论对数律和Kármán常数

本文从小涡球的随机运动出发研究壁湍流的流动特征。建立了这个随机过程所满足的随机微分方程。解此方程再次导出了著名的对数律,并从理论上决定了对数律中的Kármán常数,K=1/6~(1/2)=0.408。     

One-dimensional von Kármán models for elastic ribbons

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von Kármán plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the “linearized” von Kármán energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von Kármán energy is a non-linear model that comprises stretching, bendings, and twisting. The “constrained” von Kármán energy, instead, leads to a new Sadowsky type of model.     

Partial roll-up of a viscoelastic Kármán street

The instantaneous velocity field of a circular cylinder wake is built using a PIV technique when a small amount of viscoelastic liquid is introduced through the cylinder. It is shown that the viscoelastic fluid slows down the vorticity concentration and produces a street of partially rolled-up vortices. The underlying mechanism appears very analogous to that of a surface tension in the Kelvin–Helmholtz instability. The partial roll-up is studied in terms of the Weiss determinant. This quantity is a local measurement of the spatial separation between the strain and vorticity. In the viscoelastic wake, the Weiss determinant reaches much lower values than in the Newtonian wake. This result shows that the elasticity prevents the clear separation between vorticity and strain during the roll-up process. Since the Weiss determinant is directly related to the pressure field, it suggests that elasticity can drastically modify the pressure levels even when vorticity and strain levels are unaffected.     

An Existence Theorem for Generalized von Kármán Equations

Using techniques from formal asymptotic analysis, the first two authors have recently identified generalized von Kármán equations, which constitute a two-dimensional model for a nonlinearly elastic plate where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type, the remaining portion being free. In this paper, we establish an existence theorem for these equations. To this end, we first reduce them to a single equation, which generalizes a cubic operator equation introduced by M.S. Berger and P. Fife. We then directly solve this equation, notably by adapting a crucial compactness method due to J.-L. Lions. Résumé. En utilisant les techniques de l'analyse asymptotique formelle, les deux premiers auteurs ont récemment identifié des équations de von Kármán généralisées, qui constituent un modèle bi-dimensionnel de plaque non linéairement élastique dont une partie seulement de la face latérale est soumise à des conditions aux limites de von Kármán, la partie restante étant libre. Dans cet article, on établit un théorème d'existence pour ces équations. À cette fin, elles sont d'abord réduites à une seule équation, qui généralise une équation faisant intervenir un opérateur cubique, introduite par M.S. Berger et P. Fife. On résout ensuite directement cette équation, en adaptant notamment une méthode cruciale de compacité due à J.-L. Lions.     

On the refined first-order shear deformation plate theory of Kármán type

A new refined first-order shear-deformation plate theory of the Kármán type is presented for engineering applications and a new version of the generalized Kármán large deflection equations with deflection and stress functions as two unknown variables is formulated for nonlinear analysis of shear-deformable plates of composite material and construction, based on the Mindlin/Reissner theory. In this refined plate theory two rotations that are constrained out in the formulation are imposed upon overall displacements of the plates in an implicit role. Linear and nonlinear investigations may be made by the engineering theory to a class of shear-deformation plates such as moderately thick composite plates, orthotropic sandwich plates, densely stiffened plates, and laminated shear-deformable plates. Reduced forms of the generalized Kármán equations are derived consequently, which are found identical to those existe in the literature. Foundation item: the National Natural Science Foundation of China (59675027) Biography: Zhang Jianwu (1954-)     

含孔von Kármán板中的高次谐波散射现象

基于vonKarman板大挠度弯曲理论，利用谐波平衡法及小参数振动法，研究了含孔vonKarman板的非线性波散射问题．通过分析发现：由于弯曲应力与中面力的非线性耦合，vonKarman板中会出现高次谐波散射现象．     

含孔von kármán板中的高次谐波散射现象

基于vonKarman板大挠度弯曲理论,利用谐波平衡法及小参数振动法,研究了含孔vonKarman板的非线性波散射问题．通过分析发现：由于弯曲应力与中面力的非线性耦合,vonKarman板中会出现高次谐波散射现象．     

Existence Result for a Dynamical Equations of Generalized Marguerre-von Kármán Shallow Shells

In a recent work in the static case, Gratie (Appl. Anal. 81:1107–1126, 2002) has generalized the classical Marguerre-von Kármán equations studied by Ciarlet and Paumier in (Comput. Mech. 1:177–202, 1986), where only a portion of the lateral face of the shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is subjected to boundary conditions of free edge. Then Ciarlet and Gratie (Math. Mech. Solids 11:83–100, 2006) have established an existence theorem for these equations. In Chacha et al. (Rev. ARIMA 13:63–76, 2010), we extended formally these studies to the dynamical case. More precisely, we considered a three-dimensional dynamical model for a nonlinearly elastic shallow shell with a specific class of boundary conditions of generalized Marguerre-von Kármán type. Using technics from formal asymptotic analysis, we showed that the scaled three-dimensional solution still leads to two-dimensional dynamical boundary value problem called the dynamical equations of generalized Marguerre-von Kármán shallow shells. In this paper, we establish the existence of solutions to these equations using a compactness method of Lions (Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969).     

Modified von Kármán equations for elastic nanoplates with surface tension and surface elasticity

In this paper, modified von Kármán equations are derived for Kirchhoff nanoplates with surface tension and surface tension-induced residual stresses. The simplified Gurtin-Murdoch model which does not contain non-strain displacement gradients in surface stress-strain relations is adopted, so that the von Kármán strain-compatibility equation can be expressed in terms of the stress function and deflection. The modified von Kármán equations derived here are different than the existing related models especially for elastic plates with in-plane movable edges. Unlike the existing models which predict a surface tension-induced tensile pre-stress for an elastic plate with in-plane movable edges, the present model predicts that this tensile pre-stress is actually cancelled by the surface tension-induced residual compressive stress. Our this result is consistent with recent clarification on similar issue for cantilever beams with surface tension, which implies that the existing models have incorrectly predicted an invalid tensile pre-stress for an elastic plate with in-plane movable edges which leads to significant overestimation of postbuckling load and free vibration frequencies. In addition, our numerical examples indicated that surface stresses can moderately increase or decrease postbuckling load and free vibration frequency of Kirchhoff nanoplate with all in-plane movable edges, depending on the surface elasticity parameters and the geometrical dimensions of nanoplates.     

Stability of Slender Bodies under Compression and Validity of the von Kármán Theory

Since their formulation almost 100 years ago, the von Kármán (vK) plate equations have been frequently used both by engineers and by analysts to study thin elastic bodies, in particular their stability behaviour under applied loads. At the same time the derivation of these equations met some harsh criticism and their precise mathematical status has been unclear until very recently. Following up on a recent variational derivation of the vK theory by Friesecke, James and Müller from three-dimensional nonlinear elasticity we study the predictions and the validity of the vK equation in the presence of in-plane compressive forces. The first main result is a stability alternative: either the load leads to instability already in the nonlinear bending theory of plates (Kirchhoff–Love theory), or it leads to an instability in a geometrically linear KL theory (‘linearized instability’), or vK theory is valid. The second main result states that under suitable conditions the critical loads for nonlinear stability and linearized instability coincide. The third main result asserts this critical load also agrees with the load beyond which the infimum of the vK functional is −∞. The main ingredients are a sharp rigidity estimate for maps with low elastic energy and a study of the properties of isometric immersions from a set in to and their geometrically linear counterparts.     

Transient flow of magnetized Maxwell nanofluid: Buongiorno model perspective of Cattaneo-Christov theory

The present research article is devoted to studying the characteristics of Cattaneo-Christov heat and mass fluxes in the Maxwell nanofluid flow caused by a stretching sheet with the magnetic field properties. The Maxwell nanofluid is investigated with the impact of the Lorentz force to examine the consequence of a magnetic field on the flow characteristics and the transport of energy. The heat and mass transport mechanisms in the current physical model are analyzed with the modified versions of Fourier’s and Fick’s laws, respectively. Additionally, the well-known Buongiorno model for the nanofluids is first introduced together with the Cattaneo-Christov heat and mass fluxes during the transient motion of the Maxwell fluid. The governing partial differential equations (PDEs) for the flow and energy transport phenomena are obtained by using the Maxwell model and the Cattaneo-Christov theory in addition to the laws of conservation. Appropriate transformations are used to convert the PDEs into a system of nonlinear ordinary differential equations (ODEs). The homotopic solution methodology is applied to the nonlinear differential system for an analytic solution. The results for the time relaxation parameter in the flow, thermal energy, and mass transport equations are discussed graphically. It is noted that higher values of the thermal and solutal relaxation time parameters in the Cattaneo-Christov heat and mass fluxes decline the thermal and concentration fields of the nanofluid. Further, larger values of the thermophoretic force enhance the heat and mass transport in the nanoliquid. Moreover, the Brownian motion of the nanoparticles declines the concentration field and increases the temperature field. The validation of the results is assured with the help of numerical tabular data for the surface velocity gradient.     

Rotational flow of Oldroyd-B nanofluid subject to Cattaneo-Christov double diffusion theory

A nanofluid is composed of a base fluid component and nanoparticles, in which the nanoparticles are dispersed in the base fluid. The addition of nanoparticles into a base fluid can remarkably improve the thermal conductivity of the nanofluid, and such an increment of thermal conductivity can play an important role in improving the heat transfer rate of the base fluid. Further, the dynamics of non-Newtonian fluids along with nanoparticles is quite interesting with numerous industrial applications. The present predominately predictive modeling studies the flow of the viscoelastic Oldroyd-B fluid over a rotating disk in the presence of nanoparticles. A progressive amendment in the heat and concentration equations is made by exploiting the Cattaneo-Christov heat and mass flux expressions. The characteristic of the Lorentz force due to the magnetic field applied normal to the disk is studied. The Buongiorno model together with the Cattaneo-Christov theory is implemented in the Oldroyd-B nanofluid flow to investigate the heat and mass transport mechanism. This theory predicts the characteristics of the fluid thermal and solutal relaxation time on the boundary layer flow. The von K′arm′an similarity functions are utilized to convert the partial differential equations(PDEs) into ordinary differential equations(ODEs). A homotopic approach for obtaining the analytical solutions to the governing nonlinear problem is carried out. The graphical results are obtained for the velocity field, temperature, and concentration distributions. Comparisons are made for a limiting case between the numerical and analytical solutions, and the results are found in good agreement. The results reveal that the thermal and solutal relaxation time parameters diminish the temperature and concentration distributions, respectively. The axial flow decreases in the downward direction for higher values of the retardation time parameter. The impact of the thermophoresis parameter boosts the temperature distribution.     

Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium

The melting phenomenon in two-dimensional (2D) flow of fourth-grade material over a stretching surface is explored. The flow is created via a stretching surface. A Darcy-Forchheimer (D-F) porous medium is considered in the flow field. The heat transport is examined with the existence of the Cattaneo-Christov (C-C) heat flux. The fourth-grade material is electrically conducting subject to an applied magnetic field. The governing partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by appropriate transformations. The solutions are constructed analytically through the optimal homotopy analysis method (OHAM). The fluid velocity, temperature, and skin friction are examined under the effects of various involved parameters. The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter, porosity parameter, and Forchheimer number. The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number, magnetic parameter, and thermal relaxation parameter. Furthermore, the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters, thermal relaxation parameter, and Forchheimer number.

     

Investigation of generalized Fick’s and Fourier’s laws in the second-grade fluid flow

The second-grade fluid flow due to a rotating porous stretchable disk is modeled and analyzed. A porous medium is characterized by the Darcy relation. The heat and mass transport are characterized through Cattaneo-Christov double diffusions. The thermal and solutal stratifications at the surface are also accounted. The relevant nonlinear ordinary differential systems after using appropriate transformations are solved for the solutions with the homotopy analysis method (HAM). The effects of various involved variables on the temperature, velocity, concentration, skin friction, mass transfer rate, and heat transfer rate are discussed through graphs. From the obtained results, decreasing tendencies for the radial, axial, and tangential velocities are observed. Temperature is a decreasing function of the Reynolds number, thermal relaxation parameter, and Prandtl number. Moreover, the mass diffusivity decreases with the Schmidt number.     

Thermal analysis in swirl motion of Maxwell nanofluid over a rotating circular cylinder

In this paper, the mechanism of thermal energy transport in swirling flow of the Maxwell nanofluid induced by a stretchable rotating cylinder is studied. The rotation of the cylinder is kept constant in order to avoid the induced axially secondary flow. Further, the novel features of heat generation/absorption, thermal radiation, and Joule heating are studied to control the rate of heat transfer. The effects of Brownian and thermophoretic forces exerted by the Maxwell nanofluid to the transport of thermal energy are investigated by utilizing an effective model for the nanofluid proposed by Buongiorno. The whole physical problem of fluid flow and thermal energy transport is modelled in the form of partial differential equations(PDEs) and transformed into nonlinear ordinary differential equations(ODEs) with the help of the suitable flow ansatz.Numerically acquired results through the technique bvp4c are reported graphically with physical explanation. Graphical analysis reveals that there is higher transport of heat energy in the Maxwell nanoliquid for a constant wall temperature(CWT) as compared with the prescribed surface temperature(PST). Both thermophoretic and Brownian forces enhance the thermal energy transport in the flowing Maxwell nanofluid. Moreover, the temperature distribution increases with increasing values of the radiation parameter and the Eckert number. It is also noted that an increase in Reynolds number reduces the penetration depth, and as a result the flow and transport of energy occur only near the surface of the cylinder.     

Darcy-Forchheimer flow of variable conductivity Jeffrey liquid with Cattaneo-Christov heat flux theory

The role of the Cattaneo-Christov heat flux theory in the two-dimensional laminar flow of the Jeffrey liquid is discussed with a vertical sheet. The salient feature in the energy equation is accounted due to the implementation of the Cattaneo-Christov heat flux. A liquid with variable thermal conductivity is considered in the Darcy-Forchheimer porous space. The mathematical expressions of momentum and energy are coupled due to the presence of mixed convection. A highly nonlinear coupled system of equations is tackled with the homotopic algorithm. The convergence of the homotopy expressions is calculated graphically and numerically. The solutions of the velocity and temperature are expressed for various values of the Deborah number, the ratio of the relaxation time to the retardation time, the porosity parameter, the mixed convective parameter, the Darcy-Forchheimer parameter, and the conductivity parameter. The results show that the velocity and temperature are higher in Fourier's law of heat conduction cases in comparison with the Cattaneo-Christov heat flux model.     

Stratified magnetohydrodynamic flow of tangent hyperbolic nanofluid induced by inclined sheet

This paper studies stratified magnetohydrodynamic(MHD) flow of tangent hyperbolic nanofluid past an inclined exponentially stretching surface. The flow is subjected to velocity, thermal, and solutal boundary conditions. The partial differential systems are reduced to ordinary differential systems using appropriate transformations.The reduced systems are solved for convergent series solutions. The velocity, temperature,and concentration fields are discussed for different physical parameters. The results indicate that the temperature and the thermal boundary layer thickness increase noticeably for large values of Brownian motion and thermophoresis effects. It is also observed that the buoyancy parameter strengthens the velocity field, showing a decreasing behavior of temperature and nanoparticle volume fraction profiles.     

Dynamics of Sutterby fluid flow due to a spinning stretching disk with non-Fourier/Fick heat and mass flux models

The magnetohydrodynamic Sutterby fluid flow instigated by a spinning stretchable disk is modeled in this study. The Stefan blowing and heat and mass flux aspects are incorporated in the thermal phenomenon. The conventional models for heat and mass flux, i.e., Fourier and Fick models, are modified using the Cattaneo-Christov(CC)model for the more accurate modeling of the process. The boundary layer equations that govern this problem are solved using the apt similarity variables. The subsequent system of equations is tackled by the Runge-Kutta-Fehlberg(RKF) scheme. The graphical visualizations of the results are discussed with the physical significance. The rates of mass and heat transmission are evaluated for the augmentation in the pertinent parameters. The Stefan blowing leads to more species diffusion which in turn increases the concentration field of the fluid. The external magnetism is observed to decrease the velocity field. Also,more thermal relaxation leads to a lower thermal field which is due to the increased time required to transfer the heat among fluid particles. The heat transport is enhanced by the stretching of the rotating disk.     

Cattaneo-Christov heat and mass flux model for 3D hydrodynamic flow of chemically reactive Maxwell liquid

This research focuses on the Cattaneo-Christov theory of heat and mass flux for a three-dimensional Maxwell liquid towards a moving surface. An incompressible laminar flow with variable thermal conductivity is considered. The flow generation is due to the bidirectional stretching of sheet. The combined phenomenon of heat and mass transport is accounted. The Cattaneo-Christov model of heat and mass diffusion is used to develop the expressions of energy and mass species. The first-order chemical reaction term in the mass species equation is considered. The boundary layer assumptions lead to the governing mathematical model. The homotopic simulation is adopted to visualize the results of the dimensionless flow equations. The graphs of velocities, temperature, and concentration show the effects of different arising parameters. A numerical benchmark is presented to visualize the convergent values of the computed results. The results show that the concentration and temperature fields are decayed for the Cattaneo-Christov theory of heat and mass diffusion.     

Numerical and statistical approach for Casson-Maxwell nanofluid flow with Cattaneo-Christov theory

The rheological features of an incompressible axi-symmetric Casson-Maxwell nanofluid flow between two stationary disks are examined. The lower permeable disk is located at z =-a, while the upper disk is placed at z = a. Both the disks are porous and subjected to uniform injection. The fluid properties such as thermal conductivity vary with temperature. The Cattaneo-Christov thermal expression is implemented along with the Buongiorno nanofluid theory. By operating the similarity functions, the reduced form of the fluid model in terms of ordinary differential equations is obtained and solved by the bvp4 c numerical technique. The physical quantities are demonstrated graphically on the velocity and temperature fields. Three-dimensional flow arrangements and twodimensional contour patterns against several dimensionless variables are also sketched.The numerical values of the local Nusselt and Sherwood numbers for various quantities are presented in tabular set-up. The intensity of the linear relationship between the Nusselt and Sherwood numbers is assessed through Pearson's product-moment correlation technique. The statistical implication of the linear association between variables is also examined by the t-test statistic approach.