Two hoses are connected to the same outlet using a Y-...

Two hoses are connected to the same outlet using a Y- connector, as the drawing shows. The hoses $A$ and $B$ have the same length, but hose $B$ has the larger radius. Each is open to the atmosphere at the end where the water exits. Water flows through both hoses as a viscous fluid, and Poiseuille's law $\left.Q=\pi R^{4}\left(P_{2}-P_{1}\right) /(8 \eta L)\right]$ applies to each. In this law, $P_{2}$ is the pressure upstream $P_{1}$ is the pressure downstream, and $Q$ is the volume flow rate. (a) For hoses $A$ and $B$, is the value for the term $P_{2}-P_{1}$ the same or different? (b) How is $Q$ related to the radius of a hose and the speed of the water in the hose? Account for your answers. he ratio of the radius of hose $\mathrm{B}$ to the radius of hose $\mathrm{A}$ is $R_{\mathrm{B}} / R_{\mathrm{A}}=1.50 .$ Find the ratio of the speed of the water in hose $\mathrm{B}$ to that in hose $\mathrm{A}$.

Two hoses are connected to the same outlet using a Y- connector, as the drawing shows. The hoses $A$ and $B$ have the same length, but hose $B$ has the larger radius. Each is open to the atmosphere at the end where the water exits. Water flows through both hoses as a viscous fluid, and Poiseuille's law $\left.Q=\pi R^{4}\left(P_{2}-P_{1}\right) /(8 \eta L)\right]$ applies to each. In this law, $P_{2}$ is the pressure upstream $P_{1}$ is the pressure downstream, and $Q$ is the volume flow rate.
(a) For hoses $A$ and $B$, is the value for the term $P_{2}-P_{1}$ the same or different? (b) How is $Q$ related to the radius of a hose and the speed of the water in the hose? Account for your answers.
he ratio of the radius of hose $\mathrm{B}$ to the radius of hose $\mathrm{A}$ is $R_{\mathrm{B}} / R_{\mathrm{A}}=1.50 .$ Find the ratio of the speed of the water in hose $\mathrm{B}$ to that in hose $\mathrm{A}$.

Key Concepts

Poiseuille's Law

This law describes the flow rate of a viscous fluid through a cylindrical pipe under laminar flow conditions. It establishes that the volumetric flow rate is proportional to the fourth power of the pipe's radius and the pressure difference, and inversely proportional to the fluid viscosity and the length of the pipe. This relationship is critical in predicting how changes in geometry and pressure conditions affect flow in pipes.

Pressure Difference (Pressure Drop)

The pressure drop across a segment of a pipe (or hose) is the driving force for fluid flow. In situations where multiple paths originate from a common source or outlet, such as with a Y-connector, the effective pressure difference in each branch is determined by the common upstream and downstream pressures. This means that if the hoses are connected in such a way that they share the same inlet and outlet conditions, the pressure drop driving the flow remains the same in each hose, regardless of their radii.

Relationship Between Flow Rate, Radius, and Speed

The volumetric flow rate (Q) is linked to the speed of the fluid (v) and the cross-sectional area (A) of the hose by the relation Q = A×v, where the area is proportional to the square of the radius. Because Poiseuille's law tells us that Q is proportional to the radius raised to the fourth power (R^4), when we solve for v by dividing Q by A (which is proportional to R^2), we find that the fluid speed is proportional to the square of the radius (v ? R^2). This scaling illustrates how small changes in the radius can have a significant effect on the speed of the fluid within the hose.

Dependence on Hosing Geometry

The geometry of the hose, particularly its radius, plays a critical role in determining both the volumetric flow rate and the fluid speed. Since the flow rate scales with the fourth power of the radius, two hoses with different radii but the same pressure drop and length will have significantly different flow rates. When comparing the speeds, because speed is derived by dividing the flow rate by the cross-sectional area (which scales as R^2), the speed dependence on the radius reduces to a quadratic relationship. This concept explains why, for example, a hose with a 1.5 times larger radius will have a water speed that is (1.5)^2, or 2.25, times that in a smaller hose under identical pressure conditions.

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