The Equation of Continuity — Question 9

Question 9

A local velocity field must preserve parcel volume

A candidate incompressible two-dimensional flow is $v_x=ax$ and $v_y=by$, where $a=3.0\,\mathrm{s^{-1}}$ and $b$ is unknown. There is no motion or variation in the third direction. Consider a small rectangle of fluid with side lengths $\delta x$ and $\delta y$.

1. Find $b$ by requiring that the rectangle's area remain constant to first order in a short time $dt$.

2. If instead $b=-1.0\,\mathrm{s^{-1}}$, find the instantaneous fractional rate of volume change and the density rate for a material parcel of density $1000\,\mathrm{kg/m^3}$.

Question 9 - Solution

A local velocity field must preserve parcel volume

(a) Derivation and calculation

Over a short interval,$$\delta x^{\prime}=\delta x(1+a\,dt),\quad\delta y^{\prime}=\delta y(1+b\,dt).$$$$\frac{\delta A^{\prime}}{\delta A}=(1+a\,dt)(1+b\,dt)=1+(a+b)dt+O(dt^2).$$$$D V/Dt=0\Rightarrow a+b=0\Rightarrow\boxed{b=-3.00\,\mathrm{s^{-1}}}.$$

(b) Result and interpretation

For $b=-1.00\,\mathrm{s^{-1}}$,$$\boxed{\frac1V\frac{DV}{Dt}=a+b=2.00\,\mathrm{s^{-1}}}.$$$$m=\rho V=\text{constant}\Rightarrow V\frac{D\rho}{Dt}+\rho\frac{DV}{Dt}=0.$$$$\boxed{\frac{D\rho}{Dt}=-\rho(a+b)=-(1000)(2)=-2000\,\mathrm{kg\,m^{-3}\,s^{-1}}}.$$

Physical check

The alternate field describes expansion with decreasing density and cannot represent an incompressible parcel.