Geodesic
Second-order accurate method with splitting of boundary conditions for solving the stationary axially symmetric Navier–Stokes problem in spherical gaps
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 12, pp. 2232-2250 
B. V. Pal'tsev; I. I. Chechel'. Second-order accurate method with splitting of boundary conditions for solving the stationary axially symmetric Navier–Stokes problem in spherical gaps. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 12, pp. 2232-2250. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_12_a11/
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     author = {B. V. Pal'tsev and I. I. Chechel'},
     title = {Second-order accurate method with splitting of boundary conditions for solving the stationary axially symmetric {Navier{\textendash}Stokes} problem in spherical gaps},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {2232--2250},
     year = {2005},
     volume = {45},
     number = {12},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_12_a11/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A numerical method for solving the axisymmetric Dirichlet boundary value problem for the stationary incompressible Navier–Stokes system in spherical gaps is constructed by using the method of successive approximations and the second-order accurate finite-element implementation developed by the authors in the axisymmetric case for a fast converging iterative method with splitting of boundary conditions for the Stokes system in spherical gaps. Numerical experiments on exact solutions showed that the method is second-order accurate up to the axis of symmetry and the gap boundaries. A numerical study (on fine grids) is conducted to determine the convergence domain of the method, and its results are compared with available experimental data for spherical Couette flows driven by the rotation of a single sphere. The method proves to be highly efficient as applied to the computation of basic viscous incompressible flows in narrow spherical gaps.

[1] Paltsev B. V., Chechel I. I., “Konechno-elementnye realizatsii iteratsionnykh metodov s rasschepleniem granichnykh uslovii dlya sistem Stoksa i tipa Stoksa v sharovom sloe, obespechivayuschie 2-i poryadok tochnosti vplot do osi simmetrii”, Zh. vychisl. matem. i matem. fiz., 45:5 (2005), 846–889 | MR

[2] Paltsev B. V., Chechel I. I., “O konechno-elementnykh tipa lineinykh, vtorogo poryadka tochnosti vplot do polyusov, approksimatsiyakh operatorov Laplasa-Beltrami, gradienta i divergentsii na sfere v $\mathbb{R}^3$ v osesimmetrichnom sluchae”, Dokl. RAN, 395:3 (2004), 308–315 | MR

[3] Paltsev B. V., “Ob usloviyakh skhodimosti iteratsionnykh metodov s polnym rasschepleniem granichnykh uslovii dlya sistemy Stoksa v share i sharovom sloe”, Zh. vychisl. matem. i matem. fiz., 35:6 (1995), 935–963 | MR

[4] Paltsev B. V., “Optimizatsiya znachenii relaksatsionnykh parametrov odnoshagovogo varianta iteratsionnogo metoda s rasschepleniem granichnykh uslovii dlya sistemy Stoksa v sharovom sloe”, Vestn. RUDN, 8:2 (2001), 74–90

[5] Meller N. A., Paltsev B. V., Khlyupina E. G., “O konechno-elementnykh realizatsiyakh iteratsionnykh metodov s rasschepleniem granichnykh uslovii dlya sistem Stoksa i tipa Stoksa v sharovom sloe. Osesimmetrichnyi sluchai”, Zh. vychisl. matem. i matem. fiz., 39:1 (1999), 98–123 | MR | Zbl

[6] Meller N. A., Paltsev B. V., Khlyupina E. G., “O chislennom metode s rasschepleniem granichnykh uslovii dlya statsionarnoi sistemy Nave-Stoksa v sharovom sloe v sluchae osevoi simmetrii”, Differential Equations and Related Topics, Tezisy Mezhdunar. konf., posvyaschennoi 100-letiyu so dnya rozhdeniya akad. I. G. Petrovskogo, MGU, M., 2001, 311–313

[7] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Nauka, M., 1970 | MR

[8] Temam R., Uravneniya Nave-Stoksa. Teoriya i chislennyi analiz, Mir, M., 1981 | MR | Zbl

[9] Girault V., Raviart P., Finite element methods for Navier-Stokes equations, Springer, Berlin, 1986 | MR | Zbl

[10] Kobelkov G. M., “O chislennykh metodakh resheniya uravnenii Nave-Stoksa v peremennykh skorost-davlenie”, Vychisl. protsessy i sistemy, 8, Nauka, M., 1991, 204–236 | MR

[11] Turek S., Efficient solvers for incompressible flow problems: an algorithmic approach in view of computation problems, Springer, Berlin, 1998 | MR

[12] Fedorenko R. P., “Iteratsionnye metody resheniya raznostnykh ellipticheskikh uravnenii”, Uspekhi matem. nauk, 28:2 (1973), 121–182 | MR | Zbl

[13] Mc Cormick S. F., Ruge J. W., “Multigrid methods for variational problems”, SIAM J. Numer. Analys., 19:5 (1982), 924–929 | DOI | MR

[14] Belyaev Yu. N., Yavorskaya I. M., “Techeniya vyazkoi zhidkosti vo vraschayuschikhsya sfericheskikh sloyakh i ikh ustoichivost”, Itogi nauki i tekhn. Ser. MZhG, 15, VINITI, M., 1980, 3–80

[15] Junk M., Egbers Ch., “Isothermal spherical Coutte flow”, Phys. Rotat. Fluids, Selected Topics of the 11-th Internat. Couette-Taylor Workshop (Bremen, Germany), Lect. Nothes in Phys., Springer, Berlin–Heidelberg, 1999, 215–233

[16] Dzhozef D., Ustoichivost dvizhenii zhidkosti, Mir, M., 1981

[17] Finn R., “On the exterior stationary problem for the Navier-Stokes equation and associated perturbation problems”, Arch. Ration. Mech. and Analys., 19:5 (1965), 365–406 | MR