# Archived Syringe Problem (Fluid Dynamics)

#### Mintea

1. Homework Statement
A hypodermic syringe contains a medicine with the density of water. The barrel of the syringe has a cross-sectional area of 2.5*10^-5 m^2. In the absense of a force on the plunger, making the medicine squirt from the needle. If the syringe is horizontal and the pressure within the syringe remains 1 atm, what is the medicine's flow speed through the needle?

2. Homework Equations

Pressure = Force/Area
Flow Rate: A1v1=A2v2
P + ρv2/2 + ρgy = constant for a whole pipe
((( ρ (the last two p-looking things) = rho (density) )))

3. The Attempt at a Solution

I tried dividing the force on the syringe by the area to get a pressure, but I'm not sure how to work with that pressure. And I think you have to use the flow rate equation to calculate the velocity out of the needle, because it looks as if the fluid would have different speeds in different parts of the syringe. However, we're not given a second area... how can we find the speed at the end of the syringe without the second area?

I'm pretty lost.

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#### VTSC87

I'm tipping zero being that you have created vacuum and assuming said medicine has the same surface tension properties as water.
That said surface evaporation could occur over extended periods of time!

#### mukundpa

Homework Helper
When the pressure in the syringe is 1 atm (equal to atmospheric pressure) what is the driving force to make the medicine move?

#### TSny

Homework Helper
Gold Member
Is the problem stated correctly? Looks like a problem out of Serway

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#### James R

Homework Helper
Gold Member
I think you're right, TSny. It's most likely the Serway problem you gave, and the opening post is incomplete. So, we should have something like:

"A hypodermic syringe contains a medicine with the density of water. The barrel of the syringe has a cross-sectional area of 2.5*10^-5 m^2. In the absense of a force on the plunger the pressure everywhere is 1 atm. A force of 2.00 N is exerted on the plunger, making the medicine squirt from the needle. If the syringe is horizontal and the pressure within the needle remains at 1 atm, what is the medicine's flow speed through the needle?"​

The relevant equations are correct in the opening post.

Let $p_0$ be atmospheric pressure. The force on the plunger, $F$, adds an excess pressure in the barrel of the syringe of $p=F/A_b$, where $A_b$ is the given cross-sectional area of the barrel.

Applying Bernoulli's equation to the horizontal syringe we have
$$p_0+p+\frac{1}{2}\rho v_b^2 = p_0 + \frac{1}{2}\rho v_n^2,$$
where $v_b$ is the speed of the flow in the barrel and $v_n$ is the speed in the needle. The speeds are related by the continuity equation:
$$A_b v_b = A_n v_n.$$
We are asked about the speed through the needle, so putting the pieces together we have
$$\frac{F}{A_b}+\frac{1}{2}\rho(\frac{A_n}{A_b}v_n)^2 = \frac{1}{2}\rho v_n^2$$
or
$$\rho \left(1-\left(\frac{A_n}{A_b}\right)^2\right) v_n^2 = \frac{2F}{A_b}$$
from which we can determine $v_n$ as long as we know the cross-sectional area of the needle. However, that information seems to be missing from the question statement.

#### TSny

Homework Helper
Gold Member
$$\rho \left(1-\left(\frac{A_n}{A_b}\right)^2\right) v_n^2 = \frac{2F}{A_b}$$
from which we can determine $v_n$ as long as we know the cross-sectional area of the needle. However, that information seems to be missing from the question statement.
Yes, nice. To a good approximation, $\left(\frac{A_n}{A_b} \right)^2$ can be neglected relative to $1$.

#### James R

Homework Helper
Gold Member
Yes, nice. To a good approximation, $\left(\frac{A_n}{A_b} \right)^2$ can be neglected relative to $1$.
That's true.

"Syringe Problem (Fluid Dynamics)"

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