A velocity field assigns a velocity vector to each spatial position and time in a continuum. Following a material particle means solving with its initial position. The resulting Lagrangian trajectory samples the field at changing positions.
The field's spatial variation is the velocity gradient, whose symmetric part gives local strain rate and whose trace gives velocity divergence. Even when , a moving particle can have convective acceleration . A steady flow narrowing through a pipe demonstrates this: fixed sensors give unchanging readings while each passing parcel speeds up. Streamlines are curves tangent to the instantaneous velocity field; they coincide with particle paths only under the appropriate steady-flow conditions.
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