_{1}

^{*}

The Navier-Stokes equations for incompressible fluid flows with impervious boundary and free surface are analyzed by means of a perturbation procedure involving dimensionless variables and a dimensionless perturbation parameter which is composed of kinematic viscosity of fluid, the acceleration of gravity and a characteristic length. The new dimensionless variables are introduced into the equation system. In addition, the perturbation parameter is introduced into terms for deriving approximations systems of different orders. Such systems are obtained by equating coefficients of like powers of perturbation parameter for the successive coefficients in the series. In these systems several terms are analyzed with regards to size and significance. Based on those systems, suitable solutions of NS equations can be found for different boundary conditions. For example, a relation for stationary channel flow is obtained as approximation to the NS equations of the lowest order after transformation back to dimensional variables.

The classical Navier-Stokes equations, which were formulated by Stokes and Navier independently of each other in 1827 and 1845, are analyzed with the perturbation theory, which is a method for solving partial differential equations [

In further studies, the focus is laid on incompressible flows bounded with free surfaces and a solid wall with the no-slip condition which is experimentally well-detected.

The Euler approach of the incompressible flows in Cartesian coordinates yields with the components of velocity

In the general case for the solution of the Navier-Stokes equations it is necessary to describe velocity components and pressure values at the boundaries of an observed flow. If necessary, specifications of temperature or heat flux are required.

In the considered incompressible flows (

and as dynamic condition

applies to the fixed boundary (bottom)

The basic differential equation system for the flows under consideration is thus from Equations (01) to (05).

For the futher developments a formal perturbation procedure is used, which is developed in this form of Friedrichs [

For the application of the disturbance producer, the definition of a perturbation parameter is of crucial importance. For the present partial differential equation system, it is advisable to use the kinematic viscosity of fluid ^{2}/s] as the decisive fluid property, the acceleration due to gravity g [m/s^{2}] and an arbitrary depth

which can be considered as a dimensionless kinematic viscosity in an earthly gravity field. Taking into account the developments in [

To form the dimensionless pressure, the density

For the transformation of the variables, the following relationships are used:

In the next step, the newly defined variables are inserted in the system of differential Equations (01) to (05). In addition, is dispensed with the identification of new variables by (*).

From the continuity Equation (01) results in

respectively

From the momentum Equations (02a) to (02c), we obtain

respectively

respectively

respectively

Equations for the boundary conditions maintain the form

respectively

and

respectively

As a result, the basic system of differential equation takes the form

The selected perturbation parameter represent, for example, for liquid water with

The next step is to assume power series developments for

and insert them in the Equations (01)' to (05)'. The aim is to obtain, by equating coefficients of like powers of

For example, the terms of zero order

It turns out that only with mathematical results of the approximation of lowest order, such as with the result that the pressure in

In the next step we are considering the first order

From these equations of the first approximation Equation (02b)'_{1} can easily be integrated at once. With the boundary condition that the pressure is zero at

which is obviously the hydrostatic pressure relation in dimensionless form.

The terms of second order

It can be seen that in the second approach no linear pressure distribution is available.

Finally, it is clear that the method presented new systems of equations arise which offer new possibilities of solutions for physical problems that are related to the Navier-Stokes equations.

The application of the developed approximation systems shows the following, the derivation of an equation for the stationary open channel flow (

For this flow the Equation (02b)'_{1} gives

Similarly, it follows from the equations

If

(56) and (57)

From Equation (02a)'_{1} result

since from Equation (01)'_{0} follow

Combining Equations (58) and (59), we obtain

In the following, the channel flow will now be considered in the coordinate system

and with

Considering these relationships the integration of Equation (61) yields

As a boundary condition at the free surface of water at

and air

respectively

The integration yields

and with the no-slip condition, according to

Finally, the back transformation of Equation (67) to variables with dimension yields

It is apparent that equation for laminar channel flow can be viewed as an approximation of the lowest order of the Navier-Stokes equations for incompressible flows.

Partial differential equation systems consisting of the Navier-Stokes equations, the equation of continuity and equations of the relevant boundary conditions can be represented by the perturbation theory in dependence of a dimensionless parameter, which is introduced as a reciprocal of the Reynolds number. From the relations arising from the comparison of the coefficients of this parameter, approximate solutions of different orders can be developed for the initial system.