(D) According to Poiseuille's equation,the rate of flow $V$ through a capillary tube is given by $V = \frac{\pi P r^4}{8 \eta l}$.Since the two tubes

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(D) According to Poiseuille's equation,the rate of flow $V$ through a capillary tube is given by $V = \frac{\pi P r^4}{8 \eta l}$.Since the two tubes are connected in parallel,the total rate of flow $V$ is the sum of the rates of flow through each tube: $V = V_1 + V_2$.Substituting the expression for flow rate: $\frac{\pi P r^4}{8 \eta l} = \frac{\pi P r_1^4}{8 \eta l} + \frac{\pi P r_2^4}{8 \eta l}$.Canceling the common terms $\frac{\pi P}{8 \eta l}$ from both sides,we get $r^4 = r_1^4 + r_2^4$.Therefore,the radius of the single tube is $r = (r_1^4 + r_2^4)^{1/4}$.

Class 11 Physics Mechanical Properties of Fluids Reynold's Number and Poiseuille's Equation

Difficult

Two capillary tubes of the same length but different radii $r_1$ and $r_2$ are fitted in parallel to the bottom of a vessel. The pressure head is $P$. What should be the radius of a single tube that can replace the two tubes so that the rate of flow is the same as before?

A
$r_1 + r_2$
B
$r_1^2 + r_2^2$
C
$r_1^4 + r_2^4$
D
$(r_1^4 + r_2^4)^{1/4}$

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Mechanical Properties of Fluids

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Reynold's Number and Poiseuille's Equation

The rate of steady volume flow of water through a capillary tube of length $l$ and radius $r$,under a pressure difference of $p$ is $V$. This tube is connected with another tube of the same length but half the radius,in series. Then,the rate of steady volume flow through them is (The pressure difference across the combination is $p$.)

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Water is conveyed through a uniform tube of $8 \text{ cm}$ in diameter and $3140 \text{ m}$ in length at the rate of $2 \times 10^{-3} \text{ m}^3/\text{s}$. The pressure required to maintain the flow is (Viscosity of water $= 10^{-3} \text{ SI units}$):

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What is the value of Reynolds number for turbulent flow?

Medium

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$(a)$ What is the largest average velocity of blood flow in an artery of radius $2 \times 10^{-3} \; m$ if the flow must remain laminar?
$(b)$ What is the corresponding flow rate? (Take viscosity of blood to be $2.084 \times 10^{-3} \; Pa \; s$).

Medium

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Water from a pipe is coming at a rate of $100\, L/min$. If the radius of the pipe is $5\, cm$, the Reynolds number for the flow is of the order of: (density of water $= 1000\, kg/m^3$, coefficient of viscosity of water $= 1\, mPa\, s$)

DifficultJEE MAIN 2019

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Two capillary tubes of the same length but different radii $r_1$ and $r_2$ are fitted in parallel to the bottom of a vessel. The pressure head is $P$. What should be the radius of a single tube that can replace the two tubes so that the rate of flow is the same as before?

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Water is conveyed through a uniform tube of $8 \text{ cm}$ in diameter and $3140 \text{ m}$ in length at the rate of $2 \times 10^{-3} \text{ m}^3/\text{s}$. The pressure required to maintain the flow is (Viscosity of water $= 10^{-3} \text{ SI units}$):

What is the value of Reynolds number for turbulent flow?

$(a)$ What is the largest average velocity of blood flow in an artery of radius $2 \times 10^{-3} \; m$ if the flow must remain laminar?$(b)$ What is the corresponding flow rate? (Take viscosity of blood to be $2.084 \times 10^{-3} \; Pa \; s$).

Water from a pipe is coming at a rate of $100\, L/min$. If the radius of the pipe is $5\, cm$, the Reynolds number for the flow is of the order of: (density of water $= 1000\, kg/m^3$, coefficient of viscosity of water $= 1\, mPa\, s$)

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$(a)$ What is the largest average velocity of blood flow in an artery of radius $2 \times 10^{-3} \; m$ if the flow must remain laminar?
$(b)$ What is the corresponding flow rate? (Take viscosity of blood to be $2.084 \times 10^{-3} \; Pa \; s$).