A THEORY OF THE STABILITY OF LAMINAR FLOW 
ALONG COMPLIANT PLATES 
F. W. Boggs and N. Tokita 
Research Center, U.S. Rubber Company, 
Wayne, New Jersey 
1. INTRODUCTION 
A great deal of theoretical work has been done in recent years on viscous flow in the 
neighborhood of a rigid wall. This has made it possible to explain the initiation of turbu- 
lence through the development of flow instability. Laminar flow has been shown by the 
theory to be stable under conditions when it is actually observed, and it has been shown to 
be theoretically unstable under conditions when turbulent flow is known to prevail. These 
results developed by Tollmien, Schlichting, Lin, and many others [1-7] have been verified by 
Schubauer [8] at the National Bureau of Standards. Today the controversy which existed for 
many years over the nature of turbulent flow and its relationship to laminar flow seems to be 
settled. 
The recent work of Kramer [9-11] showing the effect of a flexible boundary wall on fluid 
flow has emphasized the need to extend the boundary-layer theory to include this interesting 
and important case. This present paper undertakes the development of a theory which will 
predict the stability conditions of the boundary layer in contact with a flexible wall under 
the same general assumptions regarding the flow that were made by previous authors for the 
rigid wall and to the same degree of approximation. The result should, therefore, have the 
same kind of validity. 
In this treatment the properties of the flexible wall are expressed in terms of its acous- 
tical compliances. 
One of the significant conclusions which follows from this work is that the conditions 
of stabilizing a flow are fairly critical and that all coatings which are flexible do not neces- 
sarily have a favorable effect on the flow conditions. 
In subsequent papers we will attempt to extend this theory to some of the special cases 
of practical importance, but here we will confine ourselves to some very general conclusions. 
The theories of boundary-layer stabilization are based on the supposition that the Navier- 
Stokes equations hold at all times and that transition from laminar to turbulent flow results 
when the laminar flow is unstable for some arbitrarily small perturbation. Stable laminar flow 
exists only if the flow is stable for every possible infinitesimal perturbation. It assumes but 
does not really prove that when a laminar flow fulfills these conditions, it will be the domi- 
nant flow pattern and that the drag coefficient calculated from such a laminar flow is the one 
applicable. 
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