[Math] Streamlines tangent to velocity field

fluid dynamics

As from the title, I'm not too sure how they are related. Definition is that streamlines are instantaneously tangential to the velocity vector of the field. Why would a steamline that shows direction be a tangent to the velocity? Thanks!

Best Answer

The velocity field $V(x)$ says what velocity fluid particles have at each particular point.

Streamline is a curve along which said particle flow. Denote its parametric form by $\gamma(t)$, where $t$ is time.

Since the velocity is the derivative of position, $V(\gamma(t))=\gamma'(t)$. This equation relates velocity field to streamline. The derivative $\gamma'(t)$ is also known as the tangent vector. It is usually drawn as an arrow beginning at $\gamma(t)$, not at $0$ - hence tangent, not just parallel to tangent. (There is a good reason for it, too: in the language of manifolds, $\gamma'(t)$ is an element of the tangent space to $\gamma$ at $\gamma(t)$).