On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

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.     

Layered petroleum reservoirs with cross-flows

Studied is a cylindrical reservoir consisting of three layers: a water-containing bottom layer, and two oil-containing top layers from whose upper layer oil is produced. For its solution, a corrected version of the finite Hankel transform for Neumann-Neumann boundary conditions was used together with numerical inversion of the Laplace transform. The effects of the water zone on the unsteady state pressure in the reservoir were evaluated at distances away from the well and at the well-bore itself. We found that the vertical pressure drop increases gradually with time and is more significant in the vicinity of the well-bore. For constant production and at any time t, smaller reservoirs experience higher pressure drops than larger ones. For the reservoir investigated, we found that for nondimensional time t _Dw <10⁴ the presence of a second fluid (water) has no effect on the pressure drop. Of all the formation and fluid properties investigated, porosity has the largest effect on pressure.Nomenclature c ₁, c ₂ Oil and water compressibilities, vol/vol/atm, vol/vol/psi - h Height of water zone from bottom of reservoir, cm, ft - h _D h/r _w , non-dimensional - H Height of reservoir, cm, ft - H _D H/r _w, non-dimensional - J ₀, J ₁ Bessel functions of the first kind, zero and first-order - K _r2, K _r1 Oil and water zones, horizontal permeabilities, darcies, md - K _z2, K _z1 Oil and water zones, vertical permeabilities, darcies, md - k ₁ n=1, 2, 3... - k ₂ n=1, 2, 3... - k _1,0 - k _2,0 - p(r, z, t) P(r, z, 0)–P(r, z, t), atm, psi - P(r, z, t) Pressure at any layer in the reservoir, atm, psi - P(r, z, 0) Initial pressure at any layer in the reservoir, atm, psi - P _D , non-dimensional - q Constant production rate of well, cc/sec, barrels/day - r Radius of reservoir, cm, ft - r _D r/r _w , non-dimensional - r _e Drainage radius, cm, ft - r _eD re/r _w , non-dimensional - r _w Well-bore radius, cm, ft - t Time, sec, hr - _Dw (k _r2 t)/( _{2 2} c ₂ r _w ² ), non-dimensional - Y ₀, Y ₁ Bessel functions of the second kind, zero and first-order - z Distance z measured vertically upward from bottom of reservoir, cm, ft - Z _D z/r _w , non-dimensional - z ₁ Height of the bottom of the producing layer, cm, ft - z _1D z ₁/r _w , non-dimensional - z ₂ Height of the top of the producing layer, cm, ft - z _2,D z ₂/r _w , non-dimensional - _n nth positive root of equation (18) - ₁ k _z1/k _r1, non-dimensional - ₂ k _z2/k _r2, non-dimensional - _{1 1 1} c ₁/k _r1, hydraulic diffusivity of layer I - _{2 2 2} c ₂/k _r2, hydraulic diffusivity of layers II and III - ₂, ₁ Viscosity of oil and water, cp, cp - _{n n} /r _w , l/cm, l/ft - ₂, ₁ Porosity of oil and water-filled zones, fraction - ( ₁/ ₂) (k _z2/k _z1), non-dimensional     

Statistical mechanics of temporary polymer networks I. The equilibrium theory

In part I of this work (the present article) the equilibrium state of temporary polymer networks is treated in the framework of thermodynamics and statistical mechanics. The network is described as an open system. Thereby we use a modified spring-bead model in which the beads represent junctions that decay and reform thus adding a viscous component to the assumed elastic behaviour of the permanent network. The relevant statistical equation — analogous to Liouville's equation — is solved. The grand-canonical probability density function and two of three equations of state are derived. Explicit formulae are given for several relevant probabilities. For instance the probabilityw (z)dz that a network chain connecting two junctions has a contour length betweenz andz +dz is given by the Wien type formulaw(z) =A z ³ exp {–B z} whereA andB do not depend onz.     

The Instationary Navier–Stokes Equations in Weighted Bessel-Potential Spaces

We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain W ì \mathbbRⁿ \Omega \subset {{\mathbb{R}}^{n}} . The class of solutions is contained in L^r(0, T; H^{b, q}_w (W))L^{r}(0, T; H^{\beta, q}_{w} (\Omega)), where H^{b, q}_w (W){H^{\beta, q}_{w}} (\Omega) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.     

Viscosity and compliance from molar mass distributions using double reptation models

Various empirical correlations between linear viscoelastic properties and molar mass distribution of linear polymers have been proposed. Many of these summarize the distribution in terms of the first few moments. This is sufficient when studying samples of limited variability. In parallel, various fundamental models that enable calculation of these rheological characteristics from the full distribution have been proposed. The advantage of the modeling approach is the ease of creating distributions, thus enabling independent control of moments up to any desired order. It is the goal of this contribution to explore this advantage and compare the findings of the single exponential (DRSE) and modified time-dependent diffusion (DRmTDD) double reptation models with the empirical relations. The models predict that η ₀ is primarily a function of the weight-average molar mass M _w, with subtle dependence on polydispersity. Furthermore, the model depends mainly on a combination of the second (M _z/M _w) and third (M _z+1/M _z) polydispersity index. The DRmTDD model shows that conventional moment-based fit equations are only valid for limited distribution parameter ranges. General fit equations are proposed based on genetic programming. The details of the predictions are sensitive to the precise physical model formulation and need to be validated from experiments.     

Boundary Regularity in Variational Problems

We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRⁿ ? \mathbbR^N{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem

minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      8. On the Wellposedness of Three-Dimensional Inhomogeneous Navier–Stokes Equations in the Critical Spaces      Hammadi Abidi  Guilong Gui  Ping Zhang 《Archive for Rational Mechanics and Analysis》2012,204(1):189-230 We prove the local wellposedness of three-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data in the critical Besov spaces, without assumptions of small density variation. Furthermore, if the initial velocity field is small enough in the critical Besov space [(B)\dot]1/22,1(\mathbbR3){\dot B^{1/2}_{2,1}(\mathbb{R}^3)} , this system has a unique global solution.      9. Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations      Juncheng Wei  Tobias Weth 《Archive for Rational Mechanics and Analysis》2008,190(1):83-106 We consider the nonlinear elliptic system where and is the unit ball. We show that, for every and , the above problem admits a radially symmetric solution (u β , v β ) such that u β − v β changes sign precisely k times in the radial variable. Furthermore, as , after passing to a subsequence, u β → w + and v β → w − uniformly in , where w = w +− w − has precisely k nodal domains and is a radially symmetric solution of the scalar equation Δw − w + w 3 = 0 in , w = 0 on . Within a Hartree–Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose–Einstein double condensates with strong repulsion.      10. Study of pulsating flow in close-coupled catalyst manifolds using phase-locked hot-wire anemometry      T.?PersoonsEmail author  E.?Van den Bulck  S.?Fausto 《Experiments in fluids》2004,36(2):217-232       11. On a Differential Equation with State-Dependent Delay      Eugen Stumpf 《Journal of Dynamics and Differential Equations》2012,24(2):197-248 We study a family of scalar differential equations with a single parameter a > 0 and delay r > 0. In the case of the constant delay r = 1 it is known that for parameters 0 < a < 1 the trivial solution of this family is asymptotically stable, whereas for a > 1 the trivial solution gets unstable, and a global center-unstable manifold connects the trivial solution to a slowly oscillating periodic orbit. Here, we consider a state-dependent delay r = r(x(t)) > 0 instead of the constant one, and generalize the result on the existence of slowly oscillating periodic solutions for parameters a > 1 under modest conditions on the delay function r.      12. The Positive Solutions of the Matukuma Equation and the Problem of Finite Radius and Finite Mass      Jürgen Batt  Yi Li 《Archive for Rational Mechanics and Analysis》2010,198(2):613-675 This work is an extensive study of the 3 different types of positive solutions of the Matukuma equation ${\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p},p >1 ,\lambda >0 }${\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p},p >1 ,\lambda >0 }: the E-solutions (regular at r = 0), the M-solutions (singular at r = 0) and the F-solutions (whose existence begins away from r = 0). An essential tool is a transformation of the equation into a 2-dimensional asymptotically autonomous system, whose limit sets (by a theorem of H. R. Thieme) are the limit sets of Emden–Fowler systems, and serve as to characterizate the different solutions. The emphasis lies on the study of the M-solutions. The asymptotic expansions obtained make it possible to apply the results to the important question of stellar dynamics, solutions to which lead to galactic models (stationary solutions of the Vlasov–Poisson system) of finite radius and/or finite mass for different p, λ.      13. Some Aspects of the Asymptotic Dynamics of Solutions of the Homogeneous Navier–Stokes Equations in General Domains      Zdeněk Skalák 《Journal of Mathematical Fluid Mechanics》2010,12(4):503-535 In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R3 and β ∈[1/2, 1) , then there exist C0 > 1 and δ0 ∈ (0, 1) such that \frac |||w(t)|||b|||w(t + d)|||b ￡ C0{\frac {|||w(t)|||_\beta}{|||w(t + \delta)|||_{\beta}}} \leq C_0      14. Inertia effects in rheometrical flow systems Part 3: Some energy considerations with respect to the flow field in the balance rheometer      H. A. Waterman  S. S. Makh 《Rheologica Acta》1981,20(5):471-477 Summary Following up a previous paper by one of the presents authors on the flow field in the balance rheometer, inertia effects being included, in this paper some energy considerations with respect to this flow field are presented. It is shown that in a frame rotating with the same angular velocity as the hemispheres the power supplied by these hemispheres equals the rate of energy dissipation in the sample, i.e. in this coordinate system there is no stress power paradox. Further it is shown that the elastic couple for a Newtonian liquid, appearing in the calculations, stems from the extra kinetic energy caused by the deviation of the actual flow field from the flow field that appears when inertia effects are ignored. Zusammenfassung Als Fortsetzung des früheren Beitrages eines der hier genannten Autoren über das Strömungsfeld in einem Képès-Rheometer unter Berücksichtigung der Flüssigkeitsträgheit werden in diesem Beitrag einige Energiebetrachtungen angestellt. Es wird gezeigt, daß in einem Koordinatensystem, das mit gleicher Winkelgeschwindigkeit wie die Halbkugeln rotiert, die durch diese Halbkugeln zugeführte Leistung der in der Probe dissipierten Leistung gleich ist, d. h. daß in diesem Koordinatensystem das sogenannte Spannungsenergieparadox nicht vorliegt. Es wird weiter gezeigt, daß das bei einer newtonschen Flüssigkeit auftretende elastische Drehmoment seinen Ursprung in der zusätzlichen kinetischen Energie hat, die der Abweichung des tatsächlichen Strömungsfeldes von dem unter Vernachlässigung der Flüssigkeitsträgheit berechneten Strömungsfeld entspricht. a distance between rotation axes of orthogonal rheometer - b, c positive numbers (see eq. [29]) - f n (z) j n (z),y n (z),h n (1) (z) orh n (2) (z) - h distance between discs of orthogonal rheometer - h n (1) (z) =j n (z) +iy n (z) spherical Bessel functions of the third kind and ordern - h n (2) (z) =j n (z) –iy n (z) spherical Bessel functions of the third kind and ordern - i - j n (z) spherical Bessel function of the first kind and ordern - k – (/2)1/2 - n integer - p 1 j 1(r 2)y 1(r 1) –j 1(r 1)y 1(r 2) - q 1 - r spherical polar coordinate - r 1(r 2) radius of inner (outer) hemisphere - s 1 - t time - u r ,u ,u physical components of displacement - x, z variables - x, y, z cartesian coordinates in eq. [1] - y n (z) spherical Bessel function of the second kind and ordern - E kin kinetic energy - E s stored energy - F force - F n (z) J n (z),Y n (z),H n (1) (z) orH n (2) (z) - G * =G + iG complex shear modulus - H n (1) (z) =J n (z) +iY n (z) Bessel functions of the third kind and ordern - H n (2) (z) =J n (z) –iY n (z) Bessel functions of the third kind and ordern - Im imaginary part of - J n (z) Bessel function of the first kind and ordern - N - P energy supplied to the sample during one cycle - Re real part of - S strain - U - W energy dissipated in the sample during one cycle - Y n Bessel function of the second kind and ordern - (/G *)1/2-complex shear wave factor - /(r 1 +r 2) - loss angle - angle between rotation axes of balance rheometer - viscosity - spherical polar coordinate - order of cylinder function (see eq. [24]) - linear combination of spherical Bessel functions of first, second, and third kind, in which the coefficients are independent of the argument and the order - µ, v constants (see eq. [24]) - see - density - shear stress - spherical polar coordinate - cylinder functions - angular velocity With 3 figures      15. A New Approach to Equations with Memory      Mauro Fabrizio  Claudio Giorgi  Vittorino Pata 《Archive for Rational Mechanics and Analysis》2010,198(1):189-232 We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function ${G : \mathbb{R}^+ \to \mathbb{R}^+}We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G : \mathbbR+ ? \mathbbR+{G : \mathbb{R}^+ \to \mathbb{R}^+} such that $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0      16. The onset of buoyancy-driven convecion in a horizontal fluid layer subjected to evaporative cooling      Chang Kyun ChoiEmail author  Joung Hwan Park  Min Chan Kim 《Heat and Mass Transfer》2004,41(2):155-162       17. Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylene melts in shear      Julia?A.?Resch  Joachim?Kaschta  Friedrich?Wolff  Helmut?MünstedtEmail author 《Rheologica Acta》2011,50(1):53-63 The stress dependencies of the steady-state viscosity η and, particularly, that of the steady-state elastic compliance J e of various linear isotactic polypropylenes (PP) and one long-chain branched PP are investigated using creep-recovery tests. The creep stresses applied range from 2 to 10,000 Pa. In order to discuss the stress-dependent viscosity η and elastic compliance J e with respect to the influence of the weight average molar mass M w and the polydispersity factor M w/M n the PP are characterized by SEC–MALLS. For the linear PP, linear steady-state elastic compliances Je0J_{\rm e}^0 in the range of 10 − 5–10 − 3 Pa − 1 are obtained depending on the molar mass distribution. Je0J_{\rm e}^0 of the LCB-PP is distinctly higher and comes to lie at around 10 − 2 Pa − 1. Je0J_{\rm e}^0 is found to be independent of M w but strongly dependent on polydispersity. η and J e decrease with increasing stress. For the linear PP, J e as a function of the stress τ is temperature independent. The higher M w/M n the stronger is the shear thinning of η and the more pronounced is the stress dependence of J e. For the LCB-PP, the strongest stress dependence of η and J e is observed. Furthermore, for all PP J e reacts more sensitively to an increasing stress than η. A qualitative explanation for the stronger stress dependence of J e compared to η is given by analyzing the contribution of long relaxation times to the viscosity and elasticity.      18. Linearized Instability in the Magnetic Bénard Problem      B. Scarpellini 《Journal of Mathematical Fluid Mechanics》1999,1(2):187-223 In the magnetic Bénard problem, a conducting fluid moves in an infinite horizontal layer W = R2  ×(- 1/2, 1/2) \Omega = R^{2} \,\times (- {1 \over 2}, {1 \over 2}) , subject to a temperature gradient T0 - T1 T_0 - T_1 and to a magnetic field B perpendicular to the layer. The relevant equations admit a trivial equilibrium state w0 w_0 . We investigate the loss of stability of w0 w_0 under perturbations in L2(W)7 L^2(\Omega)^7 , when T0 - T1 T_0 - T_1 varies in the range (0,+￥) (0,+\infty) , and with B kept constant. Since linearized instability entails Ljapounov instability (Theorem 1 and comments), we study the nonselfadjoint linearization around w0 w_0 on L2(W)7 L^2(\Omega)^7 , and the dependence of its spectrum on T0 - T1 T_0 - T_1 . This leads via Orr-Sommerfeld to a holomorphic function whose parameter-dependent zeros completely describe the spectrum (Theorems 4, 4' and lemmas). The existence of points of loss of stability then follow (Theorem 5) together with numerical information about their location.      19. Application of dispersion theory to time domain reflectometry in soils      W. K. P. Van Loon  E. Perfect  P. H. Groenevelt  B. D. Kay 《Transport in Porous Media》1991,6(4):391-406 With time domain reflectometry (TDR) two dispersive parameters, the dielectric constant, , and the electrical conductivity, can be measured. Both parameters are nonlinear functions of the volume fractions in soil. Because the volume function of water ( w) can change widely in the same soil, empirical equations have been derived to describe these relations. In this paper, a theoretical model is proposed based upon the theory of dispersive behaviour. This is compared with the empirical equations. The agreement between the empirical and theoretical aproaches was highly significant: the ( w) relation of Topp et al. had a coefficient of determination r 2 = 0.996 and the (u) relation of Smith and Tice, for the unfrozen water content, u, at temperatures below 0°C, had an r 2 = 0.997. To obtain ( w) relations, calibration measurements were performed on two soils: Caledon sand and Guelph silt loam. For both soils, an r 2 = 0.983 was obtained between the theoretical model and the measured values. The correct relations are especially important at low water contents, where the interaction between water molecules and soil particles is strong.      20. Squeezing flows of Newtonian liquid films an analysis including fluid inertia      R. J. Grimm 《Applied Scientific Research》1976,32(2):149-166 A theoretical study is made of the flow behavior of thin Newtonian liquid films being squeezed between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations.Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.Nomenclature a half of the distance, or gap, between the two plates - a 0 the value of a at time t=0 - adot da/dt - ä d2 a/dt 2 - d3 a/dt 3 - a i components of a contravariant acceleration vector - f unknown function of z 0 and t defined in (6) - f i function defined in (9) f 1=r 0 g(z 0, t) f 2= 0 f 3=f(z 0, t) - F force applied to the plates - g unknown function of z 0 and t defined in (6) - g g/z 0 - h grid dimension in the z 0 direction (see Fig. 5) - Christoffel symbol - i, j, k, l indices - k grid dimension in the t direction (see Fig. 5) - r radial coordinate direction defined in Fig. 1 - r 0 radial convected coordinate - R radius of the circular plates - t time - v r fluid velocity in the r direction - v z fluid velocity in the z direction - v fluid velocity in the direction - x i cylindrical coordinate x 1=r x2= x3=z - z vertical coordinate direction defined in Fig. 1 - z 0 vertical convected coordinate - tangential coordinate direction - 0 tangential convected coordinate - viscosity - kinematic viscosity, / - i convected coordinate 1=r0 2=0 3=z0 - density      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On the Wellposedness of Three-Dimensional Inhomogeneous Navier–Stokes Equations in the Critical Spaces

We prove the local wellposedness of three-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data in the critical Besov spaces, without assumptions of small density variation. Furthermore, if the initial velocity field is small enough in the critical Besov space [(B)\dot]^1/2_2,1(\mathbbR³){\dot B^{1/2}_{2,1}(\mathbb{R}^3)} , this system has a unique global solution.     

Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations

We consider the nonlinear elliptic system

On a Differential Equation with State-Dependent Delay

We study a family of scalar differential equations with a single parameter a > 0 and delay r > 0. In the case of the constant delay r = 1 it is known that for parameters 0 < a < 1 the trivial solution of this family is asymptotically stable, whereas for a > 1 the trivial solution gets unstable, and a global center-unstable manifold connects the trivial solution to a slowly oscillating periodic orbit. Here, we consider a state-dependent delay r = r(x(t)) > 0 instead of the constant one, and generalize the result on the existence of slowly oscillating periodic solutions for parameters a > 1 under modest conditions on the delay function r.     

The Positive Solutions of the Matukuma Equation and the Problem of Finite Radius and Finite Mass

This work is an extensive study of the 3 different types of positive solutions of the Matukuma equation ${\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p},p >1 ,\lambda >0 }${\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p},p >1 ,\lambda >0 }: the E-solutions (regular at r = 0), the M-solutions (singular at r = 0) and the F-solutions (whose existence begins away from r = 0). An essential tool is a transformation of the equation into a 2-dimensional asymptotically autonomous system, whose limit sets (by a theorem of H. R. Thieme) are the limit sets of Emden–Fowler systems, and serve as to characterizate the different solutions. The emphasis lies on the study of the M-solutions. The asymptotic expansions obtained make it possible to apply the results to the important question of stellar dynamics, solutions to which lead to galactic models (stationary solutions of the Vlasov–Poisson system) of finite radius and/or finite mass for different p, λ.     

Some Aspects of the Asymptotic Dynamics of Solutions of the Homogeneous Navier–Stokes Equations in General Domains

In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R³ and β ∈[1/2, 1) , then there exist C₀ > 1 and δ₀ ∈ (0, 1) such that

Inertia effects in rheometrical flow systems Part 3: Some energy considerations with respect to the flow field in the balance rheometer

Summary Following up a previous paper by one of the presents authors on the flow field in the balance rheometer, inertia effects being included, in this paper some energy considerations with respect to this flow field are presented. It is shown that in a frame rotating with the same angular velocity as the hemispheres the power supplied by these hemispheres equals the rate of energy dissipation in the sample, i.e. in this coordinate system there is no stress power paradox. Further it is shown that the elastic couple for a Newtonian liquid, appearing in the calculations, stems from the extra kinetic energy caused by the deviation of the actual flow field from the flow field that appears when inertia effects are ignored.

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Zusammenfassung Als Fortsetzung des früheren Beitrages eines der hier genannten Autoren über das Strömungsfeld in einem Képès-Rheometer unter Berücksichtigung der Flüssigkeitsträgheit werden in diesem Beitrag einige Energiebetrachtungen angestellt. Es wird gezeigt, daß in einem Koordinatensystem, das mit gleicher Winkelgeschwindigkeit wie die Halbkugeln rotiert, die durch diese Halbkugeln zugeführte Leistung der in der Probe dissipierten Leistung gleich ist, d. h. daß in diesem Koordinatensystem das sogenannte Spannungsenergieparadox nicht vorliegt. Es wird weiter gezeigt, daß das bei einer newtonschen Flüssigkeit auftretende elastische Drehmoment seinen Ursprung in der zusätzlichen kinetischen Energie hat, die der Abweichung des tatsächlichen Strömungsfeldes von dem unter Vernachlässigung der Flüssigkeitsträgheit berechneten Strömungsfeld entspricht.

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a distance between rotation axes of orthogonal rheometer - b, c positive numbers (see eq. [29]) - f _n (z) j _n (z),y _n (z),h _n ⁽¹⁾ (z) orh _n ⁽²⁾ (z) - h distance between discs of orthogonal rheometer - h _n ⁽¹⁾ (z) =j _n (z) +iy _n (z) spherical Bessel functions of the third kind and ordern - h _n ⁽²⁾ (z) =j _n (z) –iy _n (z) spherical Bessel functions of the third kind and ordern - i - j _n (z) spherical Bessel function of the first kind and ordern - k – (/2)^1/2 - n integer - p ₁ j ₁(r ₂)y ₁(r ₁) –j ₁(r ₁)y ₁(r ₂) - q ₁ - r spherical polar coordinate - r ₁(r ₂) radius of inner (outer) hemisphere - s ₁ - t time - u _r ,u ,u physical components of displacement - x, z variables - x, y, z cartesian coordinates in eq. [1] - y _n (z) spherical Bessel function of the second kind and ordern - E _kin kinetic energy - E _s stored energy - F force - F _n (z) J _n (z),Y _n (z),H _n ⁽¹⁾ (z) orH _n ⁽²⁾ (z) - G ^* =G + iG complex shear modulus - H _n ⁽¹⁾ (z) =J _n (z) +iY _n (z) Bessel functions of the third kind and ordern - H _n ⁽²⁾ (z) =J _n (z) –iY _n (z) Bessel functions of the third kind and ordern - Im imaginary part of - J _n (z) Bessel function of the first kind and ordern - N - P energy supplied to the sample during one cycle - Re real part of - S strain - U - W energy dissipated in the sample during one cycle - Y _n Bessel function of the second kind and ordern - (/G ^*)^1/2-complex shear wave factor - /(r ₁ +r ₂) - loss angle - angle between rotation axes of balance rheometer - viscosity - spherical polar coordinate - order of cylinder function (see eq. [24]) - linear combination of spherical Bessel functions of first, second, and third kind, in which the coefficients are independent of the argument and the order - µ, v constants (see eq. [24]) - see - density - shear stress - spherical polar coordinate - cylinder functions - angular velocity With 3 figures     

A New Approach to Equations with Memory

We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function ${G : \mathbb{R}^+ \to \mathbb{R}^+}We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G : \mathbbR⁺ ? \mathbbR⁺{G : \mathbb{R}^+ \to \mathbb{R}^+} such that

$G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0      16. The onset of buoyancy-driven convecion in a horizontal fluid layer subjected to evaporative cooling      Chang Kyun ChoiEmail author  Joung Hwan Park  Min Chan Kim 《Heat and Mass Transfer》2004,41(2):155-162       17. Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylene melts in shear      Julia?A.?Resch  Joachim?Kaschta  Friedrich?Wolff  Helmut?MünstedtEmail author 《Rheologica Acta》2011,50(1):53-63 The stress dependencies of the steady-state viscosity η and, particularly, that of the steady-state elastic compliance J e of various linear isotactic polypropylenes (PP) and one long-chain branched PP are investigated using creep-recovery tests. The creep stresses applied range from 2 to 10,000 Pa. In order to discuss the stress-dependent viscosity η and elastic compliance J e with respect to the influence of the weight average molar mass M w and the polydispersity factor M w/M n the PP are characterized by SEC–MALLS. For the linear PP, linear steady-state elastic compliances Je0J_{\rm e}^0 in the range of 10 − 5–10 − 3 Pa − 1 are obtained depending on the molar mass distribution. Je0J_{\rm e}^0 of the LCB-PP is distinctly higher and comes to lie at around 10 − 2 Pa − 1. Je0J_{\rm e}^0 is found to be independent of M w but strongly dependent on polydispersity. η and J e decrease with increasing stress. For the linear PP, J e as a function of the stress τ is temperature independent. The higher M w/M n the stronger is the shear thinning of η and the more pronounced is the stress dependence of J e. For the LCB-PP, the strongest stress dependence of η and J e is observed. Furthermore, for all PP J e reacts more sensitively to an increasing stress than η. A qualitative explanation for the stronger stress dependence of J e compared to η is given by analyzing the contribution of long relaxation times to the viscosity and elasticity.      18. Linearized Instability in the Magnetic Bénard Problem      B. Scarpellini 《Journal of Mathematical Fluid Mechanics》1999,1(2):187-223 In the magnetic Bénard problem, a conducting fluid moves in an infinite horizontal layer W = R2  ×(- 1/2, 1/2) \Omega = R^{2} \,\times (- {1 \over 2}, {1 \over 2}) , subject to a temperature gradient T0 - T1 T_0 - T_1 and to a magnetic field B perpendicular to the layer. The relevant equations admit a trivial equilibrium state w0 w_0 . We investigate the loss of stability of w0 w_0 under perturbations in L2(W)7 L^2(\Omega)^7 , when T0 - T1 T_0 - T_1 varies in the range (0,+￥) (0,+\infty) , and with B kept constant. Since linearized instability entails Ljapounov instability (Theorem 1 and comments), we study the nonselfadjoint linearization around w0 w_0 on L2(W)7 L^2(\Omega)^7 , and the dependence of its spectrum on T0 - T1 T_0 - T_1 . This leads via Orr-Sommerfeld to a holomorphic function whose parameter-dependent zeros completely describe the spectrum (Theorems 4, 4' and lemmas). The existence of points of loss of stability then follow (Theorem 5) together with numerical information about their location.      19. Application of dispersion theory to time domain reflectometry in soils      W. K. P. Van Loon  E. Perfect  P. H. Groenevelt  B. D. Kay 《Transport in Porous Media》1991,6(4):391-406 With time domain reflectometry (TDR) two dispersive parameters, the dielectric constant, , and the electrical conductivity, can be measured. Both parameters are nonlinear functions of the volume fractions in soil. Because the volume function of water ( w) can change widely in the same soil, empirical equations have been derived to describe these relations. In this paper, a theoretical model is proposed based upon the theory of dispersive behaviour. This is compared with the empirical equations. The agreement between the empirical and theoretical aproaches was highly significant: the ( w) relation of Topp et al. had a coefficient of determination r 2 = 0.996 and the (u) relation of Smith and Tice, for the unfrozen water content, u, at temperatures below 0°C, had an r 2 = 0.997. To obtain ( w) relations, calibration measurements were performed on two soils: Caledon sand and Guelph silt loam. For both soils, an r 2 = 0.983 was obtained between the theoretical model and the measured values. The correct relations are especially important at low water contents, where the interaction between water molecules and soil particles is strong.      20. Squeezing flows of Newtonian liquid films an analysis including fluid inertia      R. J. Grimm 《Applied Scientific Research》1976,32(2):149-166 A theoretical study is made of the flow behavior of thin Newtonian liquid films being squeezed between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations.Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.Nomenclature a half of the distance, or gap, between the two plates - a 0 the value of a at time t=0 - adot da/dt - ä d2 a/dt 2 - d3 a/dt 3 - a i components of a contravariant acceleration vector - f unknown function of z 0 and t defined in (6) - f i function defined in (9) f 1=r 0 g(z 0, t) f 2= 0 f 3=f(z 0, t) - F force applied to the plates - g unknown function of z 0 and t defined in (6) - g g/z 0 - h grid dimension in the z 0 direction (see Fig. 5) - Christoffel symbol - i, j, k, l indices - k grid dimension in the t direction (see Fig. 5) - r radial coordinate direction defined in Fig. 1 - r 0 radial convected coordinate - R radius of the circular plates - t time - v r fluid velocity in the r direction - v z fluid velocity in the z direction - v fluid velocity in the direction - x i cylindrical coordinate x 1=r x2= x3=z - z vertical coordinate direction defined in Fig. 1 - z 0 vertical convected coordinate - tangential coordinate direction - 0 tangential convected coordinate - viscosity - kinematic viscosity, / - i convected coordinate 1=r0 2=0 3=z0 - density

Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylene melts in shear

The stress dependencies of the steady-state viscosity η and, particularly, that of the steady-state elastic compliance J _e of various linear isotactic polypropylenes (PP) and one long-chain branched PP are investigated using creep-recovery tests. The creep stresses applied range from 2 to 10,000 Pa. In order to discuss the stress-dependent viscosity η and elastic compliance J _e with respect to the influence of the weight average molar mass M _w and the polydispersity factor M _w/M _n the PP are characterized by SEC–MALLS. For the linear PP, linear steady-state elastic compliances J_e⁰J_{\rm e}^0 in the range of 10^− 5–10^− 3 Pa^− 1 are obtained depending on the molar mass distribution. J_e⁰J_{\rm e}^0 of the LCB-PP is distinctly higher and comes to lie at around 10^− 2 Pa^− 1. J_e⁰J_{\rm e}^0 is found to be independent of M _w but strongly dependent on polydispersity. η and J _e decrease with increasing stress. For the linear PP, J _e as a function of the stress τ is temperature independent. The higher M _w/M _n the stronger is the shear thinning of η and the more pronounced is the stress dependence of J _e. For the LCB-PP, the strongest stress dependence of η and J _e is observed. Furthermore, for all PP J _e reacts more sensitively to an increasing stress than η. A qualitative explanation for the stronger stress dependence of J _e compared to η is given by analyzing the contribution of long relaxation times to the viscosity and elasticity.     

Linearized Instability in the Magnetic Bénard Problem

In the magnetic Bénard problem, a conducting fluid moves in an infinite horizontal layer W = R²  ×(- ¹/₂, ¹/₂) \Omega = R^{2} \,\times (- {1 \over 2}, {1 \over 2}) , subject to a temperature gradient T₀ - T₁ T_0 - T_1 and to a magnetic field B perpendicular to the layer. The relevant equations admit a trivial equilibrium state w₀ w_0 . We investigate the loss of stability of w₀ w_0 under perturbations in L²(W)⁷ L^2(\Omega)^7 , when T₀ - T₁ T_0 - T_1 varies in the range (0,+￥) (0,+\infty) , and with B kept constant. Since linearized instability entails Ljapounov instability (Theorem 1 and comments), we study the nonselfadjoint linearization around w₀ w_0 on L²(W)⁷ L^2(\Omega)^7 , and the dependence of its spectrum on T₀ - T₁ T_0 - T_1 . This leads via Orr-Sommerfeld to a holomorphic function whose parameter-dependent zeros completely describe the spectrum (Theorems 4, 4' and lemmas). The existence of points of loss of stability then follow (Theorem 5) together with numerical information about their location.     

Application of dispersion theory to time domain reflectometry in soils

With time domain reflectometry (TDR) two dispersive parameters, the dielectric constant, , and the electrical conductivity, can be measured. Both parameters are nonlinear functions of the volume fractions in soil. Because the volume function of water ( _w) can change widely in the same soil, empirical equations have been derived to describe these relations. In this paper, a theoretical model is proposed based upon the theory of dispersive behaviour. This is compared with the empirical equations. The agreement between the empirical and theoretical aproaches was highly significant: the ( _w) relation of Topp et al. had a coefficient of determination r ² = 0.996 and the (_u) relation of Smith and Tice, for the unfrozen water content, _u, at temperatures below 0°C, had an r ² = 0.997. To obtain ( _w) relations, calibration measurements were performed on two soils: Caledon sand and Guelph silt loam. For both soils, an r ² = 0.983 was obtained between the theoretical model and the measured values. The correct relations are especially important at low water contents, where the interaction between water molecules and soil particles is strong.     

Squeezing flows of Newtonian liquid films an analysis including fluid inertia

A theoretical study is made of the flow behavior of thin Newtonian liquid films being squeezed between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations.Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.Nomenclature a half of the distance, or gap, between the two plates - a ₀ the value of a at time t=0 - adot da/dt - ä d² a/dt ² - d³ a/dt ³ - a ⁱ components of a contravariant acceleration vector - f unknown function of z ₀ and t defined in (6) - f ⁱ function defined in (9) f ¹=r ₀ g(z ₀, t) f ²= ₀ f ³=f(z ₀, t) - F force applied to the plates - g unknown function of z ₀ and t defined in (6) - g g/z ₀ - h grid dimension in the z ₀ direction (see Fig. 5) - Christoffel symbol - i, j, k, l indices - k grid dimension in the t direction (see Fig. 5) - r radial coordinate direction defined in Fig. 1 - r ₀ radial convected coordinate - R radius of the circular plates - t time - v _r fluid velocity in the r direction - v _z fluid velocity in the z direction - v fluid velocity in the direction - x ⁱ cylindrical coordinate x ¹=r x²= x³=z - z vertical coordinate direction defined in Fig. 1 - z ₀ vertical convected coordinate - tangential coordinate direction - ₀ tangential convected coordinate - viscosity - kinematic viscosity, / - ⁱ convected coordinate ¹=r₀ ²=₀ ³=z₀ - density