共查询到20条相似文献,搜索用时 750 毫秒
1.
Dr. S. A. Beg 《Heat and Mass Transfer》1975,8(2):127-135
In order to obtain information on the effect of shape on mass transfer, overall mass transfer rates were measured from naphthalene spheroids suspended in a wind tunnel (Schmidt number 2.4). Spheroidal shapes which included spheres, oblate spheroids and spheroids with composite halves were employed for the study. The ratio of the minor to major axes of the spheroids ranged from 1∶1 to 1∶4. The data obtained from one-hundred and fifty six experimental runs were best correlated by the use of Pasternak and Gauvin's characteristic dimension defined as total surface area of the body divided by maximum perimeter normal to flow. The correlations for the ranges 200 < Re < 2000 and 2000 < Re < 32000 are as follows. $$\begin{gathered} Sh = 0.62 (Re)^{0.5} (Sc)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ Sh = 0.26 (Re)^{0.6} (Sc)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ \end{gathered}$$ which correlated the data with standard deviation of 3.75% and 3.50% respectively. 相似文献
2.
E. Weder 《Heat and Mass Transfer》1968,1(1):10-14
Simultaneous heat and mass transfer at horizontal cylinders under free convection conditions have been investigated by an electrochemical method. Over all and local measurements of the mass transfer coefficient have been executed. The over all results can be represented by two empirical equations: $$\begin{gathered} Nu = 0,858 \cdot (Gr \cdot Pr)^{0,22} \hfill \\ Sh = 0,23 \cdot [(Gr' \cdot Sc)^{1,07} + 1,49\sqrt {Sc/Pr} \cdot Gr \cdot Sc]^{0,28} \hfill \\ \end{gathered}$$ The ranges of the dimensionsless groups were as follows:Gr: 7,31 · 102 to 8,69 · 105 Gr′: 1,21 · 103 to 2,79 · 105 Pr: 10 to 7,56Sc: 3440 to 1890 相似文献
3.
Ref. [1] discussed the existence of positive solutions of quasilinear two-point boundary problems: but it restricts O相似文献
4.
Singular perturbations of two-point boundary problems for systems of ordinary differential equations
W. A. Harris Jr. 《Archive for Rational Mechanics and Analysis》1960,5(1):212-225
Asymptotic solutions of linear systems of ordinary differential equations are employed to discuss the relationship of the solution of a certain “complete” boundary problem.
$$\begin{gathered} \left\{ \begin{gathered} {\text{ }}\frac{{d{\text{ }}x_1 }}{{d{\text{ }}t}} = A_{11} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{1p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \varepsilon ^{h_2 } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{21} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{2p} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ {\text{ }} \vdots {\text{ }} \vdots {\text{ }} \vdots \hfill \\ \varepsilon ^{h_p } \frac{{d{\text{ }}x_2 }}{{d{\text{ }}t}} = A_{p1} (t,\varepsilon ){\text{ }}x_1 (t,\varepsilon ){\text{ }} + \cdots + A_{pp} (t,\varepsilon ){\text{ }}x_p (t,\varepsilon ) \hfill \\ \end{gathered} \right\} \hfill \\ {\text{ }}R(\varepsilon ){\text{ }}x(a,{\text{ }}\varepsilon ){\text{ }} + {\text{ }}S(\varepsilon ){\text{ }}x(b,{\text{ }}\varepsilon ) = c(\varepsilon ){\text{ }} \hfill \\ \end{gathered}$$ 相似文献
5.
郭艾 《应用数学和力学(英文版)》1999,20(6):683-689
1ProblemsandMainResultsInthispaper,westudythenonlinearvibrationsofinfiniterodswithviscoelasticity.Theconstitutionlawoftherods... 相似文献
6.
The article discusses elementary solutions of problems of nonlinear filtration with a piece-wise-linear resistance law, and analyzes their behavior with a relative increase in the resistance in the region of small velocities, and a transition to the law of filtration with a limiting gradient. The results obtained are applied to a determination of the dimensions of the stagnant zones in stratified strata. The law of filtration with a limiting gradient (0.1) $$\begin{gathered} w = - \frac{k}{\mu }\left( {grad p - G\frac{{grad p}}{{|grad p|}}} \right),|grad p| > G \hfill \\ w = 0,|grad p|< G \hfill \\ \end{gathered}$$ describes motion in some intermediate range of velocities w, but its satisfaction in the region of the smallest velocities, as a rule, remains unverified. It is natural to pose the problem of the degree to which a divergence between the true filtration law and its approximation (0.1) affects the accuracy of calculation of the flow fields, and the significance of a determination of the dimensions of the stagnant zones under such conditions. To answer this problem to some measure, there are considered below several simple exact (elementary) solutions obtained for a more general nonlinear filtration law (0.2) $$\begin{gathered} grad p = - (\mu /k) (w + \lambda )w/w,\lambda = kG/\mu ,w \geqslant w_0 \hfill \\ grad p = - \mu w/k\varepsilon ), w \leqslant w_0 ,\varepsilon = w_0 /w_0 + \lambda \hfill \\ \end{gathered}$$ going over into (0.1) with w0→ 0. The solutions obtained are applied also to an evaluation of the dimensions of stagnant zones, forming in stratified strata when the effects of the limiting gradient in one of the intercalations are considerable. 相似文献
7.
We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms $$\begin{gathered} \dot x(t) = H'_p (t, x(t), p(t)) + g(t), \hfill \\ \dot p(t) = - H'_x (t, x(t), p(t)) - f(t). \hfill \\ \end{gathered} $$ Among these problems is the existence of T-periodic solutions, namely those satisfying x(t+T)=x(t) and p(t+T)+p(t). A special study is made of the classical case, where H(x, p)=1/2 |p|2+V(x). In the case of parametric oscillations, where (f, g)=(0, 0) and t ? H(t, x, p) is T-periodic, we give a lower bound for the true (minimal) period of the T-periodic solution (x, p) produced by our method, and we prove the existence of an infinite number of subharmonics. 相似文献
8.
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. 相似文献
9.
Eduard Feireisl 《Journal of Dynamics and Differential Equations》1994,6(1):23-35
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. 相似文献
10.
The following is a well-known problem of statistical physics: can a dynamic system of oscillators with nonlinear coupling be described approximately by statistical laws? This problem was studied for the first time by Fermi, Ulam, and Pasta [1] for the following system of equations describing coupled oscillators:
$$\begin{gathered} x^{..} _i = x_{i + 1} - 2x_i + x_{i - 1} + \alpha [(x_{i + 1} - x_i )^2 ] - (x_i ..x_{i - 1} )^2 ], \hfill \\ (i = 1,...,N;\alpha< 1).(0.1) \hfill \\ \end{gathered} $$ 相似文献
11.
We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation $$\begin{gathered} {\text{ }}u_t = \left( {u^m } \right)_{xx} {\text{ in }}Q = \mathbb{R} \times \left( {{\text{0,}}\infty } \right){\text{,}} \hfill \\ u\left( {x{\text{,0}}} \right) = u_{\text{0}} \left( x \right){\text{ for }}x \in \mathbb{R}{\text{,}} \hfill \\ \end{gathered}$$ with m > 1 and, u 0a continuous, nonnegative function. It is well known that, across a moving interface x=ζ(t) of the solution u(x, t), the derivatives v tand v x of the pressure v = (m/(m?1)) u m?1 have jump discontinuities. We prove that each moving part of the interface is a C ∞curve and that v is C ∞on each side of the moving interface (and up to it). We also prove that for solutions with compact support the pressure becomes a concave function of x after a finite time. This fact implies sharp convergence rates for the solution and the interfaces as t→∞. 相似文献
12.
On the existence of periodic solutions to higher dimensional periodic system with delay 总被引:1,自引:0,他引:1
Consideringthehigherdimensionalperiodicsystemswithdelayoftheformx′(t)=A(t,x(t))x(t)+f(t,x(t-τ)),(1)x′(t)=gradG(x(t))+f(t,x(t-... 相似文献
13.
The thermal decomposition of CS2 highly diluted in Ar was studied behind reflected shock waves by monitoring time-dependent absorption profiles of S(3P) and S(1D) using atomic resonance absorption spectroscopy (ARAS). The rate coefficient of the reaction:
相似文献
14.
F. S. Churikov 《Fluid Dynamics》1966,1(3):70-71
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]
$$\begin{gathered} \frac{{1 - \tau ^2 }}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial ^2 \Phi }}{{\partial \theta ^2 }} + \frac{{\partial ^2 \Phi }}{{\partial \tau ^2 }} + \frac{{\tau (1 - \tau ^2 )}}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial \Phi }}{{\partial \tau }} = 0, \hfill \\ (\tau = w/c_k , w = \sqrt {u^2 + \upsilon ^2 } , \alpha = (\gamma - 1)/(\gamma + 1); \gamma = c_p /c_\upsilon ). (0.1) \hfill \\ \end{gathered} $$ 相似文献
15.
J. L. Ericksen 《Archive for Rational Mechanics and Analysis》1979,72(1):1-13
We prove existence, uniqueness, and regularity properties for a solution u of the Bellman-Dirichlet equation of dynamic programming:
16.
In this paper, exact solutions are constructed for stationary election beams that are degenerate in the Cartesian (x,y,z), axisymmetric (r,θ,z), and spiral (in the planes y=const (u,y,v)) coordinate systems. The degeneracy is determined by the fact that at least two coordinates in such a solution are cyclic or are integrals of motion. Mainly, rotational beams are considered. Invariant solutions for beams in which the presence of vorticity resulted in a linear dependence of the electric-field potential ? on the above coordinates were considered in [1], In degenerate solutions, the presence of vorticity results in a quadratic or more complex dependence of the potential on the coordinates that are integrals of motion. In [2] and in a number of papers referred to in [2], the degenerate states of irrotational beams are described. The known degenerate solutions for rotational beams apply to an axisymmetric one-dimensional (r) beam with an azimuthal velocity component [3] and to relativistic conical flow [1]. The equations used below follow from the system of electron hydrodynamic equations for a stationary relativistic beam $$\begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left[ {\sqrt \gamma g^{\beta \beta } g^{\alpha \alpha } \left( {\frac{{\partial A_\alpha }}{{\partial q^\beta }} - \frac{{\partial A_\beta }}{{\partial q^\alpha }}} \right)} \right]} = 4\pi \rho \sqrt \gamma g^{\alpha \alpha } u_\alpha ,} \\ {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left( {\sqrt \gamma g^{\beta \beta } \frac{{\partial \varphi }}{{\partial q^\beta }}} \right)} = 4\pi \rho \sqrt {\gamma u} ,\sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta ^2 + 1 = u^2 } } \\ \begin{gathered} \frac{\eta }{c}u\frac{{\partial \mathcal{E}}}{{\partial q^\alpha }} = \sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta } \left( {\frac{{\partial p_\beta }}{{\partial q^\alpha }} - \frac{{\partial p_\alpha }}{{\partial q^\beta }}} \right), \hfill \\ \begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}(\sqrt \gamma g^{\beta \beta } \rho u_\beta ) = 0,u \equiv \frac{\eta }{{c^2 }}(\varphi + \mathcal{E}) + 1,} } \\ {cu_\alpha \equiv \frac{\eta }{c}A_\alpha + p_\alpha ,\alpha ,\beta = 1,2,3,\gamma \equiv g_{11} g_{22} g_{33} } \\ \end{array} \hfill \\ \end{gathered} \\ \end{array} $$ where qβ denotes orthogonal coordinates with the metric tensor gββ (β=1,2,3); Aα is the magnetic potential; Aα = (uα/u)c is the electron velocity; ρ is the scalar space-charge density (ρ > 0); is the energy in eV; pα is the generalized momentum of an electron per unit mass; η is the electron charge-mass ratio. 相似文献
17.
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. 相似文献
18.
W. Czernous 《Nonlinear Oscillations》2011,13(4):595-612
We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the
first order
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