How Does Fluid Dynamics Affect Container Stability and Pressure Distribution?

[Lady Godiva]

Here is the question:

There is a rectangular container with the following dimensions: Length 2 m, Width 1 m, and Height 1m. This container is filled with water to 2/3 of its volume and there is a cover on the top. The cover has a hole in the center.

a) Find the maximum acceleration velocity in the horizontal direction before the water would spill out.

b) Find the hydrostatic forces (ignore the contribution from atmospheric pressure) exerted on each surface.

2) A U-tube is made of square duct of the size 1 cm X 1 cm. The two arms are separated by 20 cm. The tube arm height is 60 cm. It is filled with an indicator of specific weight as 2.00. The height of the indicator fluid initially was 25 cm measured from the bottom surface of the connecting duct.

a) If the U-tube is rotating around the center of one tube arm, what is the maximum angular velocity that one can have to prevent the indicator fluid spillage. Make necessary assumptions.

b) show the pressure variation on the bottom surface of the connecting duct as a function of the angular velocity.

The Attempt at a Solution

for (a) I used [itex]tan \theta = \frac {-a_x}{g}[/itex] and basically drew the diagram (can't show it here) I eventually got -a_x = 3.24 m/s2 is this right?

(b) not sure what to do

And the second question I have no idea.

 

[mark726]

Hello,

Thank you for your question. Let's start with the first question:

a) To find the maximum acceleration velocity in the horizontal direction before the water would spill out, we need to use the equation for hydrostatic pressure: P = ρgh, where P is the pressure, ρ is the density of the liquid (in this case water), g is the acceleration due to gravity, and h is the height of the liquid.

In this case, we can use the fact that the container is filled with water to 2/3 of its volume, which means that the height of the water is 2/3 of the total height of the container (1m). This means that the height of the water is 0.67m.

Now, we need to find the pressure at the bottom of the container, which is the maximum pressure that the container can withstand before the water spills out. Plugging in the values, we get:

P = (1000 kg/m^3)(9.8 m/s^2)(0.67m) = 6566 Pa

Next, we need to find the maximum acceleration in the horizontal direction. We can use the equation P = ma, where P is the pressure we just calculated, m is the mass of the water, and a is the maximum acceleration. The mass of the water can be calculated using the density and volume of water, which is 2/3 of the total volume of the container (2m x 1m x 1m = 2m^3). Therefore, the mass of the water is (1000 kg/m^3)(2m^3)(2/3) = 1333 kg.

Plugging in the values, we get:

6566 Pa = (1333 kg)a

a = 4.93 m/s^2

Therefore, the maximum acceleration in the horizontal direction before the water spills out is 4.93 m/s^2.

b) To find the hydrostatic forces exerted on each surface, we can use the equation F = P x A, where F is the force, P is the pressure, and A is the area of the surface. Let's label the surfaces as follows:

Top surface: A1 = 2m x 1m = 2m^2
Bottom surface: A2 = 2m x 1m = 2m^2
Side surface 1: