The effect of viscosity on the free stream in transonic flow over a corner point

An asymptotic analysis of the system of Navier–Stokes equations for describing the flow which arises from the subsonic free stream in the neighbourhood of the vertex of a convex corner with curvilinear generatrices is presented for Reynolds numbers approaching infinity. It is assumed that, in limiting non-viscous flow, the subsonic free stream reaches the velocity of sound at the vertex of the corner and, in the first approximation, is described by the Vaglio–Laurin solution. It is shown that the flow can have a different form depending on the value of the pressure gradient, which is formed in the neighbourhood of the corner point. However, irrespective of the steady form of the flow, as a result of the interaction of the Vaglio–Laurin flow with the boundary layer, the latter induces perturbations in the outer flow, which “rounds off” the vertex of the corner when there is a transonic flow around it.     

On the existence of extreme waves and the Stokes conjecture with vorticity

This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120° or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120° if the vorticity is nonnegative near the free surface.     

The Dupuit Approximation for the Rectangular Dam Problem

A porous rectangular dam is above a horizontal impermeable base.There is a steady flow in which water seeps through the damfrom one reservoir (on the left)to a lower reservoir (on theright). Because of gravity, the water does not flow throughthe entire dam and the dam is dry near its upper-right corner.The interface separating the dry and wet regions of the damis a free boundary (in the hydrology literature called the phreaticsurface). Dupuit, in 1863, derived the formal approximationthat the phreatic surface was nearly a parabola. In recent years,mathematicians have obtained existence results. Here, relationsare established between the formal approximations and the solutionsobtained from the existence proofs.     

Nonlinear Waves in a Shear Flow with a Vorticity Discontinuity

We consider nonlinear finite-amplitude progressive shear-flow waves on a basic velocity profile consisting of two coflowing layers of inviscid equal-density fluid, each of uniform but different vorticity. The problem is formulated as a nonlinear integral equation describing the shape of the vorticity discontinuity in a frame of reference in which the flow is steady. Numerical solutions to this equation are presented for a range of values of the vorticity ratio Ω. For 1 > © ≥ ? 1 the theoretical maximum wave amplitude occurs when the wave crest forms a 90° corner which just touches the appropriate critical-layer stagnation point. The linearized stability of the progressive wave states to arbitrary subharmonic isovortical disturbances is studied numerically. The results indicate stability at moderate values of the wave amplitude.     

Regions of existence of discrete spectral modes in a Blasius boundary layer

The discrete spectrum of two-dimensional perturbations of the flow in a Blasius boundary layer is investigated. It is shown that, on the front section of the surface in the flow, there are flow regions where there are no modes of the discrete spectrum. On travelling downstream a Tollmin–Schlichting mode initially arises and then, successively, other modes that emerge from under the cut of the complex plane corresponding to a continuous spectrum of vortex waves. The regions of existence of modes in the plane of the parameters of the problem are determined.     

A jump discontinuity of compressible viscous flows grazing a non-convex corner

We show existence and regularity of solution for the compressible viscous steady state Navier–Stokes system on a polygon having a grazing corner and that the density has a jump discontinuity across a curve inside the domain. There are corresponding jumps in derivatives of the velocity. The solution comes from a well-posed boundary value problem on a polygonal domain with a non-convex corner. A formula for the decay of the jump is given. The decay formula suggests that density jumps can occur in a compressible flow with a non-vanishing viscosity.     

Stability of transonic shock for supersonic flow past a wedge

When steady supersonic flow hits a slim wedge, there may appear an oblique transonic shock attached to the vertex of the wedge, if the downstream pressure is rather large. This paper studies stability in certain weighted partial Hölder spaces of the oblique transonic shock attached to the vertex of a wedge, which is against steady supersonic flows, under perturbations of the upstream flow and the profile of the wedge. We show that under reasonable conditions on the upcoming supersonic flow and the slope of the wedge, such transonic shocks are structural stable. Mathematically, we solve an elliptic–hyperbolic mixed type in an unbounded domain, and the flow field is proved to be C¹. Copyright © 2015 John Wiley & Sons, Ltd.     

Stationary corner vortex configurations

Given a stable configuration of point vortices for steady two dimensional inviscid, incompressible fluid flow in a domainD, it is shown that there is another stable configuration of stationary point vortices inD with vortices near the original vortices plus additional vortices near any of the convex corners ofD. It follows that there are steady flows which have a finite sequence, of arbitrary length, of vortices of alternating sign descending into any convex corner ofD. Several computed examples are given.     

A Proof of the Convexity of the Free Boundary for Porous Flow Through a Rectangular Dam Using the Maximum Principle

We consider the steady two-dimensional flow under gravity ofwater from one reservoir (on the left) to a lower reservoir(on the right) through a porous rectangular isotropic homogeneousdam with impervious bottom. Because of gravity the water doesnot flow through the entire dam and the dam is dry near itsupper right corner. The interface separating the dry and wetregions of the dam is a free boundary. Recently, Friedman &Jensen (1977) have proved that the free boundary is convex.We give a different proof which uses only the maximum principleand its generalizations.     

Self-similar decay of localized perturbations in the integral boundary layer equation

The integral boundary layer equation (IBLe) arises as a long wave approximation for the flow of a viscous incompressible fluid down an inclined plane. The trivial solution of the IBLe is linearly at best marginally stable, i.e., it has essential spectrum at least up to the imaginary axis. Here, we show that in the stable case this trivial solution is in fact nonlinearly stable, with a Burgers like self-similar decay of localized perturbations. The proof uses renormalization theory and the fact that in the stable case Burgers equation is the amplitude equation for long small amplitude waves in the IBLe.     

A version of the asymptotic theory of the unsteady free interaction of a boundary layer with a transonic flow

The non-linear theory of strongly perturbed flows, with a structure of the velocity fields which is characteristic of domains of so-called free interaction of the boundary layer with an external potential flow, is considered. The specific details of the transonic flow show up not only in the estimates of the amplitudes and lengths of the perturbation waves in the asymptotic analysis of the problem but, also, in the fact that the motion may turn out to be simultaneously unsteady in the part of the boundary layer close to the wall and in the external potential flow. This mechanism of the evolution of the perturbations can be described by a single integro-differential equation which, assuming that the structure of the fluctuating fields is a quadrideck structure, is derived using the Fourier-Laplace method. Examples of its non-linear solutions are given in the form of solitary or periodic waves.     

Unconditionally Stable Splitting Methods for the Shallow Water Equations

The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible CFL numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical technique is applied to four test cases, the first being an expanding bore with rotational symmetry. The second problem addresses the question of describing the time development of two constant water levels separated by a dam that breaks instantaneously. The third problem compares the front-tracking method with an explicit analytic solution of water waves rotating over a parabolic bottom profile. Finally, we study flow over an obstacle in one dimension.     

High Hartmann number flow through a rectangular duct with unequally and finitely conducting walls

Nonsymmetric Hartmann flow through a rectangular duct is investigated for thin duct walls with, generally, unequal but finite conductivities. A high Hartmann number is adopted. Consistent with known phenomena, both Hartmann layers transverse to the applied magnetic field are assumed to be separated from the two side boundary layers by four corner regions plus four inner corner regions. The method of singular perturbations and matched asymptotic expansions is applied to the coupled system. The equations governing the core and Hartmann layers are first partially resolved for leading terms. This is then followed by tackling equations governing one side layer and two adjacent corner regions. The latters' incorporation secures, for the former, only those boundary conditions that are compatible along the transverse walls. Both corner regions are denied access to non-required boundary conditions along the neighbouring side wall by the adjoining inner corner regions. However, the latters' boundary value problems need not be tackled for the acquirement of only dominant terms beyond all four inner corner regions. The complementary side layer and associated corners are accounted for by a non-symmetric reflection principle. Results reveal that a difference between conductivities in the transverse walls together with at least one finitely conducting side wall impart to disturbances within the core and Hartmann layers (i) a nontrivial dependence on the transverse coordinate relative to the magnetic field and flow in addition to the (usual) dependence on the field aligned coordinate, (ii) a dependence on side wall parameters in addition to the dependence on transverse wall parameters. Applications to related situations are considered. These include the case for a perfectly conducting lower wall, a finitely conducting upper wall, and equally and finitely conducting side walls.     

Instability of solitary waves on Euler’s elastica

Stability of solitary waves in a thin inextensible and unshearable rod of infinite length is studied. Solitary-wave profile of the elastica of such a rod without torsion has the form of a planar loop and its speed depends on a tension in the rod. The linear instability of a solitary-wave profile subject to perturbations escaping from the plane of the loop is established for a certain range of solitary-wave speeds. It is done using the properties of the Evans function, an analytic function on the right complex half-plane, that has zeros if and only if there exist the unstable modes of the linearization around a solitary-wave solution. The result follows from comparison of the behaviour of the Evans function in some neighbourhood of the origin with its asymptotic at infinity. The explicit computation of the leading coefficient of the Taylor series of the Evans function near the origin is performed by means of the symbolic computer language. Received: April 6, 2004; revised: December 12, 2004     

Stability of flow of a liquid crystal layer on an inclined plane : PMM vol. 38, n 6, 1974, pp. 1031–1040

The framework of the linear mechanics of liquid crystal media [1] is used to study propagation of waves in a layer of a nematic liquid crystal (NLC) on an inclined plane, in a magnetic field, for three different cases of orientation of the anisotropy axis, namely orthogonal to the inclined plane, parallel to the inclined plane and orthogonal to the plane of flow. Such orientations of the anisotropy axis are realized in practice in the course of special machining of solid surfaces [2]. Exact solutions of the equations of motion are obtained describing the steady flow of the layer, and the behavior of small plane perturbations is studied. It is shown that two types of plane waves can propagate in a layer of the nematic mesophase, namely, the surface and the orientational waves. In the case of long surface waves the formulas for the critical Reynolds number are obtained. For the orientational waves a sufficient criterion of stability of the flow in the layer is obtained for two cases. The influence of the magnetic field and of the rheological parameters of NLC on the character of propagation of the first and second type waves is investigated.From amongst the papers dealing with wave propagation in NLC, we draw the readers' attention to [3] which deals with the longitudinal, shear and torsional waves in a liquid crystal domain and obtains the corresponding dispersion relationships.     

A criterion of the linear long wave stability of steady magnetohydrodynamic jet flows of an ideal fluid

The problem of the linear stability of steady axisymmetric shear magnetohydrodynamic jet flows of an inviscid ideally conducting incompressible fluid with a free boundary is investigated. It is assumed that the jet is of unlimited length, there is a longitudinal constant electric current along its surface, and it is directed along the axis of a cylindrical shell with infinite conductivity, such that there is a vacuum layer between its free boundary and the inner surfaces of the shell. The necessary and sufficient condition for the stability of such flows with respect to small axisymmetric long-wave perturbations of special form is obtained by Lyapunov's direct method. Bilateral exponential estimates of the growth of small perturbations are constructed in the case when this stability condition breaks down, where the indices in their exponents are calculated from the parameters of the steady flows and the initial data for the perturbations. An example of a steady axisymmetric shear magnetohydrodynamic jet flow and of the initial small axisymmetric long-wave perturbations imposed on it is given, which, at the linear stage, will evolve in time and space in accordance with the estimates constructed.     

Forced Alveolar Flows and Mixing in the Lung

The air flows deep inside the lung are not only important in gas exchange processes but they also determine the efficiency of particle deposition and retention. The study aims at quantifying the relative influence of different flow components in the transport of small particles in alveolar geometries such as convective breathing patterns, wall movement, gravitational settling and Brownian motion. In addition, the possibility and efficiency of external forcing is studied, relying on the mechanism of internal acoustic streaming. A viscous oscillating boundary layer flow is converted into a steady, viscosity-independent bulk motion which is very efficient at low Reynolds numbers. The streaming can be controlled by external parameters (excitation amplitude, frequency, beam shape) and may thus be of diagnostic and therapeutic relevance. Numerical simulations are performed to analyze the flow patterns in 3D model geometries and to measure deposition rates. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)