16
P. Liu
d’Alembert Paradox of the steady flow of ideal fluid around arbitrary threedimensional objects without drag. In 1753, Euler proposed the continuum
hypothesis, and in 1755, he proposed the Euler method as a spatial point
method to describe fluid motion. Based on the continuum hypothesis and
the ideal fluid model, the differential equation of ideal fluid motion was
established by Newton’s second theorem, namely
du
dt
=
∂u
∂t
+ u
∂u
∂ x
+ v
∂u
∂ y
+ w
∂u
∂z
= f x −
1
ρ
∂ p
∂ x
dv
dt
=
∂v
∂t
+ u
∂v
∂ x
+ v
∂v
∂ y
+ w
∂v
∂z
= f y −
1
ρ
∂ p
∂ y
dw
dt
=
∂w
∂t
+ u
∂w
∂ x
+ v
∂w
∂ y
+ w
∂w
∂z
= f z −
1
ρ
∂ p
∂z
where u, v, and w are the velocity components of the particle, respectively;
f x , f y , and f z are the unit mass force acting on a particle, respectively. p is
the velocity of the particle. The differential equations clearly show that the
mass force acting on a fluid microelement and the pressure on the surface
of the fluid microelement change the motion behavior of the fluid microelement. That is to say, if there is no pressure gradient along a certain direction
without considering the mass force, the velocity of the fluid particle will
remain unchanged along that direction. It can be written in vector form as
d
V
dt
=
f −
1
ρ
∇ p
For steady flow of incompressible fluid with potential mass force, the
Bernoulli equation can be obtained by integrating the Euler equations along
streamlines. Further studies show that the Bernoulli equation is satisfied not
only along the same streamline but also along the same vortex line, potential
flow field, and spiral flow.
In 1781, the French scientist Joseph-Louis Lagrange (1736–1783, as
shown in Fig. 1.15) proposed the particle method for describing fluid motion
and established the relationship between particle velocity and velocity potential function and flow function. On this basis, the conservation theorem of
irrotational flow for ideal barotropic fluids with potential mass was established. In 1785, the French scientist Pierre-Simon Laplace (1749–1827, as
shown in Fig. 1.16) established the Laplace equation based on the potential
function. So far, the classical theoretical system of ideal hydrodynamics and
irrotational flow has been basically established.
P. Liu
d’Alembert Paradox of the steady flow of ideal fluid around arbitrary threedimensional objects without drag. In 1753, Euler proposed the continuum
hypothesis, and in 1755, he proposed the Euler method as a spatial point
method to describe fluid motion. Based on the continuum hypothesis and
the ideal fluid model, the differential equation of ideal fluid motion was
established by Newton’s second theorem, namely
du
dt
=
∂u
∂t
+ u
∂u
∂ x
+ v
∂u
∂ y
+ w
∂u
∂z
= f x −
1
ρ
∂ p
∂ x
dv
dt
=
∂v
∂t
+ u
∂v
∂ x
+ v
∂v
∂ y
+ w
∂v
∂z
= f y −
1
ρ
∂ p
∂ y
dw
dt
=
∂w
∂t
+ u
∂w
∂ x
+ v
∂w
∂ y
+ w
∂w
∂z
= f z −
1
ρ
∂ p
∂z
where u, v, and w are the velocity components of the particle, respectively;
f x , f y , and f z are the unit mass force acting on a particle, respectively. p is
the velocity of the particle. The differential equations clearly show that the
mass force acting on a fluid microelement and the pressure on the surface
of the fluid microelement change the motion behavior of the fluid microelement. That is to say, if there is no pressure gradient along a certain direction
without considering the mass force, the velocity of the fluid particle will
remain unchanged along that direction. It can be written in vector form as
d
V
dt
=
f −
1
ρ
∇ p
For steady flow of incompressible fluid with potential mass force, the
Bernoulli equation can be obtained by integrating the Euler equations along
streamlines. Further studies show that the Bernoulli equation is satisfied not
only along the same streamline but also along the same vortex line, potential
flow field, and spiral flow.
In 1781, the French scientist Joseph-Louis Lagrange (1736–1783, as
shown in Fig. 1.15) proposed the particle method for describing fluid motion
and established the relationship between particle velocity and velocity potential function and flow function. On this basis, the conservation theorem of
irrotational flow for ideal barotropic fluids with potential mass was established. In 1785, the French scientist Pierre-Simon Laplace (1749–1827, as
shown in Fig. 1.16) established the Laplace equation based on the potential
function. So far, the classical theoretical system of ideal hydrodynamics and
irrotational flow has been basically established.