: one stochastic in nature often delegated to the direct simulation Monte Carlo (DSMC) method and the others deterministic by virtue. Among the deterministic approaches, the discontinuous Galerkin fast spectral (DGFS) method has been recently introduced for solving the full Boltzmann equation with general collision kernels, including the variable hard/soft sphere models—necessary for simulating flows involving diffusive transport. In this work, the deterministic DGFS method, Bhatnagar-Gross-Krook (BGK), Ellipsoidal statistical BGK (ESBGK), and Shakhov kinetic models, and the widely used stochastic DSMC method, are utilized to assess the thermostress convection process in micro in-plane Knudsen radiometric actuator—a microscale compact low-power pressure sensor utilizing the Knudsen forces. The BGK model underpredicts the heat-flux, shear-stress, and flow speed; the S-model overpredicts; whereas, ESBGK comes close to the DSMC results. On the other hand, both the statistical/DSMC and deterministic/DGFS methods, segregated in perspectives, yet, yield inextricable results, bespeaking the ingenuity of Graeme Bird who laid down the foundation of practical rarefied gas dynamics for microsystems. [ABSTRACT FROM AUTHOR]

The effect of internal-kinetic energy exchange on transient spectral energy transfer in compressible turbulence is investigated. We derive the spectral evolution equations for kinetic energy and pressure fields to highlight the key mechanisms that affect the turbulence spectral evolution. Direct numerical simulations of decaying isotropic turbulence are performed from solenoidal, dilatational, and mixed velocity initial conditions. It is shown that internal-kinetic energy exchange arising due to pressure-dilatation renders the dilatational kinetic energy amplitudes at large scales of motion to be oscillatory. The oscillatory behavior of amplitude diminishes with increasing wavenumber. The dilatational energy spectrum also exhibits a wider range of scales due to its inherent tendency to form shocks. The findings are expected to lead to an improved understanding of energy dynamics in high-speed compressible flows. [ABSTRACT FROM AUTHOR]

Hysteresis is a strongly nonlinear physics phenomenon observed in many fluid mechanics flows. This paper composes evolution equations of the minimal nonlinearity and dimension which describe three hysteresis kinds related to a fold catastrophe formed by (i) two fold bifurcations, (ii) fold and transcritical bifurcations, and (iii) fold and subcritical bifurcations. [ABSTRACT FROM AUTHOR]

Rupture of thin liquid films is crucial in many industrial applications and nature such as foam stability in oil-gas separation units, coating flows, polymer processing, and tear films in the eye. In some of these situations, a liquid film may have two free surfaces (referred to here as a free film or a sheet) as opposed to a film deposited on a solid substrate that has one free surface. The rupture of such a free film or a sheet of a Newtonian fluid is analyzed under the competing influences of inertia, viscous stress, van derWaals pressure, and capillary pressure by solving a system of spatially one-dimensional evolution equations for film thickness and lateral velocity. The dynamics close to the space-time singularity where the film ruptures is asymptotically self-similar and, therefore, the problem is also analyzed by reducing the transient partial differential evolution equations to a corresponding set of ordinary differential equations in similarity space. For sheets with negligible inertia, it is shown that the dominant balance of forces involves solely viscous and van der Waals forces, with capillary force remaining negligible throughout the thinning process in a viscous regime. On the other hand, for a sheet of an inviscid fluid for which the effect of viscosity is negligible, it is shown that the dominant balance of forces is between inertial, capillary, and van der Waals forces as the film evolves towards rupture in an inertial regime. Real fluids, however, have finite viscosity. Hence, for real fluids, it is further shown that the viscous and the inertial regimes are only transitory and can only describe the initial thinning dynamics of highly viscous and slightly viscous sheets, respectively. Moreover, regardless of the fluid's viscosity, it is shown that for sheets that initially thin in either of these two regimes, their dynamics transition to a late stage or final inertial-viscous regime in which inertial, viscous, and van derWaals forces balance each other while capillary force remains negligible, in accordance with the results of Vaynblat, Lister, and Witelski. [ABSTRACT FROM AUTHOR]

A new variational method is proposed to calculate the evolution of liquid film and liquid droplet moving on a solid substrate. A simple time evolution equation is obtained for the contact angle of a liquid film that starts to move on a horizontal substrate. The equation indicates the dynamical transition at the receding side and the ridge formation at the advancing side. The same method is applied for the evolution of a droplet that starts to move on an inclined solid surface, and again the characteristic shape change of the droplet is obtained by solving a simple ordinary differential system. We will show that this method has a potential application to a wide class of problems of droplets moving on a substrate. [ABSTRACT FROM AUTHOR]