fluid_mechanics

6.4 Inviscid Flow 279
Otherwise, we will have more unknowns 1all of the stresses and velocities and the density2 than
equations. It should not be too surprising that the differential analysis of fluid motion is complicated.
We are attempting to describe, in detail, complex fluid motion.
6.4 Inviscid Flow
As is discussed in Section 1.6, shearing stresses develop in a moving fluid because of the viscosity
of the fluid. We know that for some common fluids, such as air and water, the viscosity is small,
and therefore it seems reasonable to assume that under some circumstances we may be able to
simply neglect the effect of viscosity 1and thus shearing stresses2. Flow fields in which the shearing
stresses are assumed to be negligible are said to be inviscid, nonviscous, or frictionless. These terms
are used interchangeably. As is discussed in Section 2.1, for fluids in which there are no shearing
stresses the normal stress at a point is independent of direction—that is,
In this
instance we define the pressure, p, as the negative of the normal stress so that
p s xx s yy s zz
s xx s yy s zz .
The negative sign is used so that a compressive normal stress 1which is what we expect in a fluid2
will give a positive value for p.
In Chapter 3 the inviscid flow concept was used in the development of the Bernoulli equation,
and numerous applications of this important equation were considered. In this section we will again
consider the Bernoulli equation and will show how it can be derived from the general equations of
motion for inviscid flow.
Euler’s equations
of motion apply to
an inviscid flow
field.
6.4.1 Euler’s Equations of Motion
For an inviscid flow in which all the shearing stresses are zero, and the normal stresses are replaced
by p, the general equations of motion 1Eqs. 6.502 reduce to
rg x 0p
(6.51a)
0x r a 0u
0t u 0u
0x v 0u
0y w 0u
0z b
rg y 0p
0y r a 0v
0t u 0v
0x v 0v
0y w 0v
0z b
(6.51b)
rg z 0p
(6.51c)
0z r a 0w
0t u 0w
0x v 0w
0y w 0w
0z b
These equations are commonly referred to as Euler’s equations of motion, named in honor of
Leonhard Euler 11707–17832, a famous Swiss mathematician who pioneered work on the relationship
between pressure and flow. In vector notation Euler’s equations can be expressed as
rg p r c 0V 1V 2V d
(6.52)
0t
Although Eqs. 6.51 are considerably simpler than the general equations of motion, Eqs. 6.50,
they are still not amenable to a general analytical solution that would allow us to determine the
pressure and velocity at all points within an inviscid flow field. The main difficulty arises from the
nonlinear velocity terms 1u 0u0x, v 0u0y, etc.2, which appear in the convective acceleration.
Because of these terms, Euler’s equations are nonlinear partial differential equations for which we
do not have a general method of solving. However, under some circumstances we can use them to
obtain useful information about inviscid flow fields. For example, as shown in the following section
we can integrate Eq. 6.52 to obtain a relationship 1the Bernoulli equation2 between elevation, pressure,
and velocity along a streamline.
6.4.2 The Bernoulli Equation
In Section 3.2 the Bernoulli equation was derived by a direct application of Newton’s second law
to a fluid particle moving along a streamline. In this section we will again derive this important