Question

In a horizontal water pipe that narrows to a smaller radius, the velocity of the water in the section with the smaller radius will be larger. What happens to the pressure? a) The pressure will be the same in both the wider and narrower sections of the pipe. b) The pressure will be higher in the narrower section of the pipe. c) The pressure will be higher in the wider section of the pipe. d) It is impossible to tell.

Short answer

Answer: The pressure will be higher in the wider section of the pipe.

Step-by-step solution

Identify the conservation of mass principle for a fluid in a pipe

According to the principle of the conservation of mass, the mass flow rate through a pipe must remain constant at any point in the pipe. Mathematically, mass flow rate (m_dot) can be expressed as: m_dot = ρAV, where ρ is the fluid density, A is the cross-sectional area of the pipe, and V is the fluid velocity.

Apply the conservation of mass principle to the pipe

For the wider section (1) and the narrower section (2) of the pipe, the mass flow rates must be equal. Therefore, we have: ρ₁A₁V₁ = ρ₂A₂V₂, Since the water is incompressible, ρ₁ = ρ₂, and the equation can be simplified to: A₁V₁ = A₂V₂.

Understand Bernoulli's equation

Bernoulli's equation relates pressure (P), kinetic energy (0.5ρV^2), and potential energy (ρgh) at two points in a fluid flow. The equation is given by: P₁ + 0.5ρV₁^2 + ρgh₁ = P₂ + 0.5ρV₂^2 + ρgh₂, where h is the height of the fluid above a reference point.

Apply Bernoulli's equation to the horizontal pipe

Since the pipe is horizontal, h₁ = h₂, and the height terms in Bernoulli's equation cancel out. We are left with: P₁ + 0.5ρV₁^2 = P₂ + 0.5ρV₂^2.

Substitute the conservation of mass equation into Bernoulli's equation

From Step 2, we found that A₁V₁ = A₂V₂. We can rearrange this equation for the velocity V₁: V₁ = (A₂/A₁)V₂, Substitute this expression for V₁ into the simplified Bernoulli's equation from Step 4: P₁ + 0.5ρ((A₂/A₁)V₂)^2 = P₂ + 0.5ρV₂^2.

Solve for the pressure difference

Rearrange the equation from Step 5 to find the difference between the pressures in the wider and narrower sections of the pipe: P₂ - P₁ = 0.5ρ(V₁^2 - V₂^2) = 0.5ρ((A₂^2 - A₁^2)/A₁^2)V₂^2.

Analyze the pressure difference relation

Since A₂ < A₁, the expression (A₂^2 - A₁^2) will be negative. Thus, the pressure difference (P₂ - P₁) will be negative, meaning that P₂ < P₁. This indicates that the pressure in the narrower section is lower than the pressure in the wider section of the pipe. Therefore, the correct answer is option (c): The pressure will be higher in the wider section of the pipe.

Key concepts

Conservation of Mass in Fluid Dynamics

The principle of conservation of mass is a cornerstone in the study of fluid dynamics. Imagine a busy highway where despite the varying speed of cars, no vehicle disappears into thin air or randomly pops into existence. Much like traffic flow, mass flow in fluids maintains a consistent 'number of cars', or mass, across any point in the pipe. This concept is mathematically expressed as the mass flow rate (m_dot = \rho AV\), where \(\rho\) is the fluid's density, \(A\) its cross-sectional area, and \(V\) the fluid's velocity. This equation underpins much of fluid movement analysis and tells us that if the area \(A\) goes down, as in a narrower section of the pipe, the velocity \(V\) must increase, assuming density remains constant. This principle lays the groundwork for solving countless problems in fluid mechanics, including the behavior of fluids in pipes, pumps, and even across airplane wings.

Fluid Dynamics and Motion

Fluid dynamics is the science behind the movement of liquids and gases. It reveals how fluids behave under various forces and in different environments, much like the study of chess unveils the potential moves each piece can make on the board. When we observe a fluid flowing through a pipe that narrows, fluid dynamics helps predict how the speed and pressure of the fluid will change. The narrowed section forces the fluid to 'hurry along', increasing its velocity to maintain the mass flow rate. A fundamental equation in fluid dynamics, Bernoulli's principle, describes this relationship, stating that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. The theory guides engineers to design efficient plumbing systems, calculate lift on aircraft wings, and model weather systems.

Fluid Pressure and Its Intricacies

Fluid pressure is an elemental concept that's akin to the intensity of a crowd's push against a barrier during a concert. It's the force exerted by a fluid per unit area and can be influenced by numerous factors such as gravity, outside forces, and, importantly, the fluid's velocity. Bernoulli's principle elegantly links these aspects by demonstrating how an increase in a fluid's velocity in one part of a pipe correlates with a decrease in pressure. This relationship enlightens why in our water pipe example, pressure drops as water flows faster through the narrow section. Understanding this principle is not just academic; it's practical. It's how plumbers prevent leaks, how meteorologists predict storms, and how medical professionals understand blood flow in our veins.

Incompressible Fluids in Daily Life

An incompressible fluid is a bit like a block of rubber compared to a sponge; no matter how much you squeeze it, its volume barely changes. Most liquids such as water are considered incompressible because under normal conditions, their volume and density don't change much with pressure. This unchanging density simplifies many calculations in fluid dynamics because it allows us to assume that as a fluid moves through different sections of a pipe, its velocity changes but the quantity (mass) in a given volume does not. This constancy is essential in designing various hydraulic systems, from car brakes to the fountains in a city park, ensuring these systems function predictably and efficiently under different pressures and flows.