Studies on the Motion of Viscous Flows —III 

 Bo) ^ Bo) V Bed ~ . -s 



__ + u — + _^ j^v^w , (1.9a) 



3t B? ? 3^ 



or, briefly, 



where 



(1.9b) 



The first term of Eq. (1.9a) represents local time change of vorticity. The 

 second and the third terms represent the convective change of vorticity. To- 

 gether they represent the total change of vorticity. The term on the right rep- 

 resents the rate of dissipation of vorticity due to internal friction. The form of 

 Eq. (1.9a) clearly reveals the transport and the diffusion characteristics of a 

 significant property of the flow, viz., the vorticity function. 



DIMENSIONAL BOUNDARY CONDITIONS 



Conditions at the Cylinder Wall 



The boundary of a solid circular cylinder is characterized by two proper- 

 ties. First, it is impermeable except for adsorption effects. Second, it is vir- 

 tually nondeformable with respect to the forces applied to it. By introducing the 

 two ideas of impermeability and rigidity the cylinder boundary can be idealized 

 for a simplified and yet representative mathematical formulation. The idea of 

 impermeability ascribes this property to every point on the boundary of the 

 cylinder . 



The condition of impermeability requires that, for a fluid element indefi- 

 nitely close to a surface element of the cylinder wall, their relative velocity 

 along the surface normal be zero. Since the coordinate system is fixed to the 

 cylinder, the normal and the tangential components G and v respectively of all 

 the surface elements are zero for all time t. Then the condition of imperme- 

 ability is expressed as a kinematic condition 



G(a,5,t) = (1.10a) 



for all fluid elements on the boundary of the cylinder. 



The ideas of impermeability and rigidity lead to a condition on the normal 

 velocity component u, but not on the tangential component v at the cylinder wall 

 [11,12]. To obtain a condition on v, the following three hypotheses were consid- 

 ered during the 19th century: 



1. The velocity at a solid wall is the same as that of the solid itself, and 

 changes continuously in the fluid, which has everywhere the same properties. 



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