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Colloquium FLUID DYNAMICS 2008
Institute of Thermomechanics AS CR, v.v.i., Prague, October 22 - 24, 2008
p.1
TRKALIAN FLOW FIELDS
Pavel Jonáš
Institute of Thermomechanics AS CR, v.v.i., Praha
The contribution of Professor Viktor Trkal to the fluid mechanics appearing from his paper from
1919 is the theme of this contribution. Trkal solution is briefly reproduced together with the outline how
the main result became known as Trkal flow, Trkalian flow field etc. Several examples of the application
of the Trkalian flows (sometimes known as ABC flows), extended up to present time follow in the full text
of this contribution.
1. Introduction
The story of the Viktor Trkal Note on the hydrodynamics of viscous fluids [1]
seems as very interesting in the epoch producing hundreds papers mostly after a short
time forgotten. Professor Viktor Trkal, Czech physicist (1888-1956) published the paper
[1] in Czech not long after the homecoming from Russia where he spent the wartime
from 1915 till 1919, largely in the prisoner of war status. Trkal solution of the NavierStokes equations is describing the flow field belonging to an important domain of the
fluid mechanics, to the fluid motions with nonzero vorticity. Such flows are embedding
us in nature, exist in our bodies and frequently occur in technology. Later it has been
recognised that Trkal solution of the Navier-Stokes equations forms an important
particular class of Beltrami flows [6] however there is a little probability that Trkal was
familiar with Beltrami study because Beltrami published his paper in Italian,
international communications were difficult at that time and particularly during the
wartime.
2. Beltrami flows
It is convenient introduce Beltrami flows before the interpretation of Trkal results.
The equations of Navier and Stokes are usually written in the form
P
u
  u.  u  G  grad      u
t

(1)
where u is the velocity vector,  and  are the operators of Hamilton and Laplace, G is
the resultant of external forces, P denotes the pressure and  and  are the density and
the kinematic viscosity. A bracket
 a  b  is
 a.b  indicates
the scalar product and similarly
the vector product of vectors a, b . For the sake of simplification of
derivations we suppose (if not explicitly cancelled) the constant property Newtonian
incompressible - divergence free fluid and the conservative body forces with the
potential 
 .u   0;
G  
(2)
Usually, the second term on the left hand side of the equation (1) describing the
convective derivation is useful to decompose and simultaneously the modified pressure
p can be introduced as follows
p.2
 u.  u  grad 
1 2
u    u  rot u  ;
2 
pP
(3)
Then the equation (1) simplifies to
u
1
1


  grad  p   u 2    u      rot ;   rot u
t

2


(4)
This formulation of N-S equations emphasizes the significance of vorticity  and the
non-linearity of equations. Exact solutions of the N-S equations are rare but they are
important because they solve significant problems and simultaneously they serve for
testing of some mathematical or physical models. Regardless of the nonlinearity of (4),
there exist exact solutions that are linearly superposable i.e. each of the distinct velocity
vector fields u1  t,r  and u2  t,r  meet the solution of (4) as well as their sum
u1  t,r   u2  t,r  if certain conditions would be accomplished. They must be valid
following vector equations simultaneously
u1
1
1


  grad  p1   u12    u1  1    rot1 ; 1  rot u1
t

2


(5)
u2
1
1


  grad  p2   u22    u2  2    rot2 ; 2  rot u2
t

2


(6)
  u1  u2 
t

1


grad  p12     u1  u2  .  u1  u2    

2


1
   u1  u2   rot  u1  u2     rotrot  u1  u2 
(7)
According to the vector analysis are valid following formulas
  u1  u2 
u1 u1

;   u1  u2  .  u1  u2     u1 .u1   2  u1 .u2    u2 .u2  ;
t
t
t
  u1  u2   rot  u1  u2     u1  rot u1    u1  rot u2    u2  rot u1   u2  rot u2  ;

rot rot  u1  u2   grad div  u1  u2     u1  u2   u1  u2
Hence equations (5), (6) and (7) are met simultaneously if the relation is fulfilled
0
1

grad  p12  p1  p2    u1 .u2     u1  rot u2    u2  rot u1 
(8)
Thence two different solutions u1  t,r  and u2  t,r  are linearly superposable if they
fulfil the relation
u1  rot u2   u2  rot u1   grad 12 t,r 
(9)
where 12  t,r  is a scalar function of space-time coordinates. Similarly it follows that a
self-superposable velocity field, i.e. u2  n u1 , called the generalized Beltrami flow [6]
must fulfil equation
Colloquium FLUID DYNAMICS 2008
Institute of Thermomechanics AS CR, v.v.i., Prague, October 22 - 24, 2008
p.3
u1  rot u1   grad 1 t,r 
(10)
The simplest self-superposable velocity fields satisfied the equation
rot u  t,r    t,r  u t,r 
(11)
Beltrami studied various flows of this kind and thence they are called Beltrami flows.
3. Trkal solution
Trkal [1] analysed consequences of different properties of the vector product
A   u    on the flow field behaviour. The original paper [1] was written in Czech.
Lakthakia [16] aroused English translation of the paper in 1994. So all details of Trkal
solution are not necessary reproduce here. Trkal started from the equations (1) in
component representation but assuming the compressibility of fluid (2nd viscosity) and
the potential of body forces  . Applying the operator rot (curl) to the N-S equations
(4) we receive
P 1


4
  rot grad   u 2     div u   rot  u      rot rot 
t
3
 2

(12)
Using relations from the vector analysis
rot grad  scalar   0, rot rot   grad div    , grad div   0
we rearrange the previous equation to the vorticity equation in a common form

 rot  u      
t
(13)
This equation relates to the equation (4) presented in [1]. The vector A   u    on the
right hand side of the equation is perpendicular to the plane determined by vectors of
velocity and vorticity. It plays a crucial role in fluid dynamics (vortex stretching,
Coriolis force, Magnus effect etc.). Trkal [1] analysed properties of a flow field with
vanishing vector A . Then the components of the vorticity vector are subject to
equations that are analogous to the heat convection equation (diffusion of vorticity)
A  u     0 

 
t
(14)
Further the possibilities of the pressure field determination are discussed if the vector
A   u    vanishes identically
  c  t,r  u  A   u     c  t,r  u  u   0
(15)
Initially Trkal [1] assumes the scalar c  t,r  equals zero
u     0    c.u,
c  0    0  u   grad 
(16)
p.4
then also the vorticity vanishes and the velocity potential  exists. Inserting  into the
equation (4) and then performing the integration of the equation we receive similarly to
Trkal [1] the formulae for the potential of body forces in compressible fluid flow


P 1
4
2
 T  t     grad     
t
 2
3
(17)
where T(t) is an arbitrary function of time.
Another possibility how achieve the zero value of the vector A   u    is the
configuration with vortex lines coinciding with stream lines
  rot u  c  r , t  u, c  0  u     c u  u   0
(18)
This is the most important of the investigated events - a helical motion of fluid particles
with a scalar function c  t,r  . Trkal has shown that then the equation (14) holds again
and suggested the procedure of the pressure field or the potential of body forces
calculation. However he stated that the necessary integrations could be performed only
in some very special cases. From the relations (18) follows with the regard to vector
analysis
rot   rot  cu   c.rot u   u  grad c   c2u  u  grad c   grad div u  u
This is possible to arrange in the form
u  c2u  grad div u  u  grad c 
(19)
Next the solution continues similarly to Trkal. Having in mind from the vector analysis
div   div  cu   c div u   u grad c   0
(20)
and that the difference of the substantial derivative and the local one equals the
convective derivative, we find that in an incompressible fluid flow holds the relation
 u grad c   0 
Dc c

Dt t
(21)
whence it is evident that the scalar c can be a function of time t only, if all components
of the velocity vector are nonzero. Furthermore Trkal found the solution of the velocity
vector field
u

1
f  x, y , z  exp   4 c 2dt
c

(22)
This solution must meet the relation (19). Thus must hold
f   4 c 2 f
(23)
whence Trkal [1] derived the main result: “in a viscous incompressible fluid, unless at
least one of the velocity components is identically equal to zero, c must be a constant
c  constant
Then the velocity vector follows from (22) and (24) in the form
(24)
Colloquium FLUID DYNAMICS 2008
Institute of Thermomechanics AS CR, v.v.i., Prague, October 22 - 24, 2008
p.5

u  U exp 4 c 2t

(25)
Next the boundary conditions and the limits of the validity are expressed and formulas
describing the fields of the pressure and flow velocity are derived in the paper.
It is interesting that Ballabh [3] reached the result (24) twenty years after Viktor Trkal
no doubt unaware on the Trkal paper [1] similarly as Trkal arrived at his solution no
doubt unaware on Beltrami [6].
4. Story on Trkal flow
The flow defined by the formulas (24) and (18) became known as Trkal flow or
Trkalian flow most likely since fifties of the 20th century owing to the Norwegian
physicist Björgum [4]. However, the diffusion of Trkal result begun already in thirties of
the 20th century when Dryden at al. [13 and 14] quoted the title of the paper [1] in the
list of references attached to the chapter 4. But the author did not found a reference to it
in the 4th chapter.
Björgum [4 and 5] pleads Berker [3] thesis from 1936 with cited Trkal [1].
According to the bibliography from Brdička and Trkal, Jr. [12], the name Trkal and
titles Trkalian flow or field occur frequently in the above-mentioned publications and
also in the monograph of Truesdell [22]. In the connection with Trkal results, the
publications from the turn of 20th and 21st century refer mostly the Aris [2] monograph
and the relevant chapters of the Berker [8] Handbuch der Physik and Flugge [15]
Encyclopedie of physics. Aris does not quote Trkal directly but he refers to the
Truesdell book. However Aris is repeatedly using the term Trkalian field for the vector
fields where the curl of the vector is parallel to the vector and simultaneously directly
proportional with the scalar of proportion independent of position.
The Lakhtakia paper [16] is the important milestone in the frequency of the Trkal
citations in the world literature. Lakhtakia noticed the citation of Trkal from 1919 when
reviewing the manuscript of Reed [18]. He recognized, from the Czech version, the
significance of Trkal contribution in fluid mechanics, time-harmonic electromagnetism
and astrophysics. Lakthakia [16] provoked to translate Trkal [1] in English and briefly
contextualizes the vorticity dynamics, Beltrami fields and Trkal flows in his paper.
Mechanical models utilising theory of vortex motions and ideas on vorticity are applied
since the age of Helmholz, Kelvin, Maxwell etc. Similarly are still in use Beltrami
fields. Further several examples of application of the class of the Beltrami flows that is
known as Trkal flows (sometimes called ABC flows) are briefly introduced. The
examples are meant as starting points for those who desire a deeper insight in the
problem.
Tasso [20] has shown that the force free fields can be generalized to Trkal flows
in magnetohydrodynamics if a special velocity field parallel to the magnetic field is
introduced. He established an exponentiall decay of such flows if the viscosity and
resistivity are constant. Next he derived a sufficient condition for nonlinear stability.
Another example is the paper of Tasso and Throumoulopoulos [21] from 2003, dealing
with cross-helicity and magnetized Trkal flows.
p.6
Another example is an extensive article from Mackay and Lakhtakia [17] on
electromagnetics of moving at constant velocity of an isotropic chiral medium from the
perspective of a co-moving observer published in 2006.
The contribution of Beltrami flows to theoretical study of turbulent flows and the
Trkal contribution to exact solutions of N-S equations are appreciated by Changchun
and Yongnian [19]. These authors commend the physical consequences of Trkal
solution that found in Flugge [15].
Baldwin and Townsend [3] published study on the complex Trkalian fields and
solutions to Euler´s equations in 1995. They distinguish between Trkalian fields, real
vector fields that solve the equation (18) and the complex version of the Trkalian
equation
rot h  h
where h is a three component vector function with each component in the complex field
which may be expressed in the form
h  eig grad F
with g real and F complex. Baldwin and Townsend point out the investigations of
Trkalian (ABC-) flows in the context with problems of stability and chaos.
Many other examples of the utilisation and referring to the Viktor Trkal Sr. are available
from last two decades.
5. Conclusion
The clever analysis of the Navier-Stokes equations brought Professor Viktor Trkal
to the solution that became known after several decades. Next it was also modified and
generalized to electromagnetism, magneto hydrodynamics, superconductivity and solar
physics. Trkal flow, Trkal field are a name in the fluid mechanics.
References
[1]
Trkal, V. (1919) Poznámka k hydrodynamice vazkých tekutin. Časopis pro
pěstování matematiky a fysiky 48, 302-311.English translation: Trkal, V. (1994)
A note on the hydrodynamics of viscous fluids (translated by I. Gregora) Czech. J.
Phys. 44, 97-106.
[2]
Aris, R. (1962) Vectors, tensors and the basic eqs. of fluid mechanics. Prentice –
Hall. Inc. 73.
[3]
Baldwin, P.R. and Townsend, G.M. (1995) Complex Trkalian fields and solutions
to Euler’s equations for the ideal fluid. Archiv: Chao/Dyn/9502012v1, 1-29, Phys.
Rev. E, 51 (3), 2059-2068.
[4]
Ballabh, R. (1940) Self superposale motions of the type  = .u etc.Proc. Benares
Math. Soc. (N.S.) 2, 85-89.
[5]
Ballabh, R. (1950) On coincidence of vortex and stress lines in ideal liquids.
Ganita 1, 1-4.
Colloquium FLUID DYNAMICS 2008
Institute of Thermomechanics AS CR, v.v.i., Prague, October 22 - 24, 2008
p.7
[6]
Beltrami, E. (1889) Considerazioni idrodinamiche. Rend. Inst. Lombardo Acad.
Sci. Lett. 22, 122-131, english translation by G. Filipponi: Int. J. Fusion Energy 3
(1985) no. 3, 53-57.
[7]
Berker, A.R. (1936) PhD.-Thesis, Lille.
[8]
Berker, A.R. (1963) Handbuch der Physik. Springer – Verlag, Berlin, Vol. VIII/2,
p. 161.
[9]
Bjǿrgum, O. (1951) On Beltrami vector fields and flows. Pt. I. A comparative
study of some basic types of vector fields. Universitetet i Bergen Årbok.,
Naturvitenskapeling rekke Nr. 1.
[10] Bjǿrgum, O. and Godal, T. (1952) On Beltrami vector fields and flows. Pt. II. The
case when W is constant in space. Universitetet i Bergen Årbok.,
Naturvitenskapeling rekke Nr. 13.
[11] Brdička, M. (1959) Mechanika kontinua. Nakladatelství ‚ČSAV, Praha.
[12] Brdička, M. a Trkal, V. (2007) Profesor Viktor Trkal – Pouť moderní fyzikou.
Academia, Praha.
[13] Dryden, H.L., Murnaghan, E.D. and Bateman, H. (1932) Report of the Committee
on Hydrodynamics. Bulletin of the National Research Council no. 84, 1-634.
[14] Dryden, H.L., Murnaghan, F.D. and Bateman, H. (1956) Hydrodynamics. Dover
Publications, INC. New York, N.Y., Unabridged and unaltered republication of
[2].
[15] Flugge, S. (1963) Encyclopedia of physics. Vol. VIII/2, Fluid dynamics II. New
York, Springer-Verlag, p. 161.
[16] Lakhtakia, A. (1994) Viktor Trkal, Beltrami fields and Trkalian flows. Czech. J.
Phys. 44, 89-96.
[17] Mackay, T.G. and Lakhtakia, A. (2006) On elektromagnetics of an isotropic chiral
medium moving at constant velocity. Archiv: Physics/0606158v2, 1-22.
[18] Reed, D. (1993) Archetypal vortex topology in nature. Speculat. Sci. Technol.,
accepted for publication.
[19] Shi Changchun and Huang Yongnian (1991) Some properties of threedimensional Beltrami flows. Acta Mechanika Sinica, Vol. 7, No. 4 (ISSS 05677718), 289-294.
[20] Tasso, H. (1995) Trkal flows in magnetohydrodynamics. Phys. Plasmas 2 (5),
1789-1790.
[21] Tasso, H. and Throumoulopoulos, G.N. (2003) Cross-helicity and magnetized
Trkal flows. Physics of Plasmas, Vol. 10, No. 12, 4897-4898.
[22] Truesdell, C. (1954) Kinetics of vorticity. Bloomington.