Rigorous Asymptotic Expansions for Critical Wave Speeds in a Family of Scalar Reaction-Diffusion Equations

We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.J. Needham, (1993). J. Appl. Math. Phys. (ZAMP) A, vol. 44, No. 4, 707–721] and [T.P. Witelski, K. Ono, and T.J. Kaper, (2001). Appl. Math. Lett., vol. 14, No. 1, 65–73].     

Gradients at a curved shock in reacting flow

The inviscid equations of motion for the flow at the downstream side of a curved shock are solved for the shock–normal derivatives. Combining them with the shock–parallel derivatives yields gradients and substantial derivatives. In general these consist of two terms, one proportional to the rate of removal of specific enthalpy by the reaction, and one proportional to the shock curvature. Results about the streamline curvature show that, for sufficiently fast exothermic reaction, no Crocco point exists. This leads to a stability argument for sinusoidally perturbed normal shocks that relates to the formation of the structure of a detonation wave. Application to the deflection–pressure map of a streamline emerging from a triple shock point leads to the conclusion that, for non–reacting flow, the curvature of the Mach stem and reflected shock must be zero at the triple point, if the incident shock is straight. The direction and magnitude of the gradient at the shock of any flow quantity may be written down using the results. The sonic line slope in reacting flow serves as an example. Extension of the results – derived in the first place for plane flow – to three dimensions is straightforward. Received 12 February 1997 / Accepted 10 June 1997     

Interaction of steady shear flows of a barotropic liquid

A mathematical model describing plane-parallel vortex flows of a barotropic liquid with a free boundary in a long-wave approximation is considered. For a particular class of solutions, the solvability of the problem of decay of an initial discontinuity of small amplitude is demonstrated and an algorithm of solution is proposed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 79–90, September–October, 1999.     

Aspect-ratio dependence of transient Taylor vortices close to threshold

We perform a detailed numerical study of transient Taylor vortices arising from the instability of cylindrical Couette flow with the exterior cylinder at rest for radius ratio η = 0.5 and variable aspect ratio Γ. The result of Abshagen et al. (J Fluid Mech 476:335–343, 2003) that onset transients apparently evolve on a much smaller time–scale than decay transients is recovered. It is shown to be an artefact of time scale estimations based on the Stuart–Landau amplitude equation which assumes frozen space dependence while full space–time dependence embedded in the Ginzburg–Landau formalism needs to be taken into account to understand transients already at moderate aspect ratio. Sub-critical pattern induction is shown to explain the apparently anomalous behaviour of the system at onset while decay follows the Stuart–Landau prediction more closely. The dependence of time scales on boundary effects is studied for a wide range of aspect ratios, including non-integer ones, showing general agreement with the Ginzburg–Landau picture able to account for solutions modulated by Ekman pumping at the disks bounding the cylinders.      

Plane strain wave in an isotropic layer of a viscoelastic material

The velocity and the rate of decay of a strain wave in a layer of a viscoelastic material rigidly fixed on a solid foundation are determined. The wave structure (ratio of the longitudinal to the transverse displacement) and the profiles of these displacements are analyzed. Attenuation of waves in the first mode is found to be more significant than that in an infinite space. The most intense decay is observed at resonance frequencies. A strong effect of compressibility of the medium on wave parameters is revealed. Conditions at which such a system operates as a waveguide are found. For a loss tangent higher than 0.13 (for an incompressible medium), the character of the dispersion dependence is observed to change drastically: the wave velocity decreases with decreasing frequency. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 104–111, May–June, 2006.     

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 34–47, November–December, 2006.     

Motion of a hot wedge in a melting medium

In [1–3] a series of problems of the motion of heat sources at a temperature higher than the melting point of the surrounding medium was considered. The heat source could be a laser beam or a hot body. Here, the case of a thin wedge heated to a temperature higher than the melting point of the surrounding medium and moving at a constant velocity is investigated. The velocity is high enough for the molten layer formed to be thin. The problem is solved by the method of integral relations. The shape of the molten zone, the drag on the wedge and other flow characteristics of the melt are determined. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 52–57, September–October, 1988.     

Edge effect in a laminated composite with longitudinally compressed laminas

The paper is concerned with the determination of edge effect zone in a laminated composite with laminas subject to longitudinal compression. The dependence of the maximum decay length on the ratio between the period of external loading and the structure parameter is studied. The load period depends on the number of unloaded laminas. The decay of the edge effect is analyzed by numerically solving a boundary-value problem of elasticity for piecewise-homogeneous materials and using a quantitative decay criterion for the near-edge normal stresses __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 89–96, August 2006.     

Edge effect in a fibrous composite with near-surface fibers subjected to uniform loading

The decay rate of the edge effect in a material reinforced with fibers of square cross-section and subjected to transverse uniaxial deformation is studied. The case of uniform loading of near-surface fibers is considered. The edge effect is analyzed by numerically solving a boundary-value problem of elasticity for inhomogeneous bodies and applying a quantitative decay criterion for normal stresses __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 15–23, November 2007.     

Numerical simulation of stabilization of the boundary layer on a surface with a porous coating in a supersonic separated flow

Stability of a supersonic (M _∞ = 5.373) boundary layer with local separation in a compression corner with a passive porous coating partly absorbing flow perturbations is considered by solving two-dimensional Navier-Stokes equations numerically. The second mode of disturbances of a supersonic boundary layer is demonstrated to be the most important one behind the boundary-layer reattachment point. The possibility of effective stabilization of these disturbances behind the reattachment point with the use of porous coatings is confirmed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 39–47, March–April, 2007.     

Design of airfoils providing the absence of separation in a compressible flow in a specified range of angles of attack

A problem of modification of the classical airfoils that ensure the absence of separation in a subsonic ideal-gas flow in a specified range of angles of attack is solved by a numerical-analytical method based on the quasi-solution of inverse boundary-value problems of aerohydrodynamics and Kármán-Jiang formulas. Loitsyanskii’s criterion of the non-separated flow is used to determine the boundary-layer separation point. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 99–106, November–December, 2008.     

Specific features of microparticle deformation upon impact on a rigid barrier

Based on experimental data and numerical modeling, it is shown that a lamina of melted metal of thickness of order0.01 d, in which the temperature is close to the melting point of the particle material, can be formed upon high-speed impact (v ₀≈500–1200 m/sec) of a fine metal particle (d=1–50 μm) on a rigid undeformable barrier near the contact surface. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 204–209, January–February, 2000.     

Numerical study of free oscillations of a beam with oscillators

Free oscillations of a free-ended beam of variable cross section and mass with spring-suspended point masses (oscillators) are considered. It is found that parametric resonances are possible in this oscillating system. The effectiveness of the proposed calculation procedure is confirmed numerical calculations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 150–156, May–June, 2006.     

Numerical analysis of free vibrations of a beam with oscillators

The problem of free vibrations of a beam with free ends of variable cross section and mass, from which point masses (oscillators) are suspended by bars, is considered. It is shown that parametric resonances can occur in this oscillating system. Numerical examples showing the efficiency of the calculation method proposed are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 135–144, July–August, 2006.     

Stress waves in a rod subjected to a moving load

The wave processes in a semi-infinite rod located in an elastic medium and subjected to a point load moving at a constant velocity are considered. The system of two differential equations of motion of Timoshenko beam theory is solved using the Laplace transform in time. The integrals obtained are determined numerically. Variation of the bending moment on the longitudinal coordinate behind the elastic-wave front and the region of action of the point force at various times is shown. The results of the solution are influence functions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 112–122, March–April, 2007.     

Oscillating thermocapillary convection regimes driven by a point heat source

The stability of an axisymmetric thermocapillary flow driven by a point heat source located in the neighborhood of the free surface of a fluid filling a deep tank is investigated theoretically and experimentally. It is shown that for certain values of the depth and power of the heat source thermocapillary convection becomes unstable with respect to oscillating perturbations of the surface shape. Perm’, Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 92–103, March–April, 2000.     

Numerical simulation of gas dynamics in a bubble during its collapse with the formation of shock waves

The specific features of calculation of a gas in a spherical bubble located in the center of a spherical volume of weakly compressible fluid are considered. The problems of motion of a cold gas to a point and a spherical piston converging to a point are used to evaluate the algorithm. It is shown that significant errors can arise in calculation of spherical waves in the vicinity of the pole. These errors can be substantially reduced by means of artificial viscosity in the Riemann problem. Institute of Mechanics and Machine Building, Kazan’ Scientific Center, Russian Academy of Sciences, Kazan’ 420111. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 101–110, March–April, 1999.     

Origin of instability waves in a boundary layer with moderate free-stream turbulence

The characteristics of the transition mechanism in the boundary layer on a wing section are experimentally investigated for moderate free-stream turbulence. It is shown that the frequency and wavelength of the instability depend on the thickness of the boundary layer in the region of the point of inflection on the average velocity profile. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 49–54, January–February, 1994.     

Stability and nonlinear wavy regimes in downward film flows on a corrugated surface

The linear and nonlinear stability of downward viscous film flows on a corrugated surface to freesurface perturbations is analyzed theoretically. The study is performed with the use of an integral approach in ranges of parameters where the calculated results and the corresponding solutions of Navier-Stokes equations (downward wavy flow on a smooth wall and waveless flow along a corrugated surface) are in good agreement. It is demonstrated that, for moderate Reynolds numbers, there is a range of corrugation parameters (amplitude and period) where all linear perturbations of the free surface decay. For high Reynolds numbers, the waveless downward flow is unstable. Various nonlinear wavy regimes induced by varying the corrugation amplitude are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 110–120, January–February, 2007.