Colloquium FLUID DYNAMICS 2008
Institute of Thermomechanics AS CR, v.v.i., Prague, October 22  24, 2008
p.
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 (18881956) 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 NavierStokes 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
where is the velocity vector, are the operators of Hamilton and Laplace, is the resultant of external forces, P denotes the pressure and are the density and the kinematic viscosity. A bracket indicates the scalar product and similarly is the vector product of vectors . 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
Usually, the second term on the left hand side of the equation describing the convective derivation is useful to decompose and simultaneously the modified pressure p can be introduced as follows
Then the equation simplifies to
This formulation of NS equations emphasizes the significance of vorticity and the nonlinearity of equations. Exact solutions of the NS 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 , there exist exact solutions that are linearly superposable i.e. each of the distinct velocity vector fields and meet the solution of as well as their sum if certain conditions would be accomplished. They must be valid following vector equations simultaneously
According to the vector analysis are valid following formulas
Hence equations , and are met simultaneously if the relation is fulfilled
Thence two different solutions and are linearly superposable if they fulfil the relation
where is a scalar function of spacetime coordinates. Similarly it follows that a selfsuperposable velocity field, i.e. , called the generalized Beltrami flow [6] must fulfil equation
The simplest selfsuperposable velocity fields satisfied the equation
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 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 in component representation but assuming the compressibility of fluid (2^{nd }viscosity) and the potential of body forces . Applying the operator rot (curl) to the NS equations we receive
Using relations from the vector analysis
we rearrange the previous equation to the vorticity equation in a common form
This equation relates to the equation (4) presented in [1]. The vector 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 . Then the components of the vorticity vector are subject to equations that are analogous to the heat convection equation (diffusion of vorticity)
Further the possibilities of the pressure field determination are discussed if the vector vanishes identically
Initially Trkal [1] assumes the scalar equals zero
then also the vorticity vanishes and the velocity potential exists. Inserting into the equation 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
where T(t) is an arbitrary function of time.
Another possibility how achieve the zero value of the vector is the configuration with vortex lines coinciding with stream lines
This is the most important of the investigated events  a helical motion of fluid particles with a scalar function . Trkal has shown that then the equation 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 follows with the regard to vector analysis
This is possible to arrange in the form
Next the solution continues similarly to Trkal. Having in mind from the vector analysis
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
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
This solution must meet the relation . Thus must hold
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
Then the velocity vector follows from and in the form
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 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 and became known as Trkal flow or Trkalian flow most likely since fifties of the 20^{th} century owing to the Norwegian physicist Björgum [4]. However, the diffusion of Trkal result begun already in thirties of the 20^{th} 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 4^{th} 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 abovementioned publications and also in the monograph of Truesdell [22]. In the connection with Trkal results, the publications from the turn of 20^{th} and 21^{st} 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, timeharmonic 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 crosshelicity and magnetized Trkal flows.
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 comoving observer published in 2006.
The contribution of Beltrami flows to theoretical study of turbulent flows and the Trkal contribution to exact solutions of NS 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 and the complex version of the Trkalian equation
where is a three component vector function with each component in the complex field which may be expressed in the form
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 NavierStokes 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.

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