Equations in curvilinear coordinates for fluids with non ...

Equations in curvilinear coordinates for fluids with
non-constant viscosity

M. Membrado, A. F. Pacheco

Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain.

E-mail: [email protected], [email protected]

(Received 16 September; accepted 22 December 2011)

Abstract
Explicit formulae for the viscosity stress tensor, equations of motion and kinetic energy dissipation are provided in
cylindrical and spherical coordinates for Newtonian fluids with non-constant viscosity.

Keywords: Newtonian fluids, non-constant viscosity.

Resumen
Las formulas explícitas para el tensor, ecuaciones de movimiento y la disipación de la energía cinética son
proporcionadas en coordenadas cilíndricas y esféricas para fluidos Newtonianos con viscosidad no-constante.

Palabras clave: Fluidos Newtonianos, viscosidad no-constante.

PACS: 51.20.+d, 92.60.H- ISSN 1870-9095

I. INTRODUCTION deal with a fluid where viscosity is not a constant, and in
curvilinear coordinates, we needed the above mentioned
The coefficients of viscosity  and  in a viscid fluid vary equations though in a restricted form. This paper is
organized as follows. In Section 2, after mentioning several
with temperature and pressure. In general T and p, and generalities about fluids, we write the viscosity stress tensor
therefore  and  , are not constant throughout the fluid. and the equations of motion of a viscous fluid. In Section 3,
the components of the viscosity stress tensor and those of
Taking this fact into account adds new terms to the the equations of motion are deduced in cylindrical and
equations of motion and to the formula of kinetic energy spherical coordinates. In Section 4, we present the general
dissipation. As far as we know, these formulae in equation of the dissipation of kinetic energy in viscous
cylindrical and spherical coordinates have appeared neither fluids and its form in curvilinear coordinates. As an
in the specialized literature nor in physics manuals. The example, this equation is utilized in Section 5 to compute
reason for this omission is due to the fact that in most cases the dissipation of kinetic energy in the solution found in MP
where fluid mechanics is applied, the hypothesis of constant for the atmospheric rotation. Finally, in Section 6, we set
viscosity is a reasonable assumption. But in large extended out the conclusions.
physical systems such as planets and stars it might be that
this hypothesis is no longer valid. Thus, the purpose of this II. GENERALITIES ABOUT FLUIDS AND VIS-
paper is the deduction and presentation of these formulae, COSITY STRESS TENSOR
in curvilinear coordinates, for fluids when viscosity is not
constant. The motion of any fluid is subject to a kinematic constraint
based upon the conservation of mass. This is expressed by
On the other hand, the main academic interest of this means of the continuity equation:
paper lies on the fact that students are usually familiarized
with tensors in different coordinate basis but they rarely     (v)         , (1)
have dealt with the unitary vectors of these basis. In this   v v 
paper, they have the opportunity to work with these unitary t
vectors in curvilinear coordinates regarding both their
coordinate transformations and their derivatives. where  is the velocity of the fluid and  the mass
v
The motivation for this work arose from Membrado and
Pacheco [1], hereafter MP. There we studied the rotation of density. When  is constant, i.e., the fluid is
the atmosphere under the simplifying hypothesis that
molecular viscosity is solely responsible for the differential incompressible, Eq. (1) is simplified to:
rotation of the successive air layers. In MP we used the
formalism of classical fluids mechanics and, as we had to 702 http://www.lajpe.org

Lat. Am. J. Phys. Educ. Vol. 5, No. 4, Dec. 2011

M. Membrado and A. F. Pacheco (2)  v   (v        '  , (10)
v  0. )v p 

t

As is well known, in inviscid (ideal) fluids, the ith where  ' is the notation for the intrinsic form of the
component of the amount of momentum flowing in unit viscosity stress tensor. In the added term,  represents an
time through the unit area perpendicular to the xk axis is external field acting on the fluid, typically a gravitational
given, in tensorial form, by: field.

ik  p ik  vi vk . (3) This equation without  , for an incompressible fluid
and for  and  constants, becomes

This tensor ik is called the momentum flux density tensor      )v   1     2  , (11)
v (v p v
and p is pressure. The equation of motion for an ideal fluid
is known as the Euler equation and reads as follows t  
(henceforth, summation over repeated suffixes is assumed):

which is the Navier-Stokes equation.

vi    ik (4) III. VISCOSITY STRESS TENSOR AND FLUID
t xk (5) EQUATIONS IN CYLINDRICAL AND SPHER-
ICAL COORDINATES
or

  vi  vk vi    p .
 t xk  xi
  The components of  'ik in Eq. (8) correspond to Cartesian

In viscous fluids, the tensor written in (3) is generalized by coordinates. To deduce these components in cylindrical or
adding a new term which is responsible for the viscous spherical coordinates, we start by writing this tensor in its
transfer of momentum in the fluid: intrinsic form (see for example, McQuarrie [3]), namely:

   pik  vivk  'ik    vivk ; (6)  '     )     ) T ]  (  2     , (12)
ik ik [( v ( v )( v)I
3

the tensor  ik is called the stress tensor and  'ik the where  is the unity tensor, the symbol    represent
viscosity stress tensor. Thus, they are related by I a b

the dyadic product of the two vectors and the superscript T

   'ik  p ik . (7) means transposition of the operator; i.e.,
ik  
 b )T b 
(a    a .

Using general arguments (see for example, Landau and A. Cylindrical coordinates (r, , z)
Lifshitz [2]), the  'ik tensor for a Newtonian fluid is
expressed by: In these coordinates, the gradient operator and the velocity
vector have the form:
 vi  vk 2 vl  vl
 'ik    xk xi  3  ik xl    ik xl , (8)
 
   ˆ   zˆ  , (13a)
  rˆ (13b)
r r  z
which is symmetric. The quantities  and  are called   rˆvr  ˆv  zˆvz .
v
coefficients of viscosity and are both positive.
Then the equations of motion of a viscous fluid, in the Then, to compute (12) one has to bear in mind that the three
unitary vectors are orthogonal, and that the derivative
most general form, are obtained by adding  'ik / xk to the operators act not only on the components of the velocity but
right hand side of the Euler equation, given in (5). The also on the unitary vectors:
result is:

  vi  vk vi    p    vl  rˆ  ˆ , (14a)
 t xk  xi xi  xl   (14b)
  
ˆ  rˆ .
    vi  vk  2 vl  , (9) 
xk  xk xi 3 ik xl


which is equivalent to After performing the derivation of the vectors and
Lat. Am. J. Phys. Educ. Vol. 5, No. 4, Dec. 2011 comparing the result with

703 http://www.lajpe.org

'  'rr rˆ  rˆ  'r rˆ ˆ  'rz rˆ  zˆ Equations in curvilinear coordinates for fluids with non-constant viscosity
 'r ˆ  rˆ  ' ˆ ˆ  'z ˆ  zˆ
'zr zˆ  rˆ 'z zˆ ˆ 'zz zˆ  zˆ ,  vr    vr  v vr  vz vr  v2  
t vr r r  z r 
(15) 

 p      2 vr    v
r r r r 

the nine components of the tensor are identified. They are:  1   v  1 vr  v  (21)
r   r r  r 

  'rr  2 vr     2     , (16a)     v z  vr           
r  3  v (16b) z  r z  r  v 
(16c)
(16d) 3
(16e)
 'r    v  1 vr  v    'r , (16f)     2vr 1  2 vr  2vr  1 vr 2 v  vr  ;
 r r  r   r 2 r2  2 z 2 r r r2  r2 

 'rz    vz  vr    'zr ,  component:
 r z 

  '   2 v  2vr     2      ,  v  v  v v v vr v 
 r  r  3   v t vr r r  z r 

 vz v     vz  
  z 
 'z  1    'z ,  1 p     v  1 vr v 
r r  r  r  r r  r 
 
  'zz 2 vz   2   
  z    3     v ,  1   2 v  v
r   r  
 2vr   (22)
r

where    1 vz v  1       
z  r  z  r   v 
   3 

 vr  1 v  vz  vr . (17)   2v 1  2v   2v  1 v 2 vr v 
v   r 2 r2  2 z 2 r r r2  r2 
r r  z r 
  ;

The form of the continuity equation in these coordinates is

z component:

    (vr )  1 (v )  (vz )  vr  . (18)
t  r r  z r 
 vz    vz  v v z  vz vz  
t vr r r  z 

Now, as is expressed in (10), the equation otefnsmoor tio'n.  p       v z  vr 
requires the computation of the divergence of the z z r  r z 

As said above, the derivatives act also on the unitary  1  1 vz v  (23)
r   r  z 
vectors. Performing this derivation first and then grouping 

terms, one finds:  vz  v     
z z   v 
   '   2       
  'rr  1  'r   'zr z 3
 rˆ r r  z  1 ( 'rr  ' )
r  2vz 1 2vz  2vz 1 vz 
   r 2 r2  2 z 2  r r  .
 'r  1  '   'z
 ˆ  r r  z  1 ( ' r  'r ) (19)
 r

 zˆ   'rz  1  'z   'zz  1  'rz  . Note the presence of terms involving derivatives of the
 r r  z r  viscosity. These terms are absent in fluid mechanics books
and physics vade mecums.

To obtain (19), (14) has been used together with the identity B. Spherical coordinates (R, , )

aˆ  (bˆ  cˆ)  (aˆ  bˆ)  cˆ . (20) The calculus in spherical coordinates ( R , , ), where 

Finally, performing the derivation of the tensor components and  are the zenith and the azimuth angle respectively, is
given in (16), the three components of the equation of similar to that explained above for cylindrical coordinates.
motion are:
Here, the gradient operator and the velocity are
r component:
 Rˆ   ˆ  ˆ , (24a)
   (24b)
R R  R sin  

v  RˆvR  ˆv  ˆv .

Lat. Am. J. Phys. Educ. Vol. 5, No. 4, Dec. 2011 704 http://www.lajpe.org

M. Membrado and A. F. Pacheco  ˆ   'R  1  '  1  '
The derivatives of the unitary vectors are  R R  R sin 

Rˆ  ˆ,
 (25a)  cos  ( '  ' )  1 (2 ' R  'R ) (29)
(25b) R sin R
Rˆ (25c)
  ˆsin, (25d)  ˆ   'R  1  '  1  '
(25e)  R R  R sin 
ˆ  Rˆ , 

 cos  (  1 (2  
R sin R )
 ' ' )  ' R 'R .

ˆ  ˆcos, And using the tensor components collected in (26), the
 three components of the equations of motion are:

ˆ  Rˆ sin  ˆ cos.


R component:

And the list of components of the viscosity stress tensor is:

vR   vR  vR  v vR  v vR  (v2  v2 )  
R v t   vR R R  R sin   R 
  'RR   2     2  , (26a) 
 3  (26b) 
(26c)
 'R    v  1 vR  v    'R , (26d)  p     2 vR    v
 R R  R  (26e) R R R R 

 v 1 vR v  (26f)  1   v  1 vR  v 
 R R sin   R  R   R  R 
 'R       'R ,  R 

  '    2 v  2vR     2     ,  1   v  1 vR  v 
 R  R  3   v R sin    R R sin   R 

 1 v 1 v v cos  ,          (30)
 R  R sin   R sin  R 3   v 
 '         '

 2 v 2vR 2v cos      2vR  1  2vR  1  2vR  2 vR
 sin   R Rsin   R 2 R2  2 R 2 sin 2   2 R R
 '     
R
 cos vR  2 v  2 v
    2     . R2 sin  R2  R2 sin 
3   v

 2vR  2v cos   ;
R2 R2 sin  
where 

 vR 1 v 1 v  2vR  v cos . (27)  component:
v
  
R R  R sin  R R sin
v  v  v v v v
 t   vR R R   R sin 

The form of the continuity equation in these coordinates is 

   (vR ) 1 (v ) 1 (v )  vR v  v2 cos   
t  R R  R sin   R R sin  
     

 1 p       v  1 vR  v 
R  R  R  R  
 2vR  v cos  . (28)  R R 
R R sin  
1  2 v 2vR  
 R     R    v
 R
After performing the derivation of the unitary vectors, the 
divergence of the tensor reads as follows:
 1   1 v  1 v  v cos   (31)
R sin   R  R sin  R sin  
   
  'R
 '  Rˆ   'RR  1   1  'R   1         
 R R R sin   R  3   v 

 cos   'R  1 (2 'RR  '  '      2v  1  2v  1  2v  2 v
R sin R )  R 2 R2  2 R 2 sin 2   2 R R


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Equations in curvilinear coordinates for fluids with non-constant viscosity
  
 2 vR  cos  v  2 cos  v   1 v 2    1 v 2  ( v)  v 1 v2 
R2  R2 sin   R2 sin2  
t  2  v2 p    2v 
v ( 
 v  ;    ')  
sin 
R2 2    v 1     ' v  (36)
pv v
 2 v2   

 component:      (v)  'ij vi .
p  v  x j

 v    v  v v  v v This is the most general form to express the kinetic energy
t vR R R  R sin 


dissipation for unit volume in a fluid.  

 vR v  v v cos   ('No)w ,v we will express the two terms v  ' and
R R sin   in cylindrical and spherical coordinates.

 1 p       v  1 vR  v  Cylindrical coordinates:
R sin   R sin   R  R R sin   R 
 ' (vr v vz rˆ
1  1 v 1 v v cos  v 'rr 'r 'zr )
R   R  R sin   R sin 
    (vr 'r v ' vz 'z ) ˆ (37)

1  2 v 2vR 2v cos   (32)  (vr 'rz v 'z vz 'zz ) zˆ ;
R sin    R R sin   v
  R sin     

 (   2 vr 2  v  1 vr  v 2
') v r  r r  r 
1            
R sin   3   v  
  

    2v  1  2v  1  2v  2 v   vz  vr 2  21r v  vr 2
 R 2 R2  2 R2 sin 2   2 R R  r z   r 

 v 1 vz  2 2 vz 2 
 z r   z  
 cos  v  2 vR  2 cos  v    (38)
R2 sin   R2 sin   R2 sin2  
(
 v  .    2     ) vr 1 v  vr  vz  .
sin   3 v  r r  r z 

R2 2 

Spherical coordinates:

IV. EQUATION FOR THE RATE OF ENERGY  '  (vR 'RR v 'R v 'R ) Rˆ
DISSIPATION IN A FLUID v

 (vR 'R v ' v ' ) ˆ (39)

The kinetic energy in a unit volume and the total kinetic  (vR 'R v ' v ' ) ˆ ;
energy in a fluid are equal to

  1 v 2 , (33) (      2 vR 2   v  1 vR  v 2
kin 2 ') v R  R  R 
  R 


 v 1 vR v  2  1 v vR  2
 R R sin   R  2 R  R 
and      


1 (34)  1 v 1 v v cos  2
2  R  R sin   R sin 
Ekin  v 2 dV ,     

V

 1 v  vR v cos   2 
2 sin  R R sin   
respectively. Assuming that the borders of the volume are    (40)
fixed, the time variation of the kinetic energy in that
volume is R 

   2   (  v )  vR  1 v  2 vR
3   R R  R
dEkin    1 v 2  dV . (35) 

dt V t  2   1 v  v cos  .
R sin  R sin  


Using the equation of motion (10) and the continuity Eq.
(1), we find:

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M. Membrado and A. F. Pacheco so that

V. AN EXAMPLE OF KINETIC ENERGY    ' )
DISSIPATION (v

 nˆ   rv v  v2  zv v  . (48)
R r z 
As an explicit example of the use of the equation of kinetic
energy dissipation, we will apply (36), (37) and (38) to the Now, we change the derivatives from (r, z) to (r, R)
solution found in MP for the atmospheric rotation. In MP it coordinates and make use of (42). The net result for the
was assumed that the vector velocity of the air was as surface term in (45) is:
follows:

  vˆ , (41) TS   (v  ')  nˆdS
v

S

where v was a function of (r, z) or equivalently of (r, R) 82   dR   dR   dR 2 (49)
 3  Rmax R 4 Rmin R 4  R 4 
which implies that   v  0 . This equation is     R0 

mathematically equivalent to the continuity Eq. (2) of an 82 Rmax dR    dR 2
3  R4  R4 
incompressible fluid, though the atmosphere is definitely    Rmin  R0  .

not such a fluid.

The explicit solution for the velocity of rotation of the

air in MP was In the upper line of (49) the second term is positive and

represents the energy transferred to the air contained in the

 dR    dR 1 (42) volume through the bottom surface of the shell. The source
 R4   R4 
 v(r,R) r R  R0  , of this energy is the rotation of the lower atmospheric

layers. The first term is negative and represents the loss of

energy of the air in this volume through the top surface.

where  and R0 are the angular velocity and radius of The balance between these terms written in the lower line,
Earth respectively. R is the distance to the center and r the
distance to the axis of rotation. As said above, a solution of TS, provides the net kinetic energy given to that volume of
this type implies that
atmosphere. Let us consider now the volume term in (45).

In cylindrical coordinates, the velocity and gradient was

  givevn in (13). Using (38) and exploiting the fact that
 v 0 , we find
  0 , (43)

is fulfilled, and also vi  v  v 2   v  2 
x j  r r   z 
 'ij   . (50)

  (v)  0 . (44)  


Equation (44) is a consequence of assuming a steady state. Now, passing from the coordinates (r, z) to (r, R), (50)
Therefore, in this problem, Eq. (36) is simplified: adopts the form

dEkin   1 p    'ij vi  v 2 , (51)
  2   ' x j  R 
dt     v  v 2     v   dS

S  

  'ij vi dV . (45) and using the explicit form of the fluid velocity (42), the
x j result is
V

The first term in this formula has been obtained using vi   82 Rmax dR    dR 2 (52)
x j 3 
 

V
Gauss' theorem.   TV 'ij dV    .
  R4  R0  R4
The volume in these equations can be fixed at will. We Rmin 

will choose a thick spherical shell contained between Rmin This is the kinetic energy rate of dissipation in the volume
and Rmax. Both Rmin and Rmax are bigger than R0. considered. This term, TV, exactly balances TS as is required
by the principle of energy conservation.
Then, the unitary vector perpendicular to the top

surface, in cylindrical coordinates, is

nˆ  1 (r,0, z) , (46)
R

and using (41), we find VI. CONCLUSIONS

  nˆ  0 , (47) In this paper we have presented, in the most general form,
v the viscosity stress tensor and the equations of motion of
viscid fluids, for cylindrical and spherical coordinates, with
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the hypothesis that viscosity is not constant. The complete Equations in curvilinear coordinates for fluids with non-constant viscosity
set of these equations is written in Section 3. The general are considerably simplified and for example it is possible to
formula for the rate of dissipation of kinetic energy in these recognize that the energy dissipated in a fixed volume of
fluids has been also deduced and is written in Section 4. We atmosphere is the difference between the energy transferred
believe that the set of formulae contained in this paper are by the atmosphere through the bottom surface and the
original and useful for dealing with Newtonian fluids in energy lost through the top surface.
extended systems where the symmetry of the problem
imposes the use of curvilinear coordinates and where REFERENCES
viscosity cannot be considered as a constant.
[1] Membrado, M. and Pacheco, A. F., Decrease of the
Some of the formulae collected here are obtained atmospheric co-rotation with height, Eur. J. Phys. 31, 1013-
through lengthy but instructive calculations. The details of 1020 (2010).
any of them are available on request. [2] Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, 2nd
Ed. (Butterworth-Heineman, Oxford, 2003), p. 46.
As an illustrative example, we have computed the [3] McQuarrie, D. A., Statistical Mechanics, (Harper &
kinetic energy dissipation in the solution studied in MP for Row, New York, 1976), p. 385.
atmospheric rotation. There, the viscosity coefficients were
assumed to vary with T but not with p. Besides, with the
ansatz assumed for the rotation velocity (41), the formulae

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