Question

In the 2D squeezing flow, the plate at y=0 is stationary, and the plate at y=h is given a velocity of u = (0, -V). Investigate the axi-symmetric squeezing flow.

Answer 1

The system is as shown below:

insinite plate y=h TTTTTT yo Lisad plate

Consider a cylindrical control volume (CV) of radius r and length L, as shown below:

e axis of symmetry

The walls of the CV are expanding with a radial velocity of Vw as shown. All the fluid particles at |x|=r will be having same radial velocity.

Since the fluid is incompressible, therefore volume of the CV must be constant.

Vol = volume of CV = 2*pi*r*L _________ eq. 1 (here pi = 22/7)

Therefore, $\frac{\mathrm{d}vol }{\mathrm{d} t} = \frac{\mathrm{d}( 2*pi*r*L)}{\mathrm{d} t}=0$    (since volume of CV is constant)

$=>L*\frac{\mathrm{d} r}{\mathrm{d} t}+r*\frac{\mathrm{d} L}{\mathrm{d} t}=0$

$or \;L*Vw-r*V=0$ (since $\frac{\mathrm{d} r}{\mathrm{d} t}=Vw\;and\;\frac{\mathrm{d} L}{\mathrm{d} t}=-V$ )

also L = h - V*t,

$=> Vw = r*V/L=r*V/(h-V*t)$ __________________ eq. 2

Also,

$Aw = radial\; acceleration\;=\frac{\mathrm{d} Vw}{\mathrm{d} t}$

$=\frac{V}{h-V*t}*\frac{\mathrm{d} r}{\mathrm{d} t} + r*V*\frac{\mathrm{d}(\frac{1}{h-V*t}) }{\mathrm{d} t}$

(using $\frac{\mathrm{d} r}{\mathrm{d} t}=Vw$ )

$=\frac{V^2*r}{(h-V*t)^2} + \frac{r*V^2}{(h-V*t)^2}$

$=\frac{2*V^2*r}{(h-V*t)^2}$ ____________________ eq. 3

Therefore, the velocity and acceleration of the fluid particles vary with radius r from axis of symmetry and with time as well.