Till now we were discussing the various concepts and equations such as continuity equation, Euler equation, Bernoulli’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 equation, Bernoulli’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 in terms of bulk modulus.
Expression for velocity of sound in terms of bulk modulus
Before understanding the process to derive the expression for velocity of sound in terms of bulk modulus, we must have to study our previous post which shows the derivation of velocity of sound wave in a fluid.
As we are interested here to find the expression for velocity of sound in terms of bulk modulus, therefore let us brief here first the term bulk modulus.
Bulk modulus of elasticity of a substance is basically defined as the ratio of compressive stress or hydro static stress to volumetric strain and it will be displayed by the symbol K.
Bulk modulus of a substance provides the information about the resistance of substance to the uniform pressure. In simple, we can also say that Bulk modulus of a substance provides the information about the compressibility of that substance.
Bulk modulus, K = - dP/ (dV/V)
Where,
dP = Change in pressure
dV/V = Volumetric strain
As we know that according to the conservation of mass, we will have following equation as mentioned here.
Density x Volume = Constant
ρ x V = C, where C is constant
Let us differentiate the above equation and we will have following equation after differentiation
d (ρ x V) = 0
ρ d V + V dρ = 0
ρ d V = - V dρ
dV/V = -(dρ/ ρ)
Now we will use the above value of volumetric strain in equation of bulk modulus and we will have
K = ρ x dP/dρ
dP/dρ = K/ ρ
Let us recall the expression for the velocity of sound wave in a fluid and we can write the above equation as mentioned here.
C2 = K/ ρ
Because,
dP/dρ = C2
Where, C is the velocity of sound
Therefore we will have following equation, as mentioned here, which shows the expression for velocity of sound in terms of bulk modulus.
Further we will go ahead to find out the expression for velocity of sound in isothermal process, 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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