350 



3u > 3u -. 3u 

 — + u — + V — 

 3t 3x 3r 



1 3p 

 p 3x 



(12) 



Substituting (23) into (22) and using (2) and (15) 

 we find 



3v 

 3t 



. 3v _ 3v 

 u T— + - 

 3x 



3r 



1 3p r^ 

 p r j-3 



(13) 



(pr^/gr^) + p = p - op (- - - 



^ 3x2 = 



r = r,, + 6r^. 



6r^) + pU^f , 

 (24) 



ii+ 3i + V ^ 

 3x 3r r 



(14) 



We now linearize these equations with respect to 

 the undisturbed swirl flow. 



u = U + u, V = V, p = Po + Pr 



(15) 



where the perturbation quantities u(x,r,t), v(x,r, 

 t) , and p(x,r,t,) are supposed to be of 0(e). Sub- 

 stituting (15) into (12)... (14), neglecting terms 

 of O(e^) and using (2) we find 



Expanding the functions of r in (24) with respect 

 to 5r^, neglecting second order quantities, and 

 using (3) the boundary condition (24) changes into 



(25) 



p(x, r , t) = (- £^ + ^ )6r + ap ^ 3r + pU^f. 

 c 3 2c 3x^ c 



r r 

 c c 



From (16) we find, because p ^- and ij) -»■ for x 



P = - P" 3^ - 3l ' 



(26) 



(16) 



(17) 



(18) 



which is Bernoulli's law for the unstationary lin- 

 earized flow. Herewith the dynamical boundary 

 condition (25) becomes 



3<b 3* r^ a „ 3^ , 



" aT + 37 = <^ - 7 ) ^^ -°^ "^r -U2f . (27) 

 3x 3t J.3 r2 c 3^2 c 



The kinematical condition at the boundary of the 

 cavity is 



Because the (u, v) velocity field is yithout rota- 

 tion we can write 



^ 6r + U - — or = —^ . 

 3t c 3x c 3r 



(28) 



, ,3<fi 3*, 

 u, v) = (t— , r— ) 

 3x 3r 



(19) 



where (j) = i|>(x, r, t) is a scaler potential function 

 satisfied by (18) 



Hence, we must solve (20) under the conditions (27) 

 and (28) while (f) -^ for r -*- °° and for x -*- - «>. 



3. THE GREEN FUNCTION 



3x2 3r2 r 3r 



. 



(20) 



We suppose the dimensionless loading of the boundary 

 (22) to have the form 



We now suppose the disturbed cavity wall to be at 



f (x,t) = f (x) e^ , 



(29) 



r = r + 6r , 

 c c 



(21) 



where 6r (x, t) is 0(e). On this axial symmetric 

 boundary we demand the difference between the 

 pressures inside the cavity and in the fluid to be 

 in equilibrium with the effect of the surface ten- 

 sion and with some still unspecified external 

 normal loading pU f (x, t) of the cavity wall. 



11 o 

 p = p - ap (- + -)+ pu2f{x,t) , r = r + 6r , (22) ■ 

 c Ri R2 c c 



where R^and R2 are the principle radii of curvature 

 of the boundary, reckoned positive when the centers 

 of curvature are at the side of the cavity. 



Within the accuracy of our linearized theory we 

 can put 



where e is a "small" positive parameter which has 

 no connection with the linearization parameter e. 

 Because our problem is linear we assume 



<)>(x,r,t) = (|)(x,r) e^^, 6r (x,t) = 5r (x)e^ . (30) 



c c 



Then equation (20) and the boundary conditions (27) 

 and (28) change into 



3„^ 3 -^ r 3r 



(31) 



e + Ug- <Kx,r^) - —3- -2 6r^(x) + 



-c ^c 



Rl = r, + 6r^, R2 = - (l^^rj ' 



(23) 



a— -2 i5rc(x) = U^f (x). 



(32) 



