Physics
Physics
5th Edition
ISBN: 9781260487008
Author: GIAMBATTISTA, Alan
Publisher: MCGRAW-HILL HIGHER EDUCATION
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Textbook Question
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Chapter 18, Problem 140P

Poiseuille’s law [Eq. (9-41)] gives the volume flow rate of a viscous fluid through a pipe, (a) Show that Poiseuille’s law can be written in the form ΔP = IR, where I = ΔVt represents the volume flow rate and R is a constant of proportionality called the fluid flow resistance. (b) Find R in terms of the viscosity of the fluid and the length and radius of the pipe. (c) If two or more pipes are connected in series so that the volume flow rate through them is the same, do the resistances of the pipes add as for electrical resistors ( R eq = R 1 + R 2 + ) ? Explain. (d) If two or more pipes are connected in parallel, so the pressure drop across them is the same, do the reciprocals of the resistances add as for electrical resistors ( 1 / R eq = 1 / R 1 + 1 / R 2 + ) ? Explain.

(a)

Expert Solution
Check Mark
To determine

Show that Poiseuille’s law can be written in the form ΔP=IR.

Answer to Problem 140P

It is shown that the Poiseuille’s law can be written in the form ΔP=IR.

Explanation of Solution

Write the expression for the volume flow rate.

    I=ΔVΔt        (I)

Here, I is the volume flow rate, ΔV is the volume, Δt is the time.

Write the expression for the Poiseuille’s law.

    ΔVΔt=πr4ΔP8ηL        (II)

Here, ΔP is the driving pressure, η is the viscosity of the liquid, L is the length of the tube, r is the radius of the pipe.

Equate equation (I) and (II) to solve for ΔP.

    ΔP=I8ηLπr4        (III)

Conclusion:

Therefore, it is showed that the Poiseuille’s law can be written in the form ΔP=IR.

(b)

Expert Solution
Check Mark
To determine

The constant of proportionality R in terms of the viscosity of the liquid, the length and the radius of the pipe.

Answer to Problem 140P

The constant of proportionality R in terms of the viscosity of the liquid, the length and the radius of the pipe is 8ηLπr4_.

Explanation of Solution

Write the expression for the driving pressure.

    ΔP=IR        (IV)

Equate equation (III) and (IV) to solve for R.

    R=8ηLπr4        (V)

Conclusion:

Therefore, the constant of proportionality R in terms of the viscosity of the liquid, the length and the radius of the pipe is 8ηLπr4_.

(c)

Expert Solution
Check Mark
To determine

Whether the resistance of the pipes add as for electrical resistors for pipes connected in series.

Answer to Problem 140P

Yes, the resistance of the pipes add as for electrical resistors.

Explanation of Solution

If two or more pipes are connected in series, the volume flow rate remains same.

    I=ΔVΔt        (VI)

Here, ΔVΔt remains same.

Write the expression for the total driving pressure.

    ΔPTot=ΔP1+ΔP2+        (VII)

Here, ΔPTot is the total driving pressure.

Use equation (VI) and (IV) in (VII) to solve for the ΔPTot.

    ΔPTot=IR1+IR2+=I(R1+R2+)        (VIII)

Write the expression electrical resistance for the series combination of resistors.

    Req=R1+R2+        (IX)

Use equation (IX) in (VIII) to solve for ΔPTot.

    ΔPTot=IReq        (X)

Conclusion:

Therefore, the resistance of the pipes add as for electrical resistors.

(d)

Expert Solution
Check Mark
To determine

Whether the resistance of the pipes add as for electrical resistors for pipes connected in parallel.

Answer to Problem 140P

Yes, the resistance of the pipes add as for electrical resistors for pipes connected in parallel.

Explanation of Solution

If two or more pipes are connected as parallel, the driving pressure drops across all the pipes are the same.

Use equation (IV) to solve for ITot.

    ITot=ΔPR1+ΔPR2+=ΔP(1R1+1R2+)        (XI)

Here, ITot is the total volume flow rate.

Write the expression for the electrical resistance for parallel combination.

    1Req=1R1+1R2+        (XII)

Use equation (XII) in (XI) to solve for ITot.

    ITot=ΔPReq        (XIII)

Conclusion:

Therefore, the resistance of the pipes add as for electrical resistors for pipes connected in parallel.

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Chapter 18 Solutions

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