1 Foundation of Fluid Mechanics
13
The general expression of the substantial derivative is
d
dt
=
∂
∂t
+ u
∂
∂ x
+ v
∂
∂ y
+ w
∂
∂z
Note that the arbitrary derivative here is different from the total derivative
in field theory. In field theory, the total derivative of a function u is
du
dt
=
∂u
∂t
+
dx
dt
∂u
∂ x
+
dy
dt
∂u
∂ y
+
dz
dt
∂u
∂z
If it is a substantial derivative, the specified fluid particles must be tracked.
Because the coordinate increment satisfies the motion condition of the same
particle, that is, dx = udt, dy = vdt, dz = wdt. It can be seen from the
above expression that the acceleration of any fluid particle in the Eulerian
coordinate system consists of local acceleration and convective acceleration,
the former depends on the unsteady velocity field and the latter on the nonuniformity of the velocity field. Since any physical theorem is for matter,
the derivative of physical quantity following a fluid particle in the Eulerian
coordinates refers to the substantial derivative.
The Lagrangian method describes fluid motion as follows: global tracking,
full-course recording. The Euler method describes fluid motion as follows:
local tracking and full region recording.
1.3 Establishment and Application
of Differential Equations for Ideal Fluid
Motion
In the eighteenth century, driven by the mechanical industry, classical
mechanics entered the era of establishing system theory system and wide
application under the support of calculus. During this period, the classical continuum mechanics system was formed based on the combination
of the concept of calculus continuous differentiable function and the theory
of particle system mechanics. The assumption of continuum based on the
concept of particle system is the basis of introducing calculus into mechanics
to establish the theoretical system.