Solution of the linear problem of cavitation flow around a curvilinear arc

A method is proposed for the construction, without quadratures, of a solution to the linear problem of the flow of a gradientless stream around an arc of the curve y_n=f(x), whose derivativef(x)=g(x) at x=z is a meromorphic function in the plane z=x+iy. For flow around the arc of a parabola, with fully developed and partial cavitation, convenient finite formulas are obtained and numerical calculations are made. An analogous method may be used to construct, without quadratures, the solution of a number of other problems (a hydroplane, a grid, etc.).Cheboksary. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 34–38, January–February, 1972.     

Classification of the structure of positive radial solutions to\Delta u + K(|x|)u^p = 0inR n

Positive radial solutions to the semilinear elliptic equation \(\Delta u + K(|x|)u^p = 0\) inR ⁿ are studied, wherep > 1,n > 2 andK ≧ 0. It is shown that, under a general condition onK(r) andp, the structure of positive radial solutions becomes one of three types. We give sharp criteria to classify the type of the structure, and apply the result to the conformal scalar curvature equation.     

Temperature in circular tubes with azimuthal disuniform heating

This paper provides an analytical solution to the heat conduction equation in a circular tube, with internal convective boundary conditions and asymmetrical heat supply (with a diametral plane symmetry) on its cylindrical external surface. The two dimensional steady state solution is obtained by means of the finite Fourier transform and expressed in terms of the Biot number. Numerical results and graphs are given for some form of heat supply.

---

Temperaturverteilung in runden Rohren mit azimuthal ungleichförmiger Beheizung

Zusammenfassung Der Aufsatz vermittelt eine analytische Lösung der Gleichung für die Wärmeleitung in einem Rohr mit inneren konvektiven Randbedingungen und asymmetrischer Wärmezufuhr auf der äußeren Zylinderoberfläche (mit Symmetrie in der Diametralebene). Die zweidimensionale Lösung für den stationären Zustand ergab sich mit Hilfe der finiten Fourier-Transformation und wird dargestellt in Abhängigkeit der Biot-Zahl. Numerische Ergebnisse und Diagramme werden für einige Formen der Wärmezufuhr gegeben.

---

Nomenclature a dimensionless inner radius,R _i/R₀ - Bi Biot number,h R _o/K - h coolant heat transfer coefficient - K thermal conductivity - q heat flux - r dimensionless radial coordinate - R _i inside tube radius - R _o outside tube radius - T dimensionless temperature (-T _B¦(R_oq/K) - T _B coolant bulk temperature - U Heaviside step function - Dirac delta function - azimuthal coordinate - temperature     

On the flow between two counter-rotating infinite plane disks

This paper studies the boundary-value problem arising from the behaviour of a fluid occupying the region -1≦x≦1 between two rotating disks, rotating about a common axis perpendicular to their planes when the disks are rotating with the same speed Ω₀ but in the opposite sense. The equations which describe the axially symmetric similarity solutions of this problem are $$\varepsilon H^{iv} + HH''' + GG' = 0$$ $$\varepsilon G'' + HG' - H'G = 0$$ with the boundary conditions $$H( \pm 1) = H'( \pm 1) = 0$$ $$G( - 1) = - 1,{\text{ }}G(1) = 1$$ where ?=v/2Ω₀ and v is the kinematic viscosity. The existence of an odd solution is established. This particular solution satisfies many special conditions, for example, G′ (x, ?)>0. Moreover, precise estimates are obtained on the size and behaviour of the solution as ? ↓ 0.     

Differential equation for a stratified fluid and its particular solutions

The plane steady motion of a stratified ideal incompressible fluid in a gravity field is examined. Considering that the parameter characterizing the fluid particles — their density ₀ — is constant along the streamline, it is convenient to take the stream function as one of the independent variables and, in view of the presence of the gravity force, the Cartesian coordinate as the other. In this study, on the basis of Lavrent'eva's equation [1, 2, 3], the differential equation is derived for a single unknown function — the vertical displacement of the streamline y(y₀, x), where y₀ is its initial position and x is the horizontal coordinate. The particular solutions corresponding to a wave guide, cnoidal and solitary waves and, moreover, waves of the type corresponding to a smooth ascent to a new level are presented. A similar coordinate system was used in [4], but there the problem was reduced to a system of partial differential equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 83–87, September–October, 1986.The authors are grateful to A. A. Barmin for discussing their results.     

Discretized ordinary differential equations and the Conley index

LetN be a compact isolating neighborhood of an isolated invariant setK with respect to an ODEx=f(x) (C) and(h) x=x + h(x, h) be a consistent one-step-discretization of (C). It is proved in this paper that for someh ₀ > 0 and allh ]0, h₀[, the setN isolates an invariant setK(h) of(h) and the discrete Conley index ofK(h) coincides with the continuous Conley index ofK.     

A complete expression of the asymptotic solution of differential equation with three-turning points

Ｗｅｈａｖｅｄｉｓｃｕｓｓｅｄｃｏｎｃｅｐｔｏｆｅｑｕａｔｉｏｎｗｉｔｈｎ＿ｔｕｒｎｉｎｇｐｏｉｎｔｓｉｎｍｙｐａｐｅｒ［１］,ｉ．ｅ．,ａｓｅｃｏｎｄｏｒｄｅｒｌｉｎｅａｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｄ２ｙｄｘ２＋［λ２ｑ１（ｘ）＋λｑ２（ｘ,λ）］ｙ＝０,ｗｈｅｒｅｑ１（ｘ）＝（ｘ－μ１）（ｘ－μ２）…（ｘ－μｎ）ｆ（ｘ）,ｆ（ｘ）≠０,ａｎｄλｉｓａｌａｒｇｅｐａｒａｍｅｔｅｒ．Ａｌｔｈｒｏｕｇｈｔｈｅｆｉｒｓｔｔｅｒｍｏｆｔｈｅａｓｙｍｐｔｏｔｉｃｅｘｐａｎ…     

A note on the soil-water conductivity of a fractal soil

We show that for a fractal soil the soil-water conductivity, K, is given by $$\frac{K}{{K_\varepsilon }} = (\Theta /\varepsilon )^{2D/3 + 2/(3 - D)}$$ where $K_\varepsilon$ is the saturated conductivity, θ the water content, ? its saturated value and D is the fractal dimension obtained from reinterpreting Millington and Quirk's equation for practical values of the porosity ?, as $$D = 2 + 3\frac{{\varepsilon ^{4/3} + (1 - \varepsilon )^{2/3} - 1}}{{2\varepsilon ^{4/3} \ln ,{\text{ }}\varepsilon ^{ - 1} + (1 - \varepsilon )^{2/3} \ln (1 - \varepsilon )^{ - 1} }}$$ .     

Self-similar problem of the motion of a plane layer of heated material with an arbitrary equation of state

The self-similar problem of the nonstationary motion of a plane layer of material in which energy from an external source is released for values of the flux density q₀ on the boundary which are constant in time is considered. The self-similar variable is = m/t, where m is the Lagrangian mass coordinate and t is the time. The characteristic values of the velocity, density, and pressure do not vary with time. For a self-similar problem the energy flux density q must also depend only on the self-similar variable. In this case q() can be an arbitrary function of its argument and can be given by a table. Examples are presented of actual physical processes in which the mass of the energy-release zone increases linearly with time. The equation of state can have an arbitrary form, including specification by a table. The gaseous state of matter for an arbitrary variable adiabatic exponent, the condensed state, and a two-phase state can be described. A solution of the self-similar problem is presented for the heating of a half-space bounded by a vacuum for a certain specific equation of state and various flux densities q₀ and velocities M of the advance of the energy-release zone.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 5, pp. 136–145, September–October, 1975.     

On the stability of certain nonparallel flows of a viscous incompressible liquid in a channel

The stability of very simple nonparallel flows of a viscous incompressible liquid in an infinite plane channel described by the exact solutions of the Navier-Stokes equations is studied. Such solutions are realized between two parallel porous plates when the liquid (or gas) is forced in at one wall and drawn out at the same velocity at the other, with a steady flow of liquid along the channel. In this case the transverse velocity component is constant, and the profile of the longitudinal velocity component is independent of the longitudinal con-ordinate x, being an asymmetric function of the transverse coordinate y. A study of the hydrodynamic stability then reduces to the solution of an equation differing from the Orr-Sommerfeld equation by virtue of the presence of additional terms containing the transverse velocity component of the main flow. By numerically solving both this equation and the ordinary Orr-Sommerfeld equation and comparing the corresponding results for various inflowing Reynolds numbers R₀=v₀h/ (v₀ is the inflow velocity, h is the width of the channel), the effect of the nonparallel and asymmetrical nature of such flows on their stability is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–129, July–August, 1970.     

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     

A Particular Class of Partially Invariant Solutions of the Navier—Stokes Equations

One class of partially invariant solutions of the Navier—Stokes equations is studied here. This class of solutions is constructed on the basis of the four-dimensional algebra L ₄ with the generators Systematic investigation of the case, where the Monge—Ampere equation (10) is hyperbolic (Lf _z + k + l ≥ 0) is given. It is shown that this class of solutions is a particular case of the solutions with linear velocity profile with respect to one or two space variables.     

Criterion for instability of steady-state unsaturated flows

The stability of steady-state solutions to the unsaturated flow equation is examined. Conditions under which infinitesimal disturbances are amplified are determined by linear stability analysis. Uniform suction head profiles are shown to be linearly stable to three-dimensional disturbances. The stability of nonuniform suction head profiles to planar (gc ₁ ? χ ₂) disturbances is examined. When the steady-state suction head solution (Ψ) increases with depth, χ₃, (dΨ/dχ₃ > 0), a condition for the amplification of infinitesimal planar disturbances is identified as $$\frac{{d^2 K(\Psi )}}{{d\Psi ^2 }} > \frac{{\left( {\frac{{dK(\Psi )}}{{d\Psi }}} \right)^2 }}{{K(\Psi )}},$$ , where K(Ψ) is the hydraulic conductivity versus suction head characteristic of the porous medium. The same condition applies when dΨ/dχ₃ < -1. Therefore when the rate of change of the slope of the K - Ψ characteristic curves is larger than the squared slope divided by K, even small disturbances can be amplified exponentially. The smallest wavelength of unstable planar perturbations is shown to be inversely related to the coarseness of the soil. Conditions under which the instability criterion is met are delineated for some commonly employed K - Ψ curves.     

Fourier Parameterization of Attractors for Dissipative Equations in One Space Dimension

For dissipative equations of the form $$u_t + ( - 1)^m u^{(2m)} + f(x,u,u_x ,...,u^{(2m - 1)} ) = 0$$ we show that the global attractor is a Lipschitz graph over a finite dimensional Fourier eigenspace. In particular, the statement applies to the Burgers equation u _t ?u _xx +uu _x =f and the modified Burgers equation u _t ?u _xx +uu _x ?u=0.     

An investigation of one third-order equation not allowed with respect to the time derivative

This paper is devoted to an investigation of the equation ut+u_x+uu_x =u_xxt+u_xx, which is a model equation for the problem of the nonsteady filtration of two immiscible liquids. A combined problem on all axes is set up for this equation: The initial condition u(0, x) and the boundary conditions at infinity are assigned. A solution of the special form u₀(x – ct), whose propagation velocity c is determined from the boundary conditions, is analyzed. The stability of this solution in a linear approximation is demonstrated in a certain particular case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 96–103, November–December, 1978.In conclusion, the author thanks G. I. Barenblatt for constant attention to the present work.     

Small parameter method for determining motion of a viscous incompressible fluid in a thrust bearing

We consider the plane stationary motion of a viscous incompressible fluid between two surfaces. The fixed surface is given by the equation y=h[1+f(x/h)], where the functionf(x/h=h) characterizes the deviation of the fixed surface from the plane y=h(h and , are constants). The moving surface is a plane which moves with constant velocity along the x axis and remains parallel to the plane y=h. The small parameter method is used to solve the problem. The problem formulation is presented in the first section, the solvability of the linear equations obtained using the small parameter method is investigated in the second section, and the third section studies the convergence of the method and finds the radius of convergence of the constructed series.     

Modified dissipativity for a non-linear evolution equation arising in turbulence

We are concerned with the regularity properties for all times of the equation $$\frac{{\partial U}}{{\partial t}}\left( {t,x} \right) = - \frac{{\partial ^2 }}{{\partial x^2 }}\left[ {U\left( {t,{\text{0}}} \right) - U\left( {t,x} \right)} \right]^2 - v\left( { - \frac{{\partial ^2 }}{{\partial x^2 }}} \right)^\alpha U\left( {t,x} \right)$$ which arises, with α=1, in the theory of turbulence. Here U(t,·) is of positive type and the dissipativity α is a non-negative real number. It is shown that for arbitrary ν≧0 and ?>0, there exists a global solution in \(L^\infty [0,\infty ;H^{\tfrac{3}{2} - \varepsilon } (\mathbb{R})]\) . If ν>0 and \(\alpha > \alpha _{cr} = \tfrac{1}{2}\) , smoothness of initial data persists indefinitely. If 0≦α<α _cr, there exist positive constants ν₁(α) and ν₂(α), depending on the data, such that global regularity persists for ν>ν₁(α), whereas, for 0≦ν<ν₂(α), the second spatial derivative at the origin blows up after a finite time. It is conjectured that with a suitable choice of α _cr, similar results hold for the Navier-Stokes equation.     

On the fundamental linear fractional order differential equation

This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t₀s)^2m + 2z(t₀s)^m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t₀)^2m\fracd^2mx(t)dt^2m + 2z(t₀)^m\fracd^mx(t)dt^m + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.     

Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities

On the basis of some very plausible assumptions about the response of physical systems to stimuli, such as Boltzmann's superposition principle and the causality principle, Spence showed that the following characteristics obtain for the modulus and compliance functions: (i) They are analytic in the lower half of the complex frequency plane, (ii) they are limited if the frequency tends to infinity, and (iii) the real and imaginary parts are even and odd functions, respectively, of the frequencyω. It can generally be demonstrated that the real and imaginary parts of every function satisfying these three requirements and (iv) without singularities on the real frequency axis, are interrelated by Kramers-Kronig transforms. Similar relations hold between the logarithm of the modulus and the argument of the function. Under certain conditions the Kramers-Kronig relations may be approximated by rather simple equations. For linear viscoelastic materials, for instance, the following approximate relations were obtained for the components of the complex dynamic shear modulus,G ^* (iω) = G′(ω) + iG″(ω) = G _d (ω) expiδ(ω): $$\begin{gathered} G'' (\omega ) \simeq \frac{\pi }{2}\left( {\frac{{dG'(u)}}{{d In u}}} \right)_{u = \omega } , \hfill \\ G' (\omega ) - G'(o) \simeq - \frac{{\omega \pi }}{2}\left( {\frac{{d[G''(u)/u]}}{{d In u}}} \right)_{u = \omega } , \hfill \\ \delta (\omega ) \simeq \frac{\pi }{2}\left( {\frac{{d In G_d (u)}}{{d In u}}} \right)_{u = \omega } . \hfill \\ \end{gathered} $$ The first of these relations was published long ago by Staverman and Schwarzl and is useful over broad frequency ranges, as is the second relation. The last equation is the most general one, and also is better supported by experiment.