Engineering Transactions, 11, 3, pp. 509-555, 1963

### Metoda Wyznaczania Pewnych Klas Przepływów za Pomocą Modelowania na Grupach Liego

P. Kucharczyk
Instytut Podstawowych Problemów Technicznych PAN
Poland

The method described concerns a class of geometrically simple flows. It is based on an analysis of the geometrical structure of the equations considered and the research of Lie transformation groups discerned by this structure. For this purpose are applied certain notions and theorems of more recent branches of differential geometry constituting a natural development of the tensor analysis. They are explained in Sec. 2 which, therefore, plays the role of a mathematical introduction. Among other things it contains the definition of Lie groups, Lie derivative, invariance groups of linear geometric objects and interpretation of the invariance condition in canonical coordinates of one-parameter groups.
On this basis it is proved in Sec. 3 that in the geometric structure of the equations of gas dynamics the six-parameter isometry group and seven parameter space similarity group play a special role. This fact enables us to select definite classes of flow connected with sub-groups of the above groups by the conditions (3.2) and (3.4), respectively, called of conditions of geometric modelling because they determine the geometric flow pattern. If the condition (3.4) is used to model flows on similarity groups, the constants appearing in it must satisfy the relations (3.6) or (3.7) which may be inter-preied as a requirement that the local Euler number and the SO called simultaneity parameter is invariant along trajectories of the modelling group. An investigation of the connexions with the theory of modelling of flows (Sec. 3.8) shows that the flow classes introduced are modelled intrinsically.
Final equations of flows modelled on similarity and isometry groups do not contain the canonical coordinate along the trajectory of the modelling group. In the steady-state case their determination
reduces therefore to the solution of sets of equations containing only two independent variables.
In this case the streamlines have also the property of mutual transformation with finite transformations of the modelling group.
The method is illustrated in greater detail in Sec. 4 by investigating plane and stationary flows modelled on similarities of the plane (the sub-group of spiral motions). These flows are determined from a set of ordinary differential equations. In particular it is shown in Sec. 4.4 that the appearance of limiting lines in the Tollmien flow is connected with a geometric limitation of the flow inside a region.
Section 5 is devoted to the application of the method to the equations of gas dynamics in time- space in a tensor form given by J. Bonder. In this case we can only distinguish solutions on such time-space similarity groups that satisfy the condition (5.4).
In Section 6, confining ourselves to plane, steady state and incompressible potential flows it is shown that they are intrinsically modelled on suitable groups of conformal transformations.
Finally, using the modelling on a group of similarities of plane steady-state flow of a viscous liquid the equations of Oseen and Hamel flows are derived in Sec. 7. showing a relation between the exact solutions obtained by these authors and the geometrical structure of considered equation.

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