A nonviscous liquid of density p flows at speed v₀ through a horizontal pipe that expands smoothly from diameter d₀ to a larger diameter d₁. The pressure in the narrower section is p₀. Find an expression for the pressure p₁ in the wider section.

Verified step by step guidance

1. Step 1: Begin by identifying the principles governing the problem. This involves using the Bernoulli equation, which states that for an incompressible, nonviscous fluid, the total mechanical energy (pressure energy, kinetic energy, and potential energy) remains constant along a streamline. Since the pipe is horizontal, the potential energy term can be ignored.
2. Step 2: Write the Bernoulli equation for the two sections of the pipe. At the narrower section (diameter d₀), the pressure is p₀ and the speed is v₀. At the wider section (diameter d₁), the pressure is p₁ and the speed is v₁. The equation is: $p_0+\frac{1}{2}\rho v^0=p_1+\frac{1}{2}\rho v^1$, where ρ is the density of the liquid.
3. Step 3: Use the principle of conservation of mass to relate the flow speeds v₀ and v₁. The mass flow rate must be constant, which implies that the product of the cross-sectional area and the velocity is the same at both sections. The cross-sectional area of a pipe is proportional to the square of its diameter. Therefore, $A_0{v}_0=A_1{v}_1$, or equivalently, $d_0{2}^{}{v}_0=d_1{2}^{}{v}_1$. Solve for v₁ in terms of v₀, d₀, and d₁.
4. Step 4: Substitute the expression for v₁ obtained from the conservation of mass into the Bernoulli equation. This will allow you to express p₁ in terms of p₀, v₀, ρ, d₀, and d₁.
5. Step 5: Simplify the resulting equation to find the final expression for p₁. The pressure in the wider section will depend on the pressure in the narrower section, the density of the liquid, the initial velocity, and the ratio of the diameters squared.

Key Concepts

Continuity Equation

The continuity equation states that for an incompressible fluid flowing through a pipe, the mass flow rate must remain constant. This means that the product of the cross-sectional area and the fluid velocity at any two points in the pipe must be equal. Mathematically, it is expressed as A₀v₀ = A₁v₁, where A is the cross-sectional area and v is the fluid velocity. This principle helps us understand how changes in pipe diameter affect fluid speed.

Bernoulli's Principle

Bernoulli's principle relates the pressure, velocity, and height of a fluid in steady flow. It states that an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. The equation P + 0.5ρv² + ρgh = constant illustrates this relationship, where P is pressure, ρ is fluid density, v is velocity, g is acceleration due to gravity, and h is height. This principle is crucial for deriving the pressure in the wider section of the pipe.

Fluid Dynamics

Fluid dynamics is the study of fluids in motion and the forces acting on them. It encompasses various principles, including the behavior of liquids and gases, and is essential for understanding how fluids interact with their environment. Key concepts in fluid dynamics, such as laminar and turbulent flow, viscosity, and pressure gradients, help explain how fluids behave in different scenarios, including the flow through pipes of varying diameters.