3. EULER'S EQUATIONS OF MOTION 



The equations of motion for a fluid are the mathematical equivalent of Newton's second 

 law of motion, which states that the resultant force on any particle equals the product of the 

 mass of the particle by its acceleration. For convenience we may assume a fluid particle to 

 have the form of a cube whose edges are 8x, Sy, 8z parallel to the a;-, y, 2-axes as in Figure 4. 

 Considering the cube as a free body, the forces acting on it may be considered as made up of 

 three parts: compressive or tensile, shear, and external forces, such as gravity. On each 

 face there may be two shear force components parallel to the coordinate axes. 



Shear forces, in a fluid, are due to a physical property of the fluid known as viscosity, 

 by virtue of which it offers resistance to motion involving the production of shearing strain. 

 All actual fluids have viscosity, but in some fluids, such as water, the viscosity is quite 

 small. In many flow problems the viscous forces are so small as compared with other forces 

 that their effect may be neglected. This greatly simplifies the mathematical treatment of the 

 problem. Throughout this report, the assumption will be made that the viscosity is zero; this 

 is equivalent to saying that the fluid cannot sustain a shear stress, or that it is frictionless. 



Let the pressure or force per unit area at the center of the cube be p, and consider the 

 two faces of the cube normal to the a?-axis. Since the pressure will be a function of x, y, z, 

 the average pressure on the left-hand face will be 



and that on the right-hand face. 



p = V ^ 



dx 2 



'^ dx 2 



The resultant force due to pressure in the positive a;-direction will be the difference between 



the pressures on these two faces multiplied by the area of a face or 



dp 



Sx Sy 8z 



dx 



Let X be the component of the external 



forces per unit mass of fluid in the a;-direction. 



Usually the only source of external forces is 



gravity. The external force acting on the 



material in the cube in this direction is then 



pX8xSy8z 



where p is the mass density. Here p must be 



expressed in dynamical units, for example, in 



slugs per cubic foot or pounds sec^/ft"^. As 



the viscosity has been assumed zero, there can 



be no other forces acting in the a;-direction, 



and the resultant force on the material in the 



^,.--^1 



v" 



1 



/ 1 



p 





ix,y,z) 





J— 



-—-^ 



_*— ■""^ 



"^""'^ 



Figure 4 - Illustrating Euler's equation 

 of motion. 



