Group classification of one-dimensional nonisentropic equations of fluids with internal inertia

A systematic application of the group analysis method for modeling fluids with internal inertia is presented. The equations studied include models such as the nonlinear one-velocity model of a bubbly fluid (with incompressible liquid phase) at small volume concentration of gas bubbles (Iordanski Zhurnal Prikladnoj Mekhaniki i Tekhnitheskoj Fiziki 3, 102–111, 1960; Kogarko Dokl. AS USSR 137, 1331–1333, 1961; Wijngaarden J. Fluid Mech. 33, 465–474, 1968), and the dispersive shallow water model (Green and Naghdi J. Fluid Mech. 78, 237–246, 1976; Salmon 1988). These models are obtained for special types of the potential function W(r,[(r)\dot],S){W(\rho,\dot \rho,S)} (Gavrilyuk and Teshukov Continuum Mech. Thermodyn. 13, 365–382, 2001). The main feature of the present paper is the study of the potential functions with W _S  ≠ 0. The group classification separates these models into 73 different classes.     

Effect of yaw on the starting point of curvature for reflected diffracted shock wave (subsonic and sonic cases)

Lighthill (Proc. R. Soc. A 198, 454–470, 1949) considered the diffraction of a normal shock wave passing over a small bend. The bend being small Lighthill was able to linearize the flow equations and solved the problem through several mathematical techniques. Following Lighthill (Proc. R. Soc. A 198, 454–470, 1949), Srivastava and Chopra (J. Fluid Mech. 40, 821–831, 1970) extended the work to the diffraction of oblique shock waves. Srivastava (AIAAJ 33, 2230–2231, 1995) considered the problem of starting point of curvature and extended the work to yawed wedges (Srivastava in Proceedings of the 14th International Mach reflection symposium Sun Marina Hotel, Yonezawa, Japan, 1–5 October 2000, pp. 225–249, 2002). Srivastava (Shock waves 13, 323–326, 2003) considered the problem for starting point of curvature when the relative outflow behind reflected shock before diffraction has been subsonic and sonic. The present work is an extension of the work published in Srivastava (Shock waves 13, 323–326, 2003) when the wedge has been yawed through an angle. The results have been obtained for two angles χ = 60° and χ = 40° (χ is the angle of yaw).      

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

On Finite Shear

If a pair of material line elements, passing through a typical particle P in a body, subtend an angle Θ before deformation, and Θ+γ after deformation, the pair of material elements is said to be sheared by the amount γ. Here all pairs of material elements at P are considered for arbitrary deformations. Two main problems are addressed and solved. The first is the determination of all pairs of material line elements at P which are unsheared. The second is the determination of that pair of material line elements at P which suffers the maximum shear. All unsheared pairs of material elements in a given plane π(S) with normal S passing through P are considered. Provided π(S) is not a plane of central circular section of the C-ellipsoid at P (where C is the right Cauchy-Green strain tensor), it is seen that corresponding to any material element in π(S) there is, in general, one companion material element in π(S) such that the element and its companion are unsheared. There are, however, two elements in π(S) which have no companions. We call their corresponding directions \textit{limiting directions.} Equally inclined to the direction of least stretch in the plane π(S), the limiting directions play a central role. It is seen that, in a given plane π(S), the pair of material line elements which suffer the maximum shear lie along the limiting directions in π(S). If Θ _L is the acute angle subtended by the limitig directions in π(S) before deformation, then this angle is sheared into its supplement π−Θ _L so that the maximum shear γ^*;(S) is γ^*=π− 2 Θ _L . If S is given and C is known, then Θ _L may be determined immediately. Its calculation does not involve knowing the eigenvectors or eigenvalues of C. When all possible planes through P are considered, it is seen that the global maximum shear γ^* _G occurs for material elements lying along the limiting directions in the plane spanned by the eigenvectors of C corresponding to the greatest principal stretch λ₃ and the least λ₁. The limiting directions in this principal plane of C subtend the angle and . Generally the maximum shear does not occur for a pair of material elements which are originally orthogonal. For a given material element along the unit vector N, there is, in general, in each plane π(S passing through N at P, a companion vector M such that material elements along N and M are unsheared. A formula, originally due to Joly (1905), is presented for M in terms of N and S. Given an unsheared pair π(S), the limiting directions in π(S) are seen to be easily determined, either analytically or geometrically. Planar shear, the change in the angle between the normals of a pair of material planar elements at X, is also considered. The theory of planar shear runs parallel to the theory of shear of material line elements. Corresponding results are presented. Finally, another concept of shear used in the geology literature, and apparently due to Jaeger, is considered. The connection is shown between Cauchy shear, the change in the angle of a pair of material elements, and the Jaeger shear, the change in the angle between the normal N to a planar element and a material element along the normal N. Although Jaeger's shear is described in terms of one direction N, it is seen to implicitly include a second material line element orthogonal to N. Accepted: May 25, 1999     

Rigorous Asymptotic Expansions for Critical Wave Speeds in a Family of Scalar Reaction-Diffusion Equations

We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.J. Needham, (1993). J. Appl. Math. Phys. (ZAMP) A, vol. 44, No. 4, 707–721] and [T.P. Witelski, K. Ono, and T.J. Kaper, (2001). Appl. Math. Lett., vol. 14, No. 1, 65–73].     

Multibody Formulation with Large Bending Finite Elements (LBFE)

The goal of this paper is to present a flexible multibody formulation for Euler-Bernoulli beams involving large displacements. This method is based on a discretisation of internal and kinetic energies. The beam is represented by its line of centroids and each section is oriented by a frame defined by three Euler angles. We apply a finite element formulation to describe the evolution of these angles along the neutral fibre and describe the internal energy. The kinetic energy is approximated as the one of two rigid bars tangent to the neutral fibre at the ends of the beam. We derive the equations of motion from a Lagrange formulation. These equations are solved using the Newmark method or/and the Newton-Raphson technique. We solve some very classic problems taken from the literature as the curved beam presented by Simo [Simo, J. C., ‘A three-dimensional finite-strain rod model. the three-dimensional dynamic problem. Part I’, Comput. Meths. Appl. Mech. Engrg. 49, 1985, 55–70; Simo, J. C. and Vu-Quoc, L., ‘A three-dimensional finite-strain rod model, Part II: Computationals aspects’, Comput. Meths. Appl. Mech. Engrg. 58, 1988, 79–116] and Lee [Lee, Kisu, ‘Analysis of large displacements and large rotations of three-dimensional beams by using small strains and unit vectors’, Commun. Numer. Meth. Engrg. 13, 1997, 987–997] or the rotational rod presented by Avello [Avello, A., Garcia de Jalon, J., and Bayo, E., ‘Dynamics of flexible multibody systems using cartesian co-ordinates and large displacement theory’, Int. J. Num. Methods in Engineering 32, 1991, 1543–1563] and Simo [Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part I’ Jour. of Appl. Mech. 53, 1986, 849–854; Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part II’, Jour. of Appl. Mech. 53, 1986, 855–863].     

Solvability of Elliptic Problems in Domains with Conic Point

We prove the solvability of the Dirichlet problem for a quasilinear elliptic nondivergent equation of the second order in a bounded domain with conic point. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 216–223, April–June, 2005.     

A new method for calculating hysteretic <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">S</Emphasis>) relationship

A new method for calculating the hysteretic relationship between hydraulic conductivity (K) and suction (S) is proposed. This method uses the experimental (K–S) data of the main wetting and drying branches and predicts satisfactorily the scanning drying and wetting curves. The proposed method is applicable to those porous media where the hysteretic Θ–S relationship complies with the independent domain concept.     

Adaptive <Emphasis Type="Italic">Q</Emphasis>–<Emphasis Type="Italic">S</Emphasis> synchronization of non-identical chaotic systems with unknown parameters

This work investigates the adaptive Q–S synchronization of non-identical chaotic systems with unknown parameters. The sufficient conditions for achieving Q–S synchronization of two different chaotic systems (including different dimensional systems) are derived, based on Lyapunov stability theory. By the adaptive control technique, the control laws and the corresponding parameter update laws are proposed such that the non-identical chaotic systems are to have Q–S synchronization. Finally, four illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.     

Random Fiber Networks and Special Elastic Orthotropy of Paper

We consider a particular in-plane elastic orthotropy observed experimentally for various types of paper, namely: S ₁₁₁₁+S ₂₂₂₂−2S ₁₁₂₂=S ₁₂₁₂, where S _ijkm are components of the in-plane compliance tensor. This is a statement of the invariance of in-plane shear compliance S ₁₂₁₂, which has been observed in some studies but questioned in others. We present a possible explanation of this “special orthotropy” of paper, using an analysis in which paper is modeled as a quasi-planar random microstructure of interacting fiber-beams – a model especially well suited for low basis weight papers. First, it is shown analytically that without disorder a periodic fiber network fails the special orthotropy. Next, using a computational mechanics model, we demonstrate that two-scale geometric disorder in a fiber network is necessary to explain this orthotropy. Indeed, disordered networks with weak flocculation best satisfy this relationship. It is shown that no special angular distribution function of fibers is required, and that the uniform strain assumption should not be used. Finally, it follows from an analogy to the thermal conductivity problem that the kinematic boundary conditions, rather than the traction ones, lead quite rapidly to relatively scale-independent effective constitutive responses. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the (u, p) problem in the theory of elasticity

A problem of the theory of elasticity is considered for a body with vectors of displacements u and loads p simultaneously defined on one part of the body and with undefined conditions on the remaining part of the body. For a doubly connected domain, where the vectors u and p are set on one of its boundaries (inner or outer), an iterative method based on reduction of the initial problem to a sequence of mixed problems is justified. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 100–103, May–June, 2006.     

Conjugate heat transfer study of incompressible turbulent offset jet flows

In the present case, the conjugate heat transfer involving a turbulent plane offset jet is considered. The bottom wall of the solid block is maintained at an isothermal temperature higher than the jet inlet temperature. The parameters considered are the offset ratio (OR), the conductivity ratio (K), the solid slab thickness (S) and the Prandtl number (Pr). The Reynolds number considered is 15,000 because the flow becomes fully turbulent and then it becomes independent of the Reynolds number. The ranges of parameters considered are: OR = 3, 7 and 11, K = 1–1,000, S = 1–10 and Pr = 0.01–100. High Reynolds number two-equation model (k–ε) has been used for turbulence modeling. Results for the solid–fluid interface temperature, local Nusselt number, local heat flux, average Nusselt number and average heat transfer have been presented and discussed.     

Topological Invariants in a Model of a Time-Delayed Chua's Circuit

In the last 30 years, some authors have been studying several classes of boundary value problems (BVP) for partial differential equations (PDE) using the method of reduction to obtain a difference equation with continuous argument which behavior is determined by the iteration of a one-dimensional (1D) map (see, for example, Romanenko, E. Yu. and Sharkovsky, A. N., International Journal of Bifurcation and Chaos 9(7), 1999, 1285–1306; Sharkovsky, A. N., International Journal of Bifurcation and Chaos 5(5), 1995, 1419–1425; Sharkovsky, A. N., Analysis Mathematica Sil 13, 1999, 243–255; Sharkovsky, A. N., in “New Progress in Difference Equations”, Proceedings of the ICDEA'2001, Taylor and Francis, 2003, pp. 3–22; Sharkovsky, A. N., Deregel, Ph., and Chua, L. O., International Journal of Bifurcation and Chaos 5(5), 1995, 1283–1302; Sharkovsky, A. N., Maistrenko, Yu. L., and Romanenko, E. Yu., Difference Equations and Their Applications, Kluwer, Dordrecht, 1993.). In this paper we consider the time-delayed Chua's circuit introduced in (Sharkovsky, A. N., International Journal of Bifurcation and Chaos 4(5), 1994, 303–309; Sharkovsky, A. N., Maistrenko, Yu. L., Deregel, Ph., and Chua, L. O., Journal of Circuits, Systems and Computers 3(2), 1993, 645–668.) which behavior is determined by properties of one-dimensional map, see Sharkovsky, A. N., Deregel, Ph., and Chua, L. O., International Journal of Bifurcation and Chaos 5(5), 1995, 1283–1302; Maistrenko, Yu. L., Maistrenko, V. L., Vikul, S. I., and Chua, L. O., International Journal of Bifurcation and Chaos 5(3), 1995, 653–671; Sharkovsky, A. N., International Journal of Bifurcation and Chaos 4(5), 1994, 303–309; Sharkovsky, A. N., Maistrenko, Yu. L., Deregel, Ph., and Chua, L. O., Journal of Circuits, Systems and Computers 3(2), 1993, 645–668. To characterize the time-evolution of these circuits we can compute the topological entropy and to distinguish systems with equal topological entropy we introduce a second topological invariant.     

Degenerate Regularization of Forward–Backward Parabolic Equations: The Regularized Problem

We study a quasilinear parabolic equation of forward–backward type in one space dimension, under assumptions on the nonlinearity which hold for a number of important mathematical models (for example, the one-dimensional Perona–Malik equation), using a degenerate pseudoparabolic regularization proposed in Barenblatt et al. (SIAM J Math Anal 24:1414–1439, 1993), which takes time delay effects into account. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. We also study qualitative properties of such solutions, in particular concerning their decomposition into an absolutely continuous part and a singular part with respect to the Lebesgue measure. In this respect, the existence of a family of viscous entropy inequalities plays an important role.     

Heat transfer from a surface fitted with rectangular blocks at different orientation angle

 An experimental investigation was carried out to study the enhancement of the heat transfer from a heated flat plate fitted with rectangular blocks of 1 × 2 × 2 cm³ dimensions in a channel flow as a function of Reynolds number (Re_h), spacing (S _y ) of blocks in the flow direction, and the block orientation angle (α) with respect to the main flow direction. The experiments were performed in a channel of 18 cm width and 10 cm height, with air as the working fluid. For fixed S _x =3.81 cm, which is the space between the blocks in transverse to the flow direction, the experimental ranges of the parameters were S _y =3.33–4.33 cm, α=0–45°, Re_h=7625–31550 based on the hydraulic diameter and the average velocity at the beginning of the test section in the channel. Correlations for Nusselt number were developed, and the ratios of heat transfer with blocks to those with no blocks were given. The results indicated that the heat transfer could be enhanced or reduced depending on the spacing between blocks, and the block orientation angle. The maximum heat transfer rate was obtained at the orientation angle of 45°. Received on 13 December 2000 / Published online: 29 November 2001     

On weak critical posets with respect to the positive definiteness of a quadratic tits form

Let S be a finite P-critical poset, i.e., a poset that is critical with respect to the positive definiteness of a quadratic Tits form. A set S is called weakly P-critical if any infinite poset X ⊃ S contains a P-critical subset that is not isomorphic to S. We prove the existence of weakly P-critical sets. Translated from Neliniini Kolyvannya, Vol. 11, No. 3, pp. 396–407, July–September, 2008.     

Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling

To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size, network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces fitted to capillary pressure, saturation, and interfacial area (P _c –S _w –a _nw ) points as well as to relative permeability, saturation, and interfacial area (k _r –S _w –a _nw ) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different surfaces to P _c –S _w –a _nw points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation (P _c –S _w ) and the relative permeability–saturation (k _r –S _w ) curves.