In ideal flow, a liquid of density 850 kg / m^3 moves from...

In ideal flow, a liquid of density $850 \mathrm{~kg} / \mathrm{m}^{3}$ moves from a horizontal tube of radius $1.00 \mathrm{~cm}$ into a second horizontal tube of radius $0.500 \mathrm{~cm}$ at the same elevation as the first tube. The pressure differs by $\Delta P$ between the liquid in one tube and the liquid in the second tube. (a) Find the volume flow rate as a function of $\Delta P$. Evaluate the volume flow rate for (b) $\Delta P=6.00 \mathrm{kPa}$ and (c) $\Delta P=12.0 \mathrm{kPa}$.

In ideal flow, a liquid of density $850 \mathrm{~kg} / \mathrm{m}^{3}$ moves from a horizontal tube of radius $1.00 \mathrm{~cm}$ into a second horizontal
tube of radius $0.500 \mathrm{~cm}$ at the same elevation as the first
tube. The pressure differs by $\Delta P$ between the liquid in one tube and the liquid in the second tube.
(a) Find the volume flow rate as a function of $\Delta P$. Evaluate the volume flow rate for
(b) $\Delta P=6.00 \mathrm{kPa}$ and
(c) $\Delta P=12.0 \mathrm{kPa}$.

Key Concepts

Bernoulli's Principle

This concept relates the pressure, velocity, and elevation of an ideal fluid along a streamline. When considering horizontal flows where gravitational potential energy does not change, Bernoulli’s equation shows how a pressure difference can be converted into differences in kinetic energy. In this context, the pressure drop between two sections of a tube results in a change in velocity as the fluid speeds up or slows down in order to conserve energy.

Continuity Equation

The continuity equation states that the mass flow rate must remain constant along a streamline for an incompressible fluid. This imposes a relationship between the cross-sectional area of a tube and the fluid velocity. When the cross-sectional area changes, the fluid velocity adjusts inversely, ensuring that the product of area and velocity (the volume flow rate) remains constant between different sections.

Ideal Fluid Flow

Ideal fluid flow assumes the fluid has no viscosity, no turbulence, and is incompressible. These assumptions simplify the analysis by allowing the direct application of Bernoulli’s principle and the continuity equation without the need to account for frictional losses or energy dissipation. This idealization is crucial in deriving the relationship between pressure difference and flow rate in the given problem.

Volume Flow Rate

The volume flow rate, defined as the product of the cross-sectional area and the fluid velocity, quantifies the amount of fluid passing through a section of a pipe per unit time. In problems like this, where the tube areas vary, the continuity equation is used to ensure that the same volume flow rate is maintained across different sections, relating the velocity in one section to that in another.

Relationship between Pressure Difference and Velocity

Using Bernoulli’s principle, the pressure difference between two points of a fluid in motion can be related to the change in kinetic energy, which is dictated by the fluid velocities at those points. By combining this with the continuity equation, one can derive an expression that links the pressure drop directly to the volume flow rate. This relationship is a key notion in understanding how changes in pressure drive the flow through varying cross-sectional areas.

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