Ghost effect by curvature Providence, November 2011 M&MOCS - - PowerPoint PPT Presentation

Ghost effect by curvature

Providence, November 2011

M&MOCS – MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Universit` a dell’Aquila - Italy

Ghost effect by curvature – p. 1/39

Joint project with :

• L. Arkeryd
• R. Marra
• A. Nouri

Ghost effect by curvature – p. 2/39

Ghost effect

The macroscopic equations for the hydrodynamic fields are derived from the Boltzmann equation in the limit when the Knudsen number K goes to 0. von Karman relation between K, R (Reynolds number) and M (Mach number): M ∼ KR. To keep the Reynolds number R finite, one has so take M = O(K). This and the assumption that temperature T and density ρ are constant at the lowest order in K, as is well known, give the incompressible Navier-Stokes & Fourier equations as limit of the Boltzmann equation. What about ρ and T non constant at the lowest order in K?

Ghost effect by curvature – p. 3/39

Ghost effect

The formal answer can be obtained by expanding the solution to the Boltzmann equation in K. The result is somewhat unexpected: the macroscopic equations differ from the incompressible Navier-Stokes equations by the presence of a thermal stress tensor and divu = 0. Moreover, the equation for the temperature differs from the standard heat equation. The responsible for this modification is the convective motion described by the velocity field u, which is O(K), hence invisible in the limit K → 0, but affects the temperature which O(1) in K.

Ghost effect by curvature – p. 4/39

Ghost effect

For this reason Y. Sone called this phenomenon Ghost Effect. REMARK: If one starts from the standard compressible Navier-Stokes-Fourier equations, instead of the Boltzmann equation, and takes the limit M → 0 without assuming ρ and T constant, there is no ghost effect. Ghost effect is a purely kinetic feature. The term ghost effect is used by Sone to refer more generally to situations where the macroscopic limiting equations contain finite modifications due to the presence of quantities which are infinitesimal in the limit. The ghost effect by curvature is an example of such situations.

Ghost effect by curvature – p. 5/39

Ghost effect by curvature

• .

• p

X Y Z P Q r L L+D θ z

Boltzmann equation: ε ∼ K. (vr, vθ, vz) components of v in the basis (er, eθ, ez). vr ∂F ∂r +vθ r ∂F ∂θ +vz ∂F ∂z +vθ r

• vθ

∂F ∂vr − vr ∂F ∂vθ

• = 1

εQ(F, F), Q(f, g) = 1 2

• R3 dv∗
• S2

dn|n·(v−v∗)|×

• f′

∗g′+f′g′ ∗−f∗g−g∗f

• ,

v′ = v−nn·(v−v∗), v′

∗ = v∗+nn·(v−v∗).

Ghost effect by curvature – p. 6/39

Ghost effect by curvature

• .

• p

X Y Z P Q r L L+D θ z

Fix D = 1. Change of variables y = r − L, x = −Lθ, vy = vr, vx = −vθ. Scaling: 1 L = ε2 c2 , c > 0 specified later.

Ghost effect by curvature – p. 7/39

Ghost effect by curvature

ζ(y)vx ∂F ∂x +vy ∂F ∂y +vz ∂F ∂z +ε2 c2 ζ(y)vx

• vx

∂F ∂vy − vy ∂F ∂vx

• = 1

εQ(F, F), ζ(y) = 1 1 + ε2

c2 y

. For simplicity, assume: ∂F ∂x ≡ 0.

Ghost effect by curvature – p. 8/39

Boundary conditions

Rotating cylinders at fixed temperatures: Diffuse reflection boundary conditions. Notation: M(ρ, T, u; v) = ρ (2πT)3/2 e

−|v − u|2

2T On the boundaries: u · n ≡ uy = 0. ˜ M(i)(v) =

• 2π

T (i)M(1, T (i), u(i); v),

• vy>0

vy ˜ Mdv = 1, i = 0, 1.

Ghost effect by curvature – p. 9/39

Boundary conditions

F(0, z, v) = ˜ M(0)α(0)(F) vy > 0 F(1, z, v) = ˜ M(1)α(1)(F) vy < 0, α(0)(F) = −

• vy<0

vyF(0, z, v)dv, α(1)(F) =

• vy>0

vyF(1, z, v)dv, Zero mass flux at the boundary:

• dvF(y, z, v)vy = 0

for y = 0, 1.

Ghost effect by curvature – p. 10/39

Boundary conditions

The inner cylinder is at temperature T = 1 and rotates with speed U(0); the outer cylinder is also at temperature T = 1 and rotates with speed U(1). We assume ˜ M(0) = √ 2πM(1, 1, (U(0), 0, 0); v), ˜ M(1) = √ 2πM(1, 1, (U(1)0, 0); v).

Ghost effect by curvature – p. 11/39

Low Mach number

Due to the special geometry, we do not need all the components

• f the velocity field of the order of the Knudsen number but
• nly the y and z components. With the notation ˆ

v = (vy, vz), we assume

• dvˆ

vF(y, z, v) = O(ε), low Mach number. The radial flux dvvxF(y, z, v) is O(1) in ε but we will need it to be small, say of order δ: δ ≫ ε. Therefore we set U(i) = δU(i), i = 0, 1, with U(i) = O(1) both in ε and in δ.

Ghost effect by curvature – p. 12/39

Local equilbrium

A current of order δ may produce temperature changes in the fluid of the same order and, consequently, density changes. Therefore one expects the solution at 0-th order in ε to be the

local equilibrium with ρ = 1 + δr, T = 1 + δτ:

Mδ = M(1 + δr, 1 + δτ, (δU, 0, 0); v), where δU(y, z) is the tangential component of the velocity field. We seek for the solution in the form F(y, z, v) = Mδ + ε[Φ + R] with Φ a truncated expansion in ε and R a suitable remainder.

Ghost effect by curvature – p. 13/39

Expansion

Φ =

N

• n=1

εn−1Fn. Plugging in the expansion one gets conditions on the Fn which, up to the first order in ε require F1 = Mδ

• ρ1 + u · v

1 + δτ +τ1 |v − Uex|2 − 3(1 + δτ) 2(1 + δτ)2

• +B· ˆ

∇τ, 2Q(Mδ, B) ≡ LδB = ˜ B, ˜ B = Mδˆ v|v − Uex|2 − 5(1 + δτ) 2(1 + δτ)2 ,

Ghost effect by curvature – p. 14/39

Expansion

ˆ ∇P = 0, ˆ ∇ · [ˆ uρ] = 0, ˆ ∇ ≡ (∂y, ∂z) ˆ u · ˆ ∇U = ρ−1 ˆ ∇ · (η ˆ ∇U), ˆ u · ˆ ∇ˆ u − δ2 c2 U2ey = ρ−1 − ˆ ∇P2 + ˆ ∇ · (η ˆ ∇ˆ u) + ˆ ∇Σ

• P = (1 + δr)(1 + δτ)

ˆ u = ρ−1

• ˆ

vF1dv, Σ = ˆ ∇ · δ2 P

• σ1 ˆ

∇τ ⊗ ˆ ∇τ + σ2 ˆ ∇U ⊗ ˆ ∇U

• .

with η, σ1, σ2 smooth functions of T.

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Expansion

The equation for the temperature τ is 5 2 ˆ u · ˆ ∇τ = ρ−1 ˆ ∇(κ ˆ ∇τ) + δη| ˆ ∇U|2 , with κ a smooth function of T.

Ghost effect by curvature – p. 16/39

Limit for δ → 0

The macroscopic equations become much simpler in the limit δ → 0. In order to keep the correction term in the equation ˆ u · ˆ ∇ˆ u − δ2 c2 U2ey = ρ−1 − ˆ ∇P2 + ˆ ∇ · (η ˆ ∇ˆ u) + ˆ ∇Σ

• ,

we set (with 1

L = ε2 c2 )

c = δC with C and O(1) constant related to the curvature. Hence the scaling becomes 1 L = ε2 C2δ2.

Ghost effect by curvature – p. 17/39

Limiting equations

ˆ ∇ · ˆ u = 0, ˆ u · ˆ ∇U = η0 ˆ ∆U, ˆ u · ˆ ∇ˆ u + ˆ ∇P2 − 1 C2U2ey = η0 ˆ ∆ˆ u, 5 2 ˆ u · ˆ ∇τ = κ0 ˆ ∆τ, with η0 = η(1), κ0 = κ(1). P2 second order pressure.

Ghost effect by curvature – p. 18/39

Boundary conditions

We consider now the following boundary conditions: U(0, z) = U(0), U(1, z) = U(1), ˆ u(0, z) = ˆ u(1, z) = 0, τ(0, z) = τ(1, z) = 0.

Ghost effect by curvature – p. 19/39

Bifurcation δ = 0

Set β = U(0) − U(1). Laminar solution: Uℓ(y) = U(1) + β(1 − y), ˆ uℓ = 0, τℓ = 0. There is βc such that, for β > βc (but close to βc) the solution bifurcate into two non laminar solutions U±(y, z), ˆ u±(y, z), with τ = 0. U± − Uℓ = O(β − βc), ˆ u± = O(β − βc). No such a bifurcation in planar Couette flow. The term C−2U2ey is responsible for the bifurcation. It is due the curvature which on the other hand goes to 0 in this scaling

• limit. Ghost effect.

Ghost effect by curvature – p. 20/39

Bifurcation δ = 0

Linear analysis: Sone-Doi [SD] Theorem both for linear and non linear bifurcation in Arkeryd-R.E.-Marra-Nouri [AEMN] in preparation.

Ghost effect by curvature – p. 21/39

Bifurcation δ = 0

When δ = 0 the system is slightly compressible. Bifurcation persists: there are solutions (Uδ

±, ˆ

uδ

±, τδ ±) such that

(Uδ

±, ˆ

uδ

±, τδ ±) − (U±, ˆ

u±, 0) = O(δ). Perturbation argument. Extra complications due to the (small) compressibility. Theorem proved in [AEMN] in Hs-norms.

Ghost effect by curvature – p. 22/39

Boundary layer

The solution is written as F = Mδ + ε[Φ + R]. Φ computed with the expansion does not satisfy the diffuse reflection boundary conditions. We need boundary layer corrections: Φ =

N

• n=1

εn−1Φn, Φn = Fn + b(0)

n + b(1) n .

The boundary correction b(i)

n , depending on ε−1y, solve a

modified Milne problem.

Ghost effect by curvature – p. 23/39

Milne problem

Set Y (0) = ε−1y, Y (1) = ε−1(1 − y) vy ∂b(i)

n

∂Y (i) + ε3 C2δ2ζN(b(i)

n ) = L(i)b(i) n + sn

in (0, +∞). with L(i)g = 2Q(M(i), g), M(i) = M(1, 1, (δU(i), 0, 0); v). N(f) = vx

• vx

∂f ∂vy − vy ∂f ∂vx

• ,

with suitable prescribed incoming data in Y (i) = 0 and source term s. Adding some cutoff and regularization we have proved

existence, regularity (away from the boundary) and exponential decay

• f the solution in [AEMN1].

Ghost effect by curvature – p. 24/39

Expansion

Let Mδ be computed on one of the stationary solutions to the hydrodynamical equations with δ > 0, namely perturbations of the laminar or the bifurcating solution, and Φ the corresponding bulk & boundary layer expansion. It can be proved that Mδ and Φ have the smoothness and boundedness properties needed in the rest of the argument. Moreover (set M = Mδ=0) Mδ = M(1 + O(δ)), Φ = O(δ). The stationary solution to the Boltzmann equation is written as: F = Mδ + ε[Φ + R].

Ghost effect by curvature – p. 25/39

Remainder

Equation for R: vy ∂R ∂y + ε2 δ2C2ζ(y)N(R) = 1 ε

• LδR+2Q(εΦ, R)
• +εQ(R, R)+A,

LδR = 2Q(Mδ, R), The inhogeneous term A is expressed in terms of Fn’s and bn’s. It’s size is εm with m depending on the number of terms in the expansion N. Moreover it is bounded in the relevant norms.

Ghost effect by curvature – p. 26/39

Boundary conditions for R

R(0, z, v) = α(0)(R + ε−1Ψ(0)) ˜ M(0) − 1 εΨ(0)(z, v), vy > 0 R(1, z, v) = α(1)(R + ε−1Ψ(1)) ˜ M(1) − 1 εΨ(1)(z, v), vy < 0, where the inhomegeneous term Ψ(i) depends on the bn’s and are exponentially small in ε in the relevant norms.

Ghost effect by curvature – p. 27/39

Green Identity

Given H and K, consider the equation v · ∇xF − 1 εLδF − H = 0, in Ω × R3, F(x, v) = K (x, v) ∈ γ− = {(x, v) ∈ ∂Ω × R3 | n(x) · v < 0} n(x) exterior normal to ∂Ω in x. Given a positive weight function q(x, v), multiply the equation by qF and integrate on (x, v).One gets ((f, g)q := (f, qg)L2) − 1 ε(F, LδF)q + 1 2

• γ+

v · nqF 2 = (F, H)q + 1 2

• γ−

|v · n|qK2 + 1 2

• Ω×R3 F 2v · ∇xq

Ghost effect by curvature – p. 28/39

Green Identity

Such an identity is useful if the quadratic form (F, LδF)q is non positive. This is true if one chooses q = M−1

δ . Mδ local equilibrium.

But then v · ∇xq is a polynomial of degree 3 in v and the r.h.s of the Green inequality cannot be controlled directly in terms of the l.h.s. [Caflisch]. On the other hand Lδ = L + O(δ), with L = Lδ=0 = 2Q(M, · ). With q = M−1 we have −(F, LF)M −1 ≥ c(P ⊥F, νP ⊥F)M −1, with c a positive constant, P the projector onto the the null space of L, P ⊥ ≡ 1 − P and ν a known function of v.

Ghost effect by curvature – p. 29/39

Spectral inequality

If we could neglect O(δ) in the relation Lδ = L + O(δ), then we could use the Green inequality to control P ⊥F in L2. But we cannot neglect O(δ) because of the factor ε−1: In the Green identity we would have ε−1O(δ). Set W = δ−1(Mδ − M) = O(1). Then 1 ε(F, LδF) = 1 ε(F, LF) + 2δ ε(F, Q(W, F)) δ ε|(F, Q(W, P ⊥F)M −1| ≤ δ εW∞(P ⊥F, νP ⊥F)M −1 ≪ 1 εc(P ⊥F, νP ⊥F)M −1.

Ghost effect by curvature – p. 30/39

Spectral inequality

The dangerous part of the ε−1O(δ) term comes from ε−1δ(F, Q(W, PF))M −1 which may be bigger than |(F, LF)M −1|. We define −LWF = LF + 2δQ(W + δ−1Φ, PF).

Theorem[AEMN2]: There are δ0 > 0 and c > 0 such that, if

δ < δ0 then −(F, LWF)M −1 ≥ c((1 − P W )F, ν(1 − P W )F)M −1,

where P W = P + O(δ) is the projector onto the null space of LW .

Ghost effect by curvature – p. 31/39

Green inequality

Hence, the remainder R satisfies the inequality c ε(1 − P W )R, (1 − P W )R2

νM −1 + 1

2

• γ+

v · nM−1R2 ≤ (R, H)M −1 + 1 2

• γ−

|v · n|M−1K2 H = − ε2 δ2C2ζ(y)N(R) + εQ(R, R) + A, K(0, z, v) = α(0)(R + 1 εψ(0)) ˜ M(0) − 1 εΨ(0)(z, v), vy > 0 K(1, z, v) = α(1)(R + 1 εψ(1)) ˜ M(1) − 1 εΨ(1)(z, v), vy < 0

Ghost effect by curvature – p. 32/39

Boundary inequality

When ˜ M(0) = ˜ M(1) = √ 2πM, then 1 2

• ∂Ω×R3 vyM−1R2 ≥ 0

by Darrozés-Guiraud inequality. In this case ˜ M(0) = ˜ M(1) and

• ne needs to estimate it. We can show that
• ∂Ω×R3 vyM−1R2
• ≤ C
• P W R2 + δ2

ε2(1 − P W)R2

• This is useful for

δ2 ε2 = γ ε , γ ≪ 1.

Ghost effect by curvature – p. 33/39

Estimate for N(R)

Simple algebra shows that

• R3 dvM−1RN(R) = 1

2

• R3 dvM−1R2vy.

Thus ∂y

• R3 dvM−1R2vy +

ε2 δ2C2ζ(y)

• R3 dvM−1RN(R)

= 1 σ(y)∂y

• σ(y)
• R3 dvM−1R2vy
• .

for a suitable function σ. Using this one can take care of the contribution from the centrifugal force.

Ghost effect by curvature – p. 34/39

Estimate of (1 − P W)RνM−1

Set ˜ A = A + εQ(R, R) and · ≡ · 2

νM −1.

Collecting above arguments one proves that, if ε and γ are sufficiently small, then for any η > 0 (1 − P W )R2 ≤ (η + γ)εP W R2 + 1 εηP W ˜ A2 + ε(1 − P)W ˜ A2 + 1 ηO(e− 1

ε ).

To conclude the argument we need: 1) An estimate of P W R; 2) Control of the nonlinearity.

Ghost effect by curvature – p. 35/39

Estimate of PR

First consider the 1-d case. It is not obvious but true that

• R3 dvR(1, v)vy = 0.

Then one can prove that, if

R3 dv ˜

A = 0, then P W R2 ≤ c

• (I − P W) ˜

A2 + 1 ε2P W ˜ A2 + O(e− 1

ε).

Ghost effect by curvature – p. 36/39

Conclusion for d = 1

The control of the non linear collision term is based on getting L2(R3

v; L∞ x (Ω)) estimates using characteristics. Then an

iterative procedure implies existence and boundedness of the remainder for d = 1:

Theorem[AEMN1]: Assume A = O(ε4) and ε and δ2ε−1 small. Then there is a unique solution R and

R∞ = O(ε3/2).

Therefore, corresponding to the laminar solution of the macroscopic equations there is an isolated L2-solution F to the Boltzmann equation such that,

M−1[F − Mδ]L2([0,1]×R3) ≤ cε.

Ghost effect by curvature – p. 37/39

d = 2

Since U±(y, z), ˆ u±(y, z) are O(β − βc), we use perturbation arguments of the 1-d case. Following a precedure similar to the

• ne employed in [AEMN2] for the Benard problem, we can

prove the following:

Theorem[AEMN - in preparation]: Let β − βc be positive and small (independently of ε and δ). Let Mδ be the Maxwellian with parameters given by one of the δ-perturbed bifurcating solution. Then, for ε and δ2ε−1 small, there is an isolated L2-solution F to the stationary Boltzmann equation such that

M−1[F − Mδ]L2([0,1]×R3) ≤ cε.

Ghost effect by curvature – p. 38/39

• L. Arkeryd, R. Esposito, R. Marra and A. Nouri Ghost

Effect by curvature in planar Couette flow, Kinetic and Related Models, 4, (2011), 109–138.

• L. Arkeryd, R. Esposito, R. Marra and A. Nouri Stability

for Rayleigh-Benard convective solutions of the Boltzmann equation, Arch. Rat. Mech. Anal., 198 (2010), 125–187.

• Y. Sone and T. Doi, Ghost effect of infinitesimal curvature

in the plane Couette flow of a gas in the continuum limit, Physics of Fluids, 16 (2004), 952–971.

• Y. Sone, “Kinetic Theory and Fluid Dynamics," Birkhäuser

Boston, 2002

Ghost effect by curvature – p. 39/39