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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