A Frequency-Domain Approach to Optimal Fractional-Order Damping

In this paper, we will consider the single term optimal fractional-order damper for an otherwise undamped oscillator. First, we will find the single term damper that minimizes the time domain integral of the squared step error (2-norm) and the integral of the time-weighted squared error (Hilbert–Schmidt–Hankel norm). Next we will consider a more intuitive frequency domain approach that insures the maximally flat magnitude response. Time and frequency domain plots are given for comparison with the integer-order solutions. Further improvements in performance are shown to be possible using multiple active fractional-order dampers.     

Weak Solutions and Attractors for Three-Dimensional Navier–Stokes Equations with Nonregular Force

A three-dimensional Navier–Stokes equation is considered. The forcing term is the derivative of a continuous function; the case of white noise is also considered. The aim is to prove the existence of weak solutions and to construct an attractor for the corresponding shift dynamical system in path space, following an idea of Sell.     

On the propagation of pressure waves in saturated porous media

The problem of the propagation of pressure waves through compressible porous media saturated with a slightly compressible fluid is considered. By using Darcy's law the problem is reduced to a mixed initial-boundary value problem for an equation of the heat conduction type with a nonlinear term. The method of quasi-characteristics is used to solve this equation numerically. Solutions of the wave propagation problem for media with different permeability coefficients are presented. A solution of the inverse problem of determining the permeability coefficient using wave-pulse test data is constructed on the basis of a set of solutions of the direct problem.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 81–84, November–December, 1996.     

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

Existence of Solutions for Steady Detonation of Gas Suspensions

The existence of solutions of the traveling–wave type is studied for a system of equations that describes a one–dimensional motion of a suspension of evaporating particles in a viscous and heat–conducting chemically reacting gas. Using topological methods, it is shown that solutions corresponding to weak, strong, and Chapman—Jouguet detonation exist under certain restrictions on energy release and mass transfer.     

Self-similar solutions of three-dimensional laminar magnetohydrodynamic boundary-layer equations

Self-similar solutions of three-dimensional boundary-layer equations of an incompressible fluid in ordinary hydrodynamics were considered in [1–3] et al. The present work looks for self-similar solutions of three-dimensional magnetohydrodynamic boundary-layer equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 10–17, July–August, 1968.     

One-Dimensional Steady Infiltration with Water Extraction

Exact solutions of the one-dimensional steady infiltration with water extraction are obtained for a particular but realistic form of the soil–water conductivity. The results are compared with numerical solutions for slightly different forms of the soil–water conductivity function, showing the sensitivity of the shape of the profiles on the exact representation of the soil–water conductivity. The exact solutions are an extension of Warrick's (1974) solution for the case where the soil can become saturated.     

Simulation of ratcheting strain to a high number of cycles under biaxial loading

The Ohno–Wang kinematic hardening rule is modified to incorporate the Burlet–Cailletaud radial evanescence term for an improved simulation of the ratcheting behavior. The Delobelle parameter δ^′ is implemented in the modified model to compromise shakedown of the Burlet–Cailletaud hardening rule and over-prediction of the Ohno–Wang model. An evolution equation is proposed for δ^′ to simulate the ratcheting strain over an extended domain of cycles. Ratcheting tests were conducted on S45C steel under four types of nonproportional axial–torsional loading. The new model is found to yield reasonably accurate predictions of ratcheting strain to a much higher number of cycles compared with other studies.     

Group Stratification and Exact Solutions of the Equation of Transonic Gas Motions

Exact solutions of the Kármán–Guderley equation that describes spatial gas flows in the transonic approximation are considered. A group stratification of the equation with respect to the infinite-dimensional part of the admissible group is constructed. New invariant and partly invariant solutions are obtained. The possibility of existence of solutions continuous in the entire space is analyzed for invariant submodels with one independent variable. A solution of the Kármán–Guderley equation of the double-wave type is constructed.     

Asymptotic Representations of Solutions of an Ordinary Differential Equation of the <Emphasis Type="Italic">n</Emphasis>th Order with Exponential Nonlinearity

We establish necessary and sufficient conditions for the existence of one class of solutions of a differential equation with exponential nonlinearity. Asymptotic representations of these solutions are obtained.__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 562–573, October–December, 2004.     

On Global Solutions of Linear Difference Equations with Deviating Argument

We present conditions under which global solutions of linear difference equations with deviating argument are solutions of ordinary difference equations.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 29–45, January–March, 2005.     

Asymptotic solution of the axisymmetric problem of ideal fluid flow in the neighborhood of cusped cavities

A denumerable set of axisymmetric solutions of the equations of motion of an ideal fluid in the neighborhood of a cavity closure point, including an infinitely remote point, is obtained. This family of solutions is similar to the well-known family of corresponding two-dimensional solutions (1), (2).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 84–87, July–August, 1990.     

Displacement of oil by solutions of an active additive from a uniform stratum with allowance for gravity

Self-similar solutions describing the displacement of oil by solutions of an adsorbed active additive have been obtained and investigated [1–3] in the framework of a one-dimensional flow model with neglect of diffusion, capillary, and gravity effects. In the present paper, a self-similar solution is constructed for the problem of oil displacement by an aqueous solution of an active additive from a thin horizontal stratum with allowance for gravity under the assumption that there is instantaneous vertical separation of the phases. This makes it possible to estimate the effectiveness of flooding a stratum by solutions of surfactants and polymers in the cases when gravitational segregation of the phases cannot be ignored.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–92, January–February, 1984.     

Special class of solutions of the kinetic equation of a bubbly fluid

A new class of solutions is constructed for the kinetic model of bubble motion in a perfect fluid proposed by Russo and Smereka. These solutions are characterized by a linear relationship between the Riemann integral invariants. Using the expressions following from this relationship, the construction of solutions in the special class is reduced to the integration of a hyperbolic system of two differential equations with two independent variables. Exact solutions in the class of simple waves are obtained, and their physical interpretation is given.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 33–43, March–April, 2005.     

Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

Nonlinear planar oscillations of suspended cables subjected to external excitations with three-to-one internal resonances are investigated. At first, the Galerkin method is used to discretize the governing nonlinear integral–partial-differential equation. Then, the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance. The equilibrium solutions, the periodic solutions and chaotic solutions of the modulation equations are also investigated. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency/force–response curves. The supercritical Hopf bifurcations are found in these curves. Choosing these bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. Numerical simulations are used to illustrate the cascades of period-doubling bifurcations leading to chaos. At last, the nonlinear responses of the two-degree-of-freedom model are investigated.     

Integro-Differential Inclusions with Hukuhara Derivative

For an integro-differential inclusion with Hukuhara derivative, we introduce the notion of quasisolution and present conditions under which the set of quasisolutions coincides with the set of ordinary solutions. We also prove theorems on the relaxation of ordinary solutions and the compactness of the set of these solutions.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 80–88, January–March, 2005.     

Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

Some self-similar solutions of Grad's equations

It is shown that the self-similar solutions of the Navier-Stokes and Burnett equations found earlier by the authors [1–9] can be extended to the case of two-dimensional flows of a weakly rarefied gas described by Grad's equations. Examples are given of numerical realization of self-similar solutions for flow in an expanding planar channel. It is found that there are appreciable differences between the behavior of the self-similar solutions of the Navier-Stokes, Burnett, and Grad equations in the neighborhood of a channel wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 88–94, May–June, 1982.     

Shock wave incidence on a V-shaped cavity

We study theoretically and experimentally the motion of metal arising from a plane shock wave striking a V-shaped cavity. Using the functionally invariant solutions of Sobolev, we write out the acoustic approximation for this problem and determine the region of its applicability. It is shown that in the region in which the acoustic approximation is not applicable, the flow in the principal term is described by the incompressible fluid equations for which the boundary conditions are defined by the acoustic region. The experimental technique is described and a comparison of the theoretical and experimental data is made.Translated from Zhurnal PrikladnoiMekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 57–61, November–December, 1969.The authors wish to thank A. A. Deribas for discussion on the problem formulation and experimental technique, and N. S. Kozin for carrying out the numerical calculations.