by Rummler, B..

**Series:** 2001-14, Preprints

- MSC:
- 76F99 None of the above, but in this section
- 34A34 Nonlinear equations and systems, general
- 35Q30 ~Navier-Stokes equations

**Abstract:**

A general class of boundary-pressure-driven flows of

incompressible Newtonian fluids in three-dimensional pipes

with known steady laminar realizations is investigated.

Considering the laminar velocity as a 3D-vector-function

of the cross-section-circle arguments, we fix the scale

for the velocity by the $L_{2}$-norm of the laminar velocity.

The usual new variables are introduced to get

dimension-free Navier-Stokes equations. The

characteristic physical and geometrical quantities are subsumed in

the energetic Reynolds number $Re$ and a parameter $\psi$,

which involves the energetic ratio and the directions of

the boundary-driven part and the pressure-driven part

of the laminar flow.

The solution of non-stationary dimension-free Navier-Stokes

equations is sought in the form

$\underline{{\bf u}}\,=\,{\bf u}_{L}\,+\,{\bf u}$,

where ${\bf u}_{L}$ is the scaled laminar velocity and

periodical conditions in center-line-direction

are prescribed for $\bf u $. An autonomous system (S) of

ordinary differential equations for the time-dependent

coeff\/icients of the spatial Stokes eigenfunction is got by

application of the Galerkin-method

to the dimension-free Navier-Stokes equations for $\bf u$.

The finite-dimensional approximations

${\bf u}_{N(\lambda)}$ of ${\bf u}$ are defined

in the usual way. A class of timely periodical solutions

near to the laminar velocities

but different from them was found by parameter studies

for the numerical solution

of finite-dimensional subsystems of (S).

This class of timely periodical strange numerical solutions

seems to stay for one of the first links in the bifurcation chain to

turbulence.

**Keywords:**

Navier-Stokes equations, Stokes eigenfunctions, Galerkin methods, transition to turbulence