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Fluid mechanics

FAQ 1

• Can we take other calculators into the exam?
• No, sorry that you have to use the “green one”

(TI-30XB)

FAQ 3

• Will you lose marks for not quoting correct

numbers of significant figures?

• Only in extreme cases. If you quote something

similar to the data given in the question, you will

not lose marks.

Formula sheet

Formula sheet

Fluid Mechanics key facts (1/5)

• Basic definitions:
• Density 𝜌𝜌 =

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 [units:^ 𝑘𝑘𝑘𝑘 𝑚𝑚^

−3]

• Pressure 𝑃𝑃 =

𝐹𝐹𝑉𝑉𝐹𝐹𝐹𝐹𝑉𝑉 𝐴𝐴𝐹𝐹𝑉𝑉𝑀𝑀 [units:^ 𝑁𝑁 𝑚𝑚^

−2 (^) or 𝑃𝑃𝑃𝑃]

• Pressure can also be measured in atmospheres: 1 𝑃𝑃𝑎𝑎𝑚𝑚 = 1.013 × 10^5 𝑃𝑃𝑃𝑃 [on formula sheet]
• Gauge pressure means the extra pressure above the atmospheric pressure

Fluid Mechanics key facts (3/5)

• An object immersed in a fluid will feel a buoyancy

force equal to the weight of the fluid displaced

Buoyancy force = 𝑚𝑚𝑓𝑓𝑉𝑉𝑉𝑉𝑓𝑓𝑓𝑓 𝑘𝑘 = 𝜌𝜌𝑓𝑓𝑉𝑉𝑉𝑉𝑓𝑓𝑓𝑓 𝑉𝑉 𝑘𝑘

Weight force = 𝑚𝑚𝑉𝑉𝑜𝑜𝑜𝑜𝑉𝑉𝐹𝐹𝑜𝑜 𝑘𝑘 = 𝜌𝜌𝑉𝑉𝑜𝑜𝑜𝑜𝑉𝑉𝐹𝐹𝑜𝑜 𝑉𝑉 𝑘𝑘

Object has volume 𝑉𝑉

Buoyancy > Weight if 𝜌𝜌𝑉𝑉𝑜𝑜𝑜𝑜𝑉𝑉𝐹𝐹𝑜𝑜 < 𝜌𝜌𝑓𝑓𝑉𝑉𝑉𝑉𝑓𝑓𝑓𝑓

Fluid Mechanics key facts (4/5)

• Flow of an incompressible fluid obeys the

continuity equation : 𝑣𝑣 𝐴𝐴 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑃𝑃𝑐𝑐𝑎𝑎 where

𝑣𝑣 = fluid velocity and 𝐴𝐴 = pipe area

• This equation results from conservation of mass. It’s the same as saying 𝑣𝑣𝑐𝑐𝑣𝑣𝑣𝑣𝑚𝑚𝑣𝑣 𝑓𝑓𝑣𝑣𝑐𝑐𝑓𝑓 𝑟𝑟𝑃𝑃𝑎𝑎𝑣𝑣 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑃𝑃𝑐𝑐𝑎𝑎

𝑣𝑣 1 𝑣𝑣 2

Area 𝐴𝐴 1

Area 𝐴𝐴 2

𝑣𝑣 1 𝐴𝐴 1 = 𝑣𝑣 2 𝐴𝐴 2

Practice exam questions: Section A

Weight of fluid displaced = weight of boat Same weight is displaced before and after [A]

Continuity equation 𝑣𝑣 𝐴𝐴 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑃𝑃𝑐𝑐𝑎𝑎 : velocity increases / constant flow rate Bernoulli’s equation 𝑃𝑃 + 12 𝜌𝜌𝑣𝑣^2 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑃𝑃𝑐𝑐𝑎𝑎 : pressure decreases [D]

Practice exam questions: Section A

Hydrostatic equilibrium only depends on depth : A

Lake is in hydrostatic equilibrium : pressure increases with depth

Buoyancy force = weight of water displaced = constant : D

Practice exam questions: Section B

1.5 × 10−

2 = 1.77 × 10−6^ 𝑚𝑚 2

𝐹𝐹𝑐𝑐𝑟𝑟𝑐𝑐𝑣𝑣 = 𝑃𝑃𝑟𝑟𝑣𝑣𝑐𝑐𝑐𝑐𝑣𝑣𝑟𝑟𝑣𝑣 × 𝐴𝐴𝑟𝑟𝑣𝑣𝑃𝑃 = 12 × 10^6 × 1.77 × 10−6^ = 21 𝑁𝑁

Practice exam questions: Section B

Gauge pressure is the additional pressure above the atmospheric pressure

Hydrostatic equilibrium: 𝑃𝑃 = 𝑃𝑃 0 + 𝜌𝜌𝑘𝑘𝜌

Gauge pressure = 𝜌𝜌𝑉𝑉𝑓𝑓𝑉𝑉 𝑘𝑘 𝜌𝑉𝑉𝑓𝑓𝑉𝑉 + 𝜌𝜌𝑤𝑤𝑀𝑀𝑜𝑜𝑉𝑉𝐹𝐹 𝑘𝑘 𝜌𝑤𝑤𝑀𝑀𝑜𝑜𝑉𝑉𝐹𝐹 = 820 × 9.8 × 0.05 + 1000 × 9.8 × 0.05 = 890 𝑃𝑃𝑃𝑃

Practice exam questions: Section C

Volume flow rate = 𝑣𝑣 𝐴𝐴 = 5 × 10−3^ 𝑚𝑚 3 𝑐𝑐−

5 × 10−

2 × 10−^

Practice exam questions: Section C

Bernoulli’s equation : 𝑃𝑃 + 12 𝜌𝜌𝑣𝑣 2 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑃𝑃𝑐𝑐𝑎𝑎 𝑃𝑃𝐴𝐴 + 12 𝜌𝜌𝑣𝑣𝐴𝐴^2 = 𝑃𝑃𝐵𝐵 + 12 𝜌𝜌𝑣𝑣𝐵𝐵^2 𝑃𝑃𝐴𝐴 = 1.6 × 10^5 𝑃𝑃𝑃𝑃, 𝑃𝑃𝐵𝐵 = 1.2 × 10^5 𝑃𝑃𝑃𝑃, 𝑣𝑣𝐴𝐴 = 2.5 𝑚𝑚𝑐𝑐−1, 𝜌𝜌 = 1000 𝑘𝑘𝑘𝑘 𝑚𝑚 −

→ 𝑣𝑣𝐵𝐵 = 9.3 𝑚𝑚 𝑐𝑐−

Continuity equation : 𝑣𝑣 𝐴𝐴 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑃𝑃𝑐𝑐𝑎𝑎 𝑣𝑣𝐴𝐴 𝐴𝐴𝐴𝐴 = 𝑣𝑣𝐵𝐵 𝐴𝐴𝐵𝐵

𝐴𝐴𝐵𝐵 =

𝑣𝑣𝐴𝐴 𝐴𝐴𝐴𝐴 𝑣𝑣𝐵𝐵^ =^

2.5 × 2 × 10−

9. 3 =^5.^4 ×^10

−4 (^) 𝑚𝑚 2