Yortex Theory for Brdtes Moving tn Water 
' = = T Vv =o IT A = 
dF i pTa ¥, dsias p fF). ae a | Teas ie 
= -e (TAV,),, 42 (M) (7.47) 
because 
V2 (M) wey icika et Wp (M) . 
dit 
—_ 
We know that, if the motion is unsteady, d T must be re- 
placed by dF. However there exists no contradiction. The term 
- p-Ck= aS which appears when the vortex tubes belong to a sheet 
eos from the integration of -P S* through the sheet. When the 
vortex tube is isolated, there is no igcontinuity of io on the surface 
S' and the contribution of the term p OV in the integration of pVp 
on S' is 0(R'ds)), thus negligibly Sthall. This is not the case when 
one deals with a sheet) 
CASE OF A HULL EQUIPPED WITH MOVABLE APPENDAGES 
The treatment of the problem arising from the presence of 
such appendages obviously depends upon their position with respect to 
the hull. When the axis of the rudder coincides with the edge of the 
stern, this rudder may be regarded as a part of the hull. The shape of 
the hull varies with time. At each instant t there is however a vortex 
sheet adhering to this hull. The method of Section VI therefore applies 
in principle. But separation may occur at the leading edge of the rud- 
der because of lack of continuity. Furthermore the effect of the vis- 
cous boundary layer is never negligible in this region. 
When the rudder (or diving plane) is at some distance from 
the hull, the rudder behaves as a lifting surface with a small aspect 
ratio. Because of the thinness of the rudder, the concept of the ''in- 
terior'' of the rudder becomes meaningless and the thin wing theory is 
to be used. 
(1) By using (7.46) one can simplify the expression of the fictitious 
force F' inside D,; when —— #0  ~ See [13] Chapter IlI,B, art. 9. 
t 
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