CHAPTER V 



CONCLUSIONS 



The new model of a swimbladder-bearing fish as a viscous shell enclosing an air cavity with surface 

 tension at the inner interface was developed because older models, of which Andreeva's model [26] is most 

 widely employed, do not accurately predict the results obtained in experiments on the characteristics of 

 swimbladder resonance. The principal difference between the older models and the data is that the values 

 of Q predicted by the models are always higher than those obtained experimentally. A second difference is 

 that the older models can not account for the high resonant frequencies obtained for some large 

 physoclisis. The new model sought to correct these differences by explicitly including the viscosity of fish 

 tissue and by including a tension in the swimbladder wall. 



For convenience, the equations developed for the new model will be restated here: 

 Uo2a2 = SYaPo, + 2s (3Ya _ 1} _ 



a = 



■■rad 



Poi Po,a 



VPo, I 



4nW Po " 



(V-2) 



Lo^H 2 V u 2 /J 



. w p 0( c w (V-3) 



w 2 Pow a 



i_i _ ^oPo.a 2 



VIS 2E~ ' (V-4) 



and H - ,co n a / 2Po a c Pa r (, , 2s \-i 



H,h " 3( Ya - D V -TSKT) \ P 0f o) 2 a3 ) " ( v " 5 ) 



Comparable equations for an air bubble in water are available at only co = u> , where H = Q. The equations 

 for a and Q obtained for the new model are essentially the same as those for an air bubble in water, the only 

 significant difference being in the value of the viscosities of fish flesh and water. However, the equations for 

 0) for the new model and an air bubble in water for which viscosity and surface tension are included are not 

 the same. If the viscosity of fish flesh were used in the equation obtained for an air bubble, cd could be zero 

 or imaginary. However, for the new model, the effect of viscosity on co was shown to be small enough to be 

 neglected. 



Equations V-1 and V-2 are valid for an upper limit for ^ = 6 x 10 3 poise. If E, > 6 x 10 3 poise, then the 

 outer shell radius, b, appears in the o-equation. This implies that this is a boundary layer-type problem. 

 Thus, if the fish flesh-water interface is outside the boundary layer, the magnitude of b is immaterial. 



All of the models are spherical in nature. However, the swimbladders of some fish are sufficiently 

 elongated that their shapes can have a significant effect on co . Thus, equation V-1 should be modified to 

 include this effect. Hence: 



^ = ? [^Po. + ^ (3Ya _ 1} ] . (V . 6) 



Although Z, was determined for a bubble in water, its use here is quite reasonable, especially when the 

 similarity between the equations for a bubble and the new model are considered. 



Several other conclusions can be reached for the new model. One is that, for the ratios of outside to 

 inside shell diameters considered, the actual value of the outside diameter has no effect on the results. 

 Another is that thermal losses are not very significant and, in most cases, can be neglected. A third is that, 

 for low values of H, (co /u)H) 2 can be comparable to [(co 2 /co 2 ) - 1] 2 at off-resonant frequencies. Hence, 

 considering the experimental values of Q obtained, it is apparent that H, rather than Q, should be used to 

 calculate a. 



A comparison of the new model with available experimental data indicates that the new model 

 constitutes a definite improvement over previous models. The new model can predict low values of Q and 

 elevated values of co,, which the previous models could not. In addition, the new model can be used to 

 obtain the magnitude of damping at any frequency, whereas many previous models only produced the 

 value at resonance. 



33 



