8
P. Liu
cannot occupy more than two space points (to ensure that the solution does
not appear discontinuous); each space point can only be occupied by one
fluid particle at any time, but not by more than two particles (to ensure that
the solution does not have multiple values), so people will naturally introduce
single-valued continuous differentiable functions into the analysis of fluid
flow physical quantities. For the numerous fluid particles satisfying the continuity condition, when they move, how to correctly characterize the motion
characteristics of each fluid particle must answer two basic questions. One
is how to track and distinguish each fluid particle, and the other is how to
describe the motion characteristics and changes in each fluid particle. This is
the basic problem of fluid kinematics. According to the different viewpoints
of observers, the motion of fluid particles can be described by the Lagrange
method and the Euler method.
1. Lagrange Method
This method is also called the particle system method of fluid. It identifies
and confirms all fluid particles (not space points), and then records the position coordinates of each particle at different times, so as to understand the
overall flow behavior. Obviously, this method requires an observer to track
every fluid particle at any time and anywhere, and record the particle movement process (directly measuring the position of particles at different times,
leading to the concept of particle trajectory), so as to obtain the motion law
of the overall flow. Whereas, the position coordinates of particles (a, b, c ) at a
stationary time or at an initial time t 0 are used as identifiers of fluid particles
(so that the particle identification is not renamed, as shown in Fig. 1.9), at
any time t, the spatial positions of particles (a, b, c ) are x (a, b, c, t ), y (a, b, c,
t ), z (a, b, c, t ), and the whole flow can be understood by tracking the whole
process of all particles. Among them, the position record of any particle at
different times is the direct measurement data, from which the velocity and
acceleration data obtained by definition and law are indirect measurement
data.
u =
∂ x(a, b, c, t)
∂t
, v =
∂ y(a, b, c, t)
∂t
, w =
∂z(a, b, c, t)
∂t
a x =
∂u(a, b, c, t)
∂t
, a y =
∂v(a, b, c, t)
∂t
, a z =
∂w(a, b, c, t)
∂t
where u, v, and w, respectively, represent the velocity components in x, y,
and z directions, and a x , a y , and a z , respectively, represent the acceleration components in x, y, and z directions. For any fluid particle, the line
P. Liu
cannot occupy more than two space points (to ensure that the solution does
not appear discontinuous); each space point can only be occupied by one
fluid particle at any time, but not by more than two particles (to ensure that
the solution does not have multiple values), so people will naturally introduce
single-valued continuous differentiable functions into the analysis of fluid
flow physical quantities. For the numerous fluid particles satisfying the continuity condition, when they move, how to correctly characterize the motion
characteristics of each fluid particle must answer two basic questions. One
is how to track and distinguish each fluid particle, and the other is how to
describe the motion characteristics and changes in each fluid particle. This is
the basic problem of fluid kinematics. According to the different viewpoints
of observers, the motion of fluid particles can be described by the Lagrange
method and the Euler method.
1. Lagrange Method
This method is also called the particle system method of fluid. It identifies
and confirms all fluid particles (not space points), and then records the position coordinates of each particle at different times, so as to understand the
overall flow behavior. Obviously, this method requires an observer to track
every fluid particle at any time and anywhere, and record the particle movement process (directly measuring the position of particles at different times,
leading to the concept of particle trajectory), so as to obtain the motion law
of the overall flow. Whereas, the position coordinates of particles (a, b, c ) at a
stationary time or at an initial time t 0 are used as identifiers of fluid particles
(so that the particle identification is not renamed, as shown in Fig. 1.9), at
any time t, the spatial positions of particles (a, b, c ) are x (a, b, c, t ), y (a, b, c,
t ), z (a, b, c, t ), and the whole flow can be understood by tracking the whole
process of all particles. Among them, the position record of any particle at
different times is the direct measurement data, from which the velocity and
acceleration data obtained by definition and law are indirect measurement
data.
u =
∂ x(a, b, c, t)
∂t
, v =
∂ y(a, b, c, t)
∂t
, w =
∂z(a, b, c, t)
∂t
a x =
∂u(a, b, c, t)
∂t
, a y =
∂v(a, b, c, t)
∂t
, a z =
∂w(a, b, c, t)
∂t
where u, v, and w, respectively, represent the velocity components in x, y,
and z directions, and a x , a y , and a z , respectively, represent the acceleration components in x, y, and z directions. For any fluid particle, the line