A free boundary stationary magnetohydrodynamic problem in connection with the electromagnetic casting process

Tomasz Roliński

Annales Polonici Mathematici (1995)

  • Volume: 61, Issue: 3, page 195-223
  • ISSN: 0066-2216

Abstract

top
We investigate the behaviour of the meniscus of a drop of liquid aluminium in the neighbourhood of a state of equilibrium under the influence of weak electromagnetic forces. The mathematical model comprises both Maxwell and Navier-Stokes equations in 2D. The meniscus is governed by the Young-Laplace equation, the data being the jump of the normal stress. To show the existence and uniqueness of the solution we use the classical implicit function theorem. Moreover, the differentiability of the operator solving this problem is established.

How to cite

top

Tomasz Roliński. "A free boundary stationary magnetohydrodynamic problem in connection with the electromagnetic casting process." Annales Polonici Mathematici 61.3 (1995): 195-223. <http://eudml.org/doc/262269>.

@article{TomaszRoliński1995,
abstract = {We investigate the behaviour of the meniscus of a drop of liquid aluminium in the neighbourhood of a state of equilibrium under the influence of weak electromagnetic forces. The mathematical model comprises both Maxwell and Navier-Stokes equations in 2D. The meniscus is governed by the Young-Laplace equation, the data being the jump of the normal stress. To show the existence and uniqueness of the solution we use the classical implicit function theorem. Moreover, the differentiability of the operator solving this problem is established.},
author = {Tomasz Roliński},
journal = {Annales Polonici Mathematici},
keywords = {free boundary; local existence and uniqueness; implicit function theorem; steady plane magnetohydrodynamics; electromagnetic casting; Maxwell equations; meniscus; drop of liquid aluminium; Navier-Stokes equations; Young-Laplace equation; existence; uniqueness; differentiability},
language = {eng},
number = {3},
pages = {195-223},
title = {A free boundary stationary magnetohydrodynamic problem in connection with the electromagnetic casting process},
url = {http://eudml.org/doc/262269},
volume = {61},
year = {1995},
}

TY - JOUR
AU - Tomasz Roliński
TI - A free boundary stationary magnetohydrodynamic problem in connection with the electromagnetic casting process
JO - Annales Polonici Mathematici
PY - 1995
VL - 61
IS - 3
SP - 195
EP - 223
AB - We investigate the behaviour of the meniscus of a drop of liquid aluminium in the neighbourhood of a state of equilibrium under the influence of weak electromagnetic forces. The mathematical model comprises both Maxwell and Navier-Stokes equations in 2D. The meniscus is governed by the Young-Laplace equation, the data being the jump of the normal stress. To show the existence and uniqueness of the solution we use the classical implicit function theorem. Moreover, the differentiability of the operator solving this problem is established.
LA - eng
KW - free boundary; local existence and uniqueness; implicit function theorem; steady plane magnetohydrodynamics; electromagnetic casting; Maxwell equations; meniscus; drop of liquid aluminium; Navier-Stokes equations; Young-Laplace equation; existence; uniqueness; differentiability
UR - http://eudml.org/doc/262269
ER -

References

top
  1. [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. 17 (1964), 35-92. Zbl0123.28706
  2. [2] J. Bemelmans, Gleichgewichtsfiguren zäher Flüssigkeiten mit Oberflächenspannung, Analysis 1 (1981), 241-282. Zbl0561.76042
  3. [3] O. Besson, J. Bourgeois, P. A. Chevalier, J. Rappaz and R. Touzani, Numerical modelling of electromagnetic casting processes, J. Comput. Phys. 92 (2) (1991), 482-507. Zbl0721.65076
  4. [4] H. Cartan, Cours de calcul différentiel, Hermann, Paris, 1977. Zbl0408.58001
  5. [5] B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals, Springer, Berlin, 1982. 
  6. [6] G. Duvaut et J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972. Zbl0298.73001
  7. [7] V. Girault and P. A. Raviart, Finite Elements Methods for Navier-Stokes Equations. Theory and Algorithms, Springer, Berlin, 1986. Zbl0585.65077
  8. [8] J. Giroire and J. C. Nedelec, Numerical solution of the exterior Neumann problem using a double layer potential, Math. Comput. 32 (1987), 973-990. Zbl0405.65060
  9. [9] M. N. Le Roux, Méthode d'éléments finis pour la résolution numérique de problèmes extérieurs en dimension 2, RAIRO Anal. Numér. 11 (1977), 27-60. 
  10. [10] M. N. Le Roux, Résolution numérique du problème du potentiel dans le plan par une méthode variationnelle d'éléments finis, Thèse, L'Université de Rennes, U.E.R., Mathématiques et Informatique, 1974. 
  11. [11] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967. 
  12. [12] J. C. Nedelec, Approximation des équations intégrales en mécanique et en physique, Rapport du Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 1977. 
  13. [13] J. Rappaz and R. Touzani, On a two-dimensional magnetohydrodynamic problem I. Modelling and analysis, RAIRO Modél. Math. Anal. Numér. 26 (2) (1991), 347-364. Zbl0738.76086
  14. [14] V. A. Solonnikov, Free boundary problems and problems in noncompact domains for the Navier-Stokes equations, in: Proc. Internat. Congr. Math., Berkeley, California, 1986, Vol. 2, 1987, 1113-1122. 
  15. [15] V. A. Solonnikov and V. E. Shchadilov, On a boundary value problem for a stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. 125 (1973) (in Russian); English transl.: Proc. Steklov Inst. Math. 125 (1973), 186-199. 
  16. [16] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Elsevier, Amsterdam, 1985. Zbl0383.35057

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.