Existence of periodic solutions of the equations of incompressible microstretch fluid flow

The flow of incompressible microstretch fluid is governed by a system of differential equations involving the velocity vector $\text{q}$, the microprotation vector $\text{v}$ and the scalar $\text{v}$ representing the microstretch of the fluid element. Let $\text{R = R(t)}$ be a bounded domain in space and let the field ($\text{q}$, $\text{v}$, $\text{v}$) be prescribed at each point of the boundary $\text{?R(t)}$. If the domain $\text{R(t)}$ and the boundary data depend periodically on the time $\text{t}$, it is shown that under some assumptions on the initial distribution of the flow fields and the material constants of the fluid, there exists a unique, stable, periodic solution of the microstretch flow equations in $\text{R(t)}$, taking the prescribed values on the boundary $\text{?R(t)}$ (Theorem 2 of the paper). The proof rests on some relations describing the rate of decay of the energy functionals corresponding to the difference of two microstretch flows in the domain that have the same density and gyration parameters and are subject to the same boundary conditions.     

Kinetic energy of incompressible microstretch fluid in a domain bounded by rigid walls

The basic physical quantities of microstretch flow are the velocity vector (q?), the microrotation vector (v?) and the microstretch (v), the last quantity being a scalar field signifying the stretch or contraction experienced by the local fluid element. The kinetic energy T of the flow over a domain has contributions T₁, T₂, T₃ one from each of the above three quantities q?, v?, and v. It is shown that Sgn (dT/dt) = −1 and that T(t) ? T(t₀) exp [−σ(t−t₀)] for 0 < t₀ ? t. The (positive) number σ depends on the material constants of the flow and also on the geometry of the domain.     

Universal stability of magneto-micropolar fluid motions

Criteria for universal stability of the unsteady motion of an incompressible, electrically conducting linear micropolar fluid, i.e. with rigid microinclusions, in the presence of an arbitrary magnetic field, and in an arbitrary bounded time dependent domain are established. The model of the micropolar fluid employed is essentially the one proposed by Eringen. The interaction between the flow field and the magnetic field is manifested through the body force and body couple. Relativistic, Hall, and temperature effects are neglected. The stability method employed is an energy technique due to James Serrin. Certain uniqueness theorems for the unsteady and steady flows of magneto-micropolar fluid are also established.The theorems established for the stability and uniqueness are universal in the sense that they may be applied to any geometry of bounded domains and any distribution of the basic field variables. The present problem finds application in MHD generators with neutral fluid seedings in the form of rigid microinclusions.     

Universal stability of thermo-magneto-micropolar fluid motions

Criteria for universal stability of the unsteady motion of an incompressible, electrically conducting linear micropolar fluid with heat transfer in the presence of an arbitrary magnetic field, and in an arbitrary time dependent domain are established. The model of the micropolar fluid employed is essentially the one proposed by Eringen. The interaction between the flow field and the magnetic field is manifested through the Lorentz force and the coupling between the flow and temperature field arises through the Boussinesq equation of state.The stability method employed is an energy technique due to James Serrin. Certain uniqueness theorems for the unsteady and steady flows of thermo-magneto-micropolar fluid are also established.The theorems established for the stability and uniqueness are universal in the sense that they may be applied to any geometry of bound domains and any distribution of the basic field variables.     

On the microstretch piezoelectricity

The paper is concerned with the linear theory of microstretch piezoelectricity. First, a uniqueness result and a reciprocal theorem are established. The proof of reciprocal theorem avoids both the use of the Laplace transform and the alternative formulation of the problem by incorporation of initial conditions into the equations of motion. Then, a counterpart of the Boussinesq–Somigliana–Galerkin solution in the classical elastostatics is presented. The problem of an infinite body subject to concentrated sources is solved.     

Stability of motions of moored ocean vehicles

The stability in the sense of Liapunov of the horizontal plane slow motions of Single Point Mooring (SPM) systems is studied. The mathematical model consists of the manoeuvering equations of the moored vessel and a nonlinear stress-strain relation for the mooring line. Steady excitation from current, wind and drift forces is included. Six first-order nonlinear coupled differential equations describe the system dynamics The system equilibria are first found and local analysis is performed in their vicinity. A SPM system, may asymptotically converge to a stable equilibrium, diverge from an unstable equilibrium or converge to a limit cycle. Due to the dependence of the eigenvalues of the system at each equilibrium on the system, parameters, the system may exhibit codimension-one bifurcations of pitchfork or Hopf type, or bifurcations of closed orbits. Based on the results of local analysis, the global system behaviour can be assessed, and design decisions can be made for selection of the principal SPM configuration parameters to avoid undesirable response. Finally the large-amplitude low-frequency motions observed in moored vessels, and often attributed to time-dependent external excitation, are explained using the results of the stability analysis.     

Method of complex potentials in linear microstretch elasticity

The paper is concerned with the plane strain of homogeneous and isotropic microstretch elastic bodies. We give a new representation of the solution in terms of complex potentials. The method is useful for the treatment of the constitutive equations established by Eringen in [A.C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999], where the introduction of stress functions leads to difficulties. The complex variable technique is used to study Kirsch problem in the context of the theory presented in [A.C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999].     

Thermal stresses in microstretch elastic plates

We consider the linear theory of homogeneous and isotropic thermo-microstretch elastic solids. First, we present the basic equations which characterize the bending of thin plates. Then we establish a uniqueness result with no definiteness assumption on constitutive coefficients. Existence of solutions is proved under assumption that the internal energy density is positive definite. In the equilibrium theory we present a theorem of minimum potential energy. Finally, the effects of a concentrated heat source in an infinite plate are studied.     

Stability of cosserat fluid motions-II on theN-th-order cosserat fluid

Summary We obtain criteria of universal stability of the unsteady motion of an incompressiblenth-order Cosserat fluid in an arbitrary time-dependent domain employing a general energy method due to J. Serrin. This work completes the results previously obtained by the authors [1] for a first-order Cosserat fluid motions. It is shown that the original motion is stable if or if . The quantitiesR _e andC ⁽ⁱ⁾ (i=1,2, ...,N) are the Reynolds number and Cosserat number of orderi, respectively and- is the lower bound dor the eigenvalues of the strain rate tensorD _ij.The theorems established for the stability criteria are universal in the sense that they do not depend on the shape of the domain or on the distribution of the basic field variables.We have finally discussed a certain similarity between the present model and the model for turbulence.

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Zur Stabilität von Cosserat-Flüssigkeitsbewegungen-II auf einer Cosserat-FlüssigkeitN-ter Ordnung

Zusammenfassung Für die Stabilität der instationären Bewegung einer inkompressiblen Cosserat-Flüssigkeitn-ter Ordnung in einem willkürlichen zeitabhängigen Bereich werden unter Verwendung einer allgemeinen Energiemethode nach J. Serin Kriterien erhalten. Diese Arbeit vervollständigt die Ergebnisse, die vor kurzer Zeit von den Autoren [1] für Cosserat-Flüssigkeitsbewegungen erster Ordnung erhalten wurden. Es wird gezeigt, daß die ursprüngliche Bewegung stabil ist, wenn order wenn gilt. Die GrößenR _e undC ⁽ⁱ⁾ (i=1, 2,...,N) sind die Reynolds-Zahl und die Cosserat-Zahli-te·Ordnung, und- ist die untere Grenze für die Eigenwerte des VerzerrungsgeschwindigkeitstensorsD _ij.Die angegebenen Theoreme für die Stabilitätskriterien sind allgemein in dem Sinne, daß sie nicht von der Form des Bereiches, oder der Verteilung der grundlegenden Feldvariablen abhängen.Abschließend wird eine bestimmte Ähnlichkeit zwischen dem angegebenen Modell und dem Modell für turbulente Strömung diskutiert.

     

Stress concentration effects in microstretch elastic bodies

This paper is concerned with the plane strain problem of the equilibrium theory of microstretch elastic bodies. First, we study the problem of stress concentration in the neighbourhood of a circular hole located in a plane subjected to the action of constant loads at a great distance from the hole. Then, the problem of a rigid inclusion in an infinite body is studied.     

Plane strain problem in microstretch elastic solid

The eigenvalue approach is developed for the two-dimensional plane strain problem in a microstretch elastic medium. Applying Laplace and Fourier transforms, an infinite space subjected to a concentrated force is studied. The integral transforms are inverted using a numerical technique to get displacement, force stress, couple stress and first moment, which are also shown graphically. The results of micropolar elasticity are deduced as a special case from the present formulation.     

Some theorems in the theory of microstretch thermopiezoelectricity

The electromagnetic theory of microstretch thermoelasticity is an adequate tool to describe the behaviour of porous bodies, animal bones and solids with deformable microstructures. In this paper we study the linear theory of microstretch thermopiezoelectricity. First, we establish a reciprocity relation which involves two processes at different instants. This relation forms the basis of a uniqueness result and a reciprocal theorem. Then, we study the continuous dependence of solutions upon initial data and body loads. A variational characterization of solutions is also presented. Finally, we investigate the effect of a concentrated heat supply and the effect of a concentrated volume charge density in an unbounded homogeneous and isotropic body.     

On complex potentials in the theory of microstretch elastic bodies

The paper is concerned with the plane strain problem in the equilibrium theory of microstretch elastic solids. We show that the complex variable technique of the classical theory of elasticity can be extended to the theory of microstretch elastic bodies. The method is used to study the effect of the stress concentration around a circular hole.     

Long-time behaviour of fluid motions in porous media according to the Brinkman model

The paper is concerned with a binary fluid mixture in a porous horizontal layer according to the Brinkman model. The long-time behaviour of the solutions is considered: absorbing sets are shown to exist, and the stability (instability) of the rest state is studied. Following the new approach introduced in Rionero (J. Math. Anal. Appl. 333, 1036–1057 (2007)), the global stability of the rest state is rigorously reduced to the stability of the zero solution of a linear binary system of ODEs. The results obtained are extended to cover the presence of a uniform rotation about a vertical axis.     

A class of plane self-similar motions of a non-newtonian fluid with nonlinear thermophysical properties

The plane self-similar solution of a set of complete equations for the dynamics of a nonlinearly viscous fluid aad the energy equation is obtained analytically with the temperature dependence of the transfer coefficients taken into account.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 48, No. 1, pp. 129–136, January, 1985.     

Calculation of unsteady flows due to small motions of cylinders in a viscous fluid

Summary The problem of small oscillations of a cylinder of general cross-section in a viscous fluid is formulated in terms of integral equations. Numerical solutions of the integral equation are presented for the special case of a ribbon of zero thickness.This work has been carried out under the support of the Office of Naval Research, Contract N0014-67-A-0094-0011.     

Stability of periodic flow in a micropolar fluid

The stability of unidirectional periodic flow in a micropolar fluid is treated. An analytic expression is found for the critical Reynolds number of stability loss.Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 60, No. 4, pp. 670–679, April, 1991.     

Stability of two-layered fluid flows in an inclined channel

A linear stability analysis of two-layer fluid flows in an inclined channel geometry has been carried out. The onset of flow transitions and the spatio-temporal characteristics of secondary flows produced by the flow instabilities have been examined. The effects of density and viscosity stratifications and surface tension on flow structures also have been investigated at various values of Froude numbers (channel inclinations). Multi-domain Chebyshev–Tau spectral methods along with MATLAB QZ eigenvalue solver are used to determine the whole spectrum of the eigenvalues and associated eigenfunctions. The neutral stability diagrams and stability boundaries are constructed for various values of flow parameters. The onset of flow transitions and flow structures predicted by linear stability analysis are compared against experimental results and they agree reasonably well. The results presented in the present paper imply that the shear mode of flow transitions is the one likely to be identified in experiments.     

Stability of a micropolar fluid layer heated from below

The stability of a layer of micropolar fluid heated from below is studied employing a linear theory as well as an energy method. It is proved that the principle of exchange of stability holds and the critical Rayleigh number is obtained. It is observed that the micropolar fluid layer heated from below is more stable as compared with the classical viscous fluid. The energy method is then used to study the stability under finite disturbances. A variational method is applied to obtain the sharp stability limit. It is found that no subcritical instability region exists and the critical Rayleigh number as derived from the energy method is identical to that of the linear limit.