Limit Of Small Viscosity Research Articles

Asymptotic analysis for the homogeneous model of wind‐driven oceanic circulation in the small viscosity limit

In this paper, we study the asymptotic expansion for the homogeneous model of wind‐driven oceanic circulation with the nonslip boundary condition when both of the beta‐plane parameter and the Reynolds number go to infinity. By asymptotic analysis under certain constraints on wind tensor, we derive a formal expansion including the geophysical boundary layer in the large beta‐plane parameter and high Reynolds number limit, from which one sees that the external force, wind tensor, has an important effect on the behavior of the boundary layers. Moreover, when the external force and initial data satisfy certain constraints, to exclude the appearance of strong boundary layers, we construct approximate solutions with steady boundary layers to the homogeneous model of wind‐driven oceanic circulation in the large beta‐plane parameter and high Reynolds number limit. By using an energy method, we obtain the ‐stability of small perturbation around this approximate solutions, which verifies the validity of the asymptotic expansion of weak boundary layers in the large beta‐plane parameter and high Reynolds number limit.

Asymptotic Analysis of Steady Viscous Shocks in a 1-D Finite Nozzle in the Small Viscosity Limit

Asymptotic Analysis of Steady Viscous Shocks in a 1-D Finite Nozzle in the Small Viscosity Limit

Capillarity-driven Stokes flow: the one-phase problem as small viscosity limit

We consider the quasistationary Stokes flow that describes the motion of a two-dimensional fluid body under the influence of surface tension effects in an unbounded, infinite-bottom geometry. We reformulate the problem as a fully nonlinear parabolic evolution problem for the function that parameterizes the boundary of the fluid with the nonlinearities expressed in terms of singular integrals. We prove well-posedness of the problem in the subcritical Sobolev spaces Hs(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s(\\mathbb {R})$$\\end{document} up to critical regularity, and establish parabolic smoothing properties for the solutions. Moreover, we identify the problem as the singular limit of the two-phase quasistationary Stokes flow when the viscosity of one of the fluids vanishes.

Hydrodynamics of an odd active surfer in a chiral fluid

We theoretically and computationally study the low-Reynolds-number hydrodynamics of a linear active microswimmer surfing on a compressible thin fluid layer characterized by an odd viscosity. Since the underlying three-dimensional fluid is assumed to be very thin compared to any lateral size of the fluid layer, the model is effectively two-dimensional. In the limit of small odd viscosity compared to the even viscosities of the fluid layer, we obtain analytical expressions for the self-induced flow field, which includes non-reciprocal components due to the odd viscosity. On this basis, we fully analyze the behavior of a single linear swimmer, finding that it follows a circular path, the radius of which is, to leading order, inversely proportional to the magnitude of the odd viscosity. In addition, we show that a pair of swimmers exhibits a wealth of two-body dynamics that depends on the initial relative orientation angles as well as on the propulsion mechanism adopted by each swimmer. In particular, the pusher–pusher and pusher–puller-type swimmer pairs exhibit a generic spiral motion, while the puller–puller pair is found to either co-rotate in the steady state along a circular trajectory or exhibit a more complex chaotic behavior resulting from the interplay between hydrodynamic and steric interactions. Our theoretical predictions may pave the way toward a better understanding of active transport in active chiral fluids with odd viscosity, and may find potential applications in the quantitative microrheological characterization of odd-viscous fluids.

Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

Hamiltonian variational principles have provided, since the 1960s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton’s principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton’s principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton’s action, constrained by the conservation of mass and the conservation of fluid parcels’ identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis.

Interplay of viscosity and surface tension for ripple formation by laser melting

A model for ripple formation on liquid surfaces exposed to an external laser or particle beam and a variable ground is developed. The external incident beam is hereby mechanically coupled to the liquid surface due to surface roughness. Starting from the Navier Stokes equation the coupled equations for the velocity potential and the surface height are derived in a shallow-water approximation with special attention to viscosity. The resulting equations obey conservation laws for volume and momentum where characteristic potentials for gravitation and surface tension are identified analogously to conservative forces. The approximate solutions are discussed in the context of ripple formation in laser materials processing involving melting of a surface by a laser beam. Linear stability analysis provides the formation of a damped wave modified by an interplay between the external beam, the viscosity, and the surface tension. The limit of small viscosity leads to damped gravitational and the limit of high viscosity to capillary waves. The resulting wavelengths are in the order of the ripples occurring in laser welding experiments hinting to the involvement of hydrodynamic processes in their origin. By discussing the response of the system to external periodic excitations with the help of Floquet multipliers, we show that the ripple formation could be triggered by a a periodically modulated external beam, e.g. appropriate repetition rates of an incident laser beam. The weak nonlinear stability analysis provides ranges where hexagonal or stripe structures can appear. The orientation of stripe structures and ripples are shown to be dependent on the incident angle of the laser or particle beam where a minimal angle is reported. Numerical simulations confirm the findings and allow to describe the influence of variable grounds.

Onset of transient shear banding in viscoelastic shear start-up flows: Implications from linearized dynamics

We analyze transient dynamics during shear start-up in viscoelastic flows between two parallel plates, with a specific focus on the signatures for the onset of transient shear banding using the Johnson-Segalman, non-stretching Rolie-Poly and Giesekus models. We explore the dynamics of shear start-up in monotonic regions of the constitutive curves using two different methodologies: (i) the oft-used `frozen-time' linear stability (eigenvalue) analysis, wherein we examine whether infinitesimal perturbations imposed on instantaneous stress components (treated as quasi steady states) exhibit exponential growth, and (ii) the more mathematically rigorous fundamental-matrix approach that characterizes the transient growth via a numerical solution of the time-dependent linearized governing equations, wherein the linearized perturbations co-evolve with the start-up shear flow. Our results reinforce the hitherto understated point that there is no universal connection between the overshoot and subsequent decay of shear stress in the base state and the unstable eigenvalues obtained from the frozen-time stability analysis. It may therefore be difficult to subsume the occurrence of transient shear banding during shear start-up within the ambit of a single model-independent criterion. Our work also suggests that the strong transients during shear start-up seen in earlier work could well be a consequence of consideration of the limit of small solvent viscosity in the absence of otherwise negligible terms such as fluid inertia.

Proof of the zeroth law of turbulence in one-dimensional compressible magnetohydrodynamics and shock heating.

The zeroth law is one of the oldest conjectures in turbulence that is still unproven. Here, we consider weak solutions of one-dimensional compressible magnetohydrodynamics and demonstrate that the lack of smoothness of the fields introduces a dissipative term, named inertial dissipation, into the expression of energy conservation that is neither viscous nor resistive in nature. We propose exact solutions assuming that the kinematic viscosity and the magnetic diffusivity are equal, and we demonstrate that the associated inertial dissipation is positive and equal on average to the mean viscous dissipation rate in the limit of small viscosity, proving the conjecture of the zeroth law of turbulence and the existence of an anomalous dissipation. As an illustration, we evaluate the shock heating produced by discontinuities detected by Voyager in the solar wind around 5 AU. We deduce a heating rate of ∼10^{-18}Jm^{-3}s^{-1}, which is significantly higher than the value obtained from the turbulent fluctuations. This suggests that collisionless shocks can be a dominant source of heating in the outer solar wind.

Homogenization of the backward-forward mean-field games systems in periodic environments

We study the homogenization properties in the small viscosity limit and in periodic environments of the (viscous) backward-forward mean-field games system. We consider separated Hamiltonians and provide results for systems with (i) ‘‘smoothing’’ coupling and general initial and terminal data, and (ii) with ‘‘local coupling’’ but well-prepared data. The limit is a first-order forward-backward system. In the nonlocal coupling case, the averaged system is of mean field games-type, which is well-posed in some cases. For the problems with local coupling, the homogenization result is proved assuming that the formally obtained limit system has smooth solutions with well prepared initial and terminal data. It is also shown, using a very general example (potential mean field games), that the limit system is not necessarily of mean field games-type. A complete theory for such systems was developed by the authors in [28].

Sawtooth relaxation oscillations, nonlinear helical flows and steady-state m/n=1 magnetic islands in low-viscosity tokamak plasma simulations

A numerical study on the influence of plasma viscosity and of the plasma (=kinetic pressure/magnetic pressure) parameter on the nonlinear evolution of resistive internal kink modes in tokamak plasmas is presented. A new regime with relatively low viscosity is found, such that sawtooth oscillations spontaneously evolve towards states with stationary m/n = 1 magnetic islands. It is suggested that the mechanism at work in the limit of small viscosity is related to magnetic flux pumping, which, allied with the nonlinear resistive internal kink dynamics, leads to a stationary helical flow, only weakly dissipated by viscosity and entirely self-consistent with the presence of saturated m/n = 1 stationary magnetic islands. It is also found that the threshold viscosity value for the onset of the steady-state regime increases with increasing values. The newly found regime for a steady-state m/n = 1 magnetic island may be relevant for the understanding of tokamak experiments, where saturated helical structures such as the density snake and steady-state magnetic islands are sometimes observed in the core plasma region where the safety factor is close to or below unity.

Topology and stochasticity of turbulent magnetic fields.

We present a mathematical formalism for the topology and stochasticity of vector fields based on renormalization group methodology. The concept of a scale-split energy density, ψ_{l,L}=B_{l}·B_{L}/2 for vector field B(x,t) renormalized at scales l and L, is introduced in order to quantify the notion of the field topological deformation, topology change, and stochasticity level. In particular, for magnetic fields, it is shown that the evolution of the field topology is directly related to the field-fluid slippage, which has already been linked to magnetic reconnection in previous work. The magnitude and direction of stochastic magnetic fields, shown to be governed, respectively, by the parallel and vertical components of the renormalized induction equation with respect to the magnetic field, can be studied separately by dividing ψ_{l,L} into two (3+1)-dimensional scalar fields. The velocity field can be approached in a similar way. Magnetic reconnection can then be defined in terms of the extrema of the L_{p} norms of these scalar fields. This formulation in fact clarifies different definitions of magnetic reconnection, which vaguely rely on the magnetic field topology, stochasticity, and energy conversion. Our results support the well-founded yet partly overlooked picture in which magnetic reconnection in turbulent fluids occurs on a wide range of scales as a result of nonlinearities at large scales (turbulence inertial range) and nonidealities at small scales (dissipative range). Lagrangian particle trajectories, as well as magnetic field lines, are stochastic in turbulent magnetized media in the limit of small resistivity and viscosity. The magnetic field tends to reduce its stochasticity induced by the turbulent flow by slipping through the fluid, which may accelerate fluid particles. This suggests that reconnection is a relaxation process by which the magnetic field lowers both its topological entanglements induced by turbulence and its energy level.

Precessing spherical shells: flows, dissipation, dynamo and the lunar core

SUMMARYPrecession of planets or moons affects internal liquid layers by driving flows, instabilities and possibly dynamos. The energy dissipated by these phenomena can influence orbital parameters such as the planet’s spin rate. However, there is no systematic study of these flows in the spherical shell geometry relevant for planets, and the lack of scaling law prevents convincing extrapolation to celestial bodies. We have run more than 900 simulations of fluid spherical shells affected by precession, to systematically study basic flows, instabilities, turbulence and magnetic field generation. We observe no significant effects of the inner core on the onset of the instabilities. We obtain an analytical estimate of the viscous dissipation, mostly due to boundary layer friction in our simulations. We propose theoretical onsets for hydrodynamic instabilities, and document the intensity of turbulent fluctuations. We extend previous precession dynamo studies towards lower viscosities, at the limits of today’s computers. In the low viscosity regime, precession dynamos rely on the presence of large-scale vortices, and the surface magnetic fields are dominated by small scales. Interestingly, intermittent and self-killing dynamos are observed. Our results suggest that large-scale planetary magnetic fields are unlikely to be produced by a precession-driven dynamo in a spherical core. But this question remains open as planetary cores are not exactly spherical, and thus the coupling between the fluid and the boundary does not vanish in the relevant limit of small viscosity. Moreover, the fully turbulent dissipation regime has not yet been reached in simulations. Our results suggest that the melted lunar core has been in a turbulent state throughout its history. Furthermore, in the view of recent experimental results, we propose updated formulas predicting the fluid mean rotation vector and the associated dissipation in both the laminar and the turbulent regimes.

Mathematical analysis of boundary layers in two-dimensional incompressible viscous heat conducting flows

This paper is to review our recent study on the well-posedness and blowup of the boundary layer equations in small viscosity and heat conductivity limit for the two-dimensionalincompressible viscous heat conducting flows near a physical boundary. In the case that the viscosity and heat conductivity have the same scale, first we derive the boundary layer equations of the viscous layer and thermal layer for the incompressible Navier-Stokes-Fourier equations by multi-scale analysis,and then we shall review a well-posedness result established under the monotonicity condition of tangential velocity by using the Crocco transformation and the energy method.After that, when the tangential velocity does not satisfy the monotonicity assumption, we shall present a well-posedness result when the data are analytic with respect to the tangential variable,by using the Littlewood-Paley theory. We also present a blowup result in a finite time by introducing a Lyapunov functional, when the monotonicity condition is violated for the initial velocity. This shows that the analytic solution which we obtained exists in a finite time only in general.

Nonlinear effects in the bounded dust-vortex flow in plasma.

The vortex structures in a cloud of electrically suspended dust in a streaming plasma constitutes a driven system with a rich nonlinear flow regime. Experimentally recovered toroidal formations of this system have motivated study of its volumetrically driven-dissipative vortex flow dynamics using two-dimensional hydrodynamics in the incompressible Navier-Stokes regime. Nonlinear equilibrium solutions are obtained for this system where a nonuniformly driven two-dimensional dust flow exhibits distinct regions of localized accelerations and strong friction caused by stationary fluids at the confining boundaries resisting the dust flow. In agreement with observations in experiments, it is demonstrated that the nonlinear effects appear in the limit of small viscosity, where the primary vortices form scaling with the most dominant spatial scales of the domain topology and develop separated virtual boundaries along their periphery. This separation is triggered beyond a critical dust viscosity that signifies a structural bifurcation. Emergence of uniform vorticity core and secondary vortices with a newer level of identical dynamics highlights the applicability of the studied dynamics to gigantic vortex flows, such as the Jovian great red spot, to microscopic biophysical intracellular activity.

Existence of Chapman--Jouguet Detonation and Deflagration Waves

We study the existence of profiles for Chapman--Jouguet detonation and deflagration waves in the Navier--Stokes equations for a reacting gas. In the limit of small viscosity, heat conductivity, and diffusion, the profiles correspond to heteroclinic orbits of a system of singularly perturbed ordinary differential equations. The burned end state of the waves, however, is a nonhyperbolic equilibrium of the associated, purely gas dynamic layer problem, and hence standard methods from geometric singular perturbation theory fail. We show how to resolve this degeneracy by combining a center manifold reduction with the blow-up method. The main result is the existence of viscous profiles for various types of Chapman--Jouguet processes. In addition, we obtain results on the spatial decay rates of these waves which are expected to be relevant for the stability analysis of the waves.

Direct detection of linearized bursts in turbulence

The relevance of linear transitional mechanisms in fully turbulent shear flows, and in particular of the Orr-like inviscid transient amplification of disturbances, is explored in the context of the prediction of bursting behavior. Although the logarithmic layer of wall-bounded turbulence is used as the primary example, most conclusions should apply to other flows with linearly stable mean profiles that are dominated by large-scale streamwise-velocity streaks and intermittent bursts of the cross-shear velocity. When the linearised problem is solved in the limit of small viscosity, it has previously been shown that statistical properties, such as the bursting time- and length-scales, the energy fluxes between components, and the mean inclination angles, are consistent in linear and nonlinear systems. The question addressed here is whether the individual structures predicted by the linearised solution can be detected in fully nonlinear simulations, and whether the linearized approximation can be used to predict their evolution. It is found that strong bursting of the largest scales is well described linearly, comprising about 65%–70% of the total time, but that weaker fluctuations are not. It is also found that adding an eddy viscosity does not substantially improve predictions.

Linearised Structures in Shear Turbulence

The relevance of linear transitional mechanisms in fully turbulent shear flows, and in particular of the Orr-like inviscid transient amplification of disturbances, is explored in the context of the prediction of bursting behaviour. Although the logarithmic layer of wall-bounded turbulence is used as the primary example, most conclusions also apply to turbulent flows with a uniform shear, as well as probably to other flows with linearly stable mean profiles. They are dominated by large-scale streamwise-velocity streaks and intermittent bursts of the cross-shear velocity. When the linearised problem is solved in the limit of small viscosity, it has previously been shown that many properties, such as the bursting time and length scales, the energy fluxes between components, and the mean inclination angles, agree well between linear and nonlinear systems. The question addressed here is whether the structures predicted by the linearised solution can be traced in fully nonlinear simulations. It is found that the largest scales of strong bursts are well described linearly, but that the weaker fluctuations are not.

Local Well-Posedness of Prandtl Equations for Compressible Flow in Two Space Variables

In this paper, we consider the local well-posedness of the Prandtl boundary layer equations that describe the behavior of the boundary layer in the small viscosity limit of the compressible isentropic Navier--Stokes equations with nonslip boundary condition. Under the strictly monotonic assumption on the tangential velocity in the normal variable, we apply the Nash--Moser--Hörmander iteration scheme and further develop the energy method introduced in [R. Alexander et al., J. Amer. Math. Soc., DOI:S0894-0347(2014)00813-4] to obtain the well-posedness of the equations locally in time.

Derivation of Prandtl boundary layer equations for the incompressible Navier–Stokes equations in a curved domain

Abstract The proposal of this note is to derive the equations of boundary layers in the small viscosity limit for the two-dimensional incompressible Navier–Stokes equations defined in a curved bounded domain with the non-slip boundary condition. By using curvilinear coordinate system in a neighborhood of boundary, and the multi-scale analysis we deduce that the leading profiles of boundary layers of the incompressible flows in a bounded domain still satisfy the classical Prandtl equations when the viscosity goes to zero, which are the same as for the flows defined in the half space.

Viscous boundary layers in hyperbolic-parabolic systems with Neumann boundary conditions

We initiate the study of noncharacteristic boundary layers in hyperbolic-parabolic problems with Neumann boundary conditions. More generally, we study boundary layers with mixed Dirichlet--Neumann boundary conditions where the number of Dirichlet conditions is fewer than the number of hyperbolic characteristic modes entering the domain, that is, the number of boundary conditions needed to specify an outer hyperbolic solution. We have shown previously that this situation prevents the usual WKB approximation involving an outer solution with pure Dirichlet conditions. It also rules out the usual maximal estimates for the linearization of the hyperbolic-parabolic problem about the boundary layer. Here we show that for linear, constant-coefficient, hyperbolic-parabolic problems one obtains a reduced hyperbolic problem satisfying Neumann or mixed Dirichlet--Neumann rather than Dirichlet boundary conditions. When this hyperbolic problem can be solved, a unique formal boundary-layer expansion can be constructed. In the extreme case of pure Neumann conditions and totally incoming characteristics, we carry out a full analysis of the quasilinear case, obtaining a boundary-layer approximation to all orders with a rigorous error analysis. As a corollary we characterize the small viscosity limit for this problem. The analysis shows that although the associated linearized hyperbolic and hyperbolic--parabolic problems do not satisfy the usual maximal estimates for Dirichlet conditions, they do satisfy analogous versions with losses.