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1.
In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.  相似文献   

2.
In this paper,the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind.Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity.Moreover,the solution of the variational identity satisfies the saddle-point problem of the Lagrangian functional ?.Subsequently,the Uzawa algorithm is proposed to solve the solution of the saddle-point problem. We show the convergence of the algorithm and obtain the convergence rate.Finally,we give the numerical results to verify the feasibility of the Uzawa algorithm.  相似文献   

3.
Associated with the concepts of the Lp-mixed quermassintegrals, the Lp-mixed volume, and the Lp-dual mixed volume, we establish inequalities for the quermassintegrals of the Lp-projection body and the Lp-centroid body. Further, the general results for the Shephard problem of the Lp-projection body and the Lp-centroid body are obtained.  相似文献   

4.
In this paper, we analyze the Bregman iterative model using the G-norm. Firstly, we show the convergence of the iterative model. Secondly, using the source condition and the symmetric Bregman distance, we consider the error estimations between the iterates and the exact image both in the case of clean and noisy data. The results show that the Bregman iterative model using the G-norm has the similar good properties as the Bregman iterative model using the L2-norm.  相似文献   

5.
In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary Boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity,which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor,the velocity and the temperature,whereas the pressure is computed via a postprocessing formula.In addition,an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation.Regarding the discrete problem,we follow the approach employed in a previous work dealing with the Navier-Stokes equations,and couple it with a VEM for the convection-diffusion equation modelling the temperature.More precisely,we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div)and H1,respectively,whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H1.In this way,we make use of the L2-orthogonal projectors onto suitable polynomial spaces,which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations.On the other hand,in order to manipulate the bilinear form associated to the heat equations,we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor,which represents the thermal conductivity,is variable.Next,the corresponding solvability analysis is performed using again appropriate fixed-point arguments.Further,Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure.The corresponding rates of convergence are also established.Finally,several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.  相似文献   

6.
The flow boiling phenomenon in a channel with multiple rectangular heaters under a constant wall temperature was numerically studied with the lattice Boltzmann method. The effects of spacings between heaters, heater lengths and heater surface wettabilities on the bubble morphology, the bubble area and the heat flux on the heater surface, were studied. The results show that, the bubble growth rate increases with the spacing between heaters. The larger the bubble area is, the earlier the nucleated bubbles will leave the heater surface. The corresponding boiling heat transfer performance increases by 12% with the spacing between heaters growing from 250 lattices to 1 000 lattices. On the other hand, the longer the heater length is, the earlier the bubble will nucleate and leave the heater surface, and the better the boiling heat transfer performance will be. The boiling heat transfer performance increases by 13% with the heater length rising from 16 lattices to 22 lattices. In addition, the bubble nucleates later on the hydrophilic surface than on the hydrophobic surface. Compared with the hydrophilic surface, the hydrophobic surface retains residual bubbles after the leaving of bubbles from the heater. The average heat flux and the bubble area of the hydrophilic surface are less than those of the hydrophobic surface. With the contact angle changing from 77° to 120°, the heat transfer performance increases by 26%. Finally, the orthogonal test results indicate that, the wettability of the heat exchanger surface has the greatest influence on the flow boiling heat transfer performance, while the heater length has the least influence. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.  相似文献   

7.
In this paper,the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied.The solutions to the system(1.6)–(1.8) are in the class of radius-dependent solutions,i.e.,independent of the axial variable and the angular variable.In particular,the expanding rate of the moving boundary is obtained.The main difficulty of this problem lies in the strong coupling of the magnetic field,velocity,temperature and the degenerate density near the free boundary.We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates,and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable r;by weighted estimates,and also the uniform-in-time weighted estimates of the higher-order derivatives of solutions by delicate analysis.  相似文献   

8.
The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.  相似文献   

9.
This paper is concerned with the quasi-neutral limit of the bipolar NavierStokes-Poisson system. It is rigorously proved, by introducing the new modulated energy functional and using the refined energy analysis, that the strong solutions of the bipolar Navier-Stokes-Poisson system converge to the strong solution of the compressible NavierStokes equations as the Debye length goes to zero. Moreover, if we let the viscous coefficients and the Debye length go to zero simultaneously, then we obtain the convergence of the strong solutions of bipolar Navier-Stokes-Poisson system to the strong solution of the compressible Euler equations.  相似文献   

10.
We prove the global existence and stability of a wave structure containing a stationary Mach configuration,which occurs when an incident shock front hits a wall with a large incident angle.Our result shows that the data of the upstream flow and the pressure at downstream part jointly determine the whole flow,as well as the wave structure.Particularly,we show that the height of the Mach stem depends not only on the data of upstream flow,but also on the pressure at downstream flow.The flow with the assigned wave structure is governed by a free boundary value problem for the Euler system.In the problem the location of the triple point,the shock fronts and the contact discontinuity are all unknown,they are finally determined together with the solution.  相似文献   

11.
In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a 2D map u. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a 2D Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of u at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by Γ-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices.  相似文献   

12.
A class of singularly perturbed boundary value problem with singularities is considered. Introducing the stretched variables, the boundary layer corrective terms near x = 0 and x = 1 are constructed. Under suitable conditions, by using the theory of differential inequalities the existence and asymptotic behavior of solution for boundary value problem are proved, uniformly valid asymptotic expansion of solution with boundary layers are obtained,  相似文献   

13.
This Note presents the derivation of the 2nd-order asymptotic expansion of the eigenvalues and the eigenfunctions of the operator associated to an interior elliptic equation supplemented by a Dirichlet boundary condition on a domain consisting of two cavities linked by a hole of small size. The asymptotic expansion is carried out with respect to the size of the hole. The main feature of the method is to yield a robust numerical procedure making it possible to compute the eigenvalues without resorting to a refined mesh around the hole. To cite this article: A. Bendali et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).  相似文献   

14.
In this paper, we study the asymptotic behavior for the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition in a half space of ${\mathbb{R}^3}$ when the vertical viscosity goes to zero. Firstly, by multi-scale analysis, we formally deduce an asymptotic expansion of the solution to the problem with respect to the vertical viscosity, which shows that the boundary layer appears in the tangential velocity field and satisfies a nonlinear parabolic–elliptic coupled system. Also from the expansion, it is observed that away from the boundary the solution of the anisotropic Navier–Stokes equations formally converges to a solution of a degenerate incompressible Navier–Stokes equation. Secondly, we study the well-posedness of the problems for the boundary layer equations and then rigorously justify the asymptotic expansion by using the energy method. We obtain the convergence results of the vanishing vertical viscosity limit, that is, the solution to the incompressible anisotropic Navier–Stokes equations tends to the solution to degenerate incompressible Navier–Stokes equations away from the boundary, while near the boundary, it tends to the boundary layer profile, in both the energy space and the L space.  相似文献   

15.
The Dirichlet problem on an interval for quasilinear singularly perturbed parabolic convection-diffusion equation is considered. The higher order derivative of the equation is multiplied by a parameter ε that takes any values from the half-open interval (0, 1]. For this type of linear problems, the order of the ε-uniform convergence (with respect to x and t) for the well-known schemes is not higher than unity (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge ε-uniformly at the rate of O(N ?2ln2 N + N ?2 0), where N + 1 and N 0 + 1 are the numbers of the mesh points with respect to x and t, respectively. On the x axis, piecewise uniform meshes that condense in the boundary layer are used. If the parameter value is small compared to the effective step of the spatial grid, the domain decomposition method is used, which is motivated by “asymptotic constructions.” Monotone approximations of “auxiliary” subproblems describing the main terms of the asymptotic expansion of the solution outside a neighborhood of the boundary layer neighborhood are used. In the neighborhood of the boundary layer (of the width O(ε ln N)) the first derivative with respect to x is approximated by the central difference derivative. These subproblems are successively solved in the subdomains on uniform grids. If the parameter values are not sufficiently small (compared to the effective step of the mesh with respect to x), the classical implicit difference schemes approximating the first derivative with respect to x by the central difference derivative are applied. To improve the accuracy in t, the defect correction technique is used. Notice that the calculation of the solution of the constructed difference scheme (the scheme based on the method of asymptotic constructions) can be considerably simplified for sufficiently small values of the parameter ε.  相似文献   

16.
A uniformly valid aymptotic solution is obtained for a class of perturbed Volterra integral equations, in which a naive expansion breaks down as t → ∞. The procedure used is an adaption of the formal methodology presented in [1] for the construction of a uniform asymptotic solution to Volterra equations which possess a boundary layer near t = 0.  相似文献   

17.
In this work, we build on ideas of Torki (2001 [6]) and show that if a symmetric matrix-valued map t?A(t) has a one-sided asymptotic expansion at t=0+ of order K then so does t?λm(A(t)), where λm is the mth largest eigenvalue. We derive formulas for computing the coefficients A0,A1,…,AK in the asymptotic expansion. As an application of the approach we give a new proof of a classical result due to Kato (1976 [3]) about the one-sided analyticity of the ordered spectrum under analytic perturbations. Finally, as a demonstration of the derived formulas, we compute the first three terms in the asymptotic expansion of λm(A+tE) for any fixed symmetric matrices A and E.  相似文献   

18.
In this paper we are concerned with the compactness properties of remainder terms of the Dyson-Phillips expansion of perturbed semigroups on general Banach spaces. More specifically, we derive conditions which ensure the compactness of the remainder term Rn(t) for some integer n. Our result applies directly to discuss the time asymptotic behaviour (for large times) of the solution of a one-dimensional transport equation with reentry boundary conditions on L1-spaces without regularity conditions on the initial data.  相似文献   

19.
The semilinear reaction-diffusion equation −ε2Δu+b(x,u)=0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the “reduced equation” b(x,u0(x))=0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation −Δz+f(z)=0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.  相似文献   

20.
This article deals with a problem arising in localization of the principal eigenvalue (PE) of the Stokes operator under the Dirichlet condition on the fine-grained random boundary of a domain contained in a cube of size t ? 1. The random microstructure is assumed identically distributed in distinct unit cubic cells and, in essence, independent. In this setting, the asymptotic behavior of the PE as t → ∞ is deterministic: it proves possible to find nonrandom upper and lower bounds on the PE which apply with probability that converges to 1. It was proved earlier that in two dimensions the nonrandom unilateral bounds on the PE can be chosen asymptotically equivalent, which implies the convergence in probability to a nonrandom limit of the appropriately normalized PE. The present article extends this result to higher dimensions.  相似文献   

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