The Momentum Balance or Equation of Motion

For a momentum balance, we take the same flow system as presented in Fig. 5.1, but instead of submerging an imaginary frame into the melt, we take an actual fluid element of dimensions Ax x Ay x Az (Fig. 5.4) and perform a force balance with the forces acting on its surfaces. 

we used the mechanical engineering convention that takes as positive the forces that are pulling the element (tensile stresses) and negative the forces that push on the surface 

After adding the forces, dividing by the element s volume, and letting the volume go to zero, the force balance results in 

In fluid flow, however, it is necessary to split the total stress, ry, into a deviatoric stress, Tij, and a hydrostatic stress, aH. The deviatoric stress is the one that leads to deformation (Fig. 5.5) and the hydrostatic stress is the one that is described by pressure (Fig. 5.6). 

2 The chemical engineering convention has its roots at the University of Wisconsin-Madison. Professor C.L. Tucker III, the advisor of one of the authors at the University of Illinois at Urbana-Champaign, claimed that he could feel the stress tensor turn around on the windshield of his car every time he would drive across the Illinois-Wisconsin border. 

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In laminar flow with low mass-transfer rates and constant physical properties past a solid surface, as for the two-dimensional laminar boundary layer of Fig. 3.10, the momentum balance or equation of motion (Navier-Stokes equation) for the X direction becomes [7]... 

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