非牛顿粘性不可压流方程的近似惯性流形

本文利用对非牛顿粘性不可压缩流方程对时间 t的解析性和长时间渐近性估计 ,具体构造了它的近似惯性流形 ,并得出收敛阶估计 .     

On Variational Models for Quasi-static Bingham Fluids

We study a quasi-static incompressible flow of Bingham type with constituent law \[ \begin{array}{ll} T = p\left| {\cal E}u\right| ⁁{p-2}{\cal E}u+\beta \frac{{\cal E}u}{\left| {\cal E}u\right| } & \text{if }{\cal E}u\neq 0, \\ \left| T\right| \leq \beta & \text{if }{\cal E}u = 0, \end{array} \] T = p∣ℰu∣^p-2ℰu+β ℰu ∣ℰu∣ if ℰu≠0, ∣T∣⩽β if ℰu = 0, where p≥2 and β>0. Here ℰu denotes the strain velocity and T the corresponding stress. The problem admits a variational formulation in the sense that the velocity field u minimizes the energy I(u) = ∫_Ω∣ℰu∣^p+β∣ℰu∣dx in the space {v∈H^1,p(Ω,ℝⁿ): div v = 0} subject to appropriate boundary conditions. We then show smoothness of u on the set {x∈Ω: ℰu≠0}.     

Existence for a Class of Non-Newtonian Fluids with a Nonlocal Friction Boundary Condition

We deal with a variational inequality describing the motion of incompressible fluids, whose viscous stress tensors belong to the subdifferential of a functional at the point given by the symmetric part of the velocity gradient, with a nonlocal friction condition on a part of the boundary obtained by a generalized mollification of the stresses. We establish an existence result of a solution to the nonlocal friction problem for this class of non-Newtonian flows. The result is based on the Faedo-Galerkin and Moreau Yosida methods, the duality theory of convex analysis and the Tychonov-Kakutani-Glicksberg fixed point theorem for multi-valued mappings in an appropriate functional space framework.     

Estimates of Deviations for Generalized Newtonian Fluids

The paper is concerned with estimates of deviations from exact solutions for stationary models of viscous incompressible fluids. It is shown that if a function compared with exact solution is subject to the incompressibility condition, then the deviation majorant consists of terms that penalize the inaccuracy in the equilibrium equation and the rheological relation defined by a ceratin dissipative potential. If such a function does not satisfy the incompressibility condition, then the majorant includes an additional term. The factor of this term depends on the constant in the Ladyzhenskaya–Babuka–Brezzi condition. Bibliography: 27 titles.     

Numerical Modelling of Newtonian and Non-Newtonian Fluids Flow

The work deals with numerical modelling of 2D/3D laminar incompressible viscous flows for Newtonian and non–Newtonian fluids. The unsteady system of Navier–Stokes equations with steady boundary conditions in the form of an artificial compressibility method is solved by multistage Runge–Kutta finite volume method. Steady state solution is achieved for t→∞. Convergence is followed by steady residual behaviour. For unsteady solution high compressibility coefficient β² is considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Lubrication with Structured Fluids

AMS (MOS) 76 A 05 76 O 99     

Regularity of Stationary Weak Solutions in the Theory of Generalized Newtonian Fluids with Energy Transfer

In this paper, we prove regularity results for weak solutions to some stationary problems arising in the theory of generalized Newtonian fluids with energy transfer. Namely, we prove that these solutions are strong. In the two-dimensional case, we prove the Hölder continuity of the first gradient of a solution. Bibliography: 30 titles.     

(In)Finite Extent of Stationary Perfect Fluids in Newtonian Theory

For stationary, barotropic fluids in Newtonian gravity we give simple criteria on the equation of state and the “law of motion” which guarantee finite or infinite extent of the fluid region (providing a priori estimates for the corresponding stationary Newton–Euler system). Under more restrictive conditions, we can also exclude the presence of “hollow” configurations. Our main result, which does not assume axial symmetry, uses the virial theorem as the key ingredient and generalises a known result in the static case. In the axially symmetric case stronger results are obtained and examples are discussed.     

On non‐Newtonian incompressible fluids with phase transitions

A modified model for a binary fluid is analysed mathematically. The governing equations of the motion consists of a Cahn–Hilliard equation coupled with a system describing a class of non‐Newtonian incompressible fluid with p‐structure. The existence of weak solutions for the evolution problems is shown for the space dimension d=2 with p? 2 and for d=3 with p? 11/5. The existence of measure‐valued solutions is obtained for d=3 in the case 2? p< 11/5. Similar existence results are obtained for the case of nondifferentiable free energy, corresponding to the density constraint |ψ| ? 1. We also give regularity and uniqueness results for the solutions and characterize stable stationary solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

On the two-dimensional rotational body of maximal Newtonian resistance

We investigate, by means of computer simulations, shapes of nonconvex bodies that maximize resistance to their motion through a rarefied medium, considering that the bodies are moving forward and at the same time slowly rotating. A two-dimensional geometric shape that confers to the body a resistance very close to the theoretical supremum value is obtained, which improves previous results.     

Elliptical billiard table with Newtonian potential

Translated from Matematicheskie Zametki, Vol. 55, No. 3, pp. 139–140, March, 1994.     

On the Inverse Source Problem for the Newtonian Potential

The paper deals with the inverse source problems for the Newtonian potential in the sense of distribution (generalized functions). The inverse source problem is defined as follows: A domain G is given. To find is a distribution creating the potential which is known outside of the closed domain G. New results can be applied to electronical fields, because the electrical charge is distributed only on the surface of conductor.     

On the lnviscid Theory of Rotating Fluids

Conservation theorems for “energy” and mean circulation in an inviscid rotating fluid are stated in their most general forms (which do not presume the existence of inertial modes). The geostrophic motion that contains all the mean circulation of an initial velocity field is identified as the time average of the flow. A preliminary study of the in viscid motion inside a closed cone reveals many singular features and indicates that the inertial wave spectrum may be continuous.     

On Terwilliger graphs with µ = 4

A Terwilliger graph is an incomplete connected graph in which the intersection of the neighborhoods of any two vertices at distance 2 is a µ-clique for some constant µ. The local structure of Terwilliger graphs with µ = 4 is described.     

On Static Stars in Newtonian Gravity and Lane-Emden Type Equations

. The equations governing static stellar models in Newtonian gravity are equivalent to a Lane-Emden type equation. For such equations existence,uniqueness,and regularity of global solutions is shown for a large class of right-hand sides, including a subclass of non-Lipschitz continuous equations of state which is relevant if e.g. phase transitions occur. Furthermore, it is shown that for a star of finite radius the polytropic index of the equation of state is not necessarily bounded near the star's surface.     

On surfaces with p g =2, q=1 and K 2=5

We consider minimal surfaces of general type with p _g =2, q=1 and K ²=5. We provide a stratification of the corresponding moduli space \(\mathcal{M}\) and we give some bounds for the number and the dimensions of its irreducible components.     

On the Regularity of Shear Thickening Viscous Fluids

The aim of this note is to improve the regularity results obtained by H. Beirao da Veiga in 2008 for a class of p-fluid flows in a cubic domain. The key idea is exploiting the better regularity of solutions in the tangential directions with respect to the normal one, by appealing to anisotropic Sobolev embeddings.