A curve that is tangent to the velocity vector at each point in the flow field is called line of flow. Line of flow is a curve composed of different fluid particles at the same time, which gives the velocity direction of different fluid particles at that moment.
Determine the differential equation of the streamline as:
Dr×v(r,t)=0
Where v(r,t) and dr are the velocity vector and the arc element vector, respectively, t is time, and is regarded as a constant when integrating. The expression of the above equation in the Cartesian coordinate system is:
If C is a closed curve that is not streamlined in the fluid and does not self-intersect, and at each point through C at each point as a streamline, the surface formed by these streamlines is called a flow tube.
A trace is a curve drawn by a fluid particle as it moves through space. It gives the direction of velocity of the same fluid particle at different times. If the fluid motion is given in the form of Euler variables: v = v(r, t), where v is the velocity vector; r is the radial diameter, and t is the time, then the following differential equations are integrated:
And after the integration, the t in the resulting expression is eliminated to obtain the trace equation. In the above formulas, t is an independent variable; the rectangular coordinates x, y, and z are functions of t; u, v, and w are components of the velocity vector on the x, y, and z axes, respectively.
