150 BRIDGMAN. 



Applying Green's theorem to the surface S, transforming the surface 

 to vokime integrals, using the condition that Div v = because the 

 liquid is to be considered as incompressible, and removing the integral 

 sign, gives for the differential equation of heat transfer 



dr k ^ ^ , 



— - = - v-T — V • Grad r . 

 dt c 



This equation applies at points inside the liquid. There will also be 

 an equation to fix the boundary conditions. This equation is of the 

 ordinary type, independent of the motion of the liquid, and is merely 

 the statement that the heat input across the boundary is equal to the 

 conductivity of the fluid multiplied by the normal temperature 

 gradient. 



Apply these equations now to the present problem. If the motion 

 of the liquid is not turbulent, and if the lines of flow are not altered l)y 

 the heat input, then at the surface of the metal the liquid flows in 

 planes parallel to the surface, the velocity increasing from the surface. 

 The determination of the velocity distribution is a problem of hydro- 

 dynamics, and involves the viscosity of the liquid and the variables 

 which describe the mechanical roughness of the surface, but as far as 

 we are interested in the problem the elements which enter our heat 

 equations are determined if we can specify the velocity gradient at the 

 surface. The other elements which enter the equation of heat trans- 

 fer are the thermal conductivity of the liquid and its specific heat per 

 unit volume. 



Subject now to the restrictions mentioned, we may make a dimen- 

 sional analysis of the situation. Notice in the first place that since 

 the flow of water is transverse to the specimen the rise of temperature 

 etc. is independent of the length, provided only that the specimen is 

 long enough for us to neglect end effects. 



We now enumerate the elements with which we are concerned and 

 their dimensions. 



Name of Quantity Symbol Dimensional Formula 



Average rise of temperature 



Rate of heat input per unit length 



Velocity gradient in liquid 



Thermal conductivity of liquid 



Specific heat of liquid per unit volume 



Breadth of specimen 



Frequency' of impressed heat input 



