In honor of Prof. Hokee Minn

Transcription

1 In honor of Prof. Hokee Minn It is my greatest honor to be here today to deliver the nd Minn Hokee memorial lecture. I would like to thank the Prof. Ha for the nomination and the selecting committee for giving me this honor. I

2 On incompressible Navier-Stokes equation (NSE): a new approach for analysis and computation Jian-Guo Liu The title of my talk is: on... Collaborators: Weinan E, Hans Johnston, Jie Liu, Bob Pego, Cheng Wang The nd MINNHOKEE Memorial Lecture Seoul National University May 30, 007

3 Flow phenomena modeled by incompressible NSE Lift, viscous drag, boundary layers & separation, vortex shedding,... The NSE is a simplest mathematical eqn which describes a wide range of phenomena in fluid dynamics, such as lift, viscous drag, vprtex shedding... These are all fundamental phenomena in nature which have important applications in real life.

4 Here I show a movie which simulates fluid passing a cylinder with a Reynold s number of 50,000. Solving such a problem accurately is a very hard problem. We solved this using numerical method we developed. Most recently, we solved the case with the Reynold number of 1000,000. method: vorticity-stream function, 4th order compact FD, RK4, fast FFT Poisson solver, FGP (Johnston, Liu & E)

5 3 Navier-Stokes equations for incompressible flow u t + u u + p = ν u + f in Ω momentum u = 0 in Ω incompressibility u = 0 on Ω no slip Claude-Louis Navier George Gabriel Stokes Pressure is like a Lagrange multiplier to enforce incompressibility. Q: How should one compute the pressure? (stably, accurately, fast)

6 4 Navier-Stokes equations for incompressible flow u t + u u + p = ν u + f in Ω momentum u = 0 in Ω incompressibility u = 0 on Ω no slip Claude-Louis Navier George Gabriel Stokes Pressure is like a Lagrange multiplier to enforce incompressibility. Q: How should one compute the pressure? (stably, accurately, fast) A: Solve p = ( f u u) n p = n ν( ) u + n f in Ω on Ω

7 5 Navier-Stokes equations for incompressible flow u t + (l. l. o. t.) = ν u + f in Ω momentum u = 0 in Ω incompressibility u = 0 on Ω no slip Claude-Louis Navier George Gabriel Stokes Pressure is like a Lagrange multiplier to enforce incompressibility. Q: How should one compute the pressure? (stably, accurately, fast) A: Solve p = ( f u u) n p = n ν( ) u + n f in Ω on Ω Upshot: Well-posed IBVP like a perturbed diffusion equation!

8 6 Traditional implicit Stokes/explicit advection discretization (Temam s book, many mixed FEM): u n+1 u n t + p n+1 ν u n+1 = f n u n u n u n+1 = 0 u n+1 = 0 on Ω New: proof of unconditional stability and convergence for u n+1 u n t ν u n+1 = f n u n u n p n u n+1 = 0 on Ω p n = ( f n u n u n ) n p n = n ν( ) u n + n f n on Ω

9 7 Outline Derive an unconstrained formulation of NSE in terms of the Laplace-Leray commutator P P = p Stokes Why viscosity dominates pressure a new estimate ( Simple theory of existence & uniqueness for strong solutions) New interpretation of existing projection methods and gauge methods (Kim & Moin, Orszag et al, Timmermans, Henshaw- Petersson, Johnston-Liu, E-Liu, Guermond-Minev-Shen) Simple C 0 finite element solvers apparently needing no velocity/pressure compatibility restrictions for stability C 1 finite element solvers with full theory of unconditional stability, convergence, error estimates, in C 3 domains

10 Projection Method, Chorin 68, Temam 69 u u n t + (u n )u n = ν u, u Γ = 0 u n+1 u + p n+1 = 0, u n+1 = 0, u n+1 n Γ = 0 t Efficient, Stable, Numerical boundary layer of order ν t in u, p (Orszag etc 86, E-Liu, 95) nd order projections, Kim, Moin 85, van Kan 86, Orszag etc 86, Bell, Colella, Glaz 89, reduce numerical boundary layer Recent consistent projections: Timmermans etc 96, Gauge method 97, Brown, Cortez, Minion 01, Henshaw, Petersson 01, Guermond, Shen 03, Johnston, Liu 04, Liu, Liu, Pego

11 Impulse Density Formulation Oseledets (1989) rewritten NSE as { t M + (u )M + M ( u) = Re 1 M, u = PM, where u is the velocity field, P is the Helmholtz-Leray projection operator to the space of divergence-free vector fields. Buttke/Chorin 1993: Velicity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow; Magnetization Roberts 197 A Hamiltonain theory for weakly interacting vortices Summers & Chorin 1995, Cortez 1995 etc

12 Impulse density formulation E-Liu, JCP Fourier Collocation Method Finite Difference Method

13 4 Impulse Density Formulation, E-Liu, JCP 1997 Linearized system: { t ˆM + i(ūk1 + vk ) ˆM + ik 1 ( Mû + N ˆv) = 0 t ˆN + i(ūk1 + vk ) ˆN + ik ( Mû + N ˆv) = 0 ( û ˆv ) = ( I k k ) ( ˆM k ˆN Take N = 0, v = 0 and k 1 = 0, we can show that û = ˆM. Hence ( ) ( ) ( ) ˆM 0 0 ˆM t + ik M = 0 ˆN 1 0 ˆN ˆM(k, t) = ˆM(k, 0), ˆN(k, t) = ˆN(k, 0) + ik t M ˆM(k, 0) marginally ill-posed. or, weakly well posed )

14 5 Impulse Density Formulation, E-Liu, JCP 1997 A Simplified Velocity-Impulse Density Formulation/gauge method: { t M + (u )u + q = 1 Re M u = PM If we write M = u + φ, then pressure is given by p = q + t φ 1 Re φ The variable q is arbitrary and can be chosen to suit our numerical purposes. In computation we set q = 0. And we renamed this case as gauge method.

15 Impulse density formulation E-Liu, JCP Re=150, t=/pi, finite difference method Re=10000, t=1000, 18X18 Staggered grid

16 3 Gauge Method, E & Liu 97, 03 Set a = u φ, The momentum equation becomes ( ) a t + (u )u + t φ ν φ + p = ν a. Incompressibility constraint becomes φ = a, impose BC φ n leads to gauge choice = 0, a n = 0, a τ = φ τ, on Ω, t φ ν φ = p,

17 5 Gauge Method, E & Liu 97, 03 Backward Euler, u n+1 = a n+1 + φ n+1 a n+1 a n t + (u n )u n = a n+1 a n+1 n Γ = 0, a n+1 τ Γ = φn τ, φ n+1 = a n+1, φ n+1 n Γ = 0 Theorem (Wang & Liu 00). Unconditional Stability for Stokes Flow u n+1 u n t + û n + ( φ n+1 φ n ) 0.

18 6 Geometric gauge 6 6 EL gauge J. Fluid Mech. (1999), vol. 391, pp Printed in the United Kingdom c 1999 Cambridge University Press JFM, 1999 Impulse formulation of the Euler equations: general properties and numerical methods t= By G I O V A N N I R U S S O 1 AND P E T E R S M E R E K A 0 G. Russo and P. Smereka (a) t= Geometric gauge (b) t= EL gauge vorticity pressure pressure 4 6 Figure 4. PhiD results. The left-hand column shows contours of ω at times t = 15, 1, and 7 from top to bottom, the contours are separated by 0.1 The middle column shows contours of p at times t = 15, 1, and 7 from top to bottom, in the geometric gauge; the contours are separated by 0.5. The right-hand column shows the contours of p in the EL gauge; the contours are separated by Figure 5. The L, L, and L 1 norms are shown for the results in figure 4, corresponding to the geometric gauge (a) and EL gauge (b). In each plot, the upper curve is the L norm, the middle one is the L norm, and the lower curve is the L 1 norm. The norms have been normalized to be equal to 1 at t = 0.

19 4 Gauge Method, E & Liu 97, 03 Gauge formulation of NSE with u = a + φ a t + u u = ν a, a n Γ = 0, a τ Γ = τ φ, φ φ = a, n Γ gauge choice becomes: t φ ν φ = p, = 0 a t + P(u )u = ν a, a n Γ = 0, a τ Γ = τ φ, φ φ = a, = 0 n Γ gauge choice becomes: t φ ν φ = p s. Equivalent with the Pressure Poisson formulation.

20 Pressure Poisson formulation, Johnston & Liu 04 Navier-Stokes equation (NSE): u t + (u )u + p = ν u, u Γ = 0, Incompressibility u = 0 is replaced by p = (u u), n p = νn ( u) Γ, BC first used by Orszag (1986), motivated by identity u = u + ( u). t ( u) = ν ( u), ( u) t=0 = 0, n ( u) Γ = 0.

21 Pressure Poisson formulation, Johnston & Liu 04 Navier-Stokes equation (NSE): u t + (u )u + p = ν u, u Γ = 0, Incompressibility u = 0 is replaced by p = (u u), n p = νn ( u) Γ, BC first used by Orszag (1986), motivated by identity u = u + ( u). t ( u) = ν ( u), ( u) t=0 = 0, n ( u) Γ = 0. Recast as, u t + P( u u) = νp u + ν ( u), u Γ = 0

22 5 Pressure Poisson formulation, Johnston & Liu 04 Galerkin formulation (good for C 0 finite element method): u t, v + u u, v + p, v = ν u, v, v H 1 0 p, ψ = u u, ψ + ν u, n ψ Γ, ψ H 1 Finite element spaces: X h H0(Ω, 1 R N ), v h X h and q h Y h, require Y h H 1 (Ω)/R. For all 1 t ( u n+1 h, v h u n h, v h ) + ν u n+1 h, v h = p n h, v h + f n u n h u n h, v h. p n h, q h = f n u n h u n h, q h + ν u n h, n q h Γ,

23 !0.5 Driven cavity flow!4! !3 1 3!5! !1!0.5!0.5! 0!1!1!!4!!3!5!3!3!5!4!3! !!!0.5!0.5!1! ! !!1!! !!1! !1! Re= P4 elements. CN/AB time stepping

24 Backward facing step flow 0.5 0! Re= P4 elements. CN/AB time stepping 0.5 X 1 X 0 S! X 3 Re= P4 elements. CN/AB time stepping

25 Hermann von Helmholtz W.V.D. Hodge Hermann von Helmholtz Hermann von Helmholtz Helmholtz projection onto divergence-free vector fields 5 L (Ω, R N ) = PL (Ω, R N ) H 1 (Ω) Given v L (Ω, R N ), there exists q H 1 (Ω) so that v = P v q satisfies P v, φ = v + q, φ = 0 for all φ H 1 (Ω).

26 5 Helmholtz projection onto divergence-free vector fields L (Ω, R N ) = PL (Ω, R N ) H 1 (Ω) Given v L (Ω, R N ), there exists q H 1 (Ω) so that v = P v q satisfies P v, φ = v + q, φ = 0 for all φ H 1 (Ω). Then (P v) = 0 in Ω, n P v = 0 on Γ. Note P v H(div; Ω) = { f L (Ω, R N ) : f L }, so n P v H 1/ (Γ) by a standard trace theorem.

27 Jean Leray Olga A. Ladyzhenskaya Traditional unconstrained formulation of NSE mgrestosio Kato ( ) Formally t ( u) = 0 u t + P( u u f) = νp u, u Γ = 0 Perform analysis and computation in spaces of divergence-free fields (unconstrained Stokes operator P u is incompletely dissipative). Inf-Sup/LBB condition(ladyzhenskaya-babuška-brezzi)

28 10 Unconstrained formulation of NSE u t + P( u u f) = νp u+ν u No difference! as long as u = 0. Formally, w = u satisfies a heat equation with Neumann BCs: t w = ν w in Ω, n w = 0 on Ω. w = n u = 0 = exponential decay Ω Ω

29 10 Unconstrained formulation of NSE u t + P( u u f) = νp u+ν u No difference! as long as u = 0. Formally, w = u satisfies a heat equation with Neumann BCs: t w = ν w in Ω, n w = 0 on Ω. w = n u = 0 = exponential decay Ω Ω Equivalent to a reduced formulation of Grubb & Solonnikov (1991) studied as a nondegenerate parabolic pseudodifferential IBVP

30 11 Equivalent formulation with commutator Lemma For all u L (Ω, R N ), u = u P u

31 11 Equivalent formulation with commutator Lemma For all u L (Ω, R N ), u = u P u U-NSE: u t + P( u u f) = ν u ν( P P ) u

32 11 Equivalent formulation with commutator Lemma For all u L (Ω, R N ), u = u P u U-NSE: u t + P( u u f) = ν u ν( P P ) u P u = ( ) u ( P P ) u = (I P)( ) u =: p S Stokes pressure BVP!! For u H H0(Ω, 1 R N ), we get p S H(div; Ω), with p S = 0 in Ω, n p S = n ( ) u in H 1/ ( Ω). Stokes pressure arises from tangential vorticity at the boundary: 3D weak form: p S φ = ( u) ( n φ) φ H 1 (Ω) Ω Ω

33 1 Unconstrained form pressure decomposition U-NSE: u t + P( u u f) = ν u ν( P P ) u u = 0 on Ω Pressure-Poisson form: u t + u u + p = ν u + f u = 0 on Ω p = ( f u u) in Ω n p = n ν( ) u + n f on Ω The pressure p consists of two parts: p E is nonlinear but lower order p S is nd order but a commutator p = p E + ν p S

34 13 Viscosity dominates Stokes pressure gradient Theorem Let Ω R N (N ) be a bounded domain with C 3 boundary. Then, for every ε > 0 there exists C 0 such that for all u H H 1 0(Ω, R N ), Ω ( P P ) u ( ) 1 + ε Ω u + C Ω u,

35 13 Viscosity dominates Stokes pressure gradient Theorem Let Ω R N (N ) be a bounded domain with C 3 boundary. Then, for every ε > 0 there exists C 0 such that for all u H H 1 0(Ω, R N ), Ω ( P P ) u ( ) 1 + ε Ω Ingredients of the (elementary!) proof: u + C Ω u, Decompose u H H 1 0 into parts parallel and normal to Ω: Let Φ(x) = dist(x, Ω), n(x) = Φ(x), ξ a cutoff = 1 near Ω. u = u + u, u = ξ(i n n t ) u.

36 14 1. Boundary identities (i) u = 0 and (ii) u = 0 on Ω Hence p S = ( P P ) u = (I P)( )( u + 0) u = u + u, u + u (1 ε) u C u goal: ( ε) p S C u

37 15. Orthogonal decomposition of u u = p S + p S u p S = (I P)( ) u = u u + P v p S u H0(Ω) 1 + PL, since u = 0 on Ω p S PL, and p S = 0 p S, p S u = p S, u = p S, u = 0 It remains only to show p S u (1 ε) p S C u Break in parallel, perpendicular parts, and use somewhat novel...

38 16 3. Neumann-to-Dirichlet estimates on tubes Ω s = {x Ω Φ(x) < s} Lemma For s 0 > 0 small C 0 such that whenever p = 0 in Ω s0 and 0 < s < s 0 then p p C 0 s p Φ<s Φ<s 0 Morally: p Φ<s p Φ<s + junk In a half space: p = p exactly ( nˆp = k ˆp) Now: p S u s n ( p S u ) s (1 ε) p s C u ( 1 ε) p S s C u

39 17 Stability of time-difference scheme with pressure explicit (related: Ti96,Pe01,GS03,JL04) u n+1 u n t ν u n+1 = f n u n u n p n, u n+1 = 0 on Ω, p n ν u n + ν u n + u n u n f n, φ = 0 φ H 1 (Ω). p n ν p n S + u n u n f n, φ = 0 φ H 1 (Ω). Taking φ = p n gives the pressure estimate p n ν p n S + f n u n u n

40 Unconditional stability theorem for N =, 3 Liu, Liu & Pego, CPAM Theorem Take f L (0, T ; L (Ω, R N )), u 0 H H 1 0(Ω, R N ). Then positive constants T and C depending only upon Ω, ν and M 0 := u 0 + ν t u 0 + so that whenever n t T we have T 0 f, sup u k + 0 k n ( n 1 k=0 n u k t C, k=0 u k+1 u k t + u k u k ) t C.

41 Unconditional stability theorem for N =, 3 Liu, Liu & Pego, CPAM Theorem Take f L (0, T ; L (Ω, R N )), u 0 H H 1 0(Ω, R N ). Then positive constants T and C depending only upon Ω, ν and M 0 := u 0 + ν t u 0 + so that whenever n t T we have T 0 f, sup u k + 0 k n ( n 1 k=0 n u k t C, k=0 u k+1 u k t + u k u k ) t C. u in H 1 0 strong solution u L (0, T ; H ) H 1 (0, T ; L )

42 3 C 0 finite element schemes Space-continuous projection method after Orszag et al (1986): u = P u n = u n q n p n ν u n + ν u n + u u f n, φ = 0 u n+1 u t φ H 1 (Ω). ν u n+1 = f n u u p n, u n+1 Ω = 0. C 0 finite element method: v h X h H 1 0(Ω) N, φ h Y h H 1 (Ω), u n+1 h u h, v h t u n h q h, φ h = 0, u h := u n h q h p n h, φ h = f n u h u h, φ h + ν u n h, n φ h Ω, + ν u n+1 h, v h = p n h, v h + f n u h u h, v h.

43 4 Analogous C 1 finite element schemes Finite element spaces: X h H H 1 0(Ω, R N ), Y h H 1 (Ω)/R. For all v h X h and φ h Y h, require u h, v h = P u n, v h = u n h, v h u n h, v h p n h, φ h = f n u h u h, φ h + ν u n h, n φ h Ω u n+1 h u h, v h t + ν u n+1 h, v h = p n h + u h u h f n, v h

44 4 Analogous C 1 finite element schemes Finite element spaces: X h H H 1 0(Ω, R N ), Y h H 1 (Ω)/R. For all v h X h and φ h Y h, require u h, v h = P u n, v h = u n h, v h u n h, v h p n h, φ h = f n u h u h, φ h + ν u n h, n φ h Ω, u n+1 h u h, v h t Pressure estimate: Take φ h = p n h, then + ν u n+1 h, v h = p n h + u h u h f n, v h p n h f n u h u h + ν p S ( u n h) Unconditional stability and convergence follow as before!

45 5 No compatibility required between velocity and pressure spaces In the proof, we do not require the spaces X 0,h and Y h satisfy the inf-sup stability condition for weak solutions: ( vh )q h inf sup q h Y h v h X 0,h v h q h β > 0 The inf-sup condition is necessary and sufficient for stability for weak solutions to the mixed formulation of steady Stokes problems It seriously constrains kinds of discretizations that are weakly stable Our proofs are for strong solutions in C 3 domains and we compute pressure stably from a well-posed BVP. In standard test problems, the simple C 0 scheme works well. Nevertheless, the situation is not fully understood.

46 8 Stability check: C 0 FE scheme with smooth solution 10 0 u!u L p!p L u!u L p!p L t v.s. t 10 1 (u!u) L (p!p) L u x +v y L ω!ω L t v.s. t !1 10!1 10! t 10! t 1: BE/FE, T=1000, fixed mesh, t 8

47 9 Spatial accuracy check for C 0 FE scheme !1 10!1 10! 10!3 u!u L p!p L u!u L 10! (u!u) L (p!p) L u +v x y L p!p L ω!ω L 10!4 h v.s. h 10! 10!1 h 10!3 h v.s. h 10! 10!1 h : P1/P1 finite elements

48 30 Spatial accuracy check for C 0 FE scheme ! 10! 10!4 10!6 u!u L 10!4 (u!u) L (p!p) L p!p L u x +v y L 10!8 u!u L p!p L h v.s. h 5 10!6 ω!ω L h v.s. h 4 h v.s. h 5 10!10 10!1 h 10!8 10!1 h 3: P4/P4 finite elements

49 31 Stability check: C 1 FE scheme with smooth solution !1 u!u L p!p L u!u L p!p L t v.s. t (u!u) L (p!p) L (u!u) L (p!p) L u x +v y L t v.s. t 10! 10!1 10!3 10! 10! t 10! 10! 10! t 4: BE/FE, T=1000, fixed mesh, t 8

50 3 Spatial accuracy check: C 1 FE scheme with smooth solution 10! 10!1 10!3 10! 10!4 10!3 10!5 10!4 10!6 10!7 10!8 u!u L p!p L u!u L p!p L h v.s. h 4 10!5 10!6 10!7 (u!u) L (p!p) L (u!u) L (p!p) L u x +v y L h v.s. h 3 h v.s. h !9 10! 10!1 h 10!8 10! 10!1 h

51 33 Numerics: driven cavity and backward facing step (u,v)=(1,0) (u,v) given (u,v) given

52 C 0 FEM. Driven Cavity. Re=1000. P4/P elements. 34

53 35 ʯ ۿ ޱ ԏݏ ȶԏ ੴԏ ӹۿſ ϻ ϻ ÿ ۿ ۿ ۿ

54 36 C 0 FEM. Backward facing step. Re=600. P4/P elements 0.5 X 1 X 0 S! X 3

55 C 1 FEM. Driven Cavity. Re= C 1 elements 37

56 38 C 1 FEM. Backward facing step. Re= ! (with a few recycled C 1 elements near the step)

57 39 C 1 FEM. Backward facing step. Re= C 1 elements X 1 X S! X 3 (with a few recycled C 1 elements near the step)

58 33 Summary: NSE is a perturbed heat equation! u t + P( u u f) = νp u

59 33 Summary: NSE is a perturbed heat equation! u t + P( u u f) = νp u + ν ( u)

60 33 Summary: NSE is a perturbed heat equation! u t + P( u u f) = νp u + ν ( u) = νp u + ν (I P ) u

61 33 Summary: NSE is a perturbed heat equation! u t + P( u u f) = νp u + ν ( u) = νp u + ν (I P ) u = ν u ν[, P] u

62 33 Summary: NSE is a perturbed heat equation! u t + P( u u f) = νp u + ν ( u) = νp u + ν (I P ) u = ν u ν[, P] u = ν u ν p S u satisfies a heat equation with Neumann BCs The Stokes pressure term is strictly controlled by viscosity Stability, existence, uniqueness theory is greatly simplified Promise of enhanced flexibility in design of numerical schemes

63 34 Summary: NSE is a perturbed heat equation! u t + P( u u f) = νp u + ν ( u) = νp u + ν (I P ) u) = ν u ν[, P] u = ν u ν p S u satisfies a heat equation with Neumann BCs The Stokes pressure term is strictly controlled by viscosity Stability, existence, uniqueness theory is greatly simplified Promise of enhanced flexibility in design of numerical schemes

64 Thank You!

65 3 Non-homogeneous side conditions t u + u u + p = ν u + f u = h u = g u = u in (t > 0, x Ω), (t 0, x Ω), (t 0, x Γ), (t = 0, x Ω).

66 3 Non-homogeneous side conditions t u + u u + p = ν u + f u = h u = g u = u in (t > 0, x Ω), (t 0, x Ω), (t 0, x Γ), (t = 0, x Ω). Unconstrained formulation involves an inhomogeneous pressure p gh : t u + P( u u f ν u) + p gh = ν u p gh, φ = n t g, φ Γ + t h, φ + ν h, φ φ H 1 (Ω). u h satisfies a heat equation with Neumann BCs as before.

67 36 Existence and uniqueness theorem Assume u in H uin := H 1 (Ω, R N ), f H f := L (0, T ; L (Ω, R N )), g H g := H 3/4 (0, T ; L (Γ, R N )) L (0, T ; H 3/ (Γ, R N )) { g t ( n g) L (0, T ; H 1/ (Γ))}, h H h := L (0, T ; H 1 (Ω)) H 1 (0, T ; (H 1 ) (Ω)). and the compatibility conditions g = u in (t = 0, x Γ), n t g, 1 Γ = t h, 1 Ω. Then T > 0 so that a unique strong solution exists, with u L ([0, T ], H ) H 1 ([0, T ], L ) C([0, T ], H 1 ).

68 6 Kim Moin s Method vs Gauge Method Kim-Moin: u u n t u n = 0, + (u u) n+1 = ν u +u n, u τ = t pn τ, p n+1 p n+1 = u, n Γ = 0 u n+1 = u + t p n+1 Gauge Method: u n+1 = a n+1 + φ n+1 a n+1 a n + (u u) n+1/ = ν an+1 + a n t a n+1 n Γ = 0, a n+1 τ Γ = (φn p n 1 ) τ φ n+1 = a n+1, φ n+1 n Γ = 0,

69 3 Non-homogeneous side conditions t u + u u + p = ν u + f u = h u = g u = u in (t > 0, x Ω), (t 0, x Ω), (t 0, x Γ), (t = 0, x Ω).

70 3 Non-homogeneous side conditions t u + u u + p = ν u + f u = h u = g u = u in (t > 0, x Ω), (t 0, x Ω), (t 0, x Γ), (t = 0, x Ω). Unconstrained formulation involves an inhomogeneous pressure p gh : t u + P( u u f ν u) + p gh = ν u p gh, φ = n t g, φ Γ + t h, φ + ν h, φ φ H 1 (Ω). u h satisfies a heat equation with Neumann BCs as before.