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!
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I think that at an instant in time, the streamline would parallel to the velocity vector - would that be the same as the tangent? – Mike Miller Oct 26 '13 at 19:55
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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 $\gamma(t)$).
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