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91.
A. D. Fitt D. P. Mason E. A. Moss 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(6):1049-1067
The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture
is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the
fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to
the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear
diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value
problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes
a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications
in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are
considered. 相似文献
92.
In this paper one specifies the ergodic behavior of the 2D-stochastic Navier–Stokes equation by giving a Large Deviation Principle for the occupation measure for large time. It describes the exact rate of exponential convergence. The considered random force is non-degenerate and compatible with the strong Feller property. 相似文献
93.
94.
This paper develops an efficient particle tracking algorithm to be used in fluid simulations approximated by a high-order multidomain discretization of the Navier–Stokes equations. We discuss how to locate a particle's host subdomain, how to interpolate the flow field to its location, and how to integrate its motion in time. A search algorithm for the nearest subdomain and quadrature point, tuned to a typical quadrilateral isoparametric spectral subdomain, takes advantage of the inverse of the linear blending equation. We show that to compute particle-laden flows, a sixth-order Lagrangian polynomial that uses points solely within a subdomain is sufficiently accurate to interpolate the carrier phase variables to the particle position. Time integration of particles with a lower-order Adams–Bashforth scheme, rather than the fourth-order Runge–Kutta scheme often used for the integration of the carrier phase, increases computational efficiency while maintaining engineering accuracy. We verify the tracking algorithm with numerical tests on a steady channel flow and an unsteady backward-facing step flow. 相似文献
95.
In this paper we consider a singularly perturbed elliptic problem with two small parameters posed on the unit square. Its solution may have exponential, parabolic and corner layers. We give a decomposition of the solution into regular and layer components and derive pointwise bounds on the components and their derivatives. The estimates are obtained by the analysis of appropriate problems on unbounded domains. 相似文献
96.
Joseph Majdalani Sjoerd W. Rienstra 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):289-308
The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators
and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe
the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum
and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid,
rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three
possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying
the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex.
This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression
that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general
and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method
of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions
that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach
to other complex fluid motions that are easier to capture using spherical coordinates. 相似文献
97.
《Mathematische Nachrichten》2017,290(16):2524-2546
Consider the Stokes equations in a sector‐like C 3 domain . It is shown that the Stokes operator generates an analytic semigroup in for . This includes domains where the ‐Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in and L 2 by constructing a suitable non‐Helmholtz projection to solenoidal spaces. 相似文献
98.
Jae-Myoung KIM 《数学物理学报(B辑英文版)》2017,37(4):1033-1047
We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are H¨older continuous near boundary provided that the scaled mixed L_(x,t)~(p,q) -norm of the velocity vector field with 3/p + 2/q ≤ 2,2 q ∞ is sufficiently small near the boundary. Also, we will investigate that for this solution u ∈ L_(x,t)~(p,q) with 1≤3/p+2/q≤3/2, 3 p ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}(3/p+2/q-1). 相似文献
99.
Wolfgang Dahmen Birgit Gottschlich–Müller Siegfried Müller 《Numerische Mathematik》2001,88(3):399-443
Summary. In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body. Received June 24, 1998 / Revised version received February 21, 2000 / Published online November 8, 2000 相似文献
100.
Luigi?C.?BerselliEmail author Renato?ManfrinEmail author 《Journal of Evolution Equations》2004,4(2):193-211
In this paper we study some criteria for the full (space-time) regularity of weak
solutions to the Navier-Stokes equations. In particular, we generalize some
classical and very recent criteria involving the velocity, or its
derivatives. More specifically, we show with elementary tools that if a weak
solution, or its vorticity, is small in appropriate Marcinkiewicz spaces, then it
is regular. 相似文献