Fluid flow

Explain — 3 marks

A water pipe system supplies water to a village. The pipe narrows at one section before widening again further downstream. Engineers notice that the water flows faster in the narrow section than in the wide sections.

(a) Explain why the water flows faster in the narrow section of the pipe. [1 mark]
(b) The pressure in the water is lower in the narrow section than in the wide sections. Explain why. [1 mark]
(c) The same volume of water passes through every section of the pipe each second. Explain what this tells us about the relationship between the cross-sectional area of the pipe and the speed of fluid flow. [1 mark]

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(a) The cross-sectional area is smaller/narrower in this section
(a) The same volume of water must flow through a smaller area each second
(a) Therefore the water must flow faster to maintain continuity/conservation of mass
(b) Faster-moving fluids have lower pressure
(b) This is due to Bernoulli's principle
(b) The kinetic energy of the fluid increases where it moves faster, so pressure energy decreases
(c) Cross-sectional area and fluid speed are inversely proportional
(c) As area decreases, speed increases
(c) As area increases, speed decreases
(c) This is the continuity equation: A₁v₁ = A₂v₂

Explain — 3 marks

A water treatment plant uses pipes of different diameters to transport water at different stages of the filtration process. Engineers notice that the water flows more slowly through wider pipes and faster through narrower pipes, even though the same volume of water enters and leaves each section per second.

(a) Explain why the water flows at different speeds in pipes of different diameters. [1 mark]
(b) The continuity equation for fluid flow states that A₁v₁ = A₂v₂, where A is the cross-sectional area and v is the velocity. Using this relationship, explain what happens to the velocity when the cross-sectional area of the pipe doubles. [1 mark]
(c) At one point in the treatment process, the pipe narrows from a diameter of 10 cm to a diameter of 5 cm. Explain the relationship between the change in pipe diameter and the change in water velocity, using the concept of conservation of mass. [1 mark]

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(a) The same volume of fluid passes through each cross-section per unit time / continuity equation requires constant volume flow rate
(a) When the area decreases, velocity must increase to maintain the same volume flow rate (or vice versa)
(b) If area doubles, velocity is halved / velocity is inversely proportional to cross-sectional area
(b) This maintains constant volume flow rate (A₁v₁ = A₂v₂)
(c) Diameter reduces by a factor of 2, so cross-sectional area reduces by a factor of 4 (area ∝ d²)
(c) To conserve mass/maintain constant volume flow rate, velocity must increase by a factor of 4
(c) The fluid cannot accumulate, so faster flow through the narrower section balances the volume flow rate

Show — 2 marks

A garden hose is used to fill a water tank. When the tap is fully open, water flows quickly out of the hose. When the tap is partially closed, the water flows more slowly.

Show that the volume of water flowing through the hose depends on both the cross-sectional area of the hose and the speed of the water. [2 marks]

Show mark scheme

{'mark': 1, 'description': 'States that volume flow rate equals cross-sectional area multiplied by fluid speed (Q = A × v or equivalent)'}
{'mark': 1, 'description': 'Explains that increasing either the cross-sectional area OR the speed of water increases the volume of water flowing per unit time'}

State — 4 marks

A water treatment plant uses pipes of different diameters to transport water at constant volume flow rate through the system. Engineers need to understand how the water's speed changes as it moves through sections of pipe with different cross-sectional areas.

(a) State the equation that links volume flow rate, cross-sectional area, and fluid velocity. [1 mark]
(b) Water flows through a pipe at a constant volume flow rate of 0.12 m³/s. The pipe narrows from a cross-sectional area of 0.08 m² to 0.02 m². State what happens to the velocity of the water as it enters the narrower section. [1 mark]
(c) State two reasons why understanding fluid flow is important for the design and operation of water distribution systems. [2 marks]

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(a) Q = Av (or volume flow rate = area × velocity)
(a) Accept: flow rate = cross-sectional area × speed
(b) The velocity increases (as the cross-sectional area decreases)
(b) Accept: the water speeds up / moves faster
(c) 1 mark: Any one from: to prevent pipe damage / to maintain water pressure / to control water supply rates / to minimize energy loss / to prevent flooding
(c) 1 mark: Any one from: different from the first answer: to ensure adequate flow to consumers / to design appropriate pipe diameters / to predict pressure changes / to maintain system efficiency

Explain — 3 marks

Explain — 3 marks

Show — 2 marks

State — 4 marks

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