Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 268

Whereas the Lagrangian description above follows a fixed control mass as
it moves in space, the Eulerian model focuses its attention at given locations in
space. In MWD acoustic modeling, and in studying dynamic swab and surge,
both models turn out to be important. Each model provides a convenient vehicle
to model different kinds of acoustic sources. To appreciate the differences
between Lagrangian and Eulerian points of view, we must understand that
Newton' s “ F = m a ” law describes individual moving entities, e.g., a ping-pong
ball, a Frisbee or an arrow, tracked as it moves. It is, therefore, a Lagrangian
description or model, for which the upper-case derivative D/Dt is often used for
emphasis; this is also known as the “convective,” “material,” or “substantive”
derivative following the particle.
Newton' s law following a fixed differential amount of control mass can be
written, as we have above, in the form ( Adx) D u /Dt = {p(x,t) A} - {p(x,t) +
p/ x dx}A, where p is the acoustic pressure, A is the cross-sectional area, and
our italicized u is the fluid speed at a fixed point in space . The momentum
balance simplifies to D u /Dt = - p/ x. Since u = u (x,t) is a function of the two
independent variables x and t, the convective derivative can be rewritten in
terms of the partial time derivative “ u / t” in which space is fixed, and the
partial space derivative “ u / x” in which time is fixed. Now, the total
differential d u , from calculus, satisfies d u = ( u / t) dt + ( u / x) dx, so that d u /dt
= ( u / t) + ( u / x) (dx/dt). Since dx/dt = u following the mass, we obtain D u /Dt
= d u /dt = u / t + u u / x, and hence, the required Euler momentum equation
( u / t + u u / x) = - p/ x (5.31)
applicable at fixed points in space. Note that the only external force considered
in deriving Equation 5.31 is the pressure force. Thus, Euler' s equation provides
an inviscid flow model only and does not consider attenuative effects due to
viscosity or rheology . These may be added after-the-fact using imaginary
frequency concepts as outlined in Chapters 1 and 2. On the other hand, if
application lengths are short, an inviscid model may suffice. The density in
Equation 5.31, of course, need not be constant, and it may remain quite general.

5.2.1 Steady and unsteady hydraulic limits.
In a steady, constant density, incompressible flow, the time derivative term
u / t disappears and is constant. Then, Equation 5.31 simplifies to u u / x =

- p/ x, that is, {p + 1/2 u 2 }/ x = 0, so that
p(x) + 1/2 u 2 (x) = constant = p + 1/2 u 2 (5.32)
where we have evaluated the constant of integration at known ambient
conditions far upstream (represented by infinity subscripts). Equation 5.32 is
the well known “Bernoulli equation” - for further discussion, the reader is
referred to elementary fluid mechanics textbooks.