Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 65a

Chapter 14, Problem 65a

A cylindrical tank of radius 𝑅, filled to the top with a liquid, has a small hole in the side, of radius 𝓇, at distance d below the surface. Find an expression for the volume flow rate through the hole.

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Start by identifying the principle governing the flow of liquid through the hole: Torricelli's Law. This law states that the speed of efflux, 𝑣, of a liquid under gravity through a small hole is given by 𝑣 = √(2𝑔ℎ), where ℎ is the height of the liquid above the hole and 𝑔 is the acceleration due to gravity.

Relate the height ℎ to the given distance 𝑑. Since the hole is located at a distance 𝑑 below the surface of the liquid, we can write ℎ = 𝑑.

The volume flow rate, 𝑄, is defined as the product of the cross-sectional area of the hole and the velocity of the liquid through the hole. The cross-sectional area of the hole is 𝐴 = π𝓇², where 𝓇 is the radius of the hole. Thus, 𝑄 = 𝐴 × 𝑣 = π𝓇² × √(2𝑔𝑑).

Combine the expressions to write the final formula for the volume flow rate: 𝑄 = π𝓇²√(2𝑔𝑑). This expression shows that the flow rate depends on the size of the hole (𝓇), the distance below the surface (𝑑), and the gravitational acceleration (𝑔).

Verify the assumptions: This derivation assumes that the hole is small enough for the liquid's velocity profile to remain uniform and that the liquid is incompressible and non-viscous. Additionally, the tank is assumed to be large enough that the liquid's surface area does not significantly change as the liquid flows out.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bernoulli's Principle

Bernoulli's Principle states that in a fluid flow, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle is crucial for understanding how fluids behave in motion, particularly in scenarios involving openings or holes, as it helps relate the pressure difference to the velocity of the fluid exiting the hole.

Continuity Equation

The Continuity Equation is a fundamental principle in fluid dynamics that asserts that the mass flow rate must remain constant from one cross-section of a pipe to another. For incompressible fluids, this means that the product of the cross-sectional area and the fluid velocity is constant, which is essential for determining the flow rate through the hole in the tank.

Hydrostatic Pressure

Hydrostatic Pressure refers to the pressure exerted by a fluid at rest due to the weight of the fluid above it. In the context of the cylindrical tank, the pressure at the hole is determined by the height of the liquid column above it, which influences the velocity of the fluid exiting the hole according to Bernoulli's Principle.

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