Derive an expression for velocity of sound wave in a compressible fluid.

(10 Marks) May-2018

Subject Fluid Mechanics 2

Topic Compressible Flow

Difficulty Medium

• Following figure, displayed here, indicates the condition of one-dimensional propagation of the pressure waves. Let us consider a cylinder of having uniform cross-sectional area attached with a piston as displayed in following figure.

  • Let us assume that cylinder is filled with a compressible fluid and compressible fluid is at rest initially.

  • If we apply a force through the piston in right direction, force will develop a pressure as force will be applied uniformly. Due to the application of force, piston will move by a certain distance let us say x towards right direction as displayed here in following figure.

    

  • Let us consider following terms from above figure as mentioned here.

x = Distance of piston from initial position

L = Distance of sound wave from initial position

P = Pressure applied over the piston at initial position

P + dP = Pressure inside the cylinder at final position

ρ = Density of the fluid at initial position

ρ + d ρ = Density of the fluid at final position

dt = Small amount of time taken by piston to travel distance x

V = Velocity of piston

C = Velocity of pressure wave or sound wave travelling in the fluid

Distance travelled by the piston in time dt from initial position, x = v.dt Distance travelled by pressure wave or sound wave in time dt from initial position, L = C. dt

• The law of conservation of mass is Initial mass = final mass

As we know that mass will be equal to the product of density and volume. Mass = Density x Volume=Density x Area x Length

Mass at initial position, M1 = ρ A L = ρ A C. dt

Mass at final position, M2 = (ρ + dρ) A (L-x) = (ρ + dρ) A (C. dt- V. dt)

Mass at final position, M2 = (ρ + dρ) A. dt (C - V)

• Now, considering the conservation of mass, we will have following equation as mentioned here.

Mass at initial position, M1 = Mass at final position, M2

ρ A C. dt = (ρ + dρ) A. dt (C - V)

ρ C = (ρ + dρ) (C - V)

ρ C = ρ C - ρ V + C. dρ - V dρ

• the term dρ will be very small and velocity of piston will also be very small and therefore product V dρ could be neglected.

ρ V = C. dρ

C = ρ V/ dρ -------------------------------- Eq 1

• Let us determine the force at initial position of piston and final position of piston as mentioned here

F1 = P.A

F2 = (P + dP). A

Change in force, ΔF = (P + dP). A - P.A = dP. A

• From Netwon's Second law of motion, Force is

Force = Mass x (Rate of change of velocity)

dP. A = (ρ A C. dt) [(V-u)/dt]

As we know that, V is the final velocity of the piston and u is the initial velocity of the piston and initial velocity of piston will be zero.

dP. A = (ρ A C. dt) V/dt

dP = ρ C V

C = dP / (ρ V) -------------------------------- Eq 2

We will have following equation by multiplying the equation 1 and equation 2

C^2 = (ρ V/ dρ) x [dP / (ρ V)] = dP/ dρ

C^2 = dP/ dρ

Above equation represent the velocity of sound wave in fluid.