Need help with Fluid Dynamics Problem

In summary, Vivek was trying to solve a fluid dynamics problem but had some problems. He eventually figured out the solution but had to go back and fix some errors in his code.
  • #1
maverick280857
1,789
4
Hi everyone

I have a relatively simple fluid dynamics problem I need some help with:

We have two cylinders L and R of crossectional areas A and 2A respectively. Initially, the level of water in X is H and Y is empty. At t = 0, the two cylinders are joined at the bottom by a tube of cross-section a (after a hole of the same cross-section is opened in each cylinder). Find the time at which the water level is equal in both cylinders.

This is what I've done so far.

Denote the water levels in the left (L) and right (R) cylinders by [itex]y_{L}[/itex] and [itex]y_{R}[/itex] respectively. So [itex]y_{L}(t=0) = H[/itex] and [itex]y_{R}(t=0) = 0[/itex].

Mass conservation (or continuity equation) leads to [itex]AH = Ay_{L} + 2Ay_{R}[/itex] or equivalently [itex]H = y_{L} + 2y_{R}[/itex]. This gives [itex]0 = \dot{y_{L}} + 2\dot{y_{R}}[/itex].

Applying Bernoulli's Theorem to two points at the surface of each meniscus, we get

[itex]P_{atm} + \frac{1}{2}{\rho v_{1}^2} + \rho g h_{1} = P_{atm} + \frac{1}{2}{\rho v_{2}^2} + \rho g h_{2}[/itex]

where [itex]P_{atm}[/itex] is the atmospheric pressure, [itex]\rho[/itex] is the density of water, [itex]v_{1} = -\dot{y_{L}}[/itex], [itex]v_{2} = \dot{y_{R}}[/itex], [itex]h_{1} = y_{L}[/itex], [itex]h_{2} = y_{R}[/itex].

Hence,

[itex]\frac{1}{2}{\rho \dot{y_{L}}^2} + \rho g y_{L} = \frac{1}{2}{\rho \dot{y_{R}}^2} + \rho g y_{R}[/itex]

After simplifying a bit, this gives a differential equation in [itex]y_{R}[/itex] with the boundary conditions [itex]y_{R}(t=0) = 0[/itex] and [itex]y_{R}(t = T) = \frac{H}{3}[/itex] where H/3 is the equilibrium height of water level (in each cylinder--this follows from the mass conservation equation above) and T is the time when this happens.

Now, if you try this out you get an imaginary (and therefore ridiculous) solution for [itex]y_R[/itex]. As far as I think, my algebra is okay so there must be a conceptual fault somewhere. I would be very grateful if someone could offer some advice.

PLEASE NOTE: This is not a homework problem but I couldn't think of a better place to post it on PF.

Thanks and cheers,
Vivek

EDIT: The answer should involve [itex]a[/itex], the cross-section of the orifice, but it doesn't in my case.
 
Last edited:
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  • #2
Hi again

I checked my TeX code (it seems okay to me) but most of it isn't working...

Cheers
Vivek

EDIT: Its working now :-D
 
Last edited:
  • #3
I figured it out thanks.
 

What is fluid dynamics?

Fluid dynamics is the study of how fluids such as liquids and gases move and interact with their surroundings. It involves the analysis of forces, velocities, pressures, and other physical properties of fluids in motion.

Why is fluid dynamics important?

Fluid dynamics is important in many fields, including engineering, physics, meteorology, and oceanography. It helps us understand and predict the behavior of fluids, which is crucial in designing efficient systems and structures, predicting weather patterns, and studying natural phenomena such as ocean currents.

What are the main equations used in fluid dynamics?

The main equations used in fluid dynamics are the continuity equation, Euler's equation, and the Navier-Stokes equations. These equations describe the conservation of mass, momentum, and energy in a fluid.

What are some common applications of fluid dynamics?

Some common applications of fluid dynamics include designing aerodynamic vehicles, predicting and managing the flow of air and water in pipes and channels, and understanding weather patterns and ocean currents. It is also used in medical research to study blood flow and in the design of biomedical devices.

What are some common challenges in solving fluid dynamics problems?

Solving fluid dynamics problems can be challenging due to the complexity and non-linearity of the equations involved. The behavior of fluids can also be unpredictable, making it difficult to accurately predict their motion. Additionally, the use of computer simulations and numerical methods to solve these problems can also present challenges in terms of accuracy and computational resources.

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