# Show from the first principles the requirements for dynamic similarity between two fluid motions when considering: Viscous resistance Wave resistance

Answer any ONE question from this section.

a) Explain the following terms:
Geometrical similarity
Dynamic similarity (3 marks)

b) Show from the first principles the requirements for dynamic similarity between two fluid motions when considering:
Viscous resistance
Wave resistance (6 marks)

The air resistance R of a supersonic plane during flight, is a function of its length L, velocity V, air dynamic velocity µ, air density ?, and bulk modulus K. Show that the air resistance R is given by:

R=pl^2 V^2Ø{µ/pvl ,k/(pv^2 )}

where Ø means a “function of”. (11 marks)

a) i) Define the following terms with reference to fluid flow:
Critical velocity
Reynold’s number
ii) Show that the flow rate Q, of a fluid of dynamic viscosity ?, flowing under laminar conditions through a horizontal circular pipe of diameter ‘d’, length ‘L’, with a mean velocity V, when the pressure difference between the ends is P, is given by the expression:

Q=(ppd^4)/128?L
Hence show that the pressure difference p can be given by the expression

P=32?lv/d^2 (121/2 marks)

b) Oil is pumped through a pipe 120mm diameter and 900m long. The pressure difference between the ends is 420KN/m2. The dynamic viscosity of the oil is 1.42 N-S/m2 and the relative density is 0.9.

Show that the flow is viscous if Reynold’s number is 2100.
Calculate the electric power of the motor required if the mechanical efficiency between the pump and the electric motor is 80%.

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