Continuity Equation: Relationship between vA and vB in terms of d and D

In summary, the continuity equation relates the velocities of two fluids in a system to their respective cross-sectional areas and densities. In the given example, if the diameters are d = 1 cm and D = 10 cm, the ratio of speeds vB/vA is 0.0001. The equation is: A1v1=A2v2, where A is the cross-sectional area and v is the velocity.
  • #1
bcalkins
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Homework Statement


The continuity equation provides a second relation between the vA and vB, this time in terms of the diameters d and D. Numerical check: If the diameters are d = 1 cm and D = 10 cm, what is the ratio of the speeds, vB/vA?

Homework Equations


To clarify, is both d and D diameters, just one's capitalized to show the difference between the two?
The continuity equation is: Δm=ρV=ρAvΔt
my notes show: ρ2A2v2 = ρ1A1v1
Am I reading correctly that:
ρ=density
A= cross-sectional area (equal to ∏(Diameter^2/4))?
v= velocity/speed

The Attempt at a Solution


A1v1=A2v2 ===> v2=v1(A1/A2) ====> v2=v1(d1^2/d2^2) ===>
Plugging this into the above given information:
v2= v1(10cm^2/1cm^2) = v1(100cm)
v1 = v2 (1cm^2/10cm^2) = v2(.01cm)
So the ratio of speeds would be .01cm/100cm = .0001
I know I'm doing something wrong, but this is all I have in my notes that looks somewhat useable. Please help! Thank you!
 
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  • #2
Wow, after typing that all out I was seeing what I was doing wrong. I've got it figured out now. :)
 

1. What is the Continuity Equation?

The Continuity Equation is a fundamental principle in fluid dynamics that states that the amount of fluid entering a system must equal the amount of fluid leaving the system, assuming there is no accumulation or depletion of fluid within the system.

2. How is the Continuity Equation expressed mathematically?

The Continuity Equation can be expressed as vAd = vBD, where v represents the velocity of the fluid, d represents the diameter of the inlet, and D represents the diameter of the outlet.

3. What is the relationship between vA and vB in the Continuity Equation?

The Continuity Equation shows that the velocity of the fluid at the inlet, vA, is inversely proportional to the velocity of the fluid at the outlet, vB. This means that as the diameter of the outlet decreases, the velocity of the fluid at the outlet increases, and vice versa.

4. How does the diameter of the inlet and outlet affect the Continuity Equation?

The Continuity Equation shows that as the diameter of the inlet decreases, the velocity of the fluid at the inlet increases, and vice versa. Similarly, as the diameter of the outlet decreases, the velocity of the fluid at the outlet increases, and vice versa. This relationship is important in understanding how changes in the diameter of a system can impact the flow of fluid.

5. Can the Continuity Equation be applied to all types of fluids?

Yes, the Continuity Equation can be applied to all types of fluids, including gases and liquids. However, it assumes that the fluid is incompressible, meaning that its density remains constant throughout the system. It also assumes that there is no turbulence or other external factors affecting the flow of the fluid.

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