238 Mechanical Resonances of Biological Cells /1 3 : 2 



where p is the density, and p is the pressure. Substituting Equation 1 

 into Equation 3 and integrating leads to 



This is an integrated equation of motion ; it must be valid both within 

 and outside the cell membrane. 



Quantities outside the membrane will be denoted by a subscript o, 

 whereas those within will be denoted by a subscript i. Using these, one 

 may rewrite Equations 2 and 4 as 



(2') 

 and 



(4') 



In the model, the two liquids may slip freely over the cell surface but 

 still not lose contact with it. This latter condition will be satisfied if 



the component of v in the radial r direction is continuous at the cell 

 surface, r = a. Analytically, this is 



Other boundary conditions are that there is no net acceleration of the 

 center of the cell, and that the velocity outside goes to zero at long 

 distances. These become, in terms of cp, 



^=0 and I&- » fe-0 at r=0 (6) 



or r 89 r sin 9 dip v ' 



and 



d( Po 



8r 



as r — ^ oo (7) 



where 9 is the azimuthal angle and ip the latitude. 



The final boundary condition involves the cell membrane. This 

 possesses an interfacial tension and in general will support a pressure 

 difference Lp across it. Denoting by R(9, ip) the displacement of the 

 membrane in the radial direction, one readily shows that at r = a 



A P=Pi-Po = -^— ft i(sm9 d 4) 

 a z sin 9 89 \ 89 J 



T B*R 2T 2T 



a 2 sin 2 d dtp 2 a 2 a [ j 



