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.     

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.     

Numerical stability analysis of numerical methods for volterra integral equations with delay argument

The present paper deals with the stability properties of numerical methods for Volterra integral equations with delay argument. We assess the numerical stability of numerical methods with respect to the following test equations (0.1a) $$y\left( t \right) = \psi \left( 0 \right) + \int_0^t {\left( {py\left( s \right) + q\left( {s - \tau } \right)} \right)ds (0 \leqslant t \leqslant X)}$$ (0.1b) $$y\left( t \right) = \psi \left( t \right) \left( {t \in [ - \tau ,0)} \right)$$ where τ is a positive constant, and P and q are complex valued. We investigate the stability properties of reducible quadrature methods and θ-methods in the case of the above test equations     

A connection formula for the second Painlevé transcendent

We consider the second Painlevé transcendent $$\frac{{d^2 y}}{{dx^2 }} = xy + 2y^3 .$$ It is known that if y(x) ～ k Ai (x) as x → + ∞, where ?1<k<1 and Ai (x) denotes Airy's function, then $$y(x) \sim d|x|^{ - \tfrac{1}{4}} sin\{ \tfrac{2}{3}|x|^{\tfrac{3}{2}} - \tfrac{3}{4}d^2 1n|x| - c\} ,$$ where the constants d, c depend on k. This paper shows that $$d^2 = \pi ^{ - 1} 1n(1 - k^2 )$$ , which confirms a conjecture by Ablowitz & Segur.     

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.     

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

A well posed problem in singular Fickian diffusion

We prove that the problem of solving $$u_t = (u^{m - 1} u_x )_x {\text{ for }} - 1< m \leqq 0$$ with initial conditionu(x, 0)=φ(x) and flux conditions at infinity \(\mathop {\lim }\limits_{x \to \infty } u^{m - 1} u_x = - f(t),\mathop {\lim }\limits_{x \to - \infty } u^{m - 1} u_x = g(t)\) , admits a unique solution \(u \in C^\infty \{ - \infty< x< \infty ,0< t< T\} \) for every φεL¹(R), φ≧0, φ≡0 and every pair of nonnegative flux functionsf, g ε L _loc ^∞ [0, ∞) The maximal existence time is given by $$T = \sup \left\{ {t:\smallint \phi (x)dx > \int\limits_0^t {[f} (s) + g(s)]ds} \right\}$$ This mixed problem is ill posed for anym outside the above specified range.     

On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term

We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ ₀=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.     

The existence of solution of a class of two-order quasilinear boundary value problem

Ref. [1] discussed the existence of positive solutions of quasilinear two-point boundary problems: but it restricts O相似文献   

A mathematical interpretation of Dirac δ function

The two conditions (see[1] p. 58) of the Dirac function are inconsistent in standard analysis.In this paper, the author began by studying the integral of the functions on the nucleon a(o), and then, making use of the point function in infinitesimal analysis to define the Dirac function (x) so that it satisfies the condition (1.2) and δ(X)=o, for X∈R≠0 and X Some various examples of Dirac functions have been presented and some properties of the d function have been derived.     

Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

A KAM Theorem for Higher Dimensional Nonlinear Schrödinger Equations

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional nonlinear Schrödinger equation $$\begin{aligned} iu_t-\triangle u +M_\xi u+f(|u|^2)u=0, \quad t\in \mathbb{R }, x\in \mathbb{T }^d \end{aligned}$$ with periodic boundary conditions, where $M_\xi $ is a real Fourier multiplier and $f(|u|^2)$ is a real analytic function near $u=0$ with $f(0)=0$ . We obtain for the equation a Whitney smooth family of real-analytic small-amplitude linearly-stable quasi-periodic solutions with a nice linear normal form.     

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.     

Inertia effect in conical converging flow of viscoelastic fluids

The convergent flow of viscoelastic fluids in conical nozzles has been examined. The experimentally determined streamlines agreed with those obtained from calculations with an approximation up to \(\dot V^2 \) Because of the elasticity of the fluid a rotationally symmetric eddy arises. It clings to the wall whereas the output flow proceeds in the middle. When $$\frac{\varrho }{{\eta _0 }}\left( {\frac{{\dot V^2 }}{{t_0 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} > 40$$ > 40 another eddy can arise in front of the first one. This second eddy is situated in the middle and causes the output to proceed as a flow near the wall. The higher the afore-mentioned dimensionless value is, the higher is the inclination of the flow to instability.     

Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in ${\mathbb{R}^2}$ or ${\mathbb{R}^3}$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at ${x\in D}$ at time t and ${p(t,x)\in\mathbb{R}}$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l?≥ 1 is prescribed). The third equation in the system says that ${\kappa\in\mathbb{R}^l}$ is convected by the flow and the second one that ${\phi}$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)?=?x on D (and thus l?=?2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411–452, 1999) in his Eulerian–Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross ${\partial D}$ and that carry each “particle” at time t?=?0 at a prescribed location at time t?=?T?>?0, that is, κ(T, x) is prescribed in D for all ${x\in D}$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary ${\partial D}$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through ${\partial D}$ of particles labelled by each value of κ at each point of ${\partial D}$ . One of the main novelties is the introduction of a prescribed “generalized” Bernoulli’s function ${H:\mathbb{R}^l\rightarrow \mathbb{R}}$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with ${\phi,p,\kappa}$ periodic in time of prescribed period T?>?0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of “Lamb’s surfaces” and “isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier’s formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions ${\phi}$ and κ are given in terms of the stream function ψ.     

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where ${\Omega(t) \subset \mathbb{R}^{n}}$ ( ${n \geqq 2}$ ) is bounded by the free boundary ${\Gamma(t)}$ , with ${\Omega(0) = \Omega_0}$ , μ and d are given positive constants. The initial function u ₀ is positive in ${\Omega_0}$ and vanishes on ${\partial \Omega_0}$ . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary ${\Gamma(t)}$ is smooth outside the closed convex hull of ${\Omega_0}$ , and as ${t \to \infty}$ , either ${\Omega(t)}$ expands to the entire ${\mathbb{R}^n}$ , or it stays bounded. Moreover, in the former case, ${\Gamma(t)}$ converges to the unit sphere when normalized, and in the latter case, ${u \to 0}$ uniformly. When ${g(u) = au - bu^2}$ , we further prove that in the case ${\Omega(t)}$ expands to ${{\mathbb R}^n}$ , ${u \to a/b}$ as ${t \to \infty}$ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists ${\mu^* \geqq 0}$ such that ${\Omega(t)}$ expands to ${{\mathbb{R}}^n}$ exactly when ${\mu > \mu^*}$ .     

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: $$ND_{crit} = 5.012 \times 10^3 \left( {\frac{D}{{2R}}} \right)^{2.1} ,0.0111< {D \mathord{\left/ {\vphantom {D {2R_c }}} \right. \kern-\nulldelimiterspace} {2R_c }}< 0.71.$$

     

Parabolic Systems with p, q-Growth: A Variational Approach

We consider the evolution problem associated with a convex integrand ${f : \mathbb{R}^{Nn}\to [0,\infty)}$ satisfying a non-standard p, q-growth assumption. To establish the existence of solutions we introduce the concept of variational solutions. In contrast to weak solutions, that is, mappings ${u\colon \Omega_T \to \mathbb{R}^n}$ which solve $$ \partial_tu-{\rm div} Df(Du)=0 $$ weakly in ${\Omega_T}$ , variational solutions exist under a much weaker assumption on the gap q ? p. Here, we prove the existence of variational solutions provided the integrand f is strictly convex and $$\frac{2n}{n+2} < p \le q < p+1.$$ These variational solutions turn out to be unique under certain mild additional assumptions on the data. Moreover, if the gap satisfies the natural stronger assumption $$ 2\le p \le q < p+ {\rm min}\big \{1,\frac{4}{n} \big \},$$ we show that variational solutions are actually weak solutions. This means that solutions u admit the necessary higher integrability of the spatial derivative Du to satisfy the parabolic system in the weak sense, that is, we prove that $$u\in L^q_{\rm loc}\big(0,T; W^{1,q}_{\rm loc}(\Omega,\mathbb{R}^N)\big).$$