The Motion of a Body Through a Resisting Medium

 

Let us assume that it is required to know how the resistance (R) depends on the following assumed variables if all the bodies are of similar shape but of different size when flowing through a viscous medium.

                                                                                               Dimensions

         Resisting Force                                     R                          MLT^-2

         Velocity                                                v                               LT^-1

         Linear Dimensions                               d                               L

         Density of the Resisting Medium                                     ML^-3

         Viscosity of Resisting Medium                                         ML^-1T^-1

 

                                Let                    

 

  Under which circumstances,

 

                                                

 

          Hence                        

 

It becomes immediately clear that the equation cannot be completely solved, as there are four unknowns and only three equations. Progress can, however be made by letting () remain uneliminated. 

 

Then by solving in the same manner and rearranging it is seen that,

 

        Thus, the expression  is dimensionless and rearranging,

 

                                                                 

Now for those systems for which the dimensionless expression  is constant they will be dynamically similar and for bodies of similar shape with () and () constant, the resisting force (R) will be the same as long as (vd) is constant. It follows that for the same resistance to motion the velocity is inversely proportional to the linear dimensions. It is consequently better to construct large airships than small ones. A useful conclusion to be made by dimensional analysis despite the fact an explicit equation for (R) cannot be deduced.