(adsbygoogle = window.adsbygoogle || []).push({}); 1. The velocity vector which describes the motion of a particle (point) in a fluid is [tex] \vec {u} = \vec {u}(\vec{x},t), [/tex] so that the particle follows the path on which [tex] \frac {dx}{dt}=\vec{U}(t)=\vec{u}[\vec{x}(t),t]. [/tex]

Write [tex] \vec{x}\equiv (x,y,z) [/tex] and [tex] \vec{u} \equiv (u,v,w) [/tex] (in rectangular Cartesian coordinates), and hence show that the acceleration of the particle is [tex] \frac {d\vec{U}}{dt} = \frac {\partial \vec{u}}{\partial t} + (\vec {u} . \nabla) \vec {u} \equiv \frac {D\vec {u}}{Dt} [/tex], the material derivative

2. Relevant equations

Acceleration of particle = [tex] \frac{d^2 x}{dt^2} = \frac {dU}{dt}=\frac{du}{dt}[\frac {\partial x}{\partial t}, \frac {\partial y}{\partial t}, \frac {\partial z}{\partial t}, 1] [/tex]

[tex](\vec{u}.\nabla)\vec{u} = uu_x\vec{i} + uv_x\vec{j} + uw_z\vec{k} + vu_y\vec{i} + vv_y \vec{j} + vw_y \vec {k} + wu_z \vec{i} + wv_z \vec{j} + ww_z \vec {k} [/tex]

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# Homework Help: Acceleration of a fluid particle

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