How Does Bernoulli's Equation Explain Juice Flow in a Straw?

[fubag]

Jane goes to a juice bar with her friend Neil. She is thinking of ordering her favorite drink, 7/8 orange juice and 1/8 cranberry juice, but the drink is not on the menu, so she decides to order a glass of orange juice and a glass of cranberry juice and do the mixing herself. The drinks come in two identical tall glasses; to avoid spilling while mixing the two juices, Jane shows Neil something she learned that day in class. She drinks about 1/8 of the orange juice, then takes the straw from the glass containing cranberry juice, sucks up just enough cranberry juice to fill the straw, and while covering the top of the straw with her thumb, carefully bends the straw and places the end over the orange juice glass. After she releases her thumb, the cranberry juice flows through the straw into the orange juice glass. Jane has successfully designed a siphon.

Assume that the glass containing cranberry juice has a very large diameter with respect to the diameter of the straw and that the cross-sectional area of the straw is the same at all points. Let the atmospheric pressure be p_a and assume that the cranberry juice has negligible viscosity.

Consider the end of the straw from which the cranberry juice is flowing into the glass containing orange juice, and let h_0 be the distance below the surface of cranberry juice at which that end of the straw is located: View Figure: http://session.masteringphysics.com/problemAsset/1003174/23/137431.jpg . What is the initial velocity v of the cranberry juice as it flows out of the straw? Let g denote the magnitude of the acceleration due to gravity.
Express your answer in terms of g and h_0.



I am somewhat lost in starting this problem, but I know that Bernoulli's equation of p + (1/2)rho*v^2 + rho*g*y = constant..

I tried doing:

p + (1/2)rho*v^2 + rho*g*d = p + (1/2)rho*v^2 + rho*g*(h_0 - d)

but this didnt get me anywhere.

please help

 

[rl.bhat]

Juice will not flow from the straw until it reaches the height ho in the outer portion of the straw. At that instant presure difference between the two ends of straw is rho*d*g. When the flow starts, apply Bernoulli's equation. rho*d*g = 1/2(rho*Vo^2). Hence Vo^2 = 2*d*g. Before it comes out of the straw it falls through a height ho. Hence final velocity V^2 = Vo^2 + 2gh. = 2*d*g + 2*ho*g

 

[fubag]

although this makes sense,

the feedback I got was that the answer doesn't depend on the variable d...

is my initial setup somewhat close or am I missing something?

 

[stewartcs]

This may help...

http://en.wikipedia.org/wiki/Siphon

 

[rl.bhat]

If the juice is already at the tip of the straw when the thumb is released, initial velocity will be sqrt.2dg. Later on the flow is controlled by the partial vacuume created in the tube and it is indipendent of the d.

 

[fubag]

ok thanks a lot to stewartcs for the article, really helped alot.

now I understand that the initial velocity is equal to the sqrt. (2gh_0).

The second part of the problem is also giving me some trouble:

find the time t it takes to Jane to transfer enough cranberry juice into the orange juice glass to make her favorite drink if h_0 = 10.0 centimeters. Assume that the flow rate of the liquid is constant, and that the glasses are cylindrical with a diameter of 7.0 centimeters and are filled to height 14.0 centimeters. Take the diameter of the straw to be 0.4 centimeters

So far I have that the initial velocity will be sqrt.(2*9.8*.1) . From the context of the problem i understand that 1/8 of the orange juice is missing and is being replaced by cranberry juice so the distance needed to be filled up is (1/8)14cm. Can I just simply use a one-dimensional linear motion to find time using vf = vi + at?

I do not understand how to apply the diameter of the cup and straw. All i can think of is somehow equating the volume*area of the glass with the straw but I don't know if that will work.

 

[stewartcs]

Since you now know the velocity, and you are given the diameter of the straw, you can find the volumetric flow rate (Q = v*A) out of the straw. Then you should be able to determine the volume to fill based on the given height and diameter of the glasses (cylinders). Once you know the volume to fill, and the fill rate (flow rate, Q), it's simple math.

 

[fubag]

ok so if i use Q = v*A, i have that v = pi*(3.5cm^2)*((1/8)*14cm)...and I know the velocity is traveling at sqrt.(2gh_0), now I can just divide this by the volume to find the amount of time?

 

[fubag]

ok so I did a bit of research/studying

this is what I attempted for the second part of the question...I know that Q = v*A is the volume flow rate... through the straw I used the values for v = sqrt.(2gh_0) all of which are given. For area I took the radius of the pipe...

Now for determining the amount of time...this took a bit of thought but I eventually got it...

we need to know the volume of the cup needed to be filled, from the context of the story we know we need to fill up 1/8 of the cup, since they tell us its 14 cm, that h is easy to calculate. using the volume of a cylinder we can get the volume...

since Q = volume flow rate, if we divide this from volume needed to be filled we get time!thanks a lot guys for all your support, i finally got it...!

:D

 

[stewartcs]

Looks good to me!