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An iteration method similar to the thin-wing-expansion method for the compressible flow has been proposed to solve the boundary layer flow past a flat plate. Using such an iteration, the first step of which is Oseen’s approximation, the boundary layer past a flat plate is studied. As proceeding from the first approximation to the second and third approximations, it is realized that our solution approaches to a well known Howarth’s bench mark one gradually. Hence, it is concluded that the usefulness of the present method has been confirmed.

For an analytical treatment of the boundary layer flow past a flat plate, one must solve the Navier-Stokes equation under suitable boundary conditions. But unfortunately, it accompanies a great difficulty to obtain such an analytical solution. Thus, as it is well known, one has proceeded to classify the flow according to whether the Reynolds number is small or large: if the Reynolds number is small, the linearization of Oseen or Stokes type is often employed. Whereas if the Reynolds number is large, the inertia force dominates and so the viscosity is neglected except in the boundary layer [

The main purpose of the present paper is to apply the iteration method to the flow past a flat plate at high Reynolds number. Even though no exact solution of Navier-Stokes equation for a flat plate at high Reynolds number has been known, boundary layer solution for the flow past a semi-infinite flat plate by Howarth [

Consider the steady viscous flow past a semi-infinite flat plate at zero incidence placed in the uniform flow velocity, U_{∞}. As is shown in

where u and

We shall assume that the velocity components

where ε is a small parameter that may be considered as the ratio of the boundary layer thickness to the flat plate length. Substituting (2.5) and (2.6) into (2.1)-(2.4), and rearranging the terms of the same order in ε, we have

Equation (2.7) is a modified Oseen’s equation, which is regarded as the first approximation of the boundary layer equation, being obtained by simplifying the Navier-Stokes equation at high Reynolds number.

Let’s proceed to the first approximation. Introducing the Laplace transform, ū_{1}, of u_{1} with respect to x, which is defined by

together with (2.7), we can obtain the equation governing

where λ is the parameter of the Laplace transformation. The general solution of (3.2) satisfying (2.10) and (2.12), can be easily be expressed such as

By performing the inverse Laplace transformation of

where

with

These results of (3.4) and (3.6) may be obtainable by the same technique, to be used in the second and the third approximations in Sections 4 and 5.

The equation of continuity (2.9) for n = 2 can be automatically satisfied by introducing the following stream function,

where

where

Substituting (4.2) and (4.3) into (2.8) for n = 2, and using (3.4) and (3.6), we get

where

Referring (4.2), the boundary conditions (2.11) and (2.12) for n = 2 can be expressed as follows,

The solution of (4.4) satisfying the conditions (4.6) and (4.7) is obtainable as

and

It may be worth noting here that in the approximation of each order we can introduce the reduced stream function

On the other hand, the original Equations (2.1) and (2.2) suggest that the velocity component u can be expressed by

where

Adopting the similar procedure to Section 4, the equation for

where

and

The solution of (5.1) under the boundary conditions (5.2) and (5.3) is expressed by

with

and

Substituting (3.4), (4.9), (5.9) into (2.5), we obtain the x-component of the velocity. The second and the third approximations for the velocity component have been plotted concurrently in

Introducing dimensionless drag coefficient for the plate wetted on both sides, by the definition [

where R_{x} denotes

It is found that the Formula (6.2) provides the greater value than that obtained by Blasius’ one [

Application of the present iteration method to the flow past a plate having flat, but finite thickness at moderate Reynolds number is left behind for the future study, for no analytical solution of Navier-Stokes equation on this problem exists. In such a case, the parameter ε, the ratio of the boundary layer thickness to the plate length, is

not always considered to be infinitesimally small, so that much more vigorous mathematical treatment is required to get the solution.