On transport equations in thin moving domains
Transcript of On transport equations in thin moving domains
Contents 1. Problems on moving surfaces 2. Derivations of mass transport equations
2.1 Conservation law 2.2 Zero width limit
3. Derivation of diffusion equations 3.1 Energy law 3.2 Zero width limit
4. Derivation of the Euler and the Navier-Stokes equations
2
3
โข Reaction-diffusion on biological cells โข Fluid-flow on biological cell
Boussinesq (1913), L. E. Scriven (1960)
โข Fluid-flow on rotating surface geophysics
1. Problems on moving surfaces
4
ฮ ๐ก ๐กโ[0,๐): evolving surface in ๐3
(1) How does one derive equations describing phenomena on a moving surface ฮ ๐ก ?
(2) Is it related to zero-width limit of thin moving domain? ๐ท๐ ๐ก = ๐ฅ โ ๐3 dist ๐ฅ, ฮ ๐ก < ๐ .
Problems
7
๐ฮ : outer unit normal of ฮ ๐ก ๐ = ๐(๐ฆ, ๐ก) : density of substance at ๐ฆ โ ฮ(๐ก) ๐ฃ = ๐ฃ(๐ฆ, ๐ก) : velocity of substance at ๐ฆ โ ฮ ๐ก (vector field on ฮ(๐ก) not necessarily tangential)
2. Derivations of mass transport equations
2.1 Conservation law
8
Local mass conservation law
๐ฮ : normal velocity of ฮ ๐ก in the direction of ๐ฮ. โข Ansatz: Normal component of ๐ฃ equals ๐ฮ (There is no flow leaving from or coming to ฮ(๐ก).)
๐๐๐ก โซ ๐๐(๐ก) ๐โ2 = 0 for any portion ๐(๐ก) of ฮ(๐ก)
(๐(๐ก) : relatively open set)
9
Leibniz formula ๐๐๐ก๏ฟฝ ๐๐(๐ก)
๐โ2 = ๏ฟฝ (๐โ๐ + divฮ ๐ฃ ๐)๐(๐ก)
๐โ2
๐โ๐ = ๐๐ก๐ + ๐ฃ โ ๐ป ๐ (material derivative)
divฮ๐ฃ : surface divergence Ansatz implies
๐ฃ = ๐ฮ ๐ + ๐ฃ๐ . ๐ฃ๐ : tangential velocity (๐ฃ๐ โ ๐ฮ = 0)
10
Conservation law
The law
0 =๐๐๐ก๏ฟฝ ๐๐(๐ก)
๐โ2
for all ๐(๐ก) yields
๐โ๐ + divฮ ๐ฃ ๐ = 0 on ฮ(๐ก).
11
Calculation of surface divergence
divฮ๐ฃ = divฮ ๐ฮ ๐ + divฮ ๐ฃ๐ = ๐ฮ divฮ ๐ + ๐ปฮ ๐ฮ โ ๐ + divฮ ๐ฃ๐
๐ป : mean curvature
โด divฮ ๐ฃ = โ๐ป ๐ฮ + divฮ ๐ฃ๐
Note : divฮ ๐ ๐ฃ๐ = ๐ปฮ ๐ โ ๐ฃ๐ + ๐ divฮ ๐ฃ๐
โ๐ป 0
12
Second expression of conservation law Since ๐โ๐ = ๐โ๐ + ๐ฃ๐ โ ๐ป๐ with ๐โ๐ = ๐๐ก๐ + ๐ฮ ๐ โ ๐ป๐, ๐โ๐ + divฮ ๐ฃ ๐ = ๐โ๐ โ ๐ป ๐ฮ๐ + divฮ(๐๐ฃ๐).
Mass conservation: ๐โ๐ โ ๐ป ๐ฮ ๐ + divฮ ๐๐ฃ๐ = 0
First two terms depend only on motion of ฮ(๐ก), independent of ๐ฃ๐.
13
Mass conservation
๐โ๐ + divฮ ๐ฃ ๐ = 0 or
๐โ๐ โ ๐ป ๐ฮ ๐ + divฮ ๐๐ฃ๐ = 0
divฮ๐ค : surface divergence
divฮ๐ค = ๐ปฮ โ ๐ค
๐ปฮ = ๐ฮ ๐ป
๐ฮ = ๐ผ โ ๐ฮ โ ๐ฮ : projection to tangent space of ฮ(๐ก).
14
Mass conservation in a thin domain reads (CL) ๐๐ก + div ๐ ๐ข = 0 in ๐ท๐(๐ก)
(B) ๐ข โ ๐๐ท๐ = ๐๐ท๐๐ on ๐๐ท๐(๐ก).
The substance in ๐ท๐(๐ก) moves along the boundary of ๐ท๐(๐ก). It does not go into or out of ๐ท๐(๐ก).
๐๐ท๐๐ : normal velocity of ๐๐ท๐(๐ก) in the
direction of unit outer normal ๐๐ท of ๐ท๐(๐ก)
2.2 Zero width limit
15
Problem ๐ ๐ฅ, ๐ก = ๐ ๐ ๐ฅ, ๐ก , ๐ก + ๐ ๐ฅ, ๐ก ๐1 ๐ ๐ฅ, ๐ก , ๐ก
+๐ ๐2 ๐ข ๐ฅ, ๐ก = ๐ฃ ๐ ๐ฅ, ๐ก , ๐ก + ๐ ๐ฅ, ๐ก ๐ฃ1 ๐ ๐ฅ, ๐ก , ๐ก
+๐ ๐2 as ๐ โ 0
๐ ๐ฅ, ๐ก : projection to ฮ(๐ก) ๐ ๐ฅ, ๐ก : signed distance (๐ > 0 near space infinity)
๐ ๐ฅ, ๐ก = ๐ฅ โ ๐ ๐ฅ, ๐ก ๐ฮ ๐, ๐ก
If ๐ solves (CL), what equation ๐ solves?
16
Equation derived from zero width limit
Theorem 1 (T.-H. Miura, C. Liu, Y. G.). Let (๐,๐ข) solve (CL) in ๐ท๐ with (B). Let ๐, ๐ฃ be the first term of the expansion. Then
๐ฃ = ๐ฮ ๐ฮ + ๐ฃ๐ and ๐ solves
(CS) ๐โ๐ โ ๐ป ๐ฮ ๐ + divฮ ๐๐ฃ๐ = 0.
A
Differentiate ๐ ๐ฅ, ๐ก
= ๐ ๐ ๐ฅ, ๐ก , ๐ก + ๐ ๐ฅ, ๐ก ๐1 ๐ ๐ฅ, ๐ก , ๐ก + ๐(๐2) in time.
Note ๐๐ก ๐ ๐ ๐ฅ, ๐ก , ๐ก = ๐โ๐ ๐ ๐ฅ, ๐ก , ๐ก ,
๐๐ก = โ๐ฮ.
We observe ๐๐ก = ๐โ๐ ๐, ๐ก โ ๐ฮ ๐1 ๐, ๐ก + ๐ ๐ .
17
Idea of the proof
๐ฅ
ฮ(๐ก)
๐(๐ฅ, ๐ก)
18
Idea of the proof (continued) Note that
๐ป๐ ๐ฅ, ๐ก = ๐ฮ ๐, ๐ก + ๐ ๐ . One observes that
๐๐ข ๐ฅ, ๐ก = ๐๐ฃ ๐, ๐ก + ๐1๐ + ๐ ๐2
with ๐1 ๐, ๐ก = ๐๐ฃ1 ๐, ๐ก + ๐1๐ฃ ๐, ๐ก
which yields
๐ป ๐๐ข ๐ฅ, ๐ก = ๐ป๐ ๐ฅ, ๐ก ๐ป ๐๐ฃ + ๐ป๐โจ๐1 + ๐ ๐ = Pฮ๐ป ๐๐ฃ ๐ฅ, ๐ก + ๐ฮโจ๐1 + ๐ ๐ .
Here ๐ข ๐ฅ, ๐ก = ๐ฃ ๐, ๐ก + ๐ ๐ฅ, ๐ก ๐ฃ1 ๐, ๐ก + ๐ ๐2 .
A
19
Idea of the proof (continued) We are now able to conclude
div ๐๐ข ๐ฅ, ๐ก = divฮ ๐๐ฃ + ๐ฮ โ ๐1 + ๐ ๐ . Here
divฮ ๐๐ฃ = โ๐ ๐ฮ๐ป + divฮ ๐๐ฃ๐ .
Note ๐ฮ โ ๐1 = ๐1๐ฮ because ๐ฃ โ ๐ฮ = ๐ฮ , ๐ฃ1 โ ๐ฮ = 0. Thus ๐๐ก๐ + div ๐๐ข = ๐โ๐ โ ๐ป๐๐ฮ + ๐๐๐ฃฮ(๐๐ฃ๐) ๐, ๐ก + ๐ ๐ .
20
๐ฆ : Kinetic energy โฑ: Free energy
๐ฟ ๏ฟฝ ๐ฆ๐
0๐๐ก = ๏ฟฝ ๏ฟฝ(๐น๐ โ ๐ฟ๐ฅ)
๐
0๐๐ฅ๐๐ก
๐ฟ ๏ฟฝ โฑ๐
0๐๐ก = ๏ฟฝ ๏ฟฝ(๐น๐ โ ๐ฟ๐ฅ)
๐
0๐๐ฅ๐๐ก
๐ฟ : variation with respect to flow map
3. Derivation of diffusion equations 3.1 Energy law
21
Least action principle LAP ๐น๐ : inertial force ๐น๐ : conservative force
Stationary point of the action integral
๐ด ๐ฅ = ๏ฟฝ ๐ฆ(๐ฅ)๐
0๐๐ก โ ๏ฟฝ โฑ ๐ฅ
๐
0๐๐ก
๐ฅ : Flow map ๐ โฆ ๐ฅ ๐ก,๐ ๐๐๐๐ก
= ๐ข ๐ฅ, ๐ก ๐ฅ|๐ก=0 = ๐ (๐ข : velocity field)
LAP ๐ฟ๐ฟ๐ฟ๐
= 0 โ ๐น๐ โ ๐น๐ = 0
22
Maximum dissipation principle MDP
๐ : dissipation energy depending on ๐ข = ๏ฟฝฬ๏ฟฝ.
๐ฟ๐ = ๐น๐ โ ๐ฟ๐ข ๐น๐ : dissipation force
LAP + MDP โถโ ๐น๐ โ ๐น๐ = ๐น๐
23
Diffusion process
๐ฆ = 0,ใโฑ ๐ฅ = ๏ฟฝ ๐(๐ ๐ฅ )๐ท(๐ก)
๐๐ฅ
๐ =12๏ฟฝ ๐ ๐ข 2
๐ท(๐ก)๐๐ฅ ๐ : given
๐ : satisfies the transport eq (CL) ๐๐ก + div ๐๐ข = 0 ๐ฅ : ๐ท 0 โ ๐ท(๐ก) flow map with velocity ๐ข
๐ฅ ๐, 0 = ๐,ใ๐๐ฅ๐๐ก
= ๐ข ๐ฅ, ๐ก
๐ โ ๐ท(0)
24
Conservative force
Lemma 1 (T.-H. Miura, C. Liu, Y. G.). Assume that ๐ satisfies (CL). Then
๐น๐ =๐ฟ๐ฟ๐ฅ
โฑ = ๐ป๐(๐)
with ๐ ๐ = ๐โฒ ๐ ๐ โ ๐ ๐ .
Conservative force is the pressure gradient.
25
Diffusion equation
Mass conservation law (CL): ๐๐ก + div ๐๐ข = 0
Darcyโs law: ๐๐ข = โ๐ป๐ ๐
Example: ๐ ๐ = ๐ log ๐ โ ๐ ๐ = ๐ ๐๐ก โ ฮ๐ = 0 (heat equation)
26
Conservation force on a surface
Lemma 2 (MCG). Let (๐, ๐ฃ) solves (CS). Then
๐น๐ =๐ฟโฑ๐ฟ๐ฆ
= ๐ปฮ๐ ๐ .
Here
โฑ ๐ฆ = ๏ฟฝ ๐ ๐ ๐ฆฮ ๐ก
๐โ2 ๐ฆ .
27
Diffusion equations on a surface
Conservation law (CS) ๐โ๐ โ ๐ป ๐ฮ ๐ + divฮ ๐๐ฃ๐ = 0
Darcyโs law ๐๐ฃ๐ = โ๐ปฮ๐ ๐
Example: ๐ ๐ = ๐ log ๐ โ ๐ ๐ = ๐ ๐โ๐ โ ๐ป ๐ฮ ๐ โ ฮฮ๐ = 0
(different from different equation with convective term Elliot et al.)
28
Consider
(D) ๏ฟฝ ๐๐ก + div ๐๐ข = 0 ๐๐ข = โ๐ป๐ ๐ in ๐ท๐(๐ก)๐ ๐ = ๐โฒ ๐ ๐ โ ๐ ๐
with (B) ๐ข โ ๐ฮฉ = ๐ฮฉ๐ on ๐๐ท๐ ๐ก .
Expanded pressure ๐ ๐(๐ฅ, ๐ก) = ๐0 ๐, ๐ก +๐ ๐ฅ, ๐ก ๐1 ๐ฅ, ๐ก + ๐(๐2)
3.2 Zero width limit
29
Equation derived from zero width limit
Theorem 2 (MLG). Let (๐,๐ข) solves (D) in ๐ท๐(๐ก) with (B). Let ๐, ๐ฃ be the first term of the expansion. Then
๐โ๐ โ ๐ป ๐ฮ ๐ + divฮ ๐๐ฃ๐ = 0 ๐๐ฃ๐ = โ๐ปฮ๐0 on ฮ(t).
30
Energy identity For diffusion equations (D) on ๐ท(t) with (B), we have ๐๐๐ก๏ฟฝ ๐(๐)๐ท ๐ก
๐๐ฅ
= โ๏ฟฝ ๐ ๐ข 2
๐ท ๐ก๐๐ฅ โ ๏ฟฝ ๐ ๐ ๐ฮฉ๐
๐๐ท ๐ก๐โ2
i.e. ๐โฑ๐๐ก
= โ2๐ + ๏ฟฝฬ๏ฟฝ,
๏ฟฝฬ๏ฟฝ = โ๏ฟฝ ๐ ๐ ๐๐ท๐๐๐ท ๐ก
๐โ2.
๏ฟฝฬ๏ฟฝ: rate of change of work by the eternal environment
31
Energy identity on a surface
๐๐๐ก๏ฟฝ ๐(๐)ฮ ๐ก
๐โ2
= โ๏ฟฝ ๐ ๐ฃ๐ 2
ฮ ๐ก๐โ2 + ๏ฟฝ ๐ ๐ ๐ฮ๐ป
ฮ ๐ก๐โ2,
๏ฟฝฬ๏ฟฝ = โ๏ฟฝ ๐ ๐ ๐ฮ๐ปฮ ๐ก
๐โ2.
32
Commutativity integration
by parts
energetic variation
integration by parts
energetic variation
Diffusion equation in ๐ท๐(๐ก)
Diffusion equation in ฮ(๐ก)
๐ โ 0
Energy identity in ๐ท๐(๐ก)
Energy identity in ฮ(๐ก)
๐ โ 0 (MLG)
33
Rigorous results โข If ฮ(๐ก) does not move, a solution of reaction
diffusion equation in ๐ท๐ tends to a solution of equation on ฮ in some rigorous sense in various settings. (J. Hale, G. Raugel, 1992)โฆโฆ
โข However, for moving ฮ(๐ก) , the first convergence result is given by T.-H. Miura (2016, IFB to appear). (heat eq)
โข For existence of a solution for the heat equation on moving surface with convective term (A. Reusken et al.)
34
H. Koba, C. Liu, Y. G. (2016) Incompressible Euler and Navier-Stokes on moving hypersurface: Energetic derivation
H. Koba, Compressible flow
4. Derivation of the Euler and the Navier-Stokes equations
35
Energetic formulation (Euler)
๐ฆ ๐ฅ = ๏ฟฝ๐ ๐ฃ 2
2ฮ ๐ก๐๐ฅ
๐ : density divฮ๐ฃ = 0 (incompressibility)
LAPโ ๐ฟ ๏ฟฝ ๐ฆ ๐ฅ๐
0๐๐ก = 0 under divฮ๐ฃ = 0
36
Euler equation on ๐ช(๐)
๐โ๐ = 0
(I) ๐๐โ๐ฃ + gradฮ ๐ + ๐๐ป๐ = 0 divฮ๐ฃ = 0
(This is overdetermined.)
(II) ๐โ๐ = 0 ๐ฮ ๐๐โ๐ฃ + gradฮ ๐ = 0
divฮ๐ฃ = 0
(Variation is taken only in tangential direction.)
37
Dissipative energy
๐ = ๐๏ฟฝ ๐ทฮ ๐ฃ 2
ฮ(t)๐โ2,ใ๐ > 0
๐ทฮ ๐ฃ = ๐ฮ๐ท ๐ฃ ๐ฮ
๐ท ๐ฃ =12
๐๐ฃ๐
๐๐ฅ๐+๐๐ฃ๐
๐๐ฅ1
(projected strain rate)
38
Navier-Stokes equations
๐โ๐ = 0 (III) ๐๐โ๐ฃ + gradฮ ๐ + ๐๐ป๐ = 2๐ divฮ๐ทฮ ๐ฃ
divฮ๐ฃ = 0 (overdetermined)
๐โ๐ = 0 (IV) ๐๐ฮ๐โ๐ฃ + gradฮ ๐ = 2๐๐ฮdivฮ๐ทฮ(๐ฃ)
divฮ๐ฃ = 0
39
Relation to other derivation
Incompressible perfect flow
V. I. Arnold (1966) ๐ = const ๐๐ฮ ๐โ๐ฃ + gradฮ ๐ = 0
for a fixed surface (variational principle on the space of volume preserving diffeormorphism) (Least action principle)
40
M. E. Taylor (1992) Balance of momentum on manifold
D. Bothe, J. Prรผss (2010) Boussinesq-Scriven model
T. Jankuhn, M. A. Olshanskii, A. Reusken Preprint (2017)
Incompressible viscous flow on a fixed surface
41
Zero width limit
๐๐ก๐ข + ๐ข,๐ป ๐ข + ๐ป๐ = ๐0ฮ๐ข div ๐ข = 0 in ๐ท๐(๐ก)
Boundary condition (perfect slip)
๐ข โ ๐๐ท = ๐๐ท๐
๐ท ๐ข ๐ tan = 0
42
Limit equation
Result (T.-H. Miura)
๐ฮ ๐๐ก๐ฃ + ๐ฃ,๐ป ๐ฃ + ๐ปฮ๐
= 2๐0๐ฮdiv(๐ฮ๐ท ๐ฃ ๐ฮ)
This agrees with what KCG derived.