Dynamics of Branched Continuous-Discrete Systems Towed in Rough Water

A method is proposed for dynamic analysis of towed spatial branched structures. The method permits determining the kinematic and force characteristics of a structure towed either downwave or upwave. The slacking of certain elements of the structure and changed speed of its motion are taken into account. The dynamics of a specific system is analyzed in the cases of abruptly changing towing speed and towing in regularly rough water with different wave parameters __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 84–89, October 2005.     

Testing and calculations of the motion of a model ship with a direct-flow wave propulsor

A new design of the wave propulsor is presented. In this design the thrust mechanism is due to the interaction between the waves and the ship structure elements rather than to ship’s motions. To verify the possibility of using a rigidly fixed inclined plate as the ship wave propulsor a model catamaran was constructed in the Institute of Mechanics of Moscow State University. The effect of the upwave motion of the ship, whose mean velocity is a nonmonotonic function of the wavelength, is studied. As the plate edge pierces the water surface, the ship starts to move in the opposite direction, that is, downwave. The experimentally observable effects are also revealed in the numerical simulation using the XFlow software package which involves the meshless lattice Boltzmann method. On the basis of the calculated results it is shown that the upwave motion effect is due to a variation in the hydrostatic force component in the case of wave breakdown.     

Wave action on a movable plate immersed into a fluid. Physical and numerical experiments

The effect of the motion of a rigid inclined plate immersed into a fluid counter incident waves is studied both experimentally and numerically. In the wave water channel experiments the velocity of a trolley with a plate running freely on rails is determined as a function of the wave parameters, the immersion depth, the angle of inclination, and the plate dimensions. The interaction between the traveling waves and the plate having a single translational degree of freedom along the horizontal axis is numerically calculated in the time-dependent, two-dimensional formulation. The dependence of the upwave motion effect on the parameters varied in the full-scale experiment is analyzed. In the numerical experiment a regime of the downwave plate motion at a constant high velocity is found to exist. The channel bottom effect is estimated and the behavior of the plate with a flap is studied.     

Diffraction of cnoidal waves by vertical cylinders in shallow water

Diffraction of nonlinear waves by single or multiple in-line vertical cylinders in shallow water is studied by use of different nonlinear, shallow-water wave theories. The fixed, in-line, vertical circular cylinders extend from the free surface to the seafloor and are located in a row parallel to the incident wave direction. The wave–structure interaction problem is studied by use of the nonlinear generalized Boussinesq equations, the Green–Naghdi shallow-water wave equations, and the linearized version of the shallow-water wave equations. The wave-induced force and moment of the Green–Naghdi and the Boussinesq equations are presented when the incoming waves are cnoidal, and the forces are compared with the experimental data when available. Results of the linearized equations are compared with the nonlinear results. It is observed that nonlinearity is very important in the calculation of the wave loads on circular cylinders in shallow water. The variation of wave loads with wave height, wavelength and the spacing between cylinders is studied. Effect of the neighboring cylinders, and the shielding effect of upwave cylinders on the wave-induced loads on downwave cylinders are discussed.     

Steady circulatory flow about a rotating circular cylinder with uniformly distributed suction at the surface

An exact solution for steady circulatory flow about an infinite porous circular cylinder rotating with a given angular velocity in an incompressible non-Newtonian second-order fluid that is also rotating so that a given circulation is maintained at infinity, is investigated. Using the Coleman-Noll model for the fluid, it is found that when circulation, velocity, vorticity and pressure are affected by non-Newtonian effects due to second-order terms in the constitutive equation of the fluid even at the first-order approximation or the series solution used, torque is independent of these effects even when the second-order approximation is considered.     

Deformation measurements by digital image correlation: Implementation of a second-order displacement gradient

This paper outlines the procedure for refining the digital image correlation (DIC) method by implementing a second-order approximation of the displacement gradients. The second-order approximation allows the DIC method to directly measure both the first- and second-order displacement gradients resulting from nonlinear deformation. Thirteen unknown parameters, consisting of the components of displacement, the first- and second-order displacement gradients and the gray-scale value offset, are determined through optimization of a correlation coefficient. The previous DIC method assumes that the local deformation in a subset of pixels is represented by a first-order Taylor series approximation for the displacement gradient terms, so actual deformations consisting of higher order displacement gradients tend to distort the infinitesimal strain measurements. By refining the method to measure both the first- and second-order displacement gradients, more accurate strain measurements can be achieved in large-deformation situations where second-order deformations are also present. In most cases, the new refinements allow the DIC method to maintain an accuracy of ±0.0002 for the first-order displacement gradients and to reach ±0.0002 per pixel for the second-order displacement gradients.     

Analysis of heat conduction in a rarefied gas at rest with a temperature jump at the boundary by consistent-order extended thermodynamics

In the context of the first- and second-order theories of consistent-order extended thermodynamics, a systematic approach is established for analyzing the temperature jump at the boundary through studying one-dimensional stationary heat conduction in a rarefied gas at rest. Thereby an approach to the free boundary-value problem in general is explored. Boundary values of temperature are assumed to be in the form of power expansion with respect to the Knudsen number, based on which analytical expressions of the temperature jump aswell as entropy production at the boundary are derived explicitly. Dependencies of these two boundary quantities on both the Knudsen number and accommodation factor are also extensively discussed. The present analysis is expected to be the basis for the study of higher-order theories of consistent-order extended thermodynamics.      

Non-Newtonian effects of second-order fluids on double-layered porous Rayleigh-step bearings

Theoretical study of double-layered porous Rayleigh-step bearings with second-order fluid as lubricant is presented. An approximate method for the solutions of the governing fluid film equations for a porous region is proposed. The expressions for the pressure distribution, load capacity and frictional coefficient are obtained in compact form. Calculations of the dimensionless load capacity, frictional force and frictional coefficients are presented for specific values of the material parameters. It is found that the double porous layer yields an increase in the load capacity and ensures decrease in frictional force at the porous lining as compared with the conventional porous Rayleigh-step bearings. The maximum dimensionless load-carrying capacity is found to occur at a slightly larger step ratio as compared with the conventional porous Rayleigh-step bearings.     

Nonlinear free surface action with an array of vertical cylinders

Nonlinear diffraction of regular waves by an array of bottom-seated circular cylinders is investigated in frequency domain, based on a Stokes expansion approach. A complete semi-analytical solution is developed which allows an efficient evaluation of the second-order potentials in the entire fluid domain, and the wave forces on the structure. Expressions are derived for the second-order potential in the vicinity of individual cylinders. These expressions have a simple form, thus providing an effective means for investigating the wave enhancement due to nonlinear interactions with multiple cylinders. Based on the present method, the wave run-up and free-surface elevations around an array of two, three and four cylinders are investigated numerically.     

On divergence-free stress fields and zero-energy stress functions in elasticity

Applying two identities for divergence-free non-symmetric and symmetric second-order tensors, novel type of first- and second-order stress functions are proposed for three-dimensional elasticity problems. It is shown that self-equilibrated but non-symmetric 3D stress fields can be generated by one first-order stress function vector, whereas a self-equilibrated and symmetric 3D stress field can be generated by one Airy-type second-order stress function. Assuming linearly elastic materials, the zero-energy modes of the stress functions introduced are derived and investigated. It is pointed out that the structure of the zero-energy modes of the proposed first-order stress function vector is the same as that of the rigid-body displacements in the linear theory of elasticity.     

多孔介质——自由流界面应力跳跃条件下流动特性解析解

对非对称多孔介质--自由流复合通道内多孔介质内部及多孔介质与自由流体界面处复杂质量、动量输运特性进行研究. 在多孔介质区采用Brinkman-extended Darcy模型并结合速度连续,剪切应力跳跃的界面条件对此复合通道内流体的传递现象进行求解,提出了考虑界面应力跳跃时非对称复合通道各区域流体运动速度及摩擦系数的解析式,分析了界面应力跳跃系数,达西数及无量纲多孔层偏心厚度对流体速度及摩擦系数的影响. 结果表明:改变界面性质可在一定条件下明显控制各区域流体速度分布;在达西数、多孔层偏心厚度一定情况下,界面应力系数的增大会使界面流速减小,而使流体摩擦系数增大,特别是界面应力系数小于0的情况下变化更明显,此时若不考虑界面应力系数则会造成较大误差. 当界面应力系数及多孔层偏心厚度均为较小负数值时,改变多孔层偏心厚度对界面速度的影响要大于改变界面应力系数的情况;而当界面应力系数及多孔层偏心厚度为较大正数值时,情况则相反. 较大达西数下,界面应力系数及多孔层偏心厚度对流体摩擦系数的影响均较大,继续减小达西数至一定程度时,界面应力系数对流体摩擦系数的影响可忽略不计而认为只与多孔层偏心厚度相关,且对较大多孔层偏心厚度更敏感.      

Effect of Viscous Dissipation on the Boundary Layer Flow and Heat Transfer Past a Permeable Stretching Surface Embedded in a Porous Medium with a Second-Order Slip Using Chebyshev Finite Difference Method

This article investigates a theoretical and numerical study for the effect of viscous dissipation on the steady flow with heat transfer of Newtonian fluid toward a permeable stretching surface embedded in a porous medium with a second-order slip and thermal slip. The governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting ODEs are successfully solved numerically with the help of Chebyshev finite difference method. Graphically results are shown for non-dimensional velocities and temperature. The effects of the porous parameter, the suction (injection) parameter, Eckert number, first- and second-order velocity slip parameter, the thermal slip parameter and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. A comparison of numerical results is made with the earlier published results under limiting cases.     

Mass transport in morphogenetic processes: A second gradient theory for volumetric growth and material remodeling

In this work, we derive a novel thermo-mechanical theory for growth and remodeling of biological materials in morphogenetic processes. This second gradient hyperelastic theory is the first attempt to describe both volumetric growth and mass transport phenomena in a single-phase continuum model, where both stress- and shape-dependent growth regulations can be investigated. The diffusion of biochemical species (e.g. morphogens, growth factors, migration signals) inside the material is driven by configurational forces, enforced in the balance equations and in the set of constitutive relations. Mass transport is found to depend both on first- and on second-order material connections, possibly withstanding a chemotactic behavior with respect to diffusing molecules. We find that the driving forces of mass diffusion can be written in terms of covariant material derivatives reflecting, in a purely geometrical manner, the presence of a (first-order) torsion and a (second-order) curvature. Thermodynamical arguments show that the Eshelby stress and hyperstress tensors drive the rearrangement of the first- and second-order material inhomogeneities, respectively. In particular, an evolution law is proposed for the first-order transplant, extending a well-known result for inelastic materials. Moreover, we define the first stress-driven evolution law of the second-order transplant in function of the completely material Eshelby hyperstress.The theory is applied to two biomechanical examples, showing how an Eshelbian coupling can coordinate volumetric growth, mass transport and internal stress state, both in physiological and pathological conditions. Finally, possible applications of the proposed model are discussed for studying the unknown regulation mechanisms in morphogenetic processes, as well as for optimizing scaffold architecture in regenerative medicine and tissue engineering.     

Rotating flow of a second-order fluid on a porous plate

A study is made of the unsteady flow engendered in a second-order incompressible, rotating fluid by an infinite porous plate exhibiting non-torsional oscillation of a given frequency. The porous character of the plate and the non-Newtonian effect of the fluid increase the order of the partial differential equation (it increases up to third order). The solution of the initial value problem is obtained by the method of Laplace transform. The effect of material parameters on the flow is given explicitly and several limiting cases are deduced. It is found that a non-Newtonian effect is present in the velocity field for both the unsteady and steady-state cases. Once again for a second-order fluid, it is also found that except for the resonant case the asymptotic steady solution exists for blowing. Furthermore, the structure of the associated boundary layers is determined.     

Second-order model for free surface convection and interface reconstruction

To improve the numerical analysis of free surface convection and interface reconstruction, both first- and second-order algorithms are developed based on the volume-of-fluid method. The methodology applied to the second-order model is to define the second-order linear curve having both face slopes as near horizontal as possible while satisfying the cell's defined volume fraction. The second-order method is compared with the FLAIR method and the first-order method through simulation of the convection for various sizes of circular liquid shapes and solitary waves. For small curvature of the free surface, e.g. circles with large diameter, linear methods such as the FLAIR method and the first-order method show relatively good predictions. However, for large-curvature configurations, e.g. circles with relatively small diameter or solitary waves, the linear approach shows large distortion of the free surface. In contrast, the second-order model always shows powerful prediction capabilities of free surface convection. © 1998 John Wiley & Sons, Ltd.     

Second-order hydroelastic analysis of a floating plate in multidirectional irregular waves

Based on an analysis of second-order hydrodynamic forces induced by coupling of first-order wave potentials, second-order hydroelastic equations are established and solved in the frequency domain. The responses of a very large floating structure in multidirectional irregular waves are studied. The characteristics of the difference and sum frequency coordinates are discussed in detail; peaks can be found at the difference and sum frequencies close to the wet resonant frequencies of each mode. We present and analyze the maximum vertical displacement of different points as well as the time history of the vertical displacements of selected points. The differences of the combined (the summation of the linear and non-linear responses) and linear displacements of the selected points are calculated. Our results demonstrate that non-linear fluid forces influence the total responses of the referenced floating structure.     

Rayleigh-type wave propagation through liquid layer over corrugated substrate

The propagation of the Rayleigh-type wave in a fluid layer overlying a corrugated substrate is studied. The corrugated substrate is considered as a fluid saturated poroelastic substrate and a quadratically heterogeneous isotropic elastic substrate in Case I and Case II, respectively. Closed form expressions of dispersion relation for Case I and Case II are obtained. The influence of corrugation, porosity, and heterogeneity on the phase velocity of Rayleigh-type wave, for both cases, is highlighted and demonstrated through numerical computation and graphical discussion. Neglecting corrugation at the common interface, expressions of phase velocity of the Rayleigh-type wave for both cases are derived in a closed form as a special case of the problem. Comparison between the presence and the absence of both heterogeneity and poroelasticity in the substrate of the composite structure is a key in the present study.     

Study of the antiplane deformation of linear 2-D polycrystals with different microstructures

The effective behavior and the distribution of local mechanical fields of linearly viscous 2-D polycrystals under antiplane shear is investigated. Several microstructures are considered, and a full-field approach based on the Fast Fourier Transform technique is applied. First, the accuracy of this technique is evaluated on a strictly isotropic 2-phase microstructure. Voronoi tessellation is then used to generate artificial microstructures, and a real (fully recrystallized) polycrystalline microstructure is obtained by electron back-scattering diffraction. Ensemble averages over several configurations using eight crystalline orientations (phases) are performed. Although a slight anisotropy is obtained for the effective behavior of each individual configuration, statistically, the results are in very good agreement with the available analytical isotropic solution. At phase level, a marked asymmetry is obtained for the distribution of local stresses. The intraphase first- and second-order moments of the stress field, calculated for both microstructures are compared with corresponding self-consistent predictions.     

On the thermodynamics of the Swift–Hohenberg theory

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via a proper constitutive process. Finally, we present one highly resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift–Hohenberg equation.     

On co-rotational and other rate type constitutive equations

We derive expressions for the dilatational properties of suspensions of gas bubbles in incompressible fluids, using a cell model for the suspension. A cell, consisting of a gas bubble centered in a spherical shell of incompressible fluid, is subjected to a purely dilatational boundary motion and the resulting stress at the cell boundary is obtained. The same dilatational boundary motion is prescribed at the boundary of an “equivalent” cell composed of a one-phase, uniformly compressible fluid with unknown dilatational properties. By specifying that the stress at the boundary of the one-phase cell is equal to the stress at the boundary of the two-phase suspension cell, we obtain expressions for the unknown dilatational properties as a function of observable properties of the suspension. The dilatational viscosity of a suspension with a Newtonian continuous phase and the analogous properties for suspensions with non-Newtonian continuous phases are obtained as functions of the boundary motion, volume fraction of gas, and properties of the incompressible continuous phase. Results are presented for continuous phases which are Newtonian fluids, second-order fluids, and Goddard—Miller model fluids.