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.     

Asymptotic behavior of micropolar fluid flows

This paper is devoted to some recent results on several aspects of long time behavior of micropolar fluid flows. In particular, we consider such topics as existence and uniqueness of global in time solutions, their convergence to the stationary solution for large viscosity flows, existence of a global attractor and estimates of its Hausdorff and fractal dimensions, continuous dependence of solutions on microrotation viscosity perturbations and stability of the corresponding global attractor with respect to these perturbations, and flows in unbounded domains.     

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.

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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.

     

The geometry of the cartan space associated with general compressible fluid flows

In this paper, an attempt is made to develop a general theory of the motion of a compressible fluid by means of the geometry of the general space propounded by Cartan based on the concept of area. Assuming that the flow is irrotational and isentropic, it is shown in Section I, that the equation of motion of an inviscid compressible fluid is regarded as that of hypersurfaces in the Cartan space. Various hydrodynamical features of the compressible fluid motion including the “shock” phenomena are clarified by the metric properties of the space. The fluid motion in the general case, in which the above assumptions are not valid, is treated in Section 2. The physical quantities such as vorticity, entropy and the Croccian vector are expressed by means of the geometrical terminology of the Cartan space along lines parallel to the Section 1.     

rp-adaptation for compressible flows

We present a novel rp-adaptation strategy for high-fidelity simulations of compressible inviscid flows with shocks. The mesh resolution in regions of flow discontinuities is increased by using a variational optimizer to r-adapt the mesh and cluster degrees of freedom there. In regions of smooth flow, we locally increase or decrease the local resolution through increasing or decreasing the polynomial order of the elements, respectively. This dual approach allows us to take advantage of the strengths of both methods for best computational performance, thereby reducing the overall cost of the simulation. The adaptation workflow uses a sensor for both discontinuities and smooth regions that is cheap to calculate, but the framework is general and could be used in conjunction with other feature-based sensors or error estimators. We demonstrate this proof-of-concept using two geometries in transonic and supersonic flow regimes. The method has been implemented in the open-source spectral/hp element framework Nektar++, and adaptivity is performed by its dedicated high-order mesh generation tool NekMesh. The results show that the proposed rp-adaptation methodology is a reasonably cost-effective way of improving simulation accuracy.     

Decay of the kinetic energy of compressible micropolar fluids

The decay of the kinetic energy of compressible micropolar fluids is analysed employing a linear theory of micropolar fluids. It is found that upon the cessation of the energy transferred to the system, the kinetic energy of the system decays exponentially. The spectral function of the kinetic energy is also found to decay exponentially.     

Uniqueness and reciprocity theorems in generalized linear micropolar thermoviscoelasticity

A model of the equations of generalized linear micropolar thermoviscoelasticity is given. The formulation is applied to the coupled theory as well as to five generalizations, the Lord-Shulman theory with one relaxation time, the Green-Lindsay theory with two relaxation times, the Green-Naghdi theories of type II (without energy dissipation) and of type III, and the Chandrasekharaiah-Tzou theory with dual-phase-lag. Using Laplace transforms, a uniqueness theorem for this model is proved, restrictions on relaxation functions are deduced and the dynamic reciprocity theorem is derived. The cases of generalized linear micropolar thermoviscoelasticity of Kelvin-Voigt model, generalized linear micropolar thermoelasticity, generalized thermoviscoelasticity and generalized thermoelasticity can be obtained from the given general model.     

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.     

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.     

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.     

Decay estimates of linearized micropolar fluid flows in R3 space with applications to L3-strong solutions

Through analytical argument, the L_p − L_q estimate of a three-dimensional linearized micropolar fluid flow in the whole space R³ is established. This estimate is used to show the existence and uniqueness of small L₃-strong solutions of the micropolar fluid motion system. Sharp time decay estimates of the L₃-strong solutions are derived.     

Non-orthogonal stagnation-point flow of a micropolar fluid

This paper considers the problem of steady two-dimensional flow of a micropolar fluid impinging obliquely on a flat plate. The flow under consideration is a generalization of the classical modified Hiemenz flow for a micropolar fluid which occurs in the boundary layer near an orthogonal stagnation point. A coordinate decomposition transforms the full governing equations into a primary equation describing the modified Hiemenz flow for a micropolar fluid and an equation for the tangential flow coupled to the primary solution. The solution to the boundary-value problem is governed by two non-dimensional parameters: the material parameter K and the ratio of the microrotation to skin friction parameter n. The obtained ordinary differential equations are solved numerically for some values of the governing parameters. The primary consequence of the free stream obliqueness is the shift of the stagnation point toward the incoming flow.     

Petrov-Galerkin finite element model for compressible flows

A Petrov-Galerkin method for the solution of the compressible Euler and Navier-Stokes equations is presented. It is based on the introduction of an anisotropic blancing diffusion in the direction of the local direction of propagation of the scalar variables. The local direction in which the anisotropic diffusion is introduced is uniquely determined, and the magnitude of the balancing diffusion is automatically calculated locally using a criterion that is optimal for one-dimensional transport equations. The algorithm has been implemented using four-noded bilinear elements with forward Euler and second-order Runge-Kutta methods of integration in time. Several applications are presented and show the stability and approximation properties of the method.     

Finitely conducting compressible aligned magnetofluiddynamic parallel flows

A method for studying compressible, aligned MFD parallel flows is discussed. Flow equations are recast in an orthogonal streamline coordinate system, by employing some results from differential geometry. Exact solutions of these equations are obtained by assuming, a priori, certain functional forms of the speed, for straight parallel flows. Several examples are given to illustrate this method.     

Rayleigh-Benard convection in a micropolar ferromagnetic fluid

The problem of Rayleigh-Benard convection in a micropolar ferromagnetic fluid layer permeated by a uniform, vertical magnetic field is investigated analytically with free-free, isothermal, spin-vanishing, magnetic boundaries. The influence of the various micropolar and magnetization parameters on the onset of stationary convection has been analysed. It is observed that the micropolar ferromagnetic fluid layer heated from below is more stable as compared with the classical Newtonian ferromagnetic fluid. The nature of influence of the magnetization parameters on convection in the micropolar ferromagnetic fluid is similar to that in the case of Newtonian ferromagnetic fluids. The influence of the micropolar parameters on convection in the ferromagnetic case is akin to its role in the non-magnetic fluid case. The critical wave number is found to be insensitive to the changes in the micropolar fluid parameters, but sensitive to the magnetization parameters.     

Wall slip induced by a micropolar fluid

Lubrication theory is devoted to the study of thin-film flows, More often, the fluid can be considered as a Newtonian one and no-slip boundary conditions can be retained for the velocity at the fluid solid interface. With these assumptions it is possible to deduce from the (Navier) Stokes system a simplified equation describing the flow: the Reynolds equation. It allows to compute the pressure distribution inside the film and to obtain overall performances of a lubricated device such as load and friction coefficient. For very thin films, however, surface effects at the fluid solid interface become very important and no-slip conditions cannot be retained. Solid surfaces exert some influence on the liquid molecules and the effective shear viscosity along the boundary differs from the classical bulk shear viscosity. Moreover, the microstructure of the fluid cannot be ignored, especially the effects of solid-particle additives in the lubricant. Micropolar theory for fluids is often adopted to account of such microstructure and microrotation. In the present study, a thin micropolar fluid model with new boundary conditions at the fluid–solid interface is considered. This condition links velocity and microrotation at the interface by introducing a so-called “boundary viscosity”. By way of asymptotic analysis, a generalized micropolar Reynolds equation is obtained. Numerical results show the influence of the new boundary conditions for the load and friction coefficients. Comparisons are made with other works that retain the no-slip boundary conditions.