A flat plate of width L = 1 m  is pushed down with a velocity U = 0.01 m/s towards a wall resulting in the drainage of the fluid between the plate and the wall as shown in the figure. Assume two-dimensional incompressible flow and that the plate remains parallel to the wall. The average velocity, U_avg of the fluid (in m/s) draining out at the instant shows in the figure is ___________ (correct to three decimal places).

The correct answer is 0.05.

Step 1: Understand the system parameters from the description and the image
Based on the provided diagram, we have:
• Width of the flat plate, $L=1\text{m}$
• Downward velocity of the plate, $U=0.01\text{m/s}$
• The gap between the plate and the wall, $d=0.1\text{m}$ (as labeled in the image)
• The average draining velocity of the fluid escaping from both the left and right openings, $u_{\text{avg}}$

Step 2: Apply the Principle of Conservation of Mass (Continuity Equation)
For a two-dimensional, incompressible flow, the rate at which fluid volume is displaced by the downward moving plate must equal the rate at which fluid volume flows out through the side gaps.
Let us assume the length (depth) of the plate perpendicular to the page is $b$.

The rate of fluid volume displaced by the plate moving downward is:
$Q_{\text{displaced}}=L·b·U$

Since the plate remains parallel to the wall, the fluid drains out symmetrically through two symmetric side gaps (left and right), each having a flow area of $d·b$.
The total rate of fluid volume draining out is:
$Q_{\text{draining}}=2·(d·b)·u_{\text{avg}}$

Equating the displaced volume flow rate to the draining volume flow rate for the incompressible fluid:
$L·b·U=2·d·b·u_{\text{avg}}$

Dividing both sides by the depth $b$:
$L·U=2·d·u_{\text{avg}}$

Solving for the average velocity $u_{\text{avg}}$:
$u_{\text{avg}}=\frac{L·U}{2·d}$

Step 3: Substitute the numerical values
Using the given values:
$u_{\text{avg}}=\frac{1·0.01}{2·0.1}$

Simplifying the calculation:
$u_{\text{avg}}=\frac{0.01}{0.2}=0.05\text{m/s}$

Thus, the average velocity of the fluid draining out is 0.05 m/s.