Page 101 - Bird R.B. Transport phenomena

86 Chapter 3 The Equations of Change for Isothermal Systems

EXAMPLE 3.5-1 The Bernoulli equation for steady flow of inviscid fluids is one of the most famous equations
in classical fluid dynamics. Show how it is obtained from the Euler equation of motion.
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The Bernoulli Equation
for the Steady Flow of SOLUTION
Inviscid Fluids Omit the time-derivative term in Eq. 3.5-9, and then use the vector identity [v • Vv] =
iy(v • v) - [v X [V X v]] (Eq. A.4-23) to rewrite the equation as
2
pV\v - p[v X [V X v]] = -Vp - pgVh (3.5-10)
In writing the last term, we have expressed g as - УФ = —gVh, where h is the elevation in the
gravitational field.
Next we divide Eq. 3.5-10 by p and then form the dot product with the unit vector
s = v/|v| in the flow direction. When this is done the term involving the curl of the velocity
field can be shown to vanish (a nice exercise in vector analysis), and (s • V) can be replaced by
d/ds, where s is the distance along a streamline. Thus we get

-x4-h (3.5-11)

When this is integrated along a streamline from point 1 to point 2, we get

(3.5-12)
which is called the Bernoulli equation. It relates the velocity, pressure, and elevation of two
points along a streamline in a fluid in steady-state flow. It is used in situations where it can be
assumed that viscosity plays a rather minor role.

§3.6 USE OF THE EQUATIONS OF CHANGE
TO SOLVE FLOW PROBLEMS

For most applications of the equation of motion, we have to insert the expression for т
from Eq. 1.2-7 into Eq. 3.2-9 (or, equivalently, the components of т from Eq. 1.2-6 or Ap-
pendix B.I into Eqs. 3.2-5, 3.2-6, and 3.2-7). Then to describe the flow of a Newtonian
fluid at constant temperature, we need in general
The equation of continuity Eq. 3.1-4
The equation of motion Eq. 3.2-9
The components of т Eq. 1.2-6
The equation of state P = pip)
The equations for the viscosities /JL = /X(p), К = К(р)
These equations, along with the necessary boundary and initial conditions, determine
completely the pressure, density, and velocity distributions in the fluid. They are seldom
used in their complete form to solve fluid dynamics problems. Usually restricted forms
are used for convenience, as in this chapter.
If it is appropriate to assume constant density and viscosity, then we use
The equation of continuity Eq. 3.1-4 and Table B.4
The Navier-Stokes equation Eq. 3.5-6 and Tables B.5,6, 7
along with initial and boundary conditions. From these one determines the pressure and
velocity distributions.

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Daniel Bernoulli (1700-1782) was one of the early researchers in fluid dynamics and also the
kinetic theory of gases. His hydrodynamical ideas were summarized in D. Bernoulli, Hydrodynamica sive
de viribus et motibus fluidorum commentarii, Argentorati (1738), however he did not actually give Eq. 3.5-12.
The credit for the derivation of Eq. 3.5-12 goes to L. Euler, Histoires de YAcademie de Berlin (1755).