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Newtonian Viscous Fluid 391

         In the next two sections, we shall discuss the conditions under which irrotational flows are
         dynamically possible for an inviscid and viscous fluid.

         6.21   Irrotational Flow of an Inviscid Incompressible Fluid of Homogeneous
               Density

           An inviscid fluid is defined by


         obtained by setting the viscosity fi = 0 in the constitutive equation for Newtonian viscous fluid.


           The equations of motion for an inviscid fluid are





         or




         Equations (6.21.2) are known as the Euler's equation of motion. We now show that irrotationai
         flows are always dynamically possible for an inviscid, incompressible fluid with homogeneous
         density provided that the body forces acting are derivable from a potential Q by the formulas:




         For example, in the case of gravity force, with x$ axis pointing vertically upward,



         so that



           Using Eq. (6.21.3), and noting thatp = constant for a homogeneous fluid. Eq. (6.21.2) can
         be written as




         For an irrotational flow





         so that
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