Exponential stability for a one‐dimensional compressible viscous micropolar fluid

In this paper, we consider one‐dimensional compressible viscous and heat‐conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H¹(0,1))⁴, and then we establish the global existence and exponential stability of solutions in (H²(0,1))⁴ under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.     

On existence and regularity of solutions for 2‐D micropolar fluid equations with periodic boundary conditions

In this paper, we prove the existence and uniqueness of a global solution for 2‐D micropolar fluid equation with periodic boundary conditions. Then we restrict ourselves to the autonomous case and show the existence of a global attractor. Copyright © 2006 John Wiley & Sons, Ltd.     

Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem

An initial-boundary value problem for 1-D flow of a compressible viscous heat-conducting micropolar fluid is considered; the fluid is assumed thermodynamically perfect and polytropic. The original problem is transformed into homogeneous one and studied the Faedo-Galerkin method. A local-in-time existence of generalized solution is proved.      

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

This article examines the existence and uniqueness of weak solutions to the d‐dimensional micropolar equations (d=2 or d=3) with general fractional dissipation (?Δ)^αu and (?Δ)^βw. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $\alpha \ge \frac{1}{2}$ and $\beta \ge \frac{1}{2}$, any initial data (u₀,w₀) in the critical Besov space $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{1+\frac{d}{2}?2\beta }({?}^d)$ yields a unique weak solution. For α ≥ 1 and β=0, any initial data $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{\frac{d}{2}}({?}^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, α=β=1, have a unique weak solution for $(u_0,w_0)\in B_{2,1}^0$. The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.     

1-D compressible viscous micropolar fluid model with non-homogeneous boundary conditions for temperature: A local existence theorem

We consider non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity and microrotation, as well as non-homogeneous boundary conditions for temperature are assumed. Using the Faedo-Galerkin method we prove a local-in-time existence of a generalized solution.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

Pressure regularity criteria of the three‐dimensional micropolar fluid flows

In the study of the regularity criteria of weak solutions of the three‐dimensional (3D) micropolar fluid flows, the regularity of solutions are examined by imposing some critical growth conditions only on the pressure field in the Lebesgue space, Morrey space, Multiplier space, BMO space and Besov space, respectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

Unsteady MHD flow due to noncoaxial rotations of micropolar fluid and an accelerated disk with partial slip condition

An analysis is carried out to study the magnetohydrodynamic (MHD) flow of an incompressible micropolar fluid. The flow is induced by the noncoaxial rotations of constantly accelerated porous disk and a micropolar fluid. The influence of partial slip on the flow has been taken into consideration. Numerical solution of the governing flow problem is given by means of Newto's method. The important finding in this communication is the effects of partial slip on the velocity and microrotation vector. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A micropolar fluid model of blood flow through a tapered artery with a stenosis

A nonlinear two‐dimensional micropolar fluid model for blood flow in a tapered artery with a single stenosis is considered. This model takes into account blood rheology in which blood consists of microelements suspended in plasma. The classical Navier–Stokes theory is inadequate to describe the microrotations or particles' spin of such suspension in a viscous medium. The governing equations involving unsteady nonlinear partial differential equations are solved using a finite difference scheme. A quantitative analysis performed through numerical computation shows that the axial velocity profile and the flow rate decrease and the wall shear stress increases once the artery is narrower in the presence of the polar effect. Furthermore, the taper angle certainly bears the potential to influence the velocity and the flow characteristics to considerable extent. Copyright © 2010 John Wiley & Sons, Ltd.     

Exact and numerical solutions of a micropolar fluid in a vertical annulus

In this article, we have analyzed the effects of heat transfer on a peristaltic flow of a micropolar fluid in a vertical annulus. The governing equations of two‐dimensional micropolar fluid are simplified by using the assumptions of long wavelength and neglecting the wave number. A close form solutions are obtained for velocity field υ_x and microrotation component υ_θ. Further, the numerical solutions of the simplified equation of υ_θ are computed and the results are compared with the exact solution. The influence of pertinent parameters are analyzed through graphs. Trapping phenomena is also discussed for different parameters. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Existence and uniqueness of steady motions of a third-grade fluid

The global existence and uniqueness of classical solution of steady motions of a third-grade fluid provided assumptions on positivness of μ (coefficient of viscosity) and α₁, γ (material coefficients) is proved. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Blow‐up criterion for two‐dimensional magneto‐micropolar fluid equations with partial viscosity

In this paper, we derive a blow‐up criterion of smooth solutions to the incompressible magneto‐micropolar fluid equations with partial viscosity in two space dimensions. Our proof is based on careful Hölder estimates of heat and transport equations and the standard Littlewood–Paley theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Asymptotics of the solution to the point explosion problem in the case of cylindrical symmetry

For the problem of a strong point explosion with cylindrical symmetry, high-order asymptotic expansions of the solution with respect to the viscosity and thermal conductivity are constructed and justified.     

Analysis of nonlocal model of compressible fluid in 1‐D

We study a nonlocal modification of the compressible Navier–Stokes equations in mono‐dimensional case with a boundary condition characteristic for the free boundaries problem. From the formal point of view, our system is an intermediate between the Euler and Navier–Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient, we obtain the local and global existence of solutions. Particularly, we show the uniform in time bound on the density of fluid. Copyright © 2010 John Wiley & Sons, Ltd.     

Existence and regularizing rate estimates of solutions to the 3‐D generalized micropolar system in Fourier‐Besov spaces

In this paper, we study global existence and asymptotic stability of solutions for the initial value problem of the three‐dimensional (3‐D) generalized incompressible micropolar system in Fourier‐Besov spaces. Besides, we also establish some regularizing rate estimates of the higher‐order spatial derivatives of solutions, which particularly imply the spatial analyticity and the temporal decay of global solutions.