D v D F v/d F v D F η v D (3.2) (a) F=0 (b) v=const. D F v Newtonian fluid σ ė σ = ηė (2.2) ė kl σ ij = D ijkl ė kl D ijkl (2.14) ė ij (3.3) µ η visco

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1 post glacial rebound 3.1 Viscosity and Newtonian fluid f i = kx i σ ij e kl ideal fluid (1.9) irreversible process e ij u k strain rate tensor (3.1) v i u i / t e ij v F 23

2 D v D F v/d F v D F η v D (3.2) (a) F=0 (b) v=const. D F v Newtonian fluid σ ė σ = ηė (2.2) ė kl σ ij = D ijkl ė kl D ijkl (2.14) ė ij (3.3) µ η viscosity p (1.9) p λ p η (3.4) λ, η, p σ xx + σ yy + σ zz =3λė ll +2η (ė xx +ė yy +ė zz ) 3p =(3λ +2η)(ė xx +ė yy +ė zz ) 3p (1.9) 3p (3.5) p η (3.6) θ ė xx +ė yy +ė zz (1.25) µ =0 (3.6) θ 24

3 ( ) (3.7) (3.8) [ - ] η 3.2 Lagrangian versus Eulerian derivatives 3.6 (3.8) 2.23 (2.23) D/Dt / t t P X u P x x (X,t)=X + u (3.9) u X x 1. u(x,t) Lagragian description 2. u(x,t)eulerian description x X P u(x) or u(x)? P P X x O O q X q(x,t) 25

4 Dq/Dt q x q(x(x,t),t) x / t x Dq Dt ( ) q t X dq(x(x,t),t) dt = ( ) q + t x 3 j=1 q x j x j t = q t + v j q (3.10) x j v j x j / t X x D/Dt / t x t x (3.11) v (3.12) (3.12) v 3.3 Navier-Stokes equation (2.23) ρ 2 u i t 2 = σ ji,j + f i (3,6)3.2 (2.23) 2.2 u v 2.23 (3.13) (2.23) D/Dt / t (2.23) D/Dt u, v 1 (3.14) 26

5 (3.12) O(v 2 ) O(v) Dv Dt = v v + v v t t (3.15) (2.23) 3.13 (3.14) (2.23) u (3.13) v v v η (3.8) (3.13) ρ Dv ( ) i Dt = ρ vi t + v v i j = f i p + η 2 v i (3.16) x j x i ρ Dv ( ) v Dt = ρ +(v ) v = f p + η 2 v (3.17) t [] 3.16 (3.17) ρ n V S (3.18) x (3.19) [] 3.18 (3.19) [] ρv n (3.19) 27

6 (3.11) (3.20) [] 3.19 (3.20) (3.7) (3.20) (3.21) v 3.4 Non-dimensionalization of fluid dynamic equations (3.13) (3.17) f (3.13) (3.8) ρ Dv i Dt = σ ji x j, ( vi σ ij = pδ ij +2ηė ji = pδ ij + η + v ) j x j x i (3.22) v i x i ρ V Dv i T Dt = ηv ( 2 v i L 2 x j x + 2 v ) j j x j x i Re Dv i Dt = 2 v i x j x + 2 v j j x j x i Re Reynolds number (3.23) (3.24) 28

ν = η/ρ kinematic viscosity η (3.23) (3.22) 1. 2.

7 ν = η/ρ kinematic viscosity η (3.23) (3.22) L V ρ η V Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000 (a) (d) R R R R (b) (c) (e) (f) R R (a) Re (b) Re (c) (d) Re Re > 10 (e) Re (e) (d) 29

8 3.23 Re L 2 V 100 m/s 5m 1 m/s VL :1 ρ/η thermal convection Rayleigh number Ra Nusselt number Nu II - 10, 1997 [] 3.5 Two-dimensional incompressible fluid (3.21) v =0 v =(u, v, 0) (3.21) u v stream function x y ψ(x, y) (3.25) ψ(x, y) 30

9 ψ ψ(x, y) = const. dψ = ψ ψ dx + dy = vdx + udy =0 x y dx u = dy (3.26) v (a) ψ v =(u, v) stream line (a) ds dx v u dy v y O C ds (b) n^ P (c) A ψ+δψ x ψ B ψ Δψ (b) O P C ψ = dψ = ( ψ ψ dx + x y dy) = (udy vdx) = C C (b) ds ˆn ( ds x ds = s, y ) ( ) y, ˆn = s s, x s C C ( u y s v x ) ds s (3.27) u ˆn O P C ψ (c) A B rotation curl U gradient U φ(x, y) ψ(x, y) (3.25) z = x + iy f(z) φ(x, y) ψ(x, y) f(z) φ(x, y)+iψ(x, y) φ ψ u v f(z) z (2007) 31

0= p y + η ( 2 v x 2 + 2 v y 2 ) (3.28) (3.29) [] (3.29) p (3.

10 3-5 Fowler(2005) 3.13 η (3.8) i = x σ xx = p +2η u x σ yy = p +2η v y ( v σ xy = σ yx = η x + u ) y 0= σ xx x + σ yx y = p x +2η 2 u x 2 + η 2 v x y + η 2 u y 2 (3.21) 2 v/ x y = 2 u/ x 2 i = y 0= p ( 2 ) x + η u x u y 2 0= p y + η ( 2 v x v y 2 ) (3.28) (3.29) [] (3.29) p (3.28) / y (3.29) / x u, v ψ (3.30) 2 = 2 / x / y 2 2 ψ =0 harmonic function (3.30) biharmonic function) 32

11 ψ [] (3.30) post-glacial rebound Turcotte and Schubert, Geodynamics, 2nd ed., Cambridge Univ. Press, 2001, (a) h x (a) (b) (c) (d) (a) l δy x p 0,p 1 (p 1 p 0 )δy σ xy (y) σ xy (y + δ) δy σ xy (y)l + σ xy (y + δy)l dσ xy dy δyl l (p 1 p 0 )δy + dσ xy dy δyl =0 dσ xy dy = p 1 p 0 dp l dx (3.31) p p 1 >p 0 dp/dx < 0 (b) x dp/dx σ xy (3.8) u y ( u σ xy =2ηė xy = η y + v ) = η du x dy (3.31) (3.32) (3.32) dσ xy dy = η d2 u dy 2 = dp dx (3.33) dp/dx c 1,c 2 u = 1 dp 2η dx y2 + c 1 y + c 2 (3.34) 33

12 no-slip boundary condition (c) y = h y =0 u 0 u = 0 at y = h, u = u 0 at y =0 ( u = u 0 1 y ) h (3.35) Couette flow y =0 u =0 (d) u = 1 dp y(y h) (3.36) 2η dx [] (3.35) (3.36) [ ] ψ(x, y) (3.30) (3.34) 34

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Untitled II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j