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.     

Modeling and Analog Realization of the Fundamental Linear Fractional Order Differential Equation

This paper provides a rational function approximation of the irrational transfer function of the fundamental linear fra- ctional order differential equation, namely, whose transfer function is given by for 0<m<2. Simple methods, useful in system and control theory, which consists of approximating, for a given frequency band, the transfer function of this fractional order system by a rational function are presented. The impulse and step responses of this system are derived and simple analog circuit which can serve as fundamental analog fractional order system is also obtained. Illustrative examples are presented to show the exactitude and the usefulness of the approximation methods.     

Instability of a layer op lightly ionized plasma in crossed electric and magnetic fields

In a lightly ionized plasma, charged-particle drift due to collisions with neutral atoms occurs at different velocities: $$\begin{array}{*{20}c} {v_{Ea} = \mp \frac{{b_a E}}{{1 + (\omega _a \tau _a )^2 }},v_{ \bot a} = \frac{{b_a E(\omega _a \tau _a )}}{{1 + (\omega _a \tau _a )^2 }}} \\ {\left( {b_a = \frac{{|e|\tau _a }}{{m_a }},\omega _a = \frac{{|e|\tau _a }}{{m_a }}} \right),} \\ \end{array} $$ where b_a is the mobility of particles of the type a;ω_a is the Larmor frequency; the upper sign refers to electrons and the lower sign to ions. A difference in the charged-particle drift velocities can cause instability of an inhomogeneous lightly ionized plasma. Let us consider the following example. Assume that in the initial state of the plasma there is a concentration gradient along the x-axis, that the external electric field is directed along the x-axis, and that the magnetic field coincides with the z-axis. In this system, under the influence of a Lorentz force the charged particles will move in a direction opposite to the y-axis. Since electrons have a higher velocity than ions, an electric field is induced in this direction. This electric field, together with the magnetic field, causes particle drift in the negative direction of the x-axis. Consequently, if the concentration gradient in the initial state is directed opposite to the x-axis this state cannot be stable. Instability of this kind has been examined by Simon [1]. On the basis of studies by Kadomtsev and Nedospasov [2], as well as by Rosenbluth and Longmire [3], Simon developed a theory of instability of a lightly ionized plasma in crossed fields with an inhomogeneous density distribution in the direction of the external electric field. Somewhat later, Simon's theory was developed [4]. In devices with inhomogeneous plasma flow in which the plasma (conducting) layers alternate with nonconducting layers, the external electric field and concentration are normal to one another. We shall bear this case in mind below and shall examine the instability of a lightly ionized plasma in crossed fields when the concentration inhomogeneity is in a direction perpendicular to the external electric field.     

Steady flow of a viscous incompressible fluid between two co-axial circular cylinders with suction and injection

In the present paper an attempt has been made to study the steady flow of a viscous incompressible fluid between two co-axial circular cylinders with small outward and inward normal suction on the outer and inner cylinders respectively with the assumption that the pressure is uniform over a cross-section. The expressions for axial velocity, the volume of fluid flowing per unit time across a cross-section and components of stress at any point of the fluid are derived.

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Stationäre Strömung einer zähen, inkompressiblen Flüssigkeit zwischen zwei koaxialen Kreiszylindern bei Absaugen und Ausblasung

Zusammenfassung Es wird die stationäre Strömung eines zähen, inkompressiblen Fluids zwischen zwei koaxialen Kreiszylindern mit geringer Absaugung oder Ausblasung sowohl am Innen- wie am Außenzylinder unter der Annahme gleichförmiger Druckverteilung über jeden Querschnitt untersucht. Die Beziehungen für die Axialgeschwindigkeit, der Fluidstrom über jeden Querschnitt und die Spannungskomponenten in jedem Punkt des Fluidfeldes werden hergeleitet.

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Nomenclature density of the fluid - x axial coordinate - y radial coordinate - azimuthal coordinate - u axial velocity - v radial velocity - w azimuthal velocity - p pressure - coefficient of viscosity - kinematic viscosity - a radius of the inner cylinder - b radius of the outer cylinder - v₀ suction velocity on the inner cylinder - v₀ suction velocity on the outer cylinder =suction parameter for the inner cylinder =suction parameter for the outer cylinder =dimensionlessy coordinates Q=discharge per unit time     

Integrability of the constrained rigid body

The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(γ)=U(γ ₁,γ ₂,γ ₃). This motion subject to the constraint 〈ν,ω〉=0 with ν is a constant vector is known as the Suslov problem, and when ν=γ is the known Veselova problem, here ω=(ω ₁,ω ₂,ω ₃) is the angular velocity and 〈?,?〉 is the inner product of $\mathbb{R}^{3}$ . We provide the following new integrable cases. (i) The Suslov’s problem is integrable under the assumption that ν is an eigenvector of the inertial tensor I and the potential is such that $$U=-\frac{1}{2I_1I_2}\bigl(I_1\mu^2_1+I_2 \mu^2_2\bigr), $$ where I ₁,I ₂, and I ₃ are the principal moments of inertia of the body, μ ₁ and μ ₂ are solutions of the first-order partial differential equation $$\gamma_3 \biggl(\frac{\partial\mu_1}{\partial\gamma_2}- \frac{\partial\mu_2}{\partial \gamma_1} \biggr)- \gamma_2\frac{\partial \mu_1}{\partial\gamma_3}+\gamma_1\frac{\partial\mu_2}{\partial \gamma_3}=0. $$ (ii) The Veselova problem is integrable for the potential $$U=-\frac{\varPsi^2_1+\varPsi^2_2}{2(I_1\gamma^2_2+I_2\gamma^2_1)}, $$ where Ψ ₁ and Ψ ₂ are the solutions of the first-order partial differential equation where $p=\sqrt{I_{1}I_{2}I_{3} (\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}} )}$ . Also it is integrable when the potential U is a solution of the second-order partial differential equation where $\tau_{2}=I_{1}\gamma^{2}_{1}+I_{2}\gamma^{2}_{2}+I_{3}\gamma^{2}_{3}$ and $\tau_{3}=\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}}$ . Moreover, we show that these integrable cases contain as a particular case the previous known results.     

A hereditary partial differential equation with applications in the theory of simple fluids

The linearized boundary-initial history value problem for simple fluids obeying the Coleman-Noll constitutive equation $$S + p\delta = 2\int\limits_0^\infty {m(s)(E(t - s} ) - E(t))ds$$ is considered. Here S is the stress tensor, δ the Kronecker delta, p the constitutively indeterminate mean normal stress, E the infinitesimal strain tensor, and m(s) a material function. The shear relaxation modulus G is defined as (i) $$G(s) = \int\limits_\infty ^s {m(\xi )d\xi .}$$ In this paper it is shown that if G satisfies the assumptions (i) $$G \in C^2 [0,\infty ),{\text{ }}G(s) \to 0{\text{ as }}s \to \infty,$$ (ii) $$( - 1)^k \frac{{d^k G(s)}}{{ds^k }} > 0,{\text{ }}k = 0,1,$$ (iii) $$G''(s) \geqq 0,$$ then the rest state of the fluid is stable in an appropriate “fading memory” norm. The additional assumption (iv) $$ - \int\limits_0^\infty {G'} (s)s^2 ds < \infty$$ yields asymptotic stability.     

Pressure drop in alternating curved tubes

Pressure drop measurements in the laminar and turbulent regions for water flowing through an alternating curved circular tube (x=h sin 2πz/λ) are presented. Using the minimum radius of curvature of this curved tube in place of that of the toroidally curved one in calculating the Dean number (ND=Re(D/2R _c )², it is found that the resulting Dean number can help in characterizing this flow. Also, the ratio between the height and length of the tube waves which represents the degree of waveness affects significantly the pressure drop and the transition Dean number. The following correlations have been found:

1. For laminar flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.03,\operatorname{Re}< 2000.$$
2. For turbulent flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.005,2000< \operatorname{Re}< 15000.$$
3. The transition Dean number: $$ND_{crit} = 5.012 \times 10^3 \left( {\frac{D}{{2R}}} \right)^{2.1} ,0.0111< {D \mathord{\left/ {\vphantom {D {2R_c }}} \right. \kern-\nulldelimiterspace} {2R_c }}< 0.71.$$

     

Singular perturbations of two-point boundary problems for systems of ordinary differential equations

Asymptotic solutions of linear systems of ordinary differential equations are employed to discuss the relationship of the solution of a certain “complete” boundary problem.

$$\begin{gathered} \left\{ \begin{gathered} {\text{ }}\frac{{d{\text{ }}x_1 }}{{d{\text{ }}t}} = A_{11} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{1p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \varepsilon ^{h_2 } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{21} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{2p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ {\text{ }} \vdots {\text{ }} \vdots {\text{ }} \vdots \hfill \\ \varepsilon ^{h_p } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{p1} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{pp} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \end{gathered} \right\} \hfill \\ {\text{ }}R(\varepsilon ){\text{ }}x(a,{\text{ }}\varepsilon ){\text{ }} + {\text{ }}S(\varepsilon ){\text{ }}x(b,{\text{ }}\varepsilon ) = c(\varepsilon ){\text{ }} \hfill \\ \end{gathered}$$     

Boundary conditions at the edge of a thin or thick plate bonded to an elastic support

At the clamped edge of a thin plate, the interior transverse deflection ω(x ₁, x₂) of the mid-plane x ₃=0 is required to satisfy the boundary conditions ω=?ω/?n=0. But suppose that the plate is not held fixed at the edge but is supported by being bonded to another elastic body; what now are the boundary conditions which should be applied to the interior solution in the plate? For the case in which the plate and its support are in two-dimensional plane strain, we show that the correct boundary conditions for ω must always have the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakqaabeqaaiaabEhacaqGTaWa% aSaaaeaacaGG0aGaae4vamaaCaaaleqabaGaamOqaaaaaOqaaiaaco% dadaqadaqaaiaacgdacqGHsislcaqG2baacaGLOaGaayzkaaaaaiaa% bIgadaahaaWcbeqaaiaackdaaaGcdaWcaaqaaiaabsgadaahaaWcbe% qaaiaackdaaaGccaqG3baabaGaaeizaiaabIhafaqabeGabaaajaaq% baqcLbkacaGGYaaajaaybaqcLbkacaGGXaaaaaaakiabgUcaRmaala% aabaGaaiinaiaabEfadaahaaWcbeqaaiaadAeaaaaakeaacaGGZaWa% aeWaaeaacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGOb% WaaWbaaSqabeaacaGGZaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaa% caGGZaaaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaa% saaiaacodaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0Jaaiimaiaa% cYcaaeaadaWcaaqaaiaabsgacaqG3baabaGaaeizaiaabIhaliaacg% daaaGccqGHsisldaWcaaqaaiaacsdacqqHyoqudaahaaWcbeqaaiaa% bkeaaaaakeaacaGGZaWaaeWaaeaacaGGXaGaeyOeI0IaaeODaaGaay% jkaiaawMcaaaaacaqGObWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGG% YaaaaOGaae4DaaqaaiaabsgacaqG4bqbaeqabiqaaaqcaauaaKqzGc% GaaiOmaaqcaawaaKqzGcGaaiymaaaaaaGccqGHRaWkdaWcaaqaaiaa% csdacqqHyoqudaahaaWcbeqaaiaabAeaaaaakeaacaGGZaWaaeWaae% aacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaWba% aSqabeaacaGGYaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGGZa% aaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaasaaiaa% codaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0JaaiimaiaacYcaaa% aa!993A!\[\begin{gathered}{\text{w - }}\frac{{4{\text{W}}^B }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4{\text{W}}^F }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^3 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4\Theta ^{\text{F}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\end{gathered}\]with exponentially small error as L/h→∞, where 2h is the plate thickness and L is the length scale of ω in the x ₁-direction. The four coefficients W ^B, W^F, Θ ^B , Θ ^F are computable constants which depend upon the geometry of the support and the elastic properties of the support and the plate, but are independent of the length of the plate and the loading applied to it. The leading terms in these boundary conditions as L/h→∞ (with all elastic moduli remaining fixed) are the same as those for a thin plate with a clamped edge. However by obtaining asymptotic formulae and general inequalities for Θ ^B , W ^F, we prove that these constants take large values when the support is ‘soft’ and so may still have a strong influence even when h/L is small. The coefficient W ^F is also shown to become large as the size of the support becomes large but this effect is unlikely to be significant except for very thick plates. When h/L is small, the first order corrected boundary conditions are w=0,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaabsgacaqG% 3baabaGaaeizaiaabIhaliaacgdaaaGccqGHsisldaWcaaqaaiaacs% dacqqHyoqudaahaaWcbeqaaiaabkeaaaaakeaacaGGZaWaaeWaaeaa% caGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaSaaae% aacaqGKbWaaWbaaSqabeaacaGGYaaaaOGaae4DaaqaaiaabsgacaqG% 4bqbaeqabiqaaaqcaauaaKqzGcGaaiOmaaqcaawaaKqzGcGaaiymaa% aaaaGccqGH9aqpcaGGWaGaaiilaaaa!5DD4!\[\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} = 0,\]which correspond to a hinged edge with a restoring couple proportional to the angular deflection of the plate at the edge.     

Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type

The paper is concerned with the existence and stability of an almost periodic solution of the system with deviating argument .The Wexler inequality for the Cauchy matrix is used. Conditions for the stability of the solution are given.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 295–301, July–September, 2004.     

Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data ${a_0 \in L^\infty (\mathbb{R}^d), u_0 = (u_0^h, u_0^d) \in \dot{B}^{-1+\frac{d}{p}}_{p, r} (\mathbb{R}^d)}$ , which satisfy ${(\mu \| a_0 \|_{L^\infty} + \|u_0^h\|_{\dot{B}^{-1+\frac{d}{p}}_{p, r}}) {\rm exp}(C_r{\mu^{-2r}}\|u_0^d\|_{\dot{B}^{-1+\frac{d}{p}}_{p,r}}^{2r}) \leqq c_0\mu}$ for some positive constants c ₀, C _r and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity fields. Furthermore, with additional regularity assumptions on the initial velocity or on the initial density, we can also prove the uniqueness of such a solution. We should mention that the classical maximal L ^p (L ^q ) regularity theorem for the heat kernel plays an essential role in this context.     

Approximating equations of a plane gas flow and their integration

It is known that the nonlinear system of equations of plane steady isentropic potential gas flow can be linearized and transformed to a single equivalent linear differential equation of second order. For the case of a perfect gas this equation has the form [1]

$$\begin{gathered} \frac{{1 - \tau ^2 }}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial ^2 \Phi }}{{\partial \theta ^2 }} + \frac{{\partial ^2 \Phi }}{{\partial \tau ^2 }} + \frac{{\tau (1 - \tau ^2 )}}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial \Phi }}{{\partial \tau }} = 0, \hfill \\ (\tau = w/c_k , w = \sqrt {u^2 + \upsilon ^2 } , \alpha = (\gamma - 1)/(\gamma + 1); \gamma = c_p /c_\upsilon ). (0.1) \hfill \\ \end{gathered} $$     

Experimental analysis of perforated torispherical shells under external pressure

Load tests were conducted on torispherical shells at room temperature for the purpose of predicting mechanical behavior of a reactor-vessel head at 1200°F. These shells were models of a calandria-vessel head designed for use in the sodium-reactor experiment. Principles of dimensional analysis were used in designing the tests and relating experimental results to behavior of the full-scale structure. External pressure and point loads were imposed to simulate service conditions caused by submersion of the calandria in sodium and transient thermal contraction of process tubes. Deflections and strains were monitored at several locations throughout various loading sequences. The stability limit of one 36-in. diam shell was reached at a strain number of:

$$\frac{{p(external{\text{ }}pressure)}}{{E(modulus{\text{ }}of{\text{ }}elasticity)}} = 3.07 \times 10^{ - 6} $$     

Improved theoretical model for wavy filmwise condensation

This paper presents a numerical solution for wavy laminar film-wise condensation on vertical walls. Integral method is achieved based on the recently developed simple wave equations. Solutions are obtained for ranges of dimensionless groups as follows: $$1.5 \leqslant \left( {Pr = \frac{{^{\mu C} p}}{k}} \right) \leqslant 6.0$$ $$10 \leqslant \left( {G = \frac{{^h fg}}{{^{C_p \Delta T} }}} \right) \leqslant 400$$ $$100 \leqslant \left( {S = \left( {\frac{{\sigma ^2 \rho }}{{g_\rho \mu ^4 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-\nulldelimiterspace} 5}} } \right) \leqslant 400$$ $$1000 \leqslant \left( {L = \frac{{{\rm H}_t }}{{^\delta cr}}} \right) \leqslant 10000$$ . Such ranges cover the expected situations in industrial applications. It is found that the Reynolds number (Re=h_LΔTH_t/h_fg) is a linear function of L on the log-log plane. It is also relatively insensitive to small variations of Pr at high values of this number. At situations where G less than 200 the Re appears to be dependent on S. Agreement with experimental observation is improved over that obtained from previous analytical theories.     

Dynamic crack curving—A photoelastic evaluation

A dynamic-crack-curving criterion, which is valid under pure Mode I or combined Modes I and II loadings and which is based on either the maximum circumferential stress or minimum strain-energy-density factor at a reference distance ofr ₀ from the crack tip, is verified with dynamic-photoelastic experiments. Directional stability of a Mode I crack propagation is attained when \(r_o= \frac{1}{{128\pi }}(\frac{{K_r }}{{\sigma _{ox} }})^2 V_o^2 (C,C_1 ,C_2 ) > r_c \) wherer _c =1.3 mm for Homalite-100 used in the dynamic-photoelastic experiments.     

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 $$D = 2 + 3\frac{{\varepsilon ^{4/3} + (1 - \varepsilon )^{2/3} - 1}}{{2\varepsilon ^{4/3} \ln ,{\text{ }}\varepsilon ^{ - 1} + (1 - \varepsilon )^{2/3} \ln (1 - \varepsilon )^{ - 1} }}$$ .     

Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain ${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$ , and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$ .     

Infinitely Many Stability Switches in a Problem with Sublinear Oscillatory Boundary Conditions

We consider the elliptic equation \(-\Delta u +u =0\) with nonlinear boundary condition \(\frac{\partial u}{\partial n}= \lambda u + g(\lambda ,x,u), \) where \(\frac{g(\lambda ,x,s)}{s} \rightarrow 0, \hbox { as }|s|\rightarrow \infty \) and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.     

Distribution of turbulent intensity in a gravel-bed flume

Detailed measurements have been taken for the longitudinal turbulent intensities of flow in a gravel-bed flume. The experimental results indicate that the distribution of turbulent intensity greatly depends on the relative roughness. In comparison with the smooth-bed results, the roughness makes the flow turbulence become well-distributed, especially in the region near the bed and in the case of smaller H/K _s values. In addition, the cross sectional average of turbulent intensity is also discussed in this paper, and the results show that the roughness makes flow turbulence much more intense.List of symbols D _u empirical constant - H flow depth - K _s roughness height - N , the mean turbulence intensity over the cross section - Re ^* , roughness Reynolds number - u the RMS of streamwise fluctuating velocity - u _* friction velocity - mean bulk velocity - x coordinate alined with mainstream velocity (x = 0 is channel entrance) - y vertical coordinate to the rough bed (y = 0 is the top of the rough elements) - y ⁺ _u empirical constant - v kinematic viscosity