General Three-Dimensional Disturbances to Inviscid Couette Flow

The general solution to the linearized equations governing three-dimensional disturbances to inviscid Couette flow has been obtained. This result extends the Orr solution to initial conditions that do not consist of a single Fourier sine component in the cross-stream coordinate and a plane wave in the streamwise/spanwise coordinates. The time evolution of a measure of disturbance energy for some specific pulsed initial conditions is examined, and it is concluded that, while the rapid algebraic growth to large amplitude followed by decay exemplified by the Orr solution can be of importance for individual cross-stream Fourier components, more realistic initial conditions, which in general consist of the sum of an infinite number of components, often display uniform decay to zero amplitude. However, an interesting example is described in which one positive definite measure of disturbance amplitude remains constant, yet the streamwise/spanwise velocity components grow linearly in time if the initial disturbance is three-dimensional.     

The Evolution of Linearized Perturbations of Parallel Flows

The equations of an incompressible fluid are linearized for small perturbations of a basic parallel flow. The initial-value problem is then posed by use of Fourier transforms in space. Previous results are systematized in a general framework and used to solve a series of problems for prototypical examples of basic shear flow and of initial disturbance. The prototypes of shear flow are (a) plane Couette flow bounded by rigid parallel walls, (b) plane Couette flow bounded by rigid walls at constant pressure, (c) unbounded two-layer flow with linear velocity profile in each layer, (d) a piecewise linear profile of a boundary layer on a rigid wall. The prototypes of initial perturbation are the fundamental ones: (i) a point source of the field of the transverse velocity (represented by delta functions), (ii) an unbounded sinusoidal field of the transverse velocity, (iii) a point source of the lateral component of vorticity, (iv) a sinusoidal field of the lateral vorticity. Detailed solutions for an inviscid fluid are presented, but the problem for a viscous fluid is only broached.     

The point-vortex method for periodic weak solutions of the 2-D Euler equations

Almost-sure convergence of a subsequence of the vorticity to a weak solution is proven for the point-vortex method for 2-D, inviscid, incompressible fluid flow. Here “almost-sure” is with respect to sequences of random components included in the initial position and strength of each vortex. The initial vorticity is assumed to be periodic and, depending on the initialization scheme, to lie in L log L or L^p with p > 2. The randomization of the initial data is not needed when the initial vorticity is nonnegative; such initial data also need not be periodic, and is only required to be a bounded measure lying in H⁻¹. All these results are also valid for the “vortex-blob” method with the smoothing parameter vanishing at an arbitrary rate. The sense in which solutions of point-vortex dynamics are weak solutions of the Euler equations is also discussed.     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

Van Hove's “λ2t” limit in nonrelativistic and relativistic field-theoretical models

Van Hove's “λ²t” limiting procedure is analyzed in some interesting quantum field-theoretical cases, both in nonrelativistic and relativistic models. We look at the deviations from a purely exponential behavior in a decay process and discuss the subtle issues of state preparation and initial time.     

Nondegeneracy,from the prospect of wave-wave regular interactions of a gasdynamic type

A parallel is considered between Burnat's “algebraic” approach [restricted to a genuine nonlinearity] and Martin's “differential” approach regarding their contribution to describing some nondegenerate one-dimensional gasdynamic regular interaction solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-amplitude Rotational Waves in Viscous Shear Flows

Growing finite-amplitude initially spanwise-independent two-dimensional rotational waves and their nonlinear interaction with unidirectional viscous shear flows of various strengths are considered. Both primary and secondary instabilities are studied, but only secondary instabilities are permitted to vary in the spanwise direction. A generalized Lagrangian-mean formulation is employed to describe wave-mean interactions, and a separate theory is constructed to account for the back effect of the developing mean flow on the wave field. Viscosity is seen to significantly complicate calculation of the back effect. The primary instability is seen to act as a platform for, and catalyst to, secondary instabilities. The analysis leads to an eigenvalue problem for the initial growth of the secondary instability, this being a generalization of the eigenvalue problem constructed by Craik for inviscid neutral waves. Two inviscid secondary instability mechanisms to longitudinal vortex form are observed: the first has as its basis the Craik–Leibovich type 2 mechanism. The second, which is as yet unproven, requires that both the wave and flow field distort in concert at all levels of shear. Both mechanisms excite exponential growth on a convective rather than diffusive scale in the presence of neutral waves, but growing waves alter that growth rate.     

Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t\approx {\epsilon }^{-2}$. Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.     

Long-Time Instability of the Couette Flow in Low Gevrey Spaces

We prove the instability of the Couette flow if the disturbances is less smooth than the Gevrey space of class 2. This shows that this is the critical regularity for this problem since it was proved in [5] that stability and inviscid damping hold for disturbances which are smoother than the Gevrey space of class 2. A big novelty is that this critical space is due to an instability mechanism which is completely nonlinear and is due to some energy cascade. © 2023 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Stabilization of periodic Stokesian Hele–Shaw flows of ferrofluids

We consider an incompressible ferrofluid in a vertical Hele–Shaw cell and develop a proper analytic framework for the free interface and the velocity potential of the fluid in a periodic geometry. The flow is assumed to obey a non-Newtonian Darcy law. The forces influencing the fluid are gravity, surface tension and the response to a magnetic field induced by a current. In addition, the flow is stabilized at the lower boundary component by an external source b. We prove a well-posedness result for the flow near flat solutions. Moreover, we find conditions on the parameters and on the slope of b for the exponential stability and instability of flat interfaces. Furthermore, we identify values for the current's intensity ι where critical bifurcation of nontrivial finger-shaped solutions from the branch of trivial (flat) solutions takes place.     

Some Inviscid “Cat's Eye” Flows

Some inviscid flows with the combined shear and periodic disturbance characteristic of the Kelvin “eat's eye” pattern are presented. The solutions have constant vorticity in each region of closed streamlines, with discontinuities in vorticity (but not velocity) allowed across boundaries.     

DNS of stochastically forced laminar plane Couette flow

Background of three dimensional hydrodynamic/vortical fluctuations in a stochastically forced, laminar, incompressible, plane Couette flow is simulated by direct numerical simulations (DNS). It was found that the fluctuating field has well pronounced peculiarities: (i) The hydrodynamic fluctuations exhibits non–exponential, transient growth; (ii) Streamwise non–constant fluctuations with the characteristic length scale of the order of the channel width are predominant in the fluctuating spectrum; (iii) Existence of coherent structures in the fluctuating background; (iv) Stochastic forcing breaks the spanwise reflection symmetry (inherent to the linear and full Navier–stokes equations in a case of the Couette flow) and inputs an asymmetry on dynamical processes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolution of nonparametric surfaces with speed depending on curvature II. The mean curvature case

We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t ≧ 0. In addition, we establish estimates of the rate at which solutions tend to zero as t → ∞.     

Asymptotic Solutions of the Korteweg-deVries Equation

The long-time asymptotic solution of the Korteweg-deVries equation, corresponding to initial data which decay rapidly as |x|→∞ and produce no solitons, is found to be considerably more complicated than previously reported. In general, the asymptotic solution consists of exponential decay, similarity, rapid oscillations and a “collisionless shock” layer. The wave amplitude in this layer decays as [(lnt)/t]^2/3. Only for very special initial conditions is the shock layer absent from the solution.     

On Howard’s conjecture in heterogeneous shear flow problem

Howard’s conjecture, which states that in the linear instability problem of inviscid heterogeneous parallel shear flow growth rate of an arbitrary unstable wave must approach zero as the wave length decreases to zero, is established in a mathematically rigorous fashion for plane parallel heterogeneous shear flows with negligible buoyancy forcegΒ ≪ 1 (Miles J W,J. Fluid Mech. 10 (1961) 496–508), where Β is the basic heterogeneity distribution function). An erratum to this article is available at .     

Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2

We analyze the two‐dimensional parabolic‐elliptic Patlak‐Keller‐Segel model in the whole Euclidean space ?². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local‐in‐time existence for any mass of “free‐energy solutions,” namely weak solutions with some free‐energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free‐energy solutions with initial data as before for the critical mass 8π/χ. Actually, we prove that solutions blow up as a delta Dirac at the center of mass when t → ∞ when their second moment is kept constant at any time. Furthermore, all moments larger than 2 blowup as t → ∞ if initially bounded. © 2007 Wiley Periodicals, Inc.     

Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid

In this paper, Adomian’s decomposition method is used to solve non-linear differential equations which arise in fluid dynamics. We study basic flow problems of a third grade non-Newtonian fluid between two parallel plates separated by a finite distance. The technique of Adomian decomposition is successfully applied to study the problem of a non-Newtonian plane Couette flow, fully developed plane Poiseuille flow and plane Couette–Poiseuille flow. The results obtained show the reliability and efficiency of this analytical method. Numerical solutions are also obtained by solving non-linear ordinary differential equations using Chebyshev spectral method. We present a comparative study between the analytical solutions and numerical solutions. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the Adomian decomposition method.     

光滑区域上二维无黏性无热传导Boussinesq方程组与三维轴对称不可压Euler方程组的指数增长全局光滑解

研究二维无黏性无热传导Boussinesq方程组和三维轴对称不可压Euler方程组光滑解的增长情况,找各种区域使其上的方程组有快增长的解。对Boussinesq方程组,通过选取初始温度和速度的一个分量,可以把方程去耦为两部分。从关于涡量的部分求出涡量、速度场和使结论成立的区域,从关于温度的部分,可见温度的高阶导的增长仅依赖于速度场的一个分量。通过适当选取该分量,得到温度高阶导有指数增长的全局光滑解。对轴对称Euler方程组做类似的处理,适当选取速度场的径向分量,可把方程组去耦,最终得到一类光滑区域,在其上方程组有指数增长全局光滑解。该研究把Chae、Constantin、Wu对一个二维锥形区域上无黏性无热传导Boussinesq方程的结果,推广到一类光滑区域上, 并把他们的方法应用到三维轴对称不可压Euler方程组, 得到了类似的结果。     

Periodic waves in rotating plane Couette flow

A class of nonlinear equations of Navier-Stokes type of the form (**) $$\frac{{dw}}{{dt}} + L_0 w + (\lambda - \lambda _0 )L_1 w + \gamma \lambda (M_1 w + M_2 w) + \gamma ^2 L_3 (\lambda ,\gamma )w + B(w) = 0$$ is investigated, where λ is a “load” parameter (i.e., a Reynolds number), γ is a “structure” parameter,L _o L ₁,M ₁,M ₂ andL ₃(λ, γ) are linear operators, andB is a quadratic operator. An equation of the form (**) describes a variety of spiral flow problems including rotating plane Couette flow which is studied here in detail. Under suitable hypotheses on the operators in (**), it is shown that Hopf bifurcation occurs for γ sufficiently small. In the problem of rotating plane Couette flow, by determining the sign of the real part of a certain “cubic” coefficient, it is shown, in addition, that the bifurcating periodic orbits are supercritical and asymptotically stable, and correspond to periodic waves.