ABSTRACT
Euler equations in the orthogonal Cartesian coordinates. In the orthogonal Cartesian coordinates (x, y, z) the Euler equations (1)–(2) are written as:
∂u
∂t + u
∂u
∂x + v
∂u
∂y + w
∂u
∂z = –
∂p
∂x ,
∂v
∂t + u
∂v
∂x + v
∂v
∂y + w
∂v
∂z = –
∂p
∂y ,
∂w
∂t + u
∂w
∂x + v
∂w
∂y + w
∂w
∂z = –
∂p
∂z ,
∂u
∂x + ∂v
∂y + ∂w
∂z = 0,
(3)
where u, v, and w are the components of the fluid velocity. Euler equations in the cylindrical coordinates. The cylindrical coordinates (r, ϕ, z)
are related to the orthogonal Cartesian coordinates (x, y, z) by
r =
√ x2 + y2, tanϕ = y/x, z = z (sinϕ = y/r);
x = r cosϕ, y = r sinϕ, z = z,
where 0 ≤ ϕ ≤ 2π.
