An Analytic Solution for the Boundary Layer Equations

© 2009

Clyde M. Davenport

cmdaven@comcast.net

Introduction | Complex Solution | Elementary Properties | Conclusions |

Introduction The boundary layer equations are a pair of nonlinear partial differential equations of second order, as follows: They are used in fluid dynamics engineering as a model of a fluid or gas flowing past a stationary boundary. Generally speaking, the fluid does not slip freely past the boundary surface. There is a flow perturbation that extends outward into the flow, the perturbed volume being known as the Below, we shall pursue a solution in terms of classical complex variables. We shall present something new in the form of the One might recall that partial differential equations, especially those used to model the behavior of some continuous material substance, typically describe the behavior of some variable, parameter, or physical effect Complex Variable Solution As mentioned earlier, our basic approach to solution is to first convert the (nonlinear) boundary layer equations to ODEs, then solve them by means of classical methods. To convert partial differentials to ordinary derivatives, we shall use the following consequences of the classical Cauchy-Riemann conditions: where [ASIDE: It so happens that laminar flow is Making the partial derivative conversions, we obtain: We can rearrange these as follows:By applying the continuity conditions (Cauchy-Riemann conditions) and converting to ODE form in this way, we are implicitly continuing both where For reasons that shall become clear, later, we multiply both sides of the above by α and rearrange to obtain:Top where where We have three arbitrary constants. We are free to set
and rearrange, to obtain the simplified form: From the solution to the first equation, we also have an expression for These are the We remind the reader that these do not represent the one and only, final solution of the boundary layer equations. Just as do the original PDE forms, they merely represent the behavior about a point. Given an external force, impulse, etc. on a small increment of material at the point, the characteristic functions that we have found indicate how the disturbance will be propagated away from that point Elementary Properties The However, there is a simple scenario that illustrates the main features of the boundary layer. Consider a flat, infinite boundary plate lying in a horizontal plane through the origin. Let a uniform fluid or gas flow In order to better interpret our result, here is where we wish to consider and put this into the Now, if we set This shows the proper behavior. Recall that To summarize, This has all of the right behavior. There is a smooth transition from zero flow at the boundary to the uniform background flow, then we see that, for a given fluid velocity
Prandtl used a In the present paper, we have obtained a solution by a different method; i.e., analytic function theory. We shall prove self-similarity by recasting our solution into the self-similar standard form. Consider the following operations on our But Next, we merely assign: Finally, we arrive at the Similar actions apply to the Conclusions We have found characteristic functions for the boundary layer equations. They can be used in place of the PDE forms in a finite-element calculation. We have shown that they have the self-similar property. We constructed a simple formula
© 2009 |