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Suppose the density $\mathbf{蚁}$ varies from point to point as well as with time, that is, $\mathbf{蚁}\mathbf{=}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{,}\mathbf{t}\mathbf{)}$. If we follow the fluid along a streamline, then $\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}$ are function of such that the fluid velocity is

$\mathbf{v}\mathbf{=}\mathbf{i}\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{j}\frac{\mathbf{dy}}{\mathbf{dt}}\mathbf{+}\mathbf{k}\frac{\mathbf{dz}}{\mathbf{dt}}$

Show that then$\mathbf{d蚁}\mathbf{/}\mathbf{dt}\mathbf{=}\mathbf{鈭�}\mathbf{蚁}\mathbf{/}\mathbf{鈭�}\mathbf{t}\mathbf{+}\mathbf{v}\mathbf{鈭�}\mathbf{鈭�}\mathbf{蚁}$ . Combine this equation with $\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{9}\mathbf{)}$to get

$\mathbf{蚁}\mathbf{鈭�}\mathbf{鈭�}\mathbf{v}\mathbf{+}\frac{\mathbf{d蚁}}{\mathbf{dt}}\mathbf{=}\mathbf{0}$

(Physically, is the rate of change of density with time as we follow the fluid along a streamline; $\mathbf{鈭�}\mathit{p}\mathbf{/}\mathbf{鈭�}\mathbf{t}$is the corresponding rate at a fixed point.) For a steady state (that is, time-independent), $\mathbf{鈭�}\mathbf{p}\mathbf{/}\mathbf{鈭�}\mathbf{t}\mathbf{}\mathbf{=}\mathbf{0}$, but $\mathbf{鈭�}\mathit{p}\mathbf{/}\mathbf{鈭�}\mathbf{t}$ is not necessarily zero. For an incompressible fluid, $\mathbf{鈭�}\mathit{p}\mathbf{/}\mathbf{鈭�}\mathbf{t}\mathbf{}\mathbf{=}\mathbf{0}$. Show that then role="math" localid="1657336080397" $\mathrm{鈭�}脳\mathbf{v}=\mathbf{0}$. (Note that incompressible does not necessarily mean constant density since $\mathbf{鈭�}\mathit{p}\mathbf{/}\mathbf{鈭�}\mathbf{t}\mathbf{}\mathbf{=}\mathbf{0}$does not imply either time or space independence of $\mathbf{蚁}$; consider, for example, a flow of watermixed with blobs of oil.)

Step by step solution

01

Given Information.

The function of fluid velocity is mentioned below.

$\mathrm{v}=\mathrm{i}\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{j}\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{k}\frac{\mathrm{dz}}{\mathrm{dt}}$

The rate of change of density with time is $\mathrm{鈭�}\mathrm{蚁}/\mathrm{鈭�}\mathrm{t}$.

The corresponding rate at a fixed point is $\mathrm{鈭�}\mathrm{蚁}/\mathrm{鈭�}\mathrm{t}$.

02

Definition of time derivative.

A time derivative refers to a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function.

03

Taking the equation of time derivative.

Since is a function of both position and time,

$d蚁=\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}t}dt+\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}x}dx+\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}y}dy+\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}z}dz$

Take the time derivative.

$\frac{d蚁}{dt}=\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}t}+\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}x}\frac{dx}{dt}+\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}y}\frac{dy}{dt}+\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}z}\frac{dz}{dt}$

But, $v脳\mathrm{鈭�}蚁=\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}x}\frac{dx}{dt}+\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}y}\frac{dy}{dt}+\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}z}\frac{dz}{dt}$.

$\frac{dp}{dt}=\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}t}+v脳\mathrm{鈭�}蚁$

04

Substitute ∇×(ρv)=−∂ρ∂t .

Substitute$\mathrm{鈭�}脳\left(蚁v\right)=鈭�\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}t}$.

$\frac{d蚁}{dt}=v脳\mathrm{鈭�}蚁鈭�\mathrm{鈭�}脳\left(蚁v\right)$

But $\mathrm{鈭�}脳\left(蚁\mathrm{v}\right)=蚁(\mathrm{鈭�}脳\mathrm{v})+\mathrm{v}脳\mathrm{鈭�}蚁$.

$\frac{\mathrm{鈭�}蚁}{\mathrm{鈭�}t}=鈭�蚁(\mathrm{鈭�}脳谓)$

Or, $\frac{\mathrm{d}蚁}{\mathrm{dt}}+蚁(\mathrm{鈭�}脳v)=0$

05

For incompressible fluid dρdt=0 .

According to the equation,$\frac{\mathrm{d蚁}}{\mathrm{dt}}+\mathrm{蚁}(\mathrm{鈭�}脳\mathrm{谓})=0$

For incompressible fluid, $\frac{\mathrm{d蚁}}{\mathrm{dt}}=0$

From the above steps, we get the equation mentioned below.

$\mathrm{鈭�}脳\mathrm{v}=0$

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