Ideal Fluids in Motion — Question 2

Question 2

A steady velocity field can accelerate every passing parcel

A two-dimensional steady velocity field is $\mathbf v=(ax)\hat i-(ay)\hat j$, where $a=2.0\,\mathrm{s^{-1}}$. A parcel is momentarily at $(x,y)=(0.30,0.20)\,\mathrm{m}$. No pressure calculation is required.

1. Find the parcel velocity and acceleration by differentiating its velocity along its motion.

2. Find $x(t)$ and $y(t)$ for a parcel starting at $(x_0,y_0)$ at $t=0$, and explain why fixed probes still see no time variation.

Question 2 - Solution

A steady velocity field can accelerate every passing parcel

(a) Derivation and calculation

$$\mathbf v=(ax,-ay)=(2(.30),-2(.20))=\boxed{(.600,-.400)\,\mathrm{m/s}}.$$$$\dot v_x=a\dot x=a^2x,\qquad\dot v_y=-a\dot y=a^2y.$$$$\boxed{\mathbf a=(a^2x,a^2y)=(1.20,.800)\,\mathrm{m/s^2}}.$$

(b) Result and interpretation

$$\frac{dx}{x}=a\,dt,\quad\frac{dy}{y}=-a\,dt\Rightarrow\boxed{x=x_0e^{at},\quad y=y_0e^{-at}}.$$$$\partial\mathbf v/\partial t=0,\qquad D\mathbf v/Dt=(\mathbf v\cdot\nabla)\mathbf v\ne0.$$A moving parcel samples different positions; a fixed probe does not.

Physical check

The product $xy=x_0y_0$ stays constant along each trajectory.