(2) Stresses 



(a) The normal and tangential stresses at both sides of the surface are equal.* 



b. Boussinesq's solution 5,6 



(1) Velocity 



(a) The normal velocity component vanishes. 



(b) The tangential velocity components at both sides of the surface are equal. 



(2) Stresses 



(a) The normal stress at the inside of the surface is larger than the stress 

 at the outside due to the dynamic surface tension.** 



(b) The tangential stress at the surface is increased across the surface due 

 to the dynamic increment of surface tension. 



The drag of a sphere in an infinite medium of uniform velocity V thus becomes: 



1. Rigid sphere: D •= 6n ixrV 



2. Fluid sphere: a. D = 6jrnrU — ^ 



3/i + 3^' 



b. D = 6 ul irU 



e + r(2n + 3pt') 

 e + 3r(/x + n') 



where D is the drag, 



[i is the coefficient of viscosity of the medium, 



t is the radius of the sphere, 



p' is the coefficient of viscosity of the fluid inside the sphere, and 



e is the coefficient of surface viscosity. 



Using the condition of equilibrium for a sphere rising under the influence of gravity, 



we obtain for the rigid case: 



4 4 



6irij.rU = — nr 3 p g -— irr 3 p'g 



2 r*9 



U -~ (p - p') Stokes' Law 



9 f< 



♦The pressure increase across the surface due to surface tension (= ■=£-) was neglected in Hadamard's analysis. 

 Inclusion of this pressure drop in the boundary condition for the normal stress does not change the results. That 

 is to say, surface tension as manifested only in a pressure increase inside the fluid sphere does not affect its 

 motion. (This result is also obtained by putting, in Boussinesq's analysis, the coefficient of surface viscosity, 

 see subsequent footnote, equal to zero.) 



**Boussinesq assumed that a dynamic surface tension exists at interfaces in motion. Its magnitude is given by 

 the sum of the usual (static) surface tension and the dynamic increment. The dynamic increment varies over the 

 surface of the sphere and at a given point is proportional to the rate of dilatation at that point. The constant of 

 proportionality is called (due to its similarity to the viscosity coefficient) the coefficient of surface viscosity. 

 (Surface viscosity has the dimensions mass/time, while the dimensions for viscosity are mass/length X time.) 



