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Fluid Dynamics Governing Equations

This appendix provides the governing equations for fluid flow - continuity, Navier-Stokes, and energy equations - in three coordinate systems: cartesian, cylindrical, and spherical. The equations are presented for incompressible, Newtonian fluids where density, viscosity, and thermal conductivity are constant. The equations relate partial derivatives of velocity, pressure, and temperature with respect to the spatial coordinates and time. Stress terms are defined in terms of velocity gradients for each coordinate system.

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Camille Crn
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75 views5 pages

Fluid Dynamics Governing Equations

This appendix provides the governing equations for fluid flow - continuity, Navier-Stokes, and energy equations - in three coordinate systems: cartesian, cylindrical, and spherical. The equations are presented for incompressible, Newtonian fluids where density, viscosity, and thermal conductivity are constant. The equations relate partial derivatives of velocity, pressure, and temperature with respect to the spatial coordinates and time. Stress terms are defined in terms of velocity gradients for each coordinate system.

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Camille Crn
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© © All Rights Reserved
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Appendix C

Governing Equations

This appendix gives the continuity, Navier-Stokes, and energy equations


together with the components of stress in the three most commonly used
coordinate systems: cartesian, cylindrical, and spherical coordinates. The
equations are valid for calorically perfect newtonian £uids in which r; m, and
k are all constants.

CARTESIAN COORDINATES

Coordinates r ¼ ðx; y; zÞ
Velocity u ¼ ðu; v; wÞ
@u @v @w
þ þ ¼0
@x @y @z
 
@u @u @u @u @p
r þu þv þw ¼  þ mH2 u þ rfx
@t @x @y @z @x
 
@v @v @v @v @p
r þu þv þw ¼  þ mH2 v þ rfy
@t @x @y @z @y

505
506 Appendix C
 
@w @w @w @w @p
r þu þv þw ¼  þmH2 wþrfz
@t @x @y @z @z
 
@T @T @T @T
rCv þu þv þw
@t @x @y @z
"      #
@u 2 @v 2 @w 2
¼ kH2 T þ2m þ þ
@x @y @z
" 2     #
@u @v @u @w 2 @v @w 2
þm þ þ þ þ þ
@y @x @z @x @z @y

where
@2 @2 @2
H2 ¼ þ þ
@x 2 @y 2 @z 2
 
@u @u @v
txx ¼ 2m txy ¼ tyx ¼ m þ
@x @y @x
 
@v @v @w
tyy ¼ 2m tyz ¼ tzy ¼ m þ
@y @z @y
 
@w @w @u
tzz ¼ 2m tzx ¼ txz ¼ m þ
@z @x @z

CYLINDRICAL COORDINATES

Coordinates r ¼ ðR; y; zÞ

Velocity u ¼ ðuR ; uy ; uz Þ

1 @ 1 @uy @uz
ðRuR Þ þ þ ¼0
R @R R @y @z
 
@uR @uR uy @uR u2y @uR
r þ uR þ  þ uz
@t @R R @y R @z
 
@p uR 2 @uy
¼ þ m H2 uR  2  2 þ rfR
@R R R @y
Governing Equations 507
 
@uy @uy uy @uy uR uy @uy
r þ uR þ þ þ uz
@t @R R @y R @z
 
1 @p uy 2 @uR
¼ þ m H uy  2 þ 2
2
þ rfy
R @y R R @y
 
@uz @uz uy @uz @uz @p
r þ uR þ þ uz ¼  þ mH2 uz þ rfz
@t @R R @y @z @z
 
@T @T uy @T @T
rCv þ uR þ þ uz
@t @R R @y @z
(    2  2 )
@uR 2 1 @uy @uz
¼ kH T þ 2m
2
þ þ uR þ
@R R @y @z
( 2     )
@uy 1 @uz @uz @uR 2 1 @uR @
uy 2
þm þ þ þ þ þR
@z R @y @R @z R @y @R R

where
 
1 @ @ 1 @2 @2
H2 ¼ R þ 2 2þ 2
R @R @R R @y @z
 
@uR @
uy 1 @uR
tRR ¼ 2m tRy ¼ tyR ¼ m R þ
@R @R R R @y
   
1 @uy uR @uy 1 @uz
tyy ¼ 2m þ tyz ¼ tzy ¼ m þ
R @y R @z R @y
 
@uz @uz @uR
tzz ¼ 2m tzR ¼ tRz ¼ m þ
@z @R @z

SPHERICAL COORDINATES

Coordinates r ¼ ðr; y; oÞ
Velocity u ¼ ður ; uy ; uo Þ
1 @ 2 1 @ 1 @uo
ðr ur Þ þ ðuy sin yÞ þ ¼0
r 2 @r r sin y @y r sin y @o
508 Appendix C

 
@ur @ur uy @ur uo @ur u2y þ u2o
r þ ur þ þ 
@t @r r @y r sin y @o r
 
@p 2 2 @uy 2 2 @uo
¼  þ m H2 ur  2 ur  2  2 uy cot y  2 þ rfr
@r r r @y r r sin y @o
 
@uy @uy uy @uy uo @uy ur uy u2o
r þ ur þ þ þ  cot y
@t @r r @y r sin y @o r r
 
1 @p 2 @ur uy 2 cot y @uo
¼ þ m H2 uy þ 2   þ rfy
r @y r @o r 2 sin2 y r 2 sin y @o
 
@uo @uo uy @uo uo @uo ur uo uy uo
r þ ur þ þ þ þ cot y
@t @r r @y r sin y @o r r

1 @p 2 @ur
¼ þ m H2 uo þ 2
r sin y @o r sin y @o

2 cos y @uy uo
þ  þ rfo
r 2 sin2 y @o r 2 sin2 y
 
@T @T uy @T uo @T
rCv þ ur þ þ
@t @r r @y r sin y @o
"   
@ur 2 1 @uy ur 2
¼ kH T þ 2m
2
þ þ
@r r @y r
 2 #
1 @uo ur uy
þ þ þ cot y
r sin y @o r r
(   
1 @ur @
uy 2 1 @ur @
uo 2
þm þr þ þr
r @y @r r r sin y @o @r r
  )
1 @uy sin y @
uo 2
þ þ
r sin y @o r @y sin y

where
   
1 @ 2 @ 1 @ @ 1 @2
H2 ¼ r þ sin y þ
r @r
2 @r r sin y @y
2 @y r 2 sin y @o2
2
Governing Equations 509

@ur
trr ¼ 2m
@r
 
1 @uy ur
tyy ¼ 2m þ
r @y r
 
1 @uo ur uy cot y
too ¼ 2m þ þ
r sin y @o r r

 
@
uy 1 @ur
try ¼ tyr ¼ m r þ
@r r r @y
 
sin y @
uo 1 @uy
tyo ¼ toy ¼ m þ
r @y sin y r sin y @o
 
@
uo 1 @ur
tor ¼ tro ¼ m r þ
@r r r sin y @o

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