Stability of convective flows in a rotating liquid layer under various heating conditions

For a rotating liquid layer with boundaries of low thermal conductivity, an amplitude equation is obtained that describes the evolution of secondary convective flows in uniform heating and above a hot spot. The dependence of the coefficients of the amplitude equation on the rotation parameter, Prandtl number, and heat-flux nonuniformity is obtained. The influence of rotation on the stability of nonlinear regimes is analyzed for uniform heating. The boundaries of flow stability are investigated for variously shaped hot spots. Perm'State University, Perm'614600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika. Vol. 39, No. 1, pp. 69–74, January–February, 1998.     

Nonlinear Multigrid Methods for Numerical Solution of the Variably Saturated Flow Equation in Two Space Dimensions

The need of accurate and efficient numerical schemes to solve Richards’ equation is well recognized. This study is carried out to examine the numerical performances of the nonlinear multigrid method for numerical solving of the two-dimensional Richards’ equation modeling water flow in variably saturated porous media. The numerical approach is based on an implicit, second-order accurate time discretization combined with a vertex centered finite volume method for spatial discretization. The test problems simulate infiltration of water in 2D saturated–unsaturated soils with hydraulic properties described by van Genuchten–Mualem models. The numerical results obtained are compared with those provided by the modified Picard–preconditioned conjugated gradient (Krylov subspace) approach.     

Wave regimes on a nonlinearly viscous fluid film flowing down a vertical plane

The flow of a thin film of a nonlinearly viscous fluid whose stress tensor is modeled by a power law, flowing down a vertical plane in the field of gravity, is considered. For the case of low flow rates, an equation that describes the evolution of surface disturbances is derived in the long-wave approximation. The domain of linear stability of the trivial solution is found, and weakly nonlinear, steady-state travelling solutions of this equation are obtained. The mechanism of branching of solution families at the singular point of the neutral curve is described. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 73–84, May–June, 2005.     

Extension of Ovsyannikov's Analytical Solutions to Transonic Flows

The solution of the equation of the velocity potential of a steady axisymmetric ideal-gas flow in the neighborhood of a given point at the axis of symmetry in the form of a double series in powers of the distance to the axis of symmetry and its logarithm is considered. Recurrent chains of equations with arbitrariness in two analytical functions of the streamwise variable are obtained for coefficients of the series. Convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution of the initial-boundary problem for this nonlinear differential equation in partial derivatives with a singularity at the axis of symmetry is obtained as an analog of Kovalevskaya's and Ovsyannikov's theorems. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 14–25, November–December, 2005.     

On the structure of characteristic surfaces related to partial differential equations of first and higher orders. II

The generalized method of characteristics is developed within the framework of the geometric Monge picture. Hopf-Lax-type extremality solutions are obtained for a broad class of Cauchy problems for nonlinear partial differential equations of the first and higher orders. A special Hamilton-Jacobi-type case is analyzed separately. An exact extremality Hopf-Lax-type solution of the Cauchy problem for the nonlinear Burgers equation is obtained, and its linearization to the Hopf-Cole expression and to the corresponding Airy-type linear partial differential equation is found and discussed. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 529–543, October–December, 2005.     

Cross-flow-induced instability and nonlinear dynamics of cylinder arrays with consideration of initial axial load

The instability and nonlinear dynamics of planar motions of a cylinder array subjected to cross-flow have been studied via a five-mode discretization of the governing partial differential equation, focusing on the effect of initial axial load externally imposed on the cylinder. Theoretical results based on a stability analysis have indicated that, with increasing initial axial load and flow velocity, the system may lose stability either via flutter or via buckling. The boundaries of these two forms of instability are predicted analytically. To explore the post-instability dynamics of the system, a Runge–Kutta scheme is used to solve the nonlinear governing equation of motion. Three typical behaviors, including limit cycle motions of the system, are obtained. It is shown that, for relatively low flow velocity, with increasing initial axial load, just beyond the pitchfork bifurcation the cylinder would settle in a buckled equilibrium position; and for high flow velocity, however, this phenomenon only occurs when the initial axial load becomes sufficiently large.     

Asymptotic form of the admixture transport equation for a channel flow with an anisotropic diffusion tensor

The problem of the asymptotically correct reduction of a 3-D mass (heat) transfer equation to a 1-D equation in a flow with anisotropic diffusion properties is considered. The convective mass (heat) transfer domain is a cylindrical channel of arbitrary cross section. The diffusion coefficient matrix is assumed to be independent of the spatial coordinates. In the equivalent diffusion equation constructed, a certain effective diffusion (dispersion [1]) coefficient is introduced. Formulas for this coefficient are obtained. A relation between the effective diffusion coefficient calculations and the problem of minimization of a certain functional is established, i. e. the possibility of calculations based on variational methods is noted. An example of an exact calculation of the effective diffusion coefficient is considered. The possibility of a generalization of the problem, in which the effective diffusion (heat conduction) equation is essentially a nonlinear equation of general form for the one-dimensional case, is indicated. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 110–123, March–April, 2000.     

Nonlinear dynamics of functionally graded pipes conveying hot fluid

For improved stability of fluid-conveying pipes operating under the thermal environment, functionally graded materials (FGMs) are recommended in a few recent studies. Besides this advantage, the nonlinear dynamics of fluid-conveying FG pipes is an important concern for their engineering applications. The present study is carried out in this direction, where the nonlinear dynamics of a vertical FG pipe conveying hot fluid is studied thoroughly. The FG pipe is considered with pinned ends while the internal hot fluid flows with the steady or pulsatile flow velocity. Based on the Euler–Bernoulli beam theory and the plug-flow model, the nonlinear governing equation of motion of the fluid-conveying FG pipe is derived in the form of the nonlinear integro-partial-differential equation that is subsequently reduced as the nonlinear temporal differential equation using Galerkin method. The solutions in the time or frequency domain are obtained by implementing the adaptive Runge–Kutta method or harmonic balance method. First, the divergence characteristics of the FG pipe are investigated and it is found that buckling of the FG pipe arises mainly because of temperature of the internal fluid. Next, the dynamic characteristics of the FG pipe corresponding to its pre- and post-buckled equilibrium states are studied. In the pre-buckled equilibrium state, higher-order parametric resonances are observed in addition to the principal primary and secondary parametric resonances, and thus the usual shape of the parametric instability region deviates. However, in the post-buckled equilibrium state of the FG pipe, its chaotic oscillations may arise through the intermittent transition route, cyclic-fold bifurcation, period-doubling bifurcation and subcritical bifurcation. The overall study reveals complex dynamics of the FG pipe with respect to some system parameters like temperature of fluid, material properties of FGM and fluid flow velocity.     

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear–nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt–Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge–Kutta solutions.     

One approximate solution of the Nekrasov problem

An approximate solution ω = A[ω, μ] of the nonlinear integral Nekrasov equation is obtained by successive replacement of the kernel of the integral operator by a close one. The solution is sought not directly at the bifurcation point μ₁ = 3 of the linearized equation ω = μL[ω] but at the point μ = 1 at which operator A[ω, μ], remaining nonlinear in ω, is linear in μ. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 50–56, November–December, 2007.     

Peristaltic flow of a Johnson-Segalman fluid through a deformable tube

To understand theoretically the flow properties of physiological fluids we have considered as a model the peristaltic motion of a Johnson–Segalman fluid in a tube with a sinusoidal wave traveling down its wall. The perturbation solution for the stream function is obtained for large wavelength and small Weissenberg number. The expressions for the axial velocity, pressure gradient, and pressure rise per wavelength are also constructed. The general solution of the governing nonlinear partial differential equation is given using a transformation method. The numerical solution is also obtained and is compared with the perturbation solution. Numerical results are demonstrated for various values of the physical parameters of interest.      

Nonlinear Multigrid Methods for Numerical Solution of the Unsaturated Flow Equation in Two Space Dimensions

Picard and Newton iterations are widely used to solve numerically the nonlinear Richards’ equation (RE) governing water flow in unsaturated porous media. When solving RE in two space dimensions, direct methods applied to the linearized problem in the Newton/Picard iterations are inefficient. The numerical solving of RE in 2D with a nonlinear multigrid (MG) method that avoids Picard/Newton iterations is the focus of this work. The numerical approach is based on an implicit, second-order accurate time discretization combined with a second-order accurate finite difference spatial discretization. The test problems simulate infiltration of water in 2D unsaturated soils with hydraulic properties described by Broadbridge–White and van Genuchten–Mualem models. The numerical results show that nonlinear MG deserves to be taken into consideration for numerical solving of RE.     

Nonlinear development of two-dimensional hydroelastic instability in a turbulent boundary layer on an elastic coating

Nonlinear evolution of hydroelastic instability arising in the flow past a coating of a rubber-type material by a turbulent boundary layer of an incompressible fluid is studied. A nonlinear dispersion equation for two-dimensional, quasi-monochromatic, low-amplitude waves is derived. The Prandtl equations for the mean (over the waviness period) boundary-layer flow are solved in the approximation of local similarity and by direct numerical integration. Evolution of unstable waves in time is studied on the basis of the Landau equation, which is derived separately for the instability of fast waves (flutter) and the quasi-static instability (divergence). The calculation results are compared with available experimental data. Institute of Applied Physics, Russian Academy of Sciences, Nizhnii Novgorod 603600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 69–80, July–August, 2000.     

Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp-function method

In this paper, the Exp-function method with the aid of the symbolic computational system Maple is used to obtain the generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics, namely, (2+1)-dimensional Konopelchenko–Dubrovsky equations, the (3+1)-dimensional Jimbo–Miwa equation, the Kadomtsev–Petviashvili (KP) equation, and the (2+1)-dimensional sine-Gordon equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

Amplitude equations for a system with thermohaline convection

The multiple scale expansion method is used to derive amplitude equations for a system with thermohaline convection in the neighborhood of Hopf and Taylor bifurcation points and at the double zero point of the dispersion relation. A complex Ginzburg-Landau equation, a Newell-Whitehead-type equation, and an equation of the ϕ⁴ type, respectively, were obtained. Analytic expressions for the coefficients of these equations and their various asymptotic forms are presented. In the case of Hopf bifurcation for low and high frequencies, the amplitude equation reduces to a perturbed nonlinear Shroedinger equation. In the high-frequency limit, structures of the type of “dark” solitons are characteristic of the examined physical system. Pacific Ocean Institute, Vladivostok 690041. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 56–66, May–June, 2000.     

Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

In the flow of liquids through porous media, nonlinear effects arise from the dependence of the fluid density, porosity, and permeability on pore pressure, which are commonly approximated by simple exponential functions. The resulting flow equation contains a squared gradient term and an exponential dependence of the hydraulic diffusivity on pressure. In the limiting case where the porosity and permeability moduli are comparable, the diffusivity is constant, and the squared gradient term can be removed by introducing a new variable y, depending exponentially on pressure. The published transformations that have been used for this purpose are shown to be special cases of the Cole–Hopf transformation, differing in the choice of integration constants. Application of Laplace transformation to the linear diffusion equation satisfied by y is considered, with particular reference to the effects of the transformation on the boundary conditions. The minimum fluid compressibilities at which nonlinear effects become significant are determined for steady flow between parallel planes and cylinders at constant pressure. Calculations show that the liquid densities obtained from the simple compressibility equation of state agree to within 1% with those obtained from the highly accurate Wagner-Pru?  equation of state at pressures to 20 MPa and temperatures approaching 600 K, suggesting possible applications to some geothermal systems.     

Finite Element-Based Characterization of Pore-Scale Geometry and Its Impact on Fluid Flow

We present a finite element (FEM) simulation method for pore geometry fluid flow. Within the pore space, we solve the single-phase Reynold’s lubrication equation—a simplified form of the incompressible Navier–Stokes equation yielding the velocity field in a two-step solution approach. (1) Laplace’s equation is solved with homogeneous boundary conditions and a right-hand source term, (2) pore pressure is computed, and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field, we estimate the effective permeability of porous media samples characterized by section micrographs or micro-CT scans. This two-step process is much simpler than solving the full Navier–Stokes equation and, therefore, provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier–Stokes’ equation. Given the realistic laminar flow field, dispersion in the medium can also be estimated. Our numerical model is verified with an analytical solution and validated on two 2D micro-CT scans from samples, the permeabilities, and porosities of which were pre-determined in laboratory experiments. Comparisons were also made with published experimental, approximate, and exact permeability data. With the future aim to simulate multiphase flow within the pore space, we also compute the radii and derive capillary pressure from the Young–Laplace’s equation. This permits the determination of model parameters for the classical Brooks–Corey and van-Genuchten models, so that relative permeabilities can be estimated.     

Numerical simulation of shock wave propagation in microchannels using continuum and kinetic approaches

Numerical simulations of shock wave propagation in microchannels and microtubes (viscous shock tube problem) have been performed using three different approaches: the Navier–Stokes equations with the velocity slip and temperature jump boundary conditions, the statistical Direct Simulation Monte Carlo method for the Boltzmann equation, and the model kinetic Bhatnagar–Gross–Krook equation with the Shakhov equilibrium distribution function. Effects of flow rarefaction and dissipation are investigated and the results obtained with different approaches are compared. A parametric study of the problem for different Knudsen numbers and initial shock strengths is carried out using the Navier–Stokes computations.      

Runup and green water velocities due to breaking wave impinging and overtopping

The present study investigates, through measurements in a 2D wave tank, the velocity fields of a plunging breaking wave impinging on a structure. As the wave breaks and overtops the structure, so-called green water is generated. The flow becomes multi-phased and chaotic as a large aerated region is formed in the flow in the vicinity of the structure while water runs up onto the structure. In this study, particle image velocimetry (PIV) and its derivative, bubble image velocimetry (BIV), were employed to measure the velocity field in front and on top of the structure. Mean and turbulence properties were obtained through ensemble averaging repeated tests. The dominant and maximum velocity of the breaking wave and associated green water are discussed for the three distinct phases of the impingement–runup–overtopping sequence. Initially the flow is mainly horizontal right before the breaking wave impinges on the structure. The flow then becomes primarily vertical and rushes upward along the front wall of the structure right after the impingement. Subsequently, the flow becomes mainly horizontal on top of the structure as the remaining momentum in the wave crest carries the green water through. The distribution of the green water velocity along the top of the structure has a nonlinear profile and the maximum velocity occurs near the front of the fast moving water. Using the measured data and applying dimensional analysis, a similarity profile for the green water flow on top of the structure was obtained, and a prediction equation was formulated. The prediction equation may be used to predict the green water velocity caused by extreme waves in a hurricane.