236 Mechanical Resonances of Biological Cells / 1 3 : 2. 



developments of Sections 2 and 3 can be regarded only as first approxima- 

 tions. In Section 4, more exact solutions are outlined which include the 

 effects of viscosity, of compressibility, and of departures from spherical 

 shape. 



2. Interfacial-Tension Model 



This first model approximates the biological cell by a spherical shell 

 lacking any rigidity but possessing an interfacial tension. The cell is 

 filled with liquid and surrounded by liquid. It makes no difference 

 whether this interfacial tension is a true liquid-liquid interfacial tension, 

 a liquid-membrane interfacial tension, or a surface-tension residual 

 in a stretched membrane. Physically, all of these may exist at the cell 

 boundary. Values of this interfacial tension T computed from static 

 experiments ranged from 0.01 to 3.0 dynes/cm. The theory discussed 

 here gives values of T from 0.03 to 15 dynes/cm for vertebrate red blood 

 cells and ciliate protozoans. 



The surface motions of this model are very similar to the resonant 

 modes of a rain drop or of an air bubble in water. The rain drop and the 

 bubble are different from cells in having liquid on only one side of the 

 boundary and also in possessing a much higher surface tension, around 

 75 dynes/cm. Nonetheless, the general forms of the motions are similar. 

 Some typical modes for this geometry are illustrated in Figure 2. These 

 types have been photographed for oscillating bubbles and for liquid 

 droplets. 



In order to describe the resonant modes of this model quantitatively, it 

 is necessary to use a little mathematics. Because the liquid is assumed 

 incompressible and its flow irrotational, the motion may be described in 

 terms of a velocity potential <p. This velocity potential is defined so 

 that the negative of its gradient is the vector velocity v; that is, 



v= -ftp (1) 



For noncompressible fluids, the divergence of v is zero; in terms of cp, 



V 2 <p = (2) 



Next, the equation of motion may be expressed in terms of this same 

 velocity potential. Newton's second law for an incompressible fluid, in 

 the absence of any external-force field, is approximately given by 



-V^„g (3) 



