density

Knudsen number

Experiment find-outs

A fluid that has no viscosity at all is called an ideal fluid

Real fluids are compressible

For fluid particles

Directly finding the partial derivative is the rate of change

For fixed points in space

random derivative

Lagrangian becomes Euler

Euler to Lagrangian

trace

streamline

system

control body

law of conservation of mass for fluids

Integral form of continuity equation

Differential form of continuity equation

Two special continuity equations

Relative linear expansion rate

Angular deformation

Vorticity

simple shear motion

free vortex motion

forced vortex motion

Find velocity components from velocity potential

The same velocity field can have different velocity potentials

Velocity potential is additive

In the irrotational flow field, the velocity loop along any curve is equal to the difference in velocity potential between the starting and end points.

Streamlines are orthogonal to equipotential surfaces

In a single connected area, the streamline cannot be closed, but in a multi-connected area, it can

Introduction (in plane)

nature

If the fluid is irrotational, the flow function satisfies the Laplace equation and is a harmonic function.

The complex potential is the characteristic function of plane divergence-free potential flow.

Find the velocity potential and stream function

Find the speed

Find the velocity loop and flow rate

uniform flow

point source point sink

point vortex

dipole

flow around corners

Euler's equation

internal energy equation

If the fluid pressure is distributed uniformly, the mechanical energy per unit mass of fluid remains unchanged.

There are four independent equations from the continuity equation and Euler's equation, but there are 5 unknown functions

incompressible homogeneous fluid

Positive pressure fluid

baroclinic fluid

Initial conditions

Boundary conditions

Lambert equation

Ideal positive pressure fluid performs irrotational motion

Complex potential, velocity field and streamlines

D'Alembert's fallacy

Cartesian coordinate system

cylindrical coordinate system

Spherical coordinate system

The vortex flux remains constant along the vortex tube

Kelvin's theorem (velocity potential conservation theorem)

Lagrange's theorem (theorem of the existence of velocity potential)

Helmholtz's second and third theorems

Generation and diffusion of vortices in viscous fluids

Generation of vortices in baroclinic fluids