- #1
asdf1
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can someone explain how to solve the bernoulli equation? I'm having a hard time understanding...
Solve Bernoulli Equation to Understanding
- Thread starter asdf1
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In summary, the Bernoulli equation is an energy conservation equation for fluid kinetics. To solve it, one must recognize the differential form and substitute a new variable in order to get a first-order ODE in terms of that variable and x. The key is to try to transform the non-linear equation into a linear one.
- #2
mezarashi
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The Bernoulli equation is an energy conservation equation for fluid kinetics. In what way are you having difficulty solving it? 
- #3
saltydog
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asdf1 said:
You mean:
[tex]y^{'}+P(x)y=Q(x)y^n[/tex]
The key to solving this is to recognize the differential form:
[tex]y^{-n}dy[/tex]
and what, when differentiated, gives this. Well that would be:
[tex]\frac{1}{1-n}y^{1-n}[/tex]
Hey, I know it's not easy. They catch me in here all the time with differential forms.
Ok then so we'll divide by [itex]y^n[/itex] up there in the first equation and take the differential form:
[tex]y^{-n}dy+Py^{1-n}dx=Qdx[/tex]
Alright then,so that's what we have right, the differential [itex]y^{-n}dy[/itex].
So, let:
[tex]z=y^{1-n}[/tex]
and then substitute the differential form of this into the original equation. Here's the first part:
We got:
[tex]y^{-n}dy+Py^{1-n}dx=Qdx[/tex]
So the [itex]y^{-n}dy[/itex] part would just be:
[tex]\frac{1}{1-n}dz[/tex]
Do the rest and then get a first-order ODE in terms of z and x.
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- #4
asdf1
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hmm... so the key is to try to get the non-linear equation into a linear equation...
saltydog, thank you very much for explaining it to me! :)
saltydog, thank you very much for explaining it to me! :)
FAQ: Solve Bernoulli Equation to Understanding
What is the Bernoulli Equation and why is it important in science?
The Bernoulli Equation is a fundamental principle in fluid dynamics that describes the relationship between pressure, velocity, and elevation in a fluid flow. It is important in science because it allows us to understand and predict the behavior of fluids in various situations, such as in pipes, pumps, and aircraft wings.
How do you solve the Bernoulli Equation?
The Bernoulli Equation can be solved by using the equation P + 1/2ρv^2 + ρgh = constant, where P is the pressure, ρ is the density of the fluid, v is the velocity, g is the acceleration due to gravity, and h is the elevation. This equation can be rearranged and manipulated to solve for any of the variables.
Can the Bernoulli Equation be used for all types of fluids?
The Bernoulli Equation is typically used for incompressible fluids, such as water and air at low speeds. It can also be used for compressible fluids, but only under certain conditions and with some modifications to the equation.
What are some real-world applications of the Bernoulli Equation?
The Bernoulli Equation has many practical applications, including calculating the lift force on an airplane wing, determining the flow rate of a fluid through a pipe, and predicting the behavior of water in hydraulic systems. It is also used in the design of turbines, pumps, and other fluid-powered devices.
Are there any limitations to the Bernoulli Equation?
The Bernoulli Equation assumes that the fluid is inviscid (has no internal friction) and that the flow is steady and incompressible. These assumptions may not hold true in all situations, and therefore the results from the equation may not be completely accurate. Additionally, the equation does not take into account factors such as turbulence and boundary effects, which can also affect the behavior of fluids.