From Newtons law to hydrodynamic equations 18.354 - L14 Goal: - PowerPoint PPT Presentation

From Newton’s law to hydrodynamic equations 18.354 - L14

Goal: derive ∂ρ ∂ t + r · ( ρ u ) = 0 . D u Dt = �r p + g . ρ D Dt = ∂ ∂ t + ( u · r )

6 13.2 From Newton’s laws to hydrodynamic equations a many-particle system governed d x i md v dt = v i , dt = F i , � � X F ( x 1 , . . . , x n ) = G ( x i ) + H ( x i � x j ) = �r x i Φ ( x 1 , . . . , x n ) j 6 = i s H ( r ) = � H ( � r ) X We define the fine-grained phase-space density N X f ( t, x , v ) = δ ( x � x i ( t )) δ ( v � v i ( t )) X i =1 where δ ( x � x i ) = δ ( x � x i ) δ ( y � y i ) δ ( z � z i )

6 X We define the fine-grained phase-space density N X f ( t, x , v ) = δ ( x � x i ( t )) δ ( v � v i ( t )) i =1 N ∂ d X = dt [ δ ( x � x i ) δ ( v � v i )] ∂ tf i =1 N X = { δ ( v � v i ) r x i δ ( x � x i ) · ˙ x i + δ ( x � x i ) r v i δ ( v � v i ) · ˙ v i } i N N δ ( x � x i ) δ ( v � v i ) · F i X X = δ ( v � v i ) δ ( x � x i ) · v i � r v �r x m i =1 i =1 where, in the last step, we inserted Newton’s equations and used that ∂ δ ( x � x i ) = � ∂ ∂ x δ ( x � x i ) ∂ x i

N N ∂ δ ( x � x i ) δ ( v � v i ) · F i X X = � v · r x δ ( v � v i ) δ ( x � x i ) � r v ∂ tf m i =1 i =1 N � v · r x f � 1 X = m r v δ ( x � x i ) δ ( v � v i ) · F i . (313) i =1 X Writing r = r x and inserting (309) for the forces, we may rewrite ✓ ∂ 2 3 N ◆ X X ∂ t + v · r = �r v δ ( x � x i ) δ ( v � v i ) · 4 G ( x i ) + H ( x i � x j ) m f 5 i =1 j 6 = i 2 3 N X X = �r v δ ( x � x i ) δ ( v � v i ) · 4 G ( x ) + H ( x � x j ) 5 i =1 x j 6 = x 2 3 X 5 · r v f = � 4 G ( x ) + H ( x � x j ) ( x j 6 = x

6 X X 4 5 r r ✓ ∂ 2 3 ◆ X 5 · r v f ∂ t + v · r m f = � 4 G ( x ) + H ( x � x j ) (314) x j 6 = x To obtain the hydrodynamic equations from (314), two additional reductions will be necessary: • We need to replace the fine-grained density f ( t, x , v ), which still depends implicitly on the (unknown) solutions x j ( t ), by a coarse-grained density h f ( t, x , v ) i . • We have to construct the relevant field variables, the mass density ρ ( t, r ) and velocity field u , from the coarse-grained density ¯ f . e { x 1 ( t ) , . . . , x N ( t ) } { } 1 N s { x 1 (0) , . . . , x N (0); v 1 (0) , . . . , v N (0) } =: Γ 0 . Z h f ( t, x , v ) i = d P ( Γ 0 ) f ( t, x , v ) . (315)

✓ @ ◆ @ t + v · r h f i = �r v · [ G ( x ) h f i + C ] (316) m where the collision-term X C ( t, x , v ) := h H ( x � x j ) f ( t, x , v ) i (317) x j 6 = x We now define the mass density ⇢ , the velocity field u , and the specific kinetic energy tensor Σ by Z d 3 v h f ( t, x , v ) i , ⇢ ( t, x ) = (318a) m Z d 3 v h f ( t, x , v ) i v . ⇢ ( t, x ) u ( t, x ) = (318b) m Z d 3 v h f ( t, x , v ) i vv . ⇢ ( t, x ) Σ ( t, x ) = (318c) m

We now define the mass density ⇢ , the velocity field u , and the specific kinetic energy tensor Σ by Z d 3 v h f ( t, x , v ) i , ⇢ ( t, x ) = (318a) m Z d 3 v h f ( t, x , v ) i v . ⇢ ( t, x ) u ( t, x ) = (318b) m Z d 3 v h f ( t, x , v ) i vv . ⇢ ( t, x ) Σ ( t, x ) = (318c) m Z h i The tensor Σ is, by construction, symmetric as can be seen from the definition of its individual components Z d 3 v h f ( t, x , v ) i v i v j , ⇢ ( t, x ) Σ ij ( t, x ) = m Z and the trace of Σ defines the local kinetic energy density ✏ ( t, x ) := 1 Z 2Tr( ⇢ Σ ) = m d 3 v h f ( t, x , v ) i | v | 2 . (319) 2

Mass conservation ✓ @ ◆ @ t + v · r h f i = �r v · [ G ( x ) h f i + C ] (316) m Z Integrating Eq. (316) over v , we get @ Z dv 3 r v · [ G ( x ) h f i + C ] , @ t ⇢ + r · ( ⇢ u ) = � (320) but the rhs. can be transformed into a surface integral (in velocity space) that vanishes since for physically reasonable interactions [ G ( x ) h f i + C ] ! 0 as | v | ! 1 . We thus recover the mass conservation equation @ @ t ⇢ + r · ( ⇢ u ) = 0 . (321)

Momentum conservation ✓ @ ◆ @ t + v · r h f i = �r v · [ G ( x ) h f i + C ] (316) m To obtain the momentum conservation law, lets multiply (316) by v and subsequently integrate over v , ✓ @ ◆ Z Z dv 3 m dv 3 v r v · [ G ( x ) h f i + C ] . @ t + v · r h f i v = � (322)

✓ @ ◆ Z Z dv 3 m dv 3 v r v · [ G ( x ) h f i + C ] . @ t + v · r h f i v = � (322) The lhs. can be rewritten as ✓ ∂ ◆ Z ∂ Z dv 3 m dv 3 m h f i vv ∂ t + v · r h f i v = ∂ t ( ρ u ) + r · ∂ = ∂ t ( ρ u ) + r · ( ρ Σ ) ∂ = ∂ t ( ρ u ) + r · ( ρ uu ) + r · [ ρ ( Σ � uu )] ρ ∂ ∂ t u + u ∂ = ∂ t ρ + u r · ( ρ u ) + ρ u · r u + r · [ ρ ( Σ � uu )] ✓ ∂ ◆ (321) = ∂ t + u · r u + r · [ ρ ( Σ � uu )] (323) ρ ✓ ◆ The rhs. of (322) can be computed by partial integration, yielding Z Z dv 3 v r v · [ G ( x ) h f i + C ] dv 3 · [ G ( x ) h f i + C ] � = = ρ g + c ( t, x ) , (324) where g ( x ) := G ( x ) /m is the force per unit mass (acceleration) and the last term Z Z dv 3 X dv 3 C = c ( t, x ) = h H ( x � x j ) f ( t, x , v ) i (325) x j 6 = x

6 X encodes the mean pair interactions. Combining (323) and (324), we find ✓ ∂ ◆ ∂ t + u · r = �r · [ ρ ( Σ � uu )] + ρ g ( x ) + c ( t, x ) . (326) ρ u The symmetric tensor Π := Σ � uu (327) � measures the covariance of the local velocity fluctuations of the molecules w related to their temperature. To estimate c , let us assume that the pair interaction force H can be derived from a pair potential ϕ , which means that H ( r ) = �r r ϕ ( r ). Assuming further that H ( 0 ) = 0 , we may write Z dv 3 X c ( t, x ) = � h [ r x ϕ ( x � x j )] f ( t, x , v ) i (328) x j ( t ) X Replacing for some function ζ ( x ) the sum over all particles by the integral ζ ( x j ) ' 1 Z X d 3 y ρ ( t, y ) ζ ( y ) (329) m x j

X Replacing for some function ζ ( x ) the sum over all particles by the integral ζ ( x j ) ' 1 Z X d 3 y ρ ( t, y ) ζ ( y ) (329) m x j X we have � 1 Z Z dv 3 d 3 y ρ ( t, y ) h [ r x ϕ ( x � y )] f ( t, x , v ) i c ( t, x ) ' m � 1 Z Z dv 3 d 3 y ρ ( t, y ) h [ �r y ϕ ( x � y )] f ( t, x , v ) i = m � 1 Z Z dv 3 d 3 y [ r ρ ( t, y )] h ϕ ( x � y ) f ( t, x , v ) i = (330) m

Z Z � 1 Z Z dv 3 d 3 y [ r ρ ( t, y )] h ϕ ( x � y ) f ( t, x , v ) i c ( t, x ) ' � m In general, it is impossible to simplify this further without some explicit assumptions about initial distribution P that determines the average h · i . There is however one exception, namely, the case when interactions are very short-range so that we can approximate the potential by a delta-function, ϕ ( r ) = φ 0 a 3 δ ( r ) , (331) where ϕ 0 is the interaction energy and a 3 the e ff ective particle volume. In this case, � ϕ 0 a 3 Z Z dv 3 d 3 y [ r ρ ( t, y )] h δ ( x � y ) f ( t, x , v ) i c ( t, x ) = m � ϕ 0 a 3 Z dv 3 h f ( t, x , v ) i = m [ r ρ ( t, x )] � ϕ 0 a 3 = m 2 [ r ρ ( t, x )] ρ ( t, x ) � ϕ 0 a 3 2 m 2 r ρ ( t, x ) 2 = (332)

� r Inserting this into (326), we have thus derived the following hydrodynamic equations ∂ ∂ t ρ + r · ( ρ u ) = 0 (333a) ✓ ∂ ◆ ρ ∂ t + u · r = r · Ξ + ρ g ( x ) , (333b) u where ρ ( Σ � uu ) + ϕ 0 a 3  � 2 m 2 ρ 2 I Ξ := � (333c) is the stress tensor with I denoting unit matrix. Closure problem a commonly adopted closure condition is the ideal isotropic gas approximation Σ � uu = kT m I , (334) where T is the temperature and k the Boltzmann constant. For this closure condition, Eqs. (333a) and (333b) become to a closed system for ρ and u .

ρ ( Σ � uu ) + ϕ 0 a 3  � Σ � uu = kT 2 m 2 ρ 2 I Ξ := � m I , Traditionally, and in most practical applications, one does not bother with microscopic derivations of Ξ ; instead one merely postulates that Ξ = � p I + µ ( r > u + r u > ) � 2 µ 3 ( r · u ) , (335) where p ( t, x ) is the pressure field and µ the dynamic viscosity, which can be a function of pressure, temperature etc. depending on the fluid. Equations (333a) and (333b) com- bined with the empirical ansatz (335) are the famous Navier-Stokes equations . The second summand in Eq. (335) contains the rate-of-strain tensor E = 1 2( r > u + r u > ) (336) and ( r · u ) is the rate-of-expansion of the flow. r · For incompressible flow, defined by ρ = const. , the Navier-Stokes equations simplify to = 0 (337a) r · u ✓ ∂ ◆ �r p + µ r 2 u + ρ g . ∂ t + u · r = (337b) ρ u In this case, one has to solve for ( p, u ).

14 The Navier-Stokes Equations X ∂ ✓ ∂ u j ◆ 1 � ∂ p + ∂ u i ρ Du i = + 2 µ 2 ∂ x i ∂ x j ∂ x i ∂ x j Dt �r i p + µ r i ( r · u ) + µ r 2 u i . = When the fluid density doesn’t change very much we have seen that r · u = 0, and under these conditions the Navier-Stokes equations for fluid motion are ρ D u Dt = �r p + µ r 2 u . (347)