The general form of the three-dimensional elastic field inside an isotropic plate with free faces

A homogeneous, isotropic plate has free faces and is stretched by tractions around its edge which are symmetrical about the mid-plane, but are otherwise generally distributed. We give a rigorous proof that the most general state of stress % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaBaaaleaacaWGPbGaamOAaaqabaaaaa!3FFD!\[\tau _{ij} \] which can be generated in the plate can be decomposed in the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaBaaaleaacaWGPbGaamOAaiabg2da9aqabaGccqaH% epaDdaqhaaWcbaGaamyAaiaadQgaaeaacaWGqbGaam4uaaaakiabgU% caRiabes8a0naaDaaaleaacaWGPbGaamOAaaqaaiaadofaaaGccqGH% RaWkcqaHepaDdaqhaaWcbaGaamyAaiaadQgaaeaacaWGqbGaamOraa% aaaaa!5277!\[\tau _{ij = } \tau _{ij}^{PS} + \tau _{ij}^S + \tau _{ij}^{PF} \] where (i) % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaDaaaleaacaWGPbGaamOAaaqaaiaadcfacaWGtbaa% aaaa!41AB!\[\tau _{ij}^{PS} \] is an (exact) plane stress state, (ii) % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaDaaaleaacaWGPbGaamOAaaqaaiaadofaaaaaaa!40D6!\[\tau _{ij}^S \] is a shear state, and (iii) % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaDaaaleaacaWGPbGaamOAaaqaaiaadcfacaWGgbaa% aaaa!419E!\[\tau _{ij}^{PF} \] is a Papkovich-Fadle state, which is a 3-dimensional generalisation of the Papkovich-Fadle eigenfunctions for the elastic strip.Furthermore, we prove that, as the plate thickness h0, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaDaaaleaacaWGPbGaamOAaaqaaiaadofaaaaaaa!40D6!\[\tau _{ij}^S \] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaDaaaleaacaWGPbGaamOAaaqaaiaadcfacaWGgbaa% aaaa!419E!\[\tau _{ij}^{PF} \] are exponentially small at points inside the plate and represent edge effects of thickness O(h).Corresponding results are also given for the case of plate bending, in which the applied tractions around the plate edge are anti-symmetrical about the mid-plane.     

A non-linear seales method for strongly non-linear oscillators

A non-linear seales method is presented for the analysis of strongly non-linear oseillators of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbdiqb-Hha4zaadaGaey4kaSIa% am4zaiaacIcacqWF4baEcaGGPaGae8xpa0JaeqyTduMaamOzaiaacI% cacqWF4baEcqWFSaalcuWF4baEgaGaaiaabMcaaaa!4FEC!\[\ddot x + g(x) = \varepsilon f(x,\dot x{\text{)}}\], where g(x) is an arbitrary non-linear function of the displacement x. We assumed that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbdiab-Hha4jaacIcacqWF0baD% cqWFSaalcqaH1oqzcaGGPaGaeyypa0Jae8hEaG3aaSbaaSqaaiaaic% daaeqaaOGaaiikaiabe67a4jaacYcacqaH3oaAcaGGPaGaey4kaSYa% aabmaeaacqaH1oqzdaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2% da9iaaigdaaeaacaWGTbGaeyOeI0IaaGymaaqdcqGHris5aOGae8hE% aG3aaSbaaSqaaiab-5gaUbqabaGccaGGOaGaeqOVdGNaaiykaiabgU% caRiaad+eacaGGOaGaeqyTdu2aaWbaaSqabeaacaWGTbaaaOGaaiyk% aaaa!67B9!\[x(t,\varepsilon ) = x_0 (\xi ,\eta ) + \sum\nolimits_{n = 1}^{m - 1} {\varepsilon ^n } x_n (\xi ) + O(\varepsilon ^m )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabsgacqaH+oaEcaGGVaGaaeizaiaadshacqGH9aqpdaaeWaqa% aiabew7aLnaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaG% ymaaqaaiaad2gaa0GaeyyeIuoakiaadkfadaWgaaWcbaGaamOBaaqa% baGccaGGOaGaeqOVdGNaaiykaaaa!4FFC!\[{\text{d}}\xi /{\text{d}}t = \sum\nolimits_{n = 1}^m {\varepsilon ^n } R_n (\xi )\], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabsgacqaH3oaAcaGGVaGaaeizaiaadshacqGH9aqpdaaeWaqa% aiabew7aLnaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaG% imaaqaaiaad2gaa0GaeyyeIuoakiaadofadaWgaaWcbaGaamOBaaqa% baGccaGGOaGaeqOVdGNaaiilaiabeE7aOjaacMcaaaa!5241!\[{\text{d}}\eta /{\text{d}}t = \sum\nolimits_{n = 0}^m {\varepsilon ^n } S_n (\xi ,\eta )\], and R _n,S _nare to be determined in the course of the analysis. This method is suitable for the systems with even non-linearities as well as with odd non-linearities. It can be viewed as a generalization of the two-variable expansion procedure. Using the present method we obtained a modified Krylov-Bogoliubov method. Four numerical examples are presented which served to demonstrate the effectiveness of the present method.     

Nonlinear combination resonances in cantilever composite plates

We experimentally investigated nonlinear combination resonances in two graphite-epoxy cantilever plates having the configurations (90/30/-30/-30/30/90)_s and (-75/75/75/-75/75/-75)_s. As a first step, we compared the natural frequencies and modes shapes obtained from the finite-element and experimental-modal analyses. The largest difference in the obtained frequencies for both plates was 6%. Then, we transversely excited the plates and obtained force-response and frequency-response curves, which were used to characterize the plate dynamics. We acquired time-domain data for specific input conditions using an A/D card and used them to generate time traces, power spectra, pseudo-state portraits, and Poincaré maps. The data were obtained with an accelerometer monitoring the excitation and a laser vibrometer monitoring the plate response. We observed the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaIYaaabeaakiab% gUcaRiabeM8a3naaBaaaleaacaaI3aaabeaaaaa!45C9!\[\Omega \approx \omega _2 + \omega _7 \] in the quasi-isotropic plate and the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGa% aiikaiabeM8a3naaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3n% aaBaaaleaacaaI1aaabeaakiaacMcaaaa!4AAD!\[\Omega \approx (1/2)(\omega _2 + \omega _5 )\] and the internal combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaI4aaabeaakiab% gIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGaaiikaiabeM8a3n% aaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3naaBaaaleaacaaI% XaGaaG4maaqabaGccaGGPaaaaa!4FDC!\[\Omega \approx \omega _8 \approx (1/2)(\omega _2 + \omega _{13} )\] in the ±75 plate, where the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeM8a3naaBaaaleaacaWGPbaabeaaaaa!3F16!\[\omega _i \] are the natural frequencies of the plate and is the excitation frequency. The results show that a low-amplitude high-frequency excitation can produce a high-amplitude low-frequency motion.     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

Global solution to the incompressible viscous-multipolar material problem

In the paper we give a proof of the global existence of the weak solution to the initial-boundary-value problem describing an incompressible elasto-viscous-multipolar material in finite geometry. A brief introduction to the physical background of viscous-multipolar materials is given. We suggest the hypothesis% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaaxadabaGaeu4OdmfaleaacaWGPbGaaiilaGqaciaa-bcacaWG% QbGaa8hiaiabg2da9iaa-bcacaaIXaaabaGaaG4maaaakiaa-bcada% abdiqcaasaaOWaaSaaaKaaGeaacqGHciITcqqHOoqwcaGGOaGaamOr% aiaacYcacqaH4oqCcaGGPaaabaGaeyOaIyRaamOraOWaaSbaaSqaai% aadMgacaWGbbaabeaaaaqcaaIaa8hiaiaadAeakmaaBaaaleaacaWG% QbGaamyqaaqabaaajaaqcaGLhWUaayjcSdGaa8hiaiabgsMiJkaado% gakmaaBaaaleaacaWFVbaabeaakiaadwgacaGGOaGaamOraiaacYca% ieaacaGFGaGaeqiUdeNaaiykaiaa+bcacqGHRaWkcaGFGaGaam4yam% aaBaaaleaacaaIXaGaa4hiaiaacYcaaeqaaaaa!686E!\[\mathop \Sigma \limits_{i, j = 1}^3 \left| {\frac{{\partial \Psi (F,\theta )}}{{\partial F_{iA} }} F_{jA} } \right| \leqslant c_o e(F, \theta ) + c_{1 ,} \] which enables one to obtain a priori estimates.     

Streamwise pseudo-vortical structures and associated vorticity in the near-wall region of a wall-bounded turbulent shear flow

Streamwise pseudo-vortical motions near the wall in a fully-developed two-dimensional turbulent channel flow are clearly visualized in the plane perpendicular to the flow direction by a sophisticated hydrogen-bubble technique. This technique utilizes partially insulated fine wires, which generate hydrogen-bubble clusters at several distances from the wall. These flow visualizations also supply quantitative data on two instantaneous velocity components, and w, as well as the streamwise vorticity, _x . The vorticity field thus obtained shows quasi-periodicity in the spanwise direction and also a double-layer structure near the wall, both of which are qualitatively in good agreement with a pseudo-vortical motion model of the viscous wall-region.List of symbols C _i ,c _i ,d _i constants in Eqs. (2), (3) and (4) - H channel width (m) - Re _H Reynolds number (= U _c H/) - Re Reynolds number (= U _c /) - T period (s) - t time (s) - U mean streamwise velocity (m/s) - U _c center-line velocity (m/s) - u friction velocity (m/s) - u, , w velocity fluctuations (m/s) - x, y, z coordinates (m) - ^* displacement thickness (m) - momentum thickness (m) - mean low-speed streak spacing (m) - kinematic viscosity (m²/s) - phase difference - _x streamwise vorticity fluctuation (1/s) - ( )⁺ normalized by u and - (^–) root mean square value - (^–) statistical average This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Laminar flow heat transfer in the entrance region of circular tubes

Summary The asymptotic solution of laminar convective heat transfer in the entrance region of a circular conduit where velocity and temperature profiles are developing simultaneously, is obtained for fluids with high Prandtl numbers. Numerical values of local and average Nusselt numbers as functions of Pr and dimensionless longitudinal distances have been evaluated and presented in graphical forms.Nomenclature A ₀, A ₁ ... A _k coefficients defined by (40) - B ₀, B ₁ ... B _k coefficients defined by (39) - C _p heat capacity of fluid - I _n (x) = i ^–n J _n (ix) where J _n is the n ^th order Bessel function - k thermal conductivity of fluid - Nu _z local Nusselt number defined by (41) - Nu _av average Nusselt number defined by (44) - P pressure - Pr Prandtl number of fluid defined as C _p /k - q heat flux - Re Reynold number, defined as PR/ - R radius of pipe - r radial distance - r ⁺ dimensionless radial distance defined by (8) - T temperature of fluid - T ₀ initial temperature of fluid - T _w wall temperature - T ⁺ dimensionless temperature defined by (11) - T ₀ ⁺ , T ₁ ⁺ , ... T _k ^/+ ... functions related to T ⁺ by (22). - u dimensionless variables defined by (20) - v _r radial component of velocity - v _z z-component of velocity - v ⁺ dimensionless velocity defined by (10) - y ⁺ dimensionless distance defined by (8) - X dimensionless parameter defined by (38) - z longitudinal distance - z ⁺ dimensionless longitudinal distance defined by (9) - thermal diffusivity - dimensionless parameter defined by (12) - a parameter appearing in (46) - (x) gamma function - density - dimensionless variable defined by (28) - parameter defined by (19) - dimensionless variable defined by (32) - viscosity of fluid - kinematic viscosity of fluid     

On the structure of continua and the mathematical properties of algebraic elastodynamic of a triclinic structural system

This paper is neither laudatory nor derogatory but it simply contrasts with what might be called elastostatic (or static topology), a proposition of the famous six equations. The extension strains and the shearing strains which were derived by A.L. Cauchy, are linearly expressed in terms of nine partial derivatives of the displacement function (u _i ,u _j ,u _h )=u(x ⁱ ,x ^j ,x ^k ) and it is impossible for the inverse proposition to sep up a system of the above six equations in expressing the nine components of matrix (∂(u _i ,u _j ,u _h )/∂(x ⁱ ,x ^j ,x ^k )). This is due to the fact that our geometrical representations of deformation at a given point are as yet incomplete^[1]. On the other hand, in more geometrical language this theorem is not true to any triangle, except orthogonal, for “squared length” in space^[2]. The purpose of this paper is to describe some mathematic laws of algebraic elastodynamics and the relationships between the above-mentioned important questions.     

On the Euler Equations in the Critical Triebel-Lizorkin Spaces

In this paper we study the initial value problem of the incompressible Euler equations in ⁿ for initial data belonging to the critical Triebel-Lizorkin spaces, i.e., v ₀ F ⁿ⁺¹ _1,q , q[1, ]. We prove the blow-up criterion of solutions in F ⁿ⁺¹ _1,q for n=2,3. For n=2, in particular, we prove global well-posedness of the Euler equations in F ³ _1,q , q[1, ]. For the proof of these results we establish a sharp Moser-type inequality as well as a commutator-type estimate in these spaces. The key methods are the Littlewood-Paley decomposition and the paradifferential calculus by J. M. Bony.     

Some properties of the solutions of wave equations

Some properties of solutions of initial value problems and mixed initial-boundary value problems of a class of wave equations are discussed. Wave modes are defined and it is shown that for the given class of wave equations there is a one to one correspondence with the roots _i (k) or k _j () of the dispersion relation W(, k)=0. It is shown that solutions of initial value problems cannot consist of single wave modes if the initial values belong to W ₂ ¹ (–, ); generally such solutions must contain all possible modes. Similar results hold for solutions of mixed initial-boundary value problems. It is found that such solutions are stable, even if some of the singularities of the functions k _j () lie in the upper half of the plane. The implications of this result for the Kramers-Kronig relations are discussed.     

Concentration-dependent diffusion in a composite slab with and without chemical reactions

Concentration-dependent diffusion of solute in a composite slab is investigated. The complex diffusion problem can be described by a set of nonlinear diffusion equations which is coupled to each other through the nonlinear interfacial boundary conditions. A two-layer diffusion is illustrated and the coupled nonlinear diffusion equations are conveniently solved by the orthogonal collocation method. Numerical simulation of the example reveals many interesting diffusion characteristics which are quite different from those in a single slab diffusion system.Nomenclature a _j expansion coefficient - A _i,j element of collocation matrix - B _i,j element of collocation matrix - C _a , C _b surface concentration - C _i concentration in the ith layer - D _i diffusion coefficient in the ith layer - D _i0 diffusion coefficient at very low concentration - k _i reaction rate in the ith layer - K _i dimensionless reaction rate, k _i l _i ² c _a ^m–1 /D ₁₀ - l _i thickness of the ith layer - m order of chemical reaction - n order of the orthogonal polynomial approximation - P _j–1(x _i ) orthogonal polynomial of order j - t time - x _i coordinate of the ith layer - X _i dimensionless coordinate of the ith layer, x _i/l _i - ratio of diffusion coefficient at low concentration, D ₂₀/D ₁₀ - ratio of thicknesses of layer, l ₁/l ₂ - _i dimensionless parameter in the concentration-dependent function of the ith layer - ratio of surface concentration, C _b /C _a - dimensionless time, tD ₁₀/l ₁ ² - _i dimensionless concentration in the ith layer, C _i /C _a      

Product inhibition in packed-bed immobilized enzyme reactors for complex processes: Modelling of two-substrate processes

An analytical model was developed for describing the performance of packed-bed enzymic reactors operating with two cosubstrates, and when one of the reaction products is inhibitory to the enzyme. To this aim, the compartmental analysis technique was used. The relevant equations obtained were solved numerically, and the effect of the main operational parameters on the reactor characteristics were studied.Notation C _infa,i ^sup* local concentration of products in the pores of stage i - C _j,i concentration of substrate j in the pores of stage i - D _infa ^sup* internal (pore) diffusion coefficient for the reaction product a - D _j internal (pore) diffusion coefficient of substrate j - J _infa,i ^sup* net flux of product a, taking place from the pores of stage i into the corresponding bulk phase - J _j,i net flux of substrate j, taking place from the bulk phase of stage i into the corresponding pores - K _b inhibition constant - K _m,1, K _m,2 Michaelis constants for substrate 1 and 2, respectively - K _q inhibition constant - n total number of elementary stages in the reactor - Q volumetric flow rate throughout the reactor - R _j,i, R _infa,i ^sup* local reaction rates in pores of stage i, in terms of concentration of substrate j and product a respectively - S _infa,i ^sup* , S _infa,i-1 ^sup* bulk concentration of the reaction product a, in the stages i and i — 1, respectively - S _j,0 concentration of substrate j in the reactor feed - S _j,i-1, S _j,i concentration of substrate j in the bulk phase leaving stages i — 1 and i, respectively - V total volume of the reactor - V _m maximal reaction rate in terms of volumetric units - y axial coordinate of the pores - y ₀ depth of the pores - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ^* dimensionless parameter, defined in Equation (22) - volumetric packing density of catalytic particles (dimensionless) - porosity of the catalytic particles (dimensionless) - V _infi ^sup* dimensionless concentration of reaction product in pores of stage i, defined in Equation (17) - j,i dimensionless concentration of substrate j in pores of stage i; defined in Equation (6) - _j,i-1, _j.i dimensionless concentration of substrate j in the bulk phase of stage i; defined in Equation (6) - dimensionless position along the pore; defined in Equation (6)     

Estimation of three-phase relative permeabilities

Starting from the statistical structural model of Alemánet al. (1988), we have developed an alternative to Stone's (1970, 1973; Aziz and Settari, 1979) methods for estimating steady-state, three-phase relative permeabilities from two sets of steady-state, two-phase relative permeabilities. Our result reduces to Stone's (1970; Aziz and Settari, 1979) first method, when the steady-state, two-phase relative permeability of the intermediate-wetting phase with respect to either the wetting phase or the nonwetting phase is a linear function of the saturation of the intermediate-wetting phase. As the curvature of either of these relative permeability functions increases, the deviation of our result from Stone's (1970; Aziz and Settari, 1979) first method increases. Currently, there are no data available that are sufficiently complete to form the basis of a comparison between our result and either of the methods of Stone (1970, 1973; Aziz and Settari, 1979).Notation a free parameter in Equation (19) - B(m, n) Beta function defined by Equation (17) - F ^(w), F^(nw) defined by Equations (31) and (27), respectively - G ⁽ⁱ⁾ defined by Equations (37) and (39) - H ⁽ⁱ⁾ defined by Equations (38) and (40) - k ⁽ⁱ⁾ three-phase relative permeability fo phasei - k ^(i)* defined by Equations (34) through (36) - k ^(i,j) relative permeability to phasei during a two-phase flow with phasej, possibly in the presence of an immobile phase - k ^(i,j)* defined by analogy with Equations (41) and (42) - k ^(i,j)** defined by Equations (49), (50), (53), and (54) - k _max ⁽ⁱ⁾ defined by Equation (11) - k ₁₉₇₀ ^(iw) defined by Equation (10) - k ₁₉₇₃ ^(iw)* defined by Equation (58) - k ₁₉₇₃ ^(iw) defined by Equation (13) - L length and diameter of cylindrical averaging surfaceS - L _t length of an individual capillary tube enclosed byS - L _t ^* defined by Equation (19) - L _t,min length of pore whose radius isR _max - N total number of pores contained within the averaging surfaceS - p ₁ ⁽ⁱ⁾ ,p ₂ ⁽ⁱ⁾ pressure of phasei at entrance and exit of averaging surfaceS, respectively - p defined by Equation (21) - p _c ^(i,j) capillary pressure function - p _c ^(i,j)* defined by Equations (23), (29), and (32) - p ⁽ⁱ⁾ intrinsic average of pressure within phasei defined by Alemánet al. (1988) - R pore radius - R ^* defined by Equation (18) - R _max maximum pore radius that occurs withinS - s ⁽ⁱ⁾ local saturation of phasei - s ^(i)* defined by Equation (7) - s _min ⁽ⁱ⁾ minimum or immobile saturation of phasei - S averaging surface introduced in local volume averaging - V ⁽ⁱ⁾ volume of phasei occupying the pore space enclosed byS Greek Letters , parameters in the Beta distribution defined by Equation (16) - ^(w), ^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (6) - ^(i,j) interfacial tension between phasesi andj - (x) Gamma function - defined by Equation (57) - , spherical coordinates in system centered upon the axis of the averaging surfaceS - _max maximum value of , 45 °, in view of assumption (9) - ^(i,j) contact angle between phasesi andj measured through the displacing phase - ^(w),^(nw) functions of only the wetting phase saturation and the non-wetting phase saturation, respectively. Introduced in Equation (12) Other gradient operator Amoco Production Company, PO Box 591 Tulsa, OK 74102, U.S.A.     

Measurement of the turbulent diffusivity in the near field of variable density jets using conditional velocimetry

Pulsed laser Mie scattering and laser Doppler velocimetry (LDV), both conditioned on the origin of the seed particles, have been successively performed in turbulent jets with variable density. In the early stages of the jet developments, significant differences are measured between the ensemble average LDV data obtained by jet seeding and those obtained by seeding the ambient air. Careful analysis of the marker statistics shows that this difference is a quantitative measure of the turbulent mixing. The good agreement with gradient–diffusion modelling suggests the validity of a general diffusion equation where the velocities involved are expressed in terms of ensemble conditional Favre averages. This operator accounts for all events (including intermittent ones) and for variations in the density of the marked fluid whose velocity is still specified by the binary origin of the marker.List of symbols D_L laminar diffusivity, m²/s - D_T turbulent diffusivity, m²/s - d diameter of the jet nozzle, m - F_r Froude number - J diffusion vector, m/s - k global sensitivity of the detection system for one particle (signal level) - N_P number of seed particles in the probe volume - N_P,i number of seed particles in sample i - N_P⁽ⁱ⁾ value of N_P in channel i - N_B number of Doppler bursts - count rate of bursts, s^–1 - N_v number of validated Doppler bursts - count rate of validated bursts, s^–1 - N_id number of ideal particles - N_id* number of marked ideal particles - P* probability that an ideal particle be marked by a seed particle - P(z) probability density function for z, m³/kg - probability to have k seed particles in the probe volume - probability of having k seed particle conditioned on a given value of z - r radial coordinate, m - R =⁽¹⁾/⁽²⁾, density ratio - S₁ local signal level with jet seeding - S₁⁽¹⁾ reference signal level in pure stream 1 with jet seeding - s₁ = S_1/S₁⁽¹⁾, normalized signal - v_c volumic capacity of the probe volume, m³ - V velocity vector, m/s - V_x axial velocity component, m/s - V_r radial velocity component, m/s - V_P particulate velocity vector, m/s - V_Pj velocity vector of particle j, m/s - V_Pij velocity vector of the jth particle in sample i, m/s - V_i velocity vector of the marked flow for realization i, m/s - V_1,i velocity vector of the flow such it is marked in realization i by particles issuing only from stream 1, m/s - x axial coordinate, m - Y_i local mass fraction of species i - Z mixture fraction:local mass fraction of jet fluid - Z_i mixture fraction for realization iGreek local density, kg/m³ - _i local density for realization i, kg/m³ - ⁽¹⁾ density in stream 1 (density of the jet fluid), kg/m³ - ₁ time of flight of jet seed particles to reach the probe volume, s - _B duration of a Doppler burst, sAverages <A> ensemble average of A - Ā time average of A - Favre average, , ( ) the present notation is only due to printing problems - A Favre fluctuation,      

A technique to measure wavenumber mismatch between quadratically interacting modes

Nonlinear energy cascade by means of three-wave resonant interactions is a characteristic feature of transitioning and turbulent flows. Resonant wavenumber mismatch between these interacting modes can arise from the dispersive characteristics of the interacting waves and from spectral broadening due to random effects. In this paper, a general technique is presented to estimate the average level of instantaneous wavenumber mismatch, k=k _m -k _i -k _j , between components whose frequencies obey the resonant selection condition, f _m -f _i-f _j =o. Cross-correlation of the auto-bispectrum is used to quantify the level of mismatch. The concept of bispectrum coupling coherency is introduced to determine the confidence level in the wavenumber mismatch estimates. These techniques are then applied to measure wavenumber mismatch in the transitioning field of a plane wake. The results show that the average of the instantaneous mismatch between the actual interacting modes k _m -k _i -k _j is in general not equal to the mismatch between the average wavenumbers of each interacting mode k _m -k _i -k _j .The authors wish to express their gratitude to Dr. Christoph P. Ritz of the Swiss National Science Foundation for his comments in the preparation of this work. This paper is based in part upon work supported by the Texas Advanced Research Program under Grant No. ARP 3280, and in part by the National Science Foundation under Grant No. MSM-82112O5. The digital signal processing techniques were developed under the Office of Naval Research under contract number N00014-88-k-0638     

Stochastic averaging using elliptic functions to study nonlinear stochastic systems

In this paper, a new scheme of stochastic averaging using elliptic functions is presented that approximates nonlinear dynamical systems with strong cubic nonlinearities in the presence of noise by a set of Itô differential equations. This is an extension of some recent results presented in deterministic dynamical systems. The second order nonlinear differential equation that is examined in this work can be expressed as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qeguuDJXwAKbacfiGaf8hEaGNbamaacqGHRaWkcaWGJbadcaaIXaGc% cqWF4baEcqGHRaWkcaWGJbadcaaIZaGccqWF4baEdaahaaWcbeqaai% aaiodaaaGccqGHRaWkcqaH1oqzcaWGMbGaaiikaiab-Hha4jaacYca% cqWFGaaicuWF4baEgaGaaiaacMcacqGHRaWkcqaH1oqzdaahaaWcbe% qaaiaaigdacaGGVaGaaGOmaaaaruWrL9MCNLwyaGGbcOGaa43zaiaa% cIcacqWF4baEcaGGSaGae8hiaaIaf8hEaGNbaiaacaGGSaGae8hiaa% IaeqOVdGNaaeikaiaadshacaqGPaGaaiykaiabg2da9iaaicdaaaa!645D!\[\ddot x + c1x + c3x^3 + \varepsilon f(x, \dot x) + \varepsilon ^{1/2} g(x, \dot x, \xi {\text{(}}t{\text{)}}) = 0\] where c ₁ and c ₃ are given constants, (t) is stationary stochastic process with zero mean and 1 is a small parameter. This method involves the laborious manipulation of Jacobian elliptic functions such as cn, dn and sn rather than the usual trigonometric functions. The use of a symbolic language such as Mathematica reduces the computational effort and allows us to express the results in a convenient form. The resulting equations are Markov approximations of amplitude and phase involving integrals of elliptic functions. Finally, this method was applied to study some standard second order systems.     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

Effect of hydrodynamic forces on the pressure-difference equation

Because of the influence of hydrodynamic forces, the difference in macroscopic pressure which exists, at static equilibrium, between two immiscible phases located in a porous medium may be different from that which pertains during flow. In this paper, the concept of relative pressure difference, together with a new pressure-difference equation, is used to investigate the impact that the hydrodynamic forces have on the difference in macroscopic pressure which pertains when two immiscible fluids flow simultaneously through a homogeneous, water-wet porous medium. This investigation reveals that, in general, the equation defining the difference in pressure between two flowing phases must include a term which takes proper account of the hydrodynamic effects. Moreover, it is pointed out that, while neglect of the hydrodynamic effects introduces only a small amount of error when the two fluids are flowing cocurrently, such neglect is not permissible during steady-state, countercurrent flow. This is because failure to include the impact of the hydrodynamic effects in the latter case makes it impossible to explain the pressure behaviour observed in steady-state, countercurrent flow. Finally, the results of this investigation are used as a basis for arguing that, during steady-state, countercurrent flow, saturation is uniform, as is the case of steady-state, cocurrent flow.Roman Letters a parameter in Equation (18) - k absolute permeability, m² - k _i effective permeability to phasei;i=1, 2, m² - k _ij generalized effective permeability for phasei;i, j=1, 2, m² - p _d p ₂–p ₁=difference in macroscopic pressure between two flowing phases, N/m² - p _i pressure for phasei;i=1, 2, N/m² - p _h hydrodynamic contribution to difference in macroscopic pressure which exists during flow, N/m² - P _c macroscopic static capillary pressure, N/m² - R ₁₂ function defined by Equation (18) - S _i saturation of phasei;i=1, 2 - S _n normalized saturation of phase 1 - t time, s - u _i flux of phasei;i=1, 2m³/m²/s - x distance in direction of flow, m Greek Letters _R relative pressure difference - _i k _i / _i =mobility of phasei;i=1, 2m²/Pa·s - _ij k _ij / _j =generalized mobility of phasei;i, j=1, 2m²/Pa·s - _i viscosity of phasei;i=1, 2, Pa·s - porosity     

Die Analogie zwischen den Verschiebungen und den Spannungsfunktionen in der Biegetheorie der Kreiszylinderschale

Zusammenfassung Für die Kreiszylinderschale wurde eine Biegetheorie aufgestellt, in der die Gleichgewichtsbedingungen (unter Voraussetzung der Symmetrie des Momententensors M _ik ) durch drei Spannungsfunktionen ₁, ₂, ₃ exakt erfüllt sind. Bei der Definition der Deformationsgrößen und der Einführung der Elastizitätsgesetze war die Reißner-Meißnersche Theorie der symmetrisch belasteten Rotationsschale das Vorbild. Die drei Differentialgleichungen für die Verschiebungen _{1 2}, ₃ unterscheiden sich von den drei Differentialgleichungen für die Spannungsfunktionen ₁, ₂, ₃ formal nur im Vorzeichen der Poissonschen Querkontraktionsziffer v. Die beiden Differentialgleichungen achter Ordnung, die man nach Eliminationsprozessen sowohl für ₃ als auch für ₃ erhält, unterscheiden sich nicht mehr voneinander. So trifft man bei der Zylinderschale die Timpe-Wieghardtsche Analogie zwischen Durchbiegung ₃ der Platte und Airyscher Spannungsfunktion ₃ der Scheibe wieder.Es konnte ferner gezeigt werden, daß unsere neue Biegetheorie der bekannten Flüggeschen Theorie an Genauigkeit nicht nachsteht.Es ist wohl nicht zu bezweifeln, daß auch bei Schalen beliebiger Gestalt unsere Analogie vorhanden ist. Sie scheint uns wertvoll als Ordnungsprinzip inmitten der Fülle von Gleichungen, die nun einmal zu einer Schalentheorie gehören.Die Formulierung des Schalenproblems mit Hilfe der drei Spannungsfunktionen ₁, ₂, ₃ wird sich immer dann empfehlen, wenn die Randbelastung vorgegeben ist. Denn dann lassen sich die Randbedingungen in den Spannungsfunktionen übersichtlicher formulieren als in den Verschiebungen. Auch die Gewißheit, daß selbst durch radikales Streichen lästiger Glieder in den Differentialgleichungen der Spannungsfunktionen die Gleichgewichtsbedingungen nicht verletzt werden, mag manchem Rechner angenehm sein.     

Conditions for noncircular whirling of nonlinear isotropic rotors

Nonlinear rotors are often considered as potential sources of chaotic vibrations. The aim of the present paper is that of studying in detail the behaviour of a nonlinear isotropic Jeffcott rotor, representing the simplest nonlinear rotor. The restoring and damping forces have been expanded in Taylor series obtaining a Duffing-type equation. The isotropic nature of the system allows circular whirling to be a solution at all rotational speeds. However there are ranges of rotational speed in which this solution is unstable and other, more complicated, solutions exist.The conditions for stability of circular whirling are first studied from closed form solutions of the mathematical model and then the conditions for the existence of solutions of other type are studied by numerical experimentation. Although attractors of the limit cycle type are often found, chaotic attractors were identified only in few very particular cases. An attractor supposedly of the last type reported in the literature was found, after a detailed analysis, to be related to a nonchaotic polyharmonic solution.As the typical unbalance response of isotropic nonlinear rotors has been shown to be a synchronous circular whirling motion, the convergence characteristics of Newton-Raphson algorithm applied to the solution of the set of nonlinear algebraic equations obtained from the differential equations of motion are studied in some detail. c damping coefficient i imaginaty unit (i=% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbqfgBHr% xAU9gimLMBVrxEWvgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvA% Tv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9% vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea% 0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabe% aadaabauaaaOqaamaakaaabaGaeyOeI0IaaGymaaWcbeaaaaa!3E66!\[\sqrt { - 1}\]) k stiffness m mass t time x _istate variables i=1, 4 z complex co-ordinate (z=x+iy)[J] Jacobian matrix Oxyz inertial co-ordinate frame Oz rotating co-ordinate frame perturbation term eccentricity complex co-ordinate (=+i) system eigenvalues nonlinearity parameter nondimensional time phase spin speed u nonrotating t rotating0 amplitude t nondimensional termsNomenclature