Properties of similarity solutions for flows in turbulent vortex cores

A class of similarity solutions of the equations for turbulent vortex cores matching an external inviscid similarity flow with a power law of circumferential velocity variationv-r ^−m near the rotation axis and constant Bernoulli function is considered. Solutions are found to exist only in a certain range of the indexm of the exponential. For each suchm there are two solutions. The authors wish to apologise for a mistake which resulted in the figures in this paper corresponding to [1] and those in [1] corresponding to this paper. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 60–64, May–June, 1998. The work was financially supported by the Russian Foundation for Basic Research (project No. 95-01-00483).     

Sufficient Conditions for the Existence of Bounded Solutions of Systems of Nonlinear Difference Equations

We obtain sufficient conditions for systems of nonlinear difference equations x(n + 1) = A(x(n))x(n) + f(n), n ∈ ℤ, where A(x) is a matrix function continuous on ℝ ^m , to have solutions in the space of bilateral number sequences. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 165–173, April–June, 2005.     

Theoretical analysis of convective heat transfer enhancement of microencapsulated phase change material slurries

This paper analyzes the convective heat transfer enhancement mechanism of microencapsulated phase change material slurries based on the analogy between convective heat transfer and thermal conduction with thermal sources. The influence of each factor affecting the heat transfer enhancement for laminar flow in a circular tube with constant wall temperature is analyzed using an effective specific heat capacity model. The model is validated with results available in the literature. The analysis and the results clarify the heat transfer enhancement mechanism and the main factors influencing the heat transfer. In addition, the conventional Nusselt number definition of phase change slurries for internal flow is modified to describe the degree of heat transfer enhancement of microencapsulated phase change material slurries. The modification is also consistent evaluation of the convective heat transfer of internal and external flows.c volumetric concentration of microcapsules - c_m mass concentration of microcapsules - c_p specific heat, kJ kg^–1 K^–1 - h_fs phase change material heat of fusion, kJ kg^–1 - h_m* modified convective heat transfer coefficient, W m^–2 K^–1 - k thermal conductivity, W m^–1 K^–1 - k_e effective thermal conductivity of slurry, W m^–1 K^–1 - k_b slurry bulk thermal conductivity, W m^–1 K^–1 - ML dimensionless initial subcooling - Mr dimensionless phase change temperature range - Nu conventional Nusselt number - Nu* improved Nusselt number - q_wⁿ wall heat flux, Wm^–2 - Pe Peclet number - Pr Prandtl number - Re Reynolds number - r radial coordinate, m - r₀ duct radius, m - r₁ dimensionless radial coordinate - Ste Stefan number - T temperature, K - T₁ lower phase change temperature limit, K - T₂ upper phase change temperature limit, K - T_i slurry inlet temperature, K - u axial velocity, m/s - v radial velocity, m/s - x axial coordinate, m - x₁ dimensionless axial coordinate - thermal diffusivity, m²/s - dimensionless temperature - dynamic viscosity, N·s/m² - kinematic viscosity, m²/s - _t width of thermal boundary, m - degree of heat transfer enhancement, = h_m*/(h_m*)_single - b bulk fluid (slurry) - b0 slurry without phase change - l liquid - m mean - s solid - f suspending fluid - p microcapsule particles - w wall - single single-phase fluid     

Study of stable and unstable jet flows on the basis of the boltzmann equation

Free supersonic underexpanded jets are studied using a direct method conservative splitting scheme for solving the Boltzmann equation. Numerical solutions for a jet flowing into a vacuum and into a fluid-filled space are presented for the following ranges of the parameters: Knudsen number 10⁻⁶<Kn<∞ and pressure ratio 10<n<∞. The solutions are compared with experimental data. Instabilities associated with free turbulence effects in the mixing layer are detected for low Kn numbers. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 153–157, March–April, 1998. The work was carried out with support from the Russian Foundation for Fundamental Research (project No. 96-01-00829).     

Specific features of explosive boiling of liquids on a film microheater

Explosive boiling of liquids on film heaters under the action of pulsed heat fluxes q = 10⁸–10⁹ W/m ² is considered. A technique of stroboscopic visualization of boiling stages with a time resolution of 100 nsec is used. Numerous scenarios of evolution of explosive boiling are demonstrated. Conditions of the thermal effect (magnitude of the heat flux, duration and repetition frequency of heat pulses) are found, which ensure single and repeated boiling, intermittent boiling, and boiling with formation of complicated multi-bubble structures. It is noted that homogeneous nucleation is a dominating mechanism of incipience of examined liquids for q > 10⁸ W/m ². __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 81–89, March–April, 2007.     

Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water at 4°C (maximum density) when the surface heat flux varies as x ^m and the velocity outside the boundary layer varies as x ^(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions when the mixed convection parameter λ is greater than some critical value λ _c (dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ _c is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and the limits as m → ∞ and as m → −1.     

On the structure of the set of continuously differentiable solutions of one boundary-value problem for a system of functional differential equations of neutral type

We study the structure of the set of solutions, continuously differentiable for t ∈ R ⁺ = [0; + ∞), of one limit problem for systems of nonlinear functional differential equations of neutral type with nonlinear deviations of argument that depend on an unknown function. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 277–289, April–June, 2007.     

Mass flow-rate control unit to calibrate hot-wire sensors

Hot-wire anemometry is a measuring technique that is widely employed in fluid mechanics research to study the velocity fields of gas flows. It is general practice to calibrate hot-wire sensors against velocity. Calibrations are usually carried out under atmospheric pressure conditions and these suggest that the wire is sensitive to the instantaneous local volume flow rate. It is pointed out, however, that hot wires are sensitive to the instantaneous local mass flow rate and, of course, also to the gas heat conductivity. To calibrate hot wires with respect to mass flow rates per unit area, i.e., with respect to (ρU), requires special calibration test rigs. Such a device is described and its application is summarized within the (ρU) range 0.1–25 kg/m² s. Calibrations are shown to yield the same hot-wire response curves for density variations in the range 1–7 kg/m³. The application of the calibrated wires to measure pulsating mass flows is demonstrated, and suggestions are made for carrying out extensive calibrations to yield the (ρU) wire response as a basis for advanced fluid mechanics research on (ρU) data in density-varying flows.     

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

Laminar-turbulent transition in the boundary layer on cones in a hypersonic flow at high Reynolds numbers per meter

The laminar-turbulent transition is experimentally studied in boundary-layer flows on cones with a rectangular axisymmetric step in the base part of the cone and without the step. The experiments are performed in an A-1 two-step piston-driven gas-dynamic facility with adiabatic compression of the working gas with Mach numbers at the nozzle exit M _∞ = 12–14 and pressures in the settling chamber P₀ = 60–600 MPa. These values of parameters allow obtaining Reynolds numbers per meter near the cone surface equal to Re _1e = (53–200) · 10⁶ m ⁻¹. The transition occurs at Reynolds numbers Re _tr = (2.3–5.7) · 10⁶. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 76–83, May–June, 2007.     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

Steady mixed convection boundary layer flow over a vertical flat plate in a porous medium filled with water at 4°C: case of variable wall temperature

The problem of steady mixed convection boundary layer flow over a vertical impermeable flat plate in a porous medium saturated with water at 4°C (maximum density) when the temperature of the plate varies as x ^m and the velocity outside boundary layer varies as x ^{2 m} , where x measures the distance from the leading edge of the plate and m is a constant is studied. Both cases of the assisting and the opposing flows are considered. The plate is aligned parallel to a free stream velocity U _∞ oriented in the upward or downward direction, while the ambient temperature is T _∞ = T _m (temperature at maximum density). The mathematical models for this problem are formulated, analyzed and simplified, and further transformed into non-dimensional form using non-dimensional variables. Next, the system of governing partial differential equations is transformed into a system of ordinary differential equations using the similarity variables. The resulting system of ordinary differential equations is solved numerically using a finite-difference method known as the Keller-box scheme. Numerical results for the non-dimensional skin friction or shear stress, wall heat transfer, as well as the temperature profiles are obtained and discussed for different values of the mixed convection parameter λ and the power index m. All the numerical solutions are presented in the form of tables and figures. The results show that solutions are possible for large values of λ and m for the case of assisting flow. Dual solutions occurred for the case of opposing flow with limited admissible values of λ and m. In addition, separation of boundary layers occurred for opposing flow, and separation is delayed for the case of water at 4°C (maximum density) compared to water at normal temperature.     

Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations

We consider reaction diffusion equations of the prototype form u _t = u _xx + λ u + |u| ^p-1 u on the interval 0 < x < π, with p > 1 and λ > m ². We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 0. In particular, sign-changing solutions are included. Specifically, we find initial conditions such that the blow-up profile u(t, x) at blow-up time t = T possesses m + 1 intervals of strict monotonicity with prescribed extremal values u ₁, . . . ,u _m . Since u _k = ± ∞ at blow-up time t = T, for some k, this exhausts the dimensional possibilities of trajectories in the m-dimensional fast unstable manifold. Alternatively, we can prescribe the locations x = x ₁, . . . ,x _m of the extrema, at blow-up time, up to a one-dimensional constraint. The proofs are based on an elementary Brouwer degree argument for maps which encode the shapes of solution profiles via their extremal values and extremal locations, respectively. Even in the linear case, such an “interpolation of shape” was not known to us. Our blow-up result generalizes earlier work by Chen and Matano (1989), J. Diff. Eq. 78, 160–190, and Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300 on multi-point blow-up for positive solutions, which were not constrained to possess global extensions for all negative times. Our results are based on continuity of the blow-up time, as proved by Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300, and Quittner (2003), Houston J. Math. 29(3), 757–799, and on a refined variant of Merle’s continuity of the blow-up profile, as addressed in the companion paper Matano and Fiedler (2007) (in preparation). Dedicated to Palo Brunovsky on the occasion of his birthday.     

Optimal Decay Rate of the Compressible Navier–Stokes–Poisson System in {\mathbb {R}^3}

The compressible Navier–Stokes–Poisson (NSP) system is considered in ${\mathbb {R}^3}The compressible Navier–Stokes–Poisson (NSP) system is considered in \mathbb R³{\mathbb {R}^3} in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 respectively as the compressible Navier–Stokes system, but the momentum of the NSP system decays at the L ²-rate (1+t)^{-\frac 14}{(1+t)^{-\frac {1}{4}}} or L ^∞-rate (1 + t)⁻¹ respectively, which is slower than the L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 for compressible Navier–Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737–758, 2007; Liu and Wang, in Comm Math Phys 196:145–173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67–104, 1980] and the L ^∞-rate (1 + t)^−p with p ? (1, 3/2){p \in (1, 3/2)} for irrotational Euler–Poisson system [Guo, in Comm Math Phys 195:249–265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

Second-order nonlinear differential equations with infinite set of periodic solutions

For the differential equation u″ = f(t, u, u′), where the function f: R × R ² → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Published in Neliniini Kolyvannya, Vol. 11, No. 4, pp. 495–500, October–December, 2008.     

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

We consider non-negative solutions of the fast diffusion equation u _t  = Δ u ^m with m ∈ (0, 1) in the Euclidean space , d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ m _c  = (d − 2)/d, or as t approaches the extinction time when m < m _c . For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≧ m _c , or close enough to the extinction time if m < m _c . Such results are new in the range m ≦ m _c where previous approaches fail. In the range m _c  < m < 1, we improve on known results.     

Loading of obstacles by explosion of a low-density sheet explosive

The dependence of the detonation velocity of aNIL-1 low-density sheet explosive on density is found in the range of charge densities0.1–0.3 g/cm ³. The equation of state of theNIL-1 detonation products with a linear dependence of the effective isentropic exponent of unloading on the density of an explosive that is acceptable for applied calculations is proposed. Calculated estimates of the mechanical action of anNIL-1 explosion on obstacles from several powerful explosive compositions are given. Institute of Experimental Physics, Sarov 607190. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 43–47, May–June, 2000.     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

Dynamical deformation of a flat liquid–liquid interface

The passage of solid spheres through a liquid–liquid interface was experimentally investigated using a high-speed video and PIV (particle image velocimetry) system. Experiments were conducted in a square Plexiglas column of 0.1 m. The Newtonian Emkarox (HV45 50 and 65% wt) aqueous solutions were employed for the dense phase, while different silicone oils of different viscosity ranging from 10 to 100 mPa s were used as light phase. Experimental results quantitatively reveal the effect of the sphere’s size, interfacial tension and viscosity of both phases on the retaining time and the height of the liquid entrained behind the sphere. These data were combined with our previous results concerning the passage of a rising bubble through a liquid–liquid interface in order to propose a general relationship for the interface breakthrough for the wide range of Mo ₁/Mo ₂ ∈ [2 × 10⁻⁵–5 × 10⁴] and Re ₁/Re ₂ ∈ [2 × 10⁻³–5 × 10²].