Some uniqueness and continuous dependence results in micropolar fluid dynamics

Thanks to a preliminary lemma, which is a strengthened version of an analogous one given in a previous article [1], some uniqueness theorems for micropolar incompressible flows are derived. These results seem to improve the previous ones even in the classical scheme of Newtonian fluids. Subsequently, some continuous dependence theorems are given by extending the methods previously introduced in [2, 3].     

Some uniqueness and continuous dependence results in the micropolar mixture theory of porous media

The present paper studies the uniqueness and continuous data dependence of solutions of the initial-boundary value problem associated with the micropolar mixture linear theory of porous media. For a binary homogeneous mixture of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid, an uniqueness result is established. Then we deduce some estimates for describing the continuous dependence of solution with respect to the changes in the body force and body couple and in the initial-boundary given data. Thus, it is shown that the general approach of a binary homogeneous mixture of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid is well posed.     

On the uniqueness and existence of Stokes flows in micropolar fluid theory

Summary The present paper addresses itself to the problem of existence and uniqueness of solutions of Stokes flows in micropolar fluid theory using the methods of potential theory. Integral representations for the velocity and microrotation vectors are derived and they lead naturally to the introduction of single-layer and double-layer potentials whose properties are stated. With the aid of these results, necessary and sufficient conditions are generated for the Stokes problem in this microcontinuum fluid mechanics theory.

---

Zur Eindeutigkeit und Existenz von Stokesschen Strömungen mikropolarer Flüssigkeiten

Zusammenfassung Die gegenständliche Arbeit befaßt sich mit der Existenz und Eindeutigkeit der Lösungen Stokesscher Strömungen mikropolarer Flüssigkeiten unter Verwendung der Methoden der Potentialtheorie. Integraldarstellungen werden für die Geschwindigkeits-und Mikrorotationsvektoren abgeleitet. Diese führen zur Einführung von Einschicht-und Zweischichtpotentialen, deren Eigenschaften festgelegt werden. Mit Hilfe dieser Ergebnisse werden notwendige und hinreichende Bedingungen für das Stokes-problem der Mikrokontinuumsflüssigkeitsmechanik abgeleitet.

     

Faxen's laws for a micropolar fluid

Summary Faxen's formulas for the drag and torque on a rigid spherical particle immersed in a Stokes flow of a viscous incompressible fluid are extended for the case of an incompressible micropolar fluid.     

A uniqueness theorem for compressible micropolar flows

Summary Using an energy integral method it is proved that the motion of a non-heat conducting compressible micropolar fluid in a bounded regionV=V(t) is uniquely determined by the initial distributions of velocity, microrotation, density and temperature, together with certain boundary conditions.     

On the motion of a micropolar fluid drop in a viscous fluid

Summary The problems of the flow of a viscous fluid past a micropolar fluid sphere and the flow of a micropolar fluid past a viscous fluid drop are discussed. The expressions for the stream functions, velocities, spins and the drag are obtained in each case and are compared with the classical (viscous fluid past a viscous fluid sphere) results. It is found that the viscosity ratios and the parameters, which arises in connection with the boundary condition, have significant effect upon the drag on the sphere in each case.     

A note on the existence and uniqueness of solutions of the micropolar fluid equations

We show that existence and uniqueness theorems, known in the theory of the Navier-Stokes equations, are valid for the incompressible micropolar equations too.     

On the linear theory of micropolar elasticity

The present paper is concerned with some theorems in the linear dynamic theory of homogeneous and anisotropic micropolar elastic solids. The reciprocity theorem is derived without using the Laplace transform. Two variational theorems which fully characterize the solution of the mixed problem are given.     

On the stability of a hot layer of micropolar fluid

A critical study on the stability of a hot layer of micropolar fluid heated from below with free boundaries has been investigated. The analysis shows that the method by which the previous investigators (Datta and Sastry, and Pérez-García and Rubí) obtained the critical Rayleigh number is not justified and the final result obtained thereby is erroneous. The correct solution to the problem has been presented. Moreover, it is found that the possibility of having an overstable marginal state which was shown by one of the previous investigators (Pérez-García and Rubí) is not justified. The correct approach proves the validity of the principle of exchange of stabilities for this problem. The results show that the criteria of micropolar stability have some interesting features having no classical analogue.     

On the slow motion of a cylinder in a micropolar fluid

Slow steady 2-dimensional motion of an incompressible micropolar fluid in the unbounded region exterior to a cylinder of arbitrary cross section is considered. The possibility of a solution in the strict sense of the Stokes' approximation is examined. It is shown that the near and far boundary conditions can be satisfied simultaneously only when the drag on the cylinder is zero and the velocity and microrotation vectors satisfy an integral condition.     

Flow and heat transfer from a continuous surface in a parallel free stream of micropolar fluid

Boundary layer solutions are presented to investigate the steady now and heat transfer characteristics from a continuous flat surface moving in a parallel free stream of micropolar fluid. Numerical results are presented for the distribution of velocity, micro-rotation and temperature profiles within the boundary layer. The groupings of the material properties of the fluid were allowed to vary over a wide range.     

Analytic solution for flow of a micropolar fluid

Summary The time-independent equations for the two dimensional incompressible micropolar fluid have been considered. Using group method the equations have been reduced to ordinary differential equations and then solved analytically. Finally the boundary value problem has been discussed, and the graphical results are in good agreement with the numerical solution.     

Penetrative convection in a micropolar fluid

This work investigates stationary penetrative convection in an incompressible micropolar fluid heated from above and with its lower boundary at a fixed temperature. Linear, energy and conditional stability results are obtained for a series of upper surface temperatures and values for the micropolar parameter. The conditional energy analysis strongly suggests that the micropolar parameter should be small for optimal results.     

On the uniqueness and continuous data dependence of solutions in the theory of swelling porous thermoelastic soils

This paper studies the uniqueness and continuous data dependence of solutions of the initial-boundary value problems associated with the linear theory of swelling porous thermoelastic soils. The formulation belongs to the theory of mixtures for porous elastic solids filled with fluid and gas with thermal conduction and by considering the time derivative of temperature as a variable in the set of constitutive equations. Some uniqueness and continuous data dependence results are established under mild assumptions on the constitutive constants. Thus, it is shown that the general approach of swelling porous thermoelastic soils is well posed. The method of proof is based on some integro-differential inequalities and some Lagrange–Brun identities.     

On the stability of a hot rotating layer of micropolar fluid

The convective stability of a horizontal layer of incompressible micropolar fluid heated from below and rotating about a vertical axis has been investigated on the basis of linear theory, using normal mode analysis. The boundaries are assumed to be free. After introducing the corrections to the basic equations considered by Sastry and Rao [1], it has been found that the rotation has a destabilizing effect which contradicts the earlier assertion presented in [1].Moreover, microinertia, which does not affect the stability of a hot horizontal layer of incompressible micropolar fluid in the absence of rotation [2], is found to have destabilizing effect.     

Boundary layer growth of a micropolar fluid

The unsteady boundary layer flow of a micropolar fluid along an infinite plate is analysed when the plate undergoes an impulsive motion in its plane. The solution is obtained by Laplace transform technique. The microstructure of the fluid is found to induce a wave-dominated flow pattern, there being two modes of wave-propagation. The characteristics of these waves, during the initial and final stages of the boundary layer growth, are discussed.     

Flat plate drag in a micropolar fluid

Summary The drag experienced in a micropolar fluid is investigated by considering uniform streaming past a flat plate. Some recent results on the fundamental solution of the Oseenlinearization of the micropolar flow equations are used to reduce the problem to that of solving a scalar integral equation. The integral equation is analyzed by the application of both asymptotic and variational methods. Results indicate that the drag experienced in a micropolar fluid always exceeds that found in the absence of any micropolarity; however one of the parameters which characterizes a micropolar fluid can be used to minimize the drag.With 2 Figures