Figure 1: Body and surface forces acting on the liquid element.
Figure 2: Forces acting on the liquid element.
ρ∆U
d~u
= ρ∆U F~ + p~n ∆S − p~x ∆Sx − p~y ∆Sy − p~z ∆Sz
dt
1
p~n = p~x cos(nx) + p~y cos(ny) + p~z cos(nz)
p~x = ~ipxx + ~jτxy + ~kτxz
p~y = ~iτyx + ~jpyy + ~kτyz
p~z = ~iτzx + ~jτzy + ~kpzz
Stress matrix:


pxx τxy τxz
 τxy pyy τyz 
τxz τyz pzz
Figure 3: Stresses acting on the liquid cube with sizes a.
Z
Z
Z
d~u
~
ρdU = F ρdU + p~n dS
dt
U
S
Z
ZU
p~n dS = (~px cos(nx) + p~y cos(ny) + p~z cos(nz)) dS
S
S
Newton
hypothesis




pxx τxy τxz
p o o
 τxy pyy τyz  = −  o p o  + 2µSij
τxz τyz pzz
o o p
2
S11 = Sxx =
∂ux
;
∂x
S12 = Sxy =
S21 = S12 , S22 = Syy =
1
2
∂uy
,
∂y
∂ux
∂y
+
∂uy
∂x
1 ∂ux
; S13 = Sxz = 2 ∂z +
S23 = Syz =
S31 = S13 , S32 = S23 , S33 = Szz =
1
2
∂uy
∂z
+
∂uz
∂y
∂uz
∂z
p - pressure
y
z
x
, pyy = −p + 2µ ∂u
, pzz = −p + 2µ ∂u
pxx = −p + 2µ ∂u
∂x
∂y
∂z
pxx +pyy +pzz
3
= −p
d~
u
dt
= F~ − %l ∇p + ν∆~u
d~
u
dt
=
∂~u
∂~u
∂~u
∂~u
+ ux
+ uy
+ uz
∂t
∂x
∂y
∂z
|{z}
|
{z
}
local
acceleration
∂ui
∂t
+
∂
(ui uj )
∂xj
convective
acceleration
= Fi −
1 ∂p
ρ ∂xi
+
ν ∂x∂ j
3
∂
u
∂xj i
∂uz
∂x
4
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