28 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



If the medium be both incompressible and homogeneous, the moving masses 

 will not change volume, and the mass-contents of every stationary volume will be 

 invariable. We thus get the special case : 



(C) Both velocity and specific momentum will be solenoidal vectors if the 

 moving medium be both homogeneous and incompressible. 



Without restricting the physical properties of the medium, we can apply theo- 

 rem 114 (A) to the infinitely small volume contained between two parallel surfaces 

 running at infinitely small distance from each other. Finite difference between the 

 normal velocity-components at adjacent points on the two surfaces would in this 

 case lead to finite expansion of an infinitely small volume. Thus the continuity 

 would be broken. Therefore a finite difference between the normal components 

 can not exist. This leads to the solenoidal surface-condition : 



(D) The normal component of velocity must have the same value on both sides 

 of any surface in a material system filling space continuously. 



This solenoidal surface-condition must be fulfilled, for instance, at the surface 

 of separation between atmosphere and hydrosphere. It applies only to velocity, not 

 in general to specific momentum. Taking the case of mercury and water in contact 

 with each other, the normal component of velocity will be the same on both sides 

 of the surface; but that of specific momentum will be 13.6 times greater on the side 

 of the mercury than on that of the water. 



If the system is at rest on the one side of the surface, there will be no velocity- 

 component normal to it on the other side ; consequently the normal component of 

 specific momentum will also be zero. Thus : 



(E) Velocity and specific momentum are directed tangentially to every resting 

 boundary. 



This condition is to be applied to the motion of the air along the ground and of 

 the water along the bottom of the sea. 



116. Examples of Volume-Transport and Mass-Transport. It will be useful 

 here to take a few examples illustrating the difference between the conditions of 

 solenoidal velocity-field and solenoidal field of specific momentum. 



Let a tube be filled partly with water and partly with mercury, both fluids being 

 considered incompressible. If the fluid column moves along the tube there will be 

 equal volume-transport through a section in the water and through one in the 

 mercury, say one cubic meter per second through each. The volume-outflow out 

 of the closed surface formed by the walls of the tube and the two cross-sections 

 will be zero, and the field of velocity will be solenoidal. But measuring the transport 

 in tons, we find a transport of one ton per second through the upper and a trans- 

 port of 13.6 tons per second through the lower section. The difference, 12.6 tons 

 per second, gives the outflow of mass through the walls of the volume, and thus 

 the decrease per unit time of the mass contained in the volume. We shall have 

 outflow or inflow of mass according to the direction of the motion. For there will 

 be a decrease of mass in the volume when water expels mercury, and an increase 



