Abstract:
The present investigation deals with study of deformation in homogeneous, isotropic thermodiffusion elastic half-space as a result of inclined load. The inclined load is assumed to be a linear combination of normal load and tangential load. The integral transform technique is used to solve the problem. As an application of the approach distributed and moving forces have been taken. The transformed components of displacement, stresses, temperature distribution and concentration are inverted using numerical inversion technique. The effect of relaxation times and response of two theories of thermoelasticity i.e. Green and Lindsay (G-L) theory and coupled theory (CT) on these quantities have been depicted graphically for a particular model. Some particular cases are also deduced.

Abstract:
the conditions for propagation of accelerating waves in a general nonlinear thermoelastic micropolar media are established. deformation of micropolar media is described by the time-varying displacement vector r(t) and tensor of microrotation r(t) at each point. we call a surface s(t) an accelerating wave (or a singular surface for a solution of the dynamic problem for the medium) if the points are points of continuity of both r(t) and h(t) and their first spatial and time derivatives while the second spatial and time derivatives (acceleration) of r(t) and h(t) have jumps on s(t) (meaning that their one-sided limits at s(t) differ). so s(t) carries jumps in the acceleration fields as it propagat es through the body. in the thermomechanics of a micropolar continuum, similar propagating surfaces of singularities can exist for the fields of temperature, heat flux, etc. we establish the kinematic and dynamic compatibility relations for the singular surface s(t) in a nonlinear micropolar thermoelastic medium. an analog of fresnel--hadamard--duhem theorem and an expression for the acoustic tensor are derived.

Abstract:
Regularity criterion for the 3D micropolar fluid equations is investigated. We prove that, for some , if , where and , then the solution can be extended smoothly beyond . The derivative can be substituted with any directional derivative of . 1. Introduction In the paper, we investigate the initial value problem for the micropolar fluid equations in : with the initial value where ,？？ , and stand for the divergence free velocity field, nondivergence free microrotation field (angular velocity of the rotation of the particles of the fluid), the scalar pressure, respectively is the Newtonian kinetic viscosity, is the dynamics microrotation viscosity, and are the angular viscosity (see, e.g., Lukaszewicz [1]). The micropolar fluid equations was first proposed by Eringen [2]. It is a type of fluids which exhibits the microrotational effects and microrotational inertia and can be viewed as a non-Newtonian fluid. Physically, micropolar fluid may represent fluids that consists of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of fluid particles is ignored. It can describe many phenomena appeared in a large number of complex fluids such as the suspensions, animal blood, and liquid crystals which cannot be characterized appropriately by the Navier-Stokes equations, and that is important to the scientists working with the hydrodynamic fluid problems and phenomena. For more background, we refer to [1] and references therein. Besides their physical applications, the micropolar fluid equations are also mathematically significant. The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero [3] and Yamaguchi [4], respectively. The convergence of weak solutions of the micropolar fluids in bounded domains of was investigated (see [5]). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research, and many interesting results have been obtained (see [6–8]). A Beale-Kato-Madja criterion (see [9]) of smooth solutions to a related model with (1.1) was established in [10]. If and , then (1.1) reduces to be the Navier-Stokes equations. Besides its physical applications, the Navier-Stokes equations are also mathematically significant. In the last century, Leray [11] and Hopf [12] constructed weak solutions to the Navier-Stokes equations. The solution is called the Leray-Hopf weak solution. Later on, much effort has been

Abstract:
This paper presents an extension of mathematical static model to dynamic problems of micropolar elastic plates, recently developed by the authors. The dynamic model is based on the generalization of Hellinger-Prange-Reissner (HPR) variational principle for the linearized micropolar (Cosserat) elastodynamics. The vibration model incorporates high accuracy assumptions of the micropolar plate deformation. The computations predict additional natural frequencies, related with the material microstructure. These results are consistent with the size-effect principle known from the micropolar plate deformation. The classic Mindlin-Reissner plate resonance frequencies appear as a limiting case for homogeneous materials with no microstructure.

Abstract:
The conditions for propagation of accelerating waves in a general nonlinear thermoelastic micropolar media are established. Deformation of micropolar media is described by the time-varying displacement vector r(t) and tensor of microrotation r(t) at each point. We call a surface S(t) an accelerating wave (or a singular surface for a solution of the dynamic problem for the medium) if the points are points of continuity of both r(t) and H(t) and their first spatial and time derivatives while the second spatial and time derivatives (acceleration) of r(t) and H(t) have jumps on S(t) (meaning that their one-sided limits at S(t) differ). So S(t) carries jumps in the acceleration fields as it propagat es through the body. In the thermomechanics of a micropolar continuum, similar propagating surfaces of singularities can exist for the fields of temperature, heat flux, etc. We establish the kinematic and dynamic compatibility relations for the singular surface S(t) in a nonlinear micropolar thermoelastic medium. An analog of Fresnel--Hadamard--Duhem theorem and an expression for the acoustic tensor are derived. Se establecen las condiciones de propagación de ondas aceleradas en un medio no lineal micropolar termoelástico. Las deformaciones del medio micropolar son descritas por las variaciones temporales del vector de desplazamiento r(t) y del tensor de microrotación r(t) en cada punto. Llamamos una superficie S(t) a una onda acelerada (o superficie singular para la solución del problema dinámico del medio) si los puntos son puntos de continuidad de r(t), H(t) y sus primeras derivas espaciales y temporales, mientras que las segundas derivadas espaciales y temporales tienen saltos en S(t). Entonces S(t) transporta los saltos en los campos acelerados cuando se propagan en el cuerpo. En la termomecánica de un continuo micropolar, superficies de propagación similares pueden existir para los campos de temperatura y de flujo de calor. Establecemos las relaciones de compatibilidad cinética y dinámica para las superficies singulares en un medio micropolar termoelástico no lineal. Un análogo del teorema Fresnel--Hadamard--Duhem y una expresión para el tensor acústico son establecidos.

Abstract:
The dynamics of distributed sources is described by nonlinear partial differential equations. Lagrangian analytical solutions of these (and associated) equations are obtained and discussed in the context of Lagrangian modeling - from the Lagrangian invariants to dynamics. Possible applications of distributed sources and sinks to geophysical fluid dynamics and to the cosmology are indicated.

Abstract:
The present work is concerned with unsteady free convection flow of an incompressible electrically conducting micropolar fluid, bounded by an infinite vertical plane surface of constant temperature. A uniform magnetic field acts perpendicularly to the plane. The state space technique is adopted for the one-dimensional problems including heat sources with one relaxation time. The resulting formulation is applied to a problem for the whole space with a plane distribution of heat sources. The reflection method together with the solution obtained for the whole space is applied to a semispace problem with a plane distribution of heat sources located inside the fluid. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for the temperature, the velocity, and the angular velocity distributions are given and illustrated graphically for the problems considered.

Abstract:
In this note we extend integrability conditions for the symmetric stretch tensor $U$ in the polar decomposition of the deformation gradient $\nabla\varphi=F=R\,U$ to the non-symmetric case. In doing so we recover integrability conditions for the first Cosserat deformation tensor. Let $F=\bar R\,\bar U$ with $\bar R:\Omega\subset\mathbb{R}^3\longrightarrow\mathrm{SO}(3)$ and $\bar U:\Omega\subset\mathbb{R}^3\longrightarrow \mathrm{GL}(3)$. Then $\mathfrak{K}:={\bar R}^T\mathrm{Grad}\,{\bar R}=\mathrm{Anti}\Big( \frac{1}{\mathrm{det} \bar U}\Big[\bar U(\mathrm{Curl} \bar U)^T-\frac{1}{2} \mathrm{tr}(\bar U(\mathrm{Curl} \bar U)^T) 1\!\!1 \Big]\bar U\Big),$ giving a connection between the first Cosserat deformation tensor $\bar U$ and the second Cosserat tensor ${\mathfrak{K}}$. (Here, Anti denotes an isomorphism between $\mathbb{R}^{3\times 3}$ and $\mathfrak{So}(3):=\{\,\mathfrak{A}\in\mathbb{R}^{3\times 3\times 3}\,|\,\mathfrak{A}.u\in\mathfrak{so}(3)\;\forall u\in \mathbb{R}^3\}$.) The formula shows that it is not possible to prescribe $\bar U$ and $\mathfrak{K}$ independent from each other. We also propose a new energy formulation of geometrically nonlinear Cosserat models which completely separate the effects of nonsymmetric straining and curvature. For very weak constitutive assumptions (no direct boundary condition on rotations, zero Cosserat couple modulus, quadratic curvature energy) we show existence of minimizers in Sobolev-spaces.

Abstract:
Distributed source coding is traditionally viewed in the block coding context -- all the source symbols are known in advance at the encoders. This paper instead considers a streaming setting in which iid source symbol pairs are revealed to the separate encoders in real time and need to be reconstructed at the decoder with some tolerable end-to-end delay using finite rate noiseless channels. A sequential random binning argument is used to derive a lower bound on the error exponent with delay and show that both ML decoding and universal decoding achieve the same positive error exponents inside the traditional Slepian-Wolf rate region. The error events are different from the block-coding error events and give rise to slightly different exponents. Because the sequential random binning scheme is also universal over delays, the resulting code eventually reconstructs every source symbol correctly with probability 1.

This article proposes a simplified way to solve solid mechanic problems in micropolar elasticity using the solution found in the classic theory of elasticity as a starting point. In this study, an analysis of the linear isotropic micropolar elasticity is conducted based on the properties imposed on the micropolar medium by the constitutive and equilibrium equations. To ascertain how the micropolar medium responses deviate from Hooke’s theory of elasticity, different loading conditions were classified. Three cases have been found so far: the rotational couple response, the quasi-classic equilibrium of momentum response and the general case. This study is the first in a series planned to explore the use of commercial packages of finite element in order to solve micropolar elasticity problems.