Tamtam Proceedings - lamsin

Spectral approximations for the N-S problem 2332. Continuous problemThe question is now what are the necessary boundary conditions to insure the existenceand unicity of the current function ψ defined by:{u r = ∂ z ψu z = − 1 r ∂ , (8)r(rψ)and then deduce its regularity from the regularity of the velocity u, the solution of theStokes problem, see [4 ] and [5] and references therein. In [5], we showed how we cangeneralize this result. We considerH 1 = (div r , Ω) = { v ∈ (L 2 1(Ω)) 2 /div r v ∈ L 2 1(Ω) }with the norm:()‖v‖ = H1(div r,Ω)‖v‖ 2 (L + ‖div rv‖ 2 1221 (Ω))2 L 2 1 (Ω)where 2 1(Ω) = { v square integrable such that ∫ Ω v2 (r, z)rdrdz ≺ ∞ } . We also definethe spaces: H 1 1 (Ω) = { v ∈ L 2 1 (Ω) /∇v = (∂ r v, ∂ z v) ∈ (L 2 1 (Ω)) 2} ,H 1 1,0 (Ω) = { v ∈ H 1 1 (Ω) v = 0 on ∂Ω/Γ 0}. Here Γ0 is the interior of the boundary Γ.Note that the boundary operator is defined and continuous from H1 1 (Ω) in H 1 21 (Γ). Notealso that H − 1 21 (Γ) coincides with H 1 21 (Γ) in the points away from Γ 0 ∩ Γ. In the sameway, we define the normal boundary operator which is continuous from H1 1 (div r , Ω) inH − 1 21 (Γ) . The dual of H 1 21 (Γ) is given by :∀v ∈ H 1 (div r , Ω) and q ∈ H 1 1 (Ω) ,(v.n, q) Γ= ∫ Ω (div rv.q)(r, z)rdrdz + ∫ (v.∇q)(r, z)rdrdzΩ3. Current-Whirpool function formulationIn this section, we introduce a new unknown called Whirpool functionω = rot(u) which gives: −∆u = rot r ω, ∆ is applied to (u r , u z ) in the two firstequations of (1). If we substitute this result in (1), we obtain{ −υ∂z ω + ∂ r p = f r−υ 1 r ∂ (9)r(rω) + ∂ z p = f z .The interesting point of this formulation is to decouple the pressure from the velocity toobtain a separate problem for the pressure (For details see [4]). We have−∆ r ω = 1 υ (∂ zf r − ∂ r f z ), (10)TAMTAM –Tunis– 2005