Navier-Stokes Equations

D # "U = # Pds + # "FdV Dt V S V Where it is assumed that stress in D the j direction is summed over all # "u = # $ ds + # "f dV three i surface elements Dt j ij j ! V S V

Recall the Reynolds Transport Theorem has the following form… ! D % $" $ ( # "dV = # ' + ("uk )* dV Dt V V& $t $xk )

By replacing α with ρu , the LHS of the top equation can be expanded… ! j $ " " ' * & (#u j ) + (#u juk )) dV = * +ijds + * #f jdV V% "t "xk ( S V

Note that we are dealing with each component of U=[u1,u2,u3] separately. !

$ " " ' * & #u j + #u juk ) dV = * +ijds + * #f jdV V% "t "xk ( S V

Recall: Divergence Theorem from Vector Calculus States: #u r r u n dS i dV ! " U • nˆ dS = " # •UdV " i i = " s V #xi S V The equivalent expression in tensor notation

$ " " ' "+ij ! ! * & #u j + #u juk ) dV = * dV + * #f jdV V% "t "xk ( V "xi V Since the volume choice is "$ " " ij arbitrary ! (#u j ) + (#u juk ) = + #f j "t "xk "xi

1 " " "$ij (#u j ) + (#u juk ) = + #f j "t "xk "xi

expanding the derivative…

! # # # # #$ij " (u j ) + u j (") + u j ("uk ) + "uk (u j ) = + "f j #t #t #xk #xk #xi

terms sum to zero because of continuity (conservation of ! mass) requirements

# # #$ij " (u j ) + "uk (u j ) = + "f j #t #xk #xi

Local rate of change !of momentum

rate of change due to advection of momentum

divergence of stress tensor or net stress over the surface element

body force (e.g., gravity)

The stress tensor (σij) can be broken down into a hydrostatic pressure term (P) that only acts normal the the surface (i = j) and a shear stress tensor (τij)

" = #P$ + % δij=1 when i=j ij ij ij δij=0 when i䍫㼍

# # # " (u j ) + "uk (u j ) = ($P%ij + &ij ) + "f j ! #t #xk #xi

! The number of unknowns in the conservation of momentum equation can be reduced by replacing the shear stress tensor (τ

ij) with an expression containing the deformation rate tensor (εij) which is a function of spatial derivatives of U.

2 du "y"t du dv dy #y#t "x"t dy du dx dv "#$1 = = #t "#2 = = "t #y dy "x dx

-ΔΘ1 d"1 du ! d"2 dv ! ! = # = dv dt dy dt dx "x"t ΔΘ2 dx

! !

deformation = (ΔΘ2 - ΔΘ1) ! d(#$2 % #$1) d#$2 d#$1 ' &v &u* deformation rate = "xy = = % = ) + , dt dt dt ( &x &y+

deformation rate tensor ! $ ' #u j #ui "ij = & + ) % #xi #x j (

Assume that the shear stress (τij) on a fluid element is linearly related to the deformation rate (εij)… % ( $u j $ui Note: there is also a normal component "ij = µ#ij = µ' + * ' $x $x * of deformation but for incompressible & i j ) fluids this terms is zero Note: This is a purely empirical by continuity & ) %u %u %u j relationship. the constant (µ) has to " = #$ k + µ( i + + ij ij x ( x x + be determined by laboratory % k ' % j % i * ! measurement. It defines the fluid as being a Newtonian Fluid # # # ! " (u j ) + "uk (u j ) = ($P%ij + &ij ) + "f j #t #xk #xi

& ) ! # # # # #u j #ui " (u j ) + "uk (u j ) = $ (P%ij ) + µ ( + + + "f j #t #xk #xi #xi ' #xi #x j *

& ) # # # # #u j #ui " (u j ) + "uk (u j ) = $ (P%ij ) + µ ( + + + "f j #t #xk #xi #xi ' #xi #x j *

δij=1 when i=j

δij=0 when i䍫㼍 continuity ! % 2 ( # # #P # # % #ui ( " (u j ) + "uk (u j ) = $ + µ' (u j ) + ' ** + "f j #t #xk #x j & #xi#xi #x j & #xi ))

Navier-Stokes Equation ! # # #P #2 " (u j ) + "uk (u j ) = $ + µ (u j ) + "f j #t #xk #x j #xi#xi Local rate of Advection of j-directed j-directed j-directed change j-directed pressure force due to body force of j-directed momentum gradient fluid stress (e.g., gravity momentum force (i.e., viscous force) !

3 Flux vector due to Fickian diffusion "C Fi = a "xi Loss of material from a volume element due to divergence of the flux vector ! " "2C (Fi ) = a "xi "xi"xi

"2 µ "2 µ (u j ) = (#u j ) ! "xi"xi # "xi"xi The viscous stress term in the Navier-Stokes equation can be seen as the divergence of the diffusive flux of momentum where the momentum diffusion coefficient is µ/ρ ! µ = dynamic viscosity µ/ρ = υ = kinematic viscosity

ˆ ˆ ˆ r R = i x + j y + k z

Knauss, page 90-94