Learning Goal:

To understand the local and convective acceleration of fluid elements.

In the Eulerian description of a fluid flow, the velocity of a fluid is a function of both space and

time, $V(x,y,z,t).$ However, Newton's second law of motion, $F=ma,$ relates the

acceleration of a Lagrangian particle to the sum of the forces on that particle. These two

descriptions can be related using the material, or total derivative of the velocity.

The material derivative of the velocity is an expression for the acceleration of a fluid particle at

a point $(x,y,z).$ This derivative can be written as

$a=\frac{Dv}{Dt}=\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{t}}+(u\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{x}}+v\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{y}}+w\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{z}}),$

where $V(x,y,z,t)=ui+vj+wk$ is the velocity vector. Each component velocity $u,v,$

and $w$ is also a function of $x,y,z,$ and $t.$

The first term of the expression for the acceleration is the local acceleration, which is nonzero

for unsteady flow. The second term is called the convective acceleration, which is nonzero for

nonuniform flow.

Consider the velocity field $V=\{-4$yti$+2xtj\}\frac{m}{s}$ for the following parts.

Part A $-$ The $x-$component of acceleration

Write an expression for the x$-$component of acceleration of the given velocity field as a function of $x,y,z,$ and$/$or $t.$

Express your answer in terms of $x,y,z$ and$/$or $t.$

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Part B $-$ The $y-$component of acceleration

Write an expression for the $y-$component of the acceleration of the given velocity field as a function of $x,y,z,$ and$/$or $t.$

Express your answer in terms of $x,y,z$ and$/$or $t.$

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$a_y=$

$\frac{m}{s^2}$

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