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Wednesday, 19 December 2018

EXPRESSION FOR VELOCITY OF SOUND WAVE IN A FLUID

We were discussing the basics of drag force & lift force and drag and lift coefficient in the subject of fluid mechanics, in our recent posts. Before going in detail discussion about compressible flow, we must have basic knowledge about various equations associated with the compressible flow. 

Till now we were discussing the various concepts and equations such as continuity equationEuler equationBernoulli’s equation and momentum equation for incompressible fluid flow. In same way we have also discussed above equations for compressible fluid flow. 

We have already seen the derivation of continuity equationBernoulli’s equation and momentum equation for compressible fluid flow in our previous posts. We will start here our discussion about the compressible fluid flow with the derivation of expression for velocity of sound wave in a fluid.  

Expression for velocity of sound wave in a fluid 

Compressible flow is basically defined as the flow where fluid density could be changed during flow. 
The disturbance in a solid, liquid or gas will be transmitted from one point to other. Velocity with which the disturbance will be transmitted will be dependent over the distance between the molecules of the medium. 

If we will consider the case of solid, molecules will be very much closely packed and therefore the disturbance will be transmitted instantaneously. 

If we will consider the case of liquid, molecules will be relatively apart and therefore disturbance will be transmitted from one molecule to the next molecule. Velocity of disturbance will be less as compared to the velocity of disturbance in case of solid. 

If we will consider the case of gases, molecules will be relatively apart and disturbance will be transmitted from one molecule to the next molecule. There will be some distance between two adjacent molecules. Each molecule will have to travel a certain distance before it can transmit the disturbance. 

Therefore, velocity of disturbance in fluids (liquid and gas) will be less as compared to the velocity of disturbance in case of solids. 

This disturbance will develop the pressure waves in fluids. These pressure waves will travel with velocity of sound waves in all the directions. Let us consider here one dimensional case only. 

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. 
Due to the application of this small amount of force, there will be generation of pressure waves inside the cylinder and pressure will be applied over the fluid contained inside the cylinder. 

In simple, we can say that disturbance will be created inside the fluid due to the movement of piston by a distance x and this disturbance will move in the form of pressure wave inside the cylinder with a velocity of sound wave as discussed above. 

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 

Let us recall the law of conservation of mass 

Initial mass = final mass 

As we know that mass will be equal to the product of density and volume. We can write here the equation as mentioned here. 

Mass = Density x Volume 
Mass = 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, M
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 

Here, 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 

As we know that force could be written by recalling the Newton’s second law of motion and we will have the following equation 

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
C2 = (ρ V/ dρ) x [dP / (ρ V)] = dP/ 
C2dP/  
Above equation represent the velocity of sound wave in fluid. 

Further, we will go ahead to find out the expression for velocity of sound in terms of bulk modulus, in the subject of fluid mechanics, with the help of our next post. 

Do you have any suggestions? Please write in comment box. 

Reference: 
Fluid mechanics, By R. K. Bansal 
Image courtesy: Google 

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