navier stokes equation in cartesian and polar coordinates
TRANSCRIPT
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Navier - Stokes equation:
We consider an incompressible , isothermal Newtonian flow (density =const, viscosity =const), with a velocity field ))()()(( x,y,z, w x,y,z, vx,y,zuV =r Incompressible continuity equation:
0=+
+
zw
yv
xu eq1.
Navier - Stokes equation:
vector form: VgPDt
VD rrr
2++= x component:
)( 22
2
2
2
2
zu
yu
xug
xP
zuw
yuv
xuu
tu
x +
+++
=
+
++
eq2. y component:
)( 22
2
2
2
2
zv
yv
xvg
yP
zvw
yvv
xvu
tv
y +
+++
=
+
++
eq3. z component:
)( 22
2
2
2
2
zw
yw
xwg
zP
zww
ywv
xwu
tw
z +
+++
=
+
++
eq4. Cylindrical coordinates ),,( zr : We consider an incompressible , isothermal Newtonian flow (density =const, viscosity =const), with a velocity field .uuuV zr ),,( =
r
Incompressible continuity equation:
0)(1)(1 =+
+
zuu
rrru
rzr
eq a)
r-component:
+
+
++
=
+
++
2
2
22
2
22
2
211zuu
ru
rru
rur
rrg
rP
zuu
ruu
ru
ruu
tu
rrrrr
rz
rrr
r
eq b)
-component:
+
++
++
=
++
++
2
2
22
2
22
2111zuu
ru
rru
rur
rrgP
r
zuu
ruuu
ru
ruu
tu
r
zr
r
eq c)
z-component:
+
+
++
=
+
++
2
2
2
2
2
11zuu
rrur
rrg
zP
zuuu
ru
ruu
tu
zzzz
zz
zzr
z
eq d)