The Equation of Continuity and the Equation of Motion inCartesian, cylindrical, and spherical coordinates

CM310 Fall 1998 Faith A. Morrison

Continuity Equation, Cartesian coordinates

@�

@t+

�vx

@�

@x+ vy

@�

@y+ vz

@�

@z

�+ �

�@vx

@x+

@vy

@y+

@vz

@z

�= 0

Continuity Equation, cylindrical coordinates

@�

@t+

1

r

@(�rvr)

@r+

1

r

@(�v�)

@�+

@(�vz)

@z= 0

Continuity Equation, spherical coordinates

@�

@t+

1

r2@(�r2vr)

@r+

1

r sin �

@(�v� sin �)

@�+

1

r sin �

@(�v�)

@�= 0

Equation of Motion for an incompressible uid, 3 components in Cartesian coordinates

�

�@vx

@t+ vx

@vx

@x+ vy

@vx

@y+ vz

@vx

@z

�= �

@P

@x�

�@�xx

@x+

@�yx

@y+

@�zx

@z

�+ �gx

�

�@vy

@t+ vx

@vy

@x+ vy

@vy

@y+ vz

@vy

@z

�= �

@P

@y�

�@�xy

@x+

@�yy

@y+

@�zy

@z

�+ �gy

�

�@vz

@t+ vx

@vz

@x+ vy

@vz

@y+ vz

@vz

@z

�= �

@P

@z�

�@�xz

@x+

@�yz

@y+

@�zz

@z

�+ �gz

Equation of Motion for an incompressible uid, 3 components in cylindrical coordinates

�

�@vr

@t+ vr

@vr

@r+

v�

r

@vr

@��

v2�r+ vz

@vr

@z

�= �

@P

@r�

�1

r

@(r�rr)

@r+

1

r

@�r�

@��

���

r+

@�rz

@z

�+ �gr

�

�@v�

@t+ vr

@v�

@r+

v�

r

@v�

@�+

v�vr

r+ vz

@v�

@z

�= �

1

r

@P

@��

�1

r2@(r2�r�)

@r+

1

r

@���

@�+

@��z

@z

�+ �g�

�

�@vz

@t+ vr

@vz

@r+

v�

r

@vz

@�+ vz

@vz

@z

�= �

@P

@z�

�1

r

@(r�rz)

@r+

1

r

@��z

@�+

@�zz

@z

�+ �gz

Equation of Motion for an incompressible uid, 3 components in spherical coordinates

��@vr@t

+ vr@vr@r

+ v�r

@vr@�

+v�

r sin �@vr@��

v2�+v2�r

�

= �@P@r�

�1r2

@(r2�rr)@r

+ 1r sin �

@(�r� sin �)@�

+ 1r sin �

@�r�@�

����+���

r

�+ �gr

��@v�@t

+ vr@v�@r

+ v�r

@v�@�

+v�

r sin �@v�@�

+ vrv�r�

v2� cot �

r

�

= � 1r@P@��

�1r2

@(r2�r�)@r

+ 1r sin �

@(��� sin �)@�

+ 1r sin �

@���@�

+ �r�r�

(cot �)���r

�+ �g�

��@v�@t

+ vr@v�@r

+ v�r

@v�@�

+v�

r sin �@v�@�

+vrv�r

+v�v� cot �

r

�

= � 1r sin �

@P@��

�1r2

@(r2�r�)@r

+ 1r

@���@�

+ 1r sin �

@���@�

+�r�r

+(2 cot �)���

r

�+ �g�

1

Equation of Motion for incompressible, Newtonian uid (Navier-Stokes equation) 3 components inCartesian coordinates

�

�@vx

@t+ vx

@vx

@x+ vy

@vx

@y+ vz

@vx

@z

�= �

@P

@x+ �

�@2vx

@x2+

@2vx

@y2+

@2vx

@z2

�+ �gx

�

�@vy

@t+ vx

@vy

@x+ vy

@vy

@y+ vz

@vy

@z

�= �

@P

@y+ �

�@2vy

@x2+

@2vy

@y2+

@2vy

@z2

�+ �gy

�

�@vz

@t+ vx

@vz

@x+ vy

@vz

@y+ vz

@vz

@z

�= �

@P

@z+ �

�@2vz

@x2+

@2vz

@y2+

@2vz

@z2

�+ �gz

Equation of Motion for incompressible, Newtonian uid (Navier-Stokes equation), 3 components incylindrical coordinates

�

�@vr

@t+ vr

@vr

@r+

v�

r

@vr

@��

v2�r+ vz

@vr

@z

�= �

@P

@r+ �

�@

@r

�1

r

@(rvr)

@r

�+

1

r2@2vr

@�2�

2

r2@v�

@�+

@2vr

@z2

�+ �gr

�

�@v�

@t+ vr

@v�

@r+

v�

r

@v�

@�+

vrv�

r+ vz

@v�

@z

�= �

1

r

@P

@�+ �

�@

@r

�1

r

@(rv�)

@r

�+

1

r2@2v�

@�2+

2

r2@vr

@�+

@2v�

@z2

�+ �g�

�

�@vz

@t+ vr

@vz

@r+

v�

r

@vz

@�+ vz

@vz

@z

�= �

@P

@z+ �

�1

r

@

@r

�r@vz

@r

�+

1

r2@2vz

@�2+

@2vz

@z2

�+ �gz

Equation of Motion for incompressible, Newtonian uid (Navier-Stokes equation), 3 components inspherical coordinates

��@vr@t

+ vr@vr@r

+ v�r

@vr@�

+v�

r sin �@vr@��

v2�+v2�r

�

= �@P@r

+ ��r2vr �

2vrr2�

2r2

v�@��

2v� cot �r2

�2

r2 sin �@v�@�

�+ �gr

��@v�@t

+ vr@v�@r

+ v�r

@v�@�

+v�

r sin �@v�@�

+ vrv�r�

v2� cot �

r

�

= � 1r@P@�

+ ��r2v� +

2r2

@vr@��

v�r2 sin2 �

�2 cos �r2 sin2 �

@v�@�

�+ �g�

��@v�@t

+ vr@v�@r

+ v�r

@v�@�

+v�

r sin �@v�@�

+vrv�r

+v�v� cot �

r

�

= � 1r sin �

@P@�

+ ��r2v� �

v�r2 sin � +

2r2 sin �

@vr@�

+ 2 cos �r2 sin2 �

@v�@�

�+ �g�

where, in these equations, r � 1r2

@@r

�r2 @

@r

�+ 1

r2 sin �@@�

�sin � @

@�

�+ 1

r2 sin2 �

�@2

@�2

�.

Reference: R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley: NY, 1960.

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