A 1.3-m length of horizontal pipe has a radius of 6.4 ×10^-3 m. Water within the pipe flows wit

step 1 Solving Part A of this problem, since the water behaves as ideal fluid here, so the pressure at

A 1.3-m length of horizontal pipe has a radius of 6.4 ×10^-3...

A $1.3-\mathrm{m}$ length of horizontal pipe has a radius of $6.4 \times 10^{-3} \mathrm{~m}$. Water within the pipe flows with a volume flow rate of $9.0 \times 10^{-3} \mathrm{~m}^{3} / \mathrm{s}$ out of the right end of the pipe and into the air. What is the pressure in the flowing water at the left end of the pipe if the water behaves as (a) an ideal fluid and (b) a viscous fluid $\left(\eta=1.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\right) ?$

Key Concepts

Laminar Flow

Laminar flow refers to a smooth, orderly fluid motion in which layers of fluid slide past one another without mixing. This type of flow is characterized by low Reynolds numbers and is often assumed in many theoretical analyses, such as those involving the Hagen-Poiseuille equation. Laminar flow simplifies the mathematical treatment of fluid motion and is critical when evaluating viscous effects in pipe flows.

Continuity Equation

The Continuity Equation in fluid dynamics is based on the principle of conservation of mass. It states that for an incompressible fluid flowing through a pipe, the product of the cross-sectional area and the fluid velocity remains constant along the flow. This key concept helps relate changes in velocity or cross-sectional area to changes in pressure and is fundamental to analyzing fluid motion.

Viscosity

Viscosity is a measure of a fluid's resistance to gradual deformation by shear stress or tensile stress. It quantifies the internal friction within the fluid, influencing how easily it flows. In fluid dynamics, viscosity plays a crucial role in determining whether the flow is laminar or turbulent and affects the energy loss in fluid transport systems.

Hagen-Poiseuille Equation

The Hagen-Poiseuille equation describes the volumetric flow rate of a viscous fluid in laminar flow through a cylindrical pipe. It shows that the flow rate is proportional to the fourth power of the radius of the pipe and the pressure difference along the pipe, and inversely proportional to the fluid's viscosity and the length of the pipe. This equation is essential for understanding how viscous forces affect pressure drop in real fluids.

Bernoulli's Principle

Bernoulli's Principle is a fundamental concept in fluid dynamics that relates the pressure, velocity, and height (potential energy) along a streamline for an ideal, incompressible and non-viscous fluid. It states that an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or potential energy of the fluid. This principle is used to analyze energy conservation in fluid flow, especially when no frictional losses are considered.

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