Two capillary tubes of same length l but radii r1 and r2 are...

Two capillary tubes of same length $l$ but 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 $r$ of the single tube that can replace the two tubes, so that the rate of flow is same as before? (A) $r=r_{1}+r_{2}$ (B) $r=r_{1}^{2}+r_{2}^{2}$ (C) $r^{4}=r_{1}^{4}+r_{2}^{4}$ (D) $\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}$

Two capillary tubes of same length $l$ but 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 $r$ of the single tube that can replace the two tubes, so that the rate of flow is same as before?
(A) $r=r_{1}+r_{2}$
(B) $r=r_{1}^{2}+r_{2}^{2}$
(C) $r^{4}=r_{1}^{4}+r_{2}^{4}$
(D) $\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Combining Parallel Flow Paths

When multiple tubes are arranged in parallel, the total flow rate is the sum of the individual flow rates through each tube. Because the flow rate through each tube is proportional to the fourth power of its radius, the effective radius of a single tube that gives the same overall flow rate is determined by adding the individual contributions raised to the fourth power.

Hagen-Poiseuille Law

This law describes the volumetric flow rate of a viscous fluid through a tube under laminar flow conditions. It states that the flow rate is directly proportional to the pressure difference and the fourth power of the tube's radius, and inversely proportional to the fluid's viscosity and the length of the tube. This relationship is key to understanding how changes in the radius of the tube affect the flow rate.

Laminar Flow in Capillary Tubes

Laminar flow refers to a smooth, orderly flow of fluid in parallel layers with minimal mixing between them. In the context of capillary tubes, it ensures that the flow can be described accurately by the Hagen-Poiseuille equation, and that the cumulative effects of multiple tubes can be predicted by adding their individual flow contributions.

Please give Ace some feedback