Influence of viscosity on the propagation of sound
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A detailed solution is then proposed to you.
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In the case of a viscous fluid, the equation satisfied by the velocity field is the Navier-Stokes :
\(\rho \left( {\frac{{\partial \vec v}}{{\partial t}} + \left( {\vec v.\overrightarrow {grad} } \right)\vec v} \right) = - \overrightarrow {grad} P + \eta \Delta \vec v\)
Where \(\eta\) is the dynamic viscosity of the fluid.
We assume that the density and pressure fluctuations are small and that evolution is isentropic.
In the following, we place in one dimension along the (Ox) axis.
Question
Establish the propagation equation :
\(\frac {\partial ^2 p}{\partial x^2}+ \frac{\eta }{{\rho _0 c_s^2 }}\frac{{\partial^2 }}{{\partial x^2}}(\frac {\partial p}{\partial t}) - \frac{1}{{c_s^2 }}\frac{{\partial ^2 p}}{{\partial t^2 }} = 0\;\;\;\;\;\;\;\;\;\;(with\;:\;\rho _0 \chi _s c_s^2 = 1)\)
Indice
Linearize the Euler equation and mass conservation equation.
Using the coefficient of isentropic compressibility.
Solution
The acoustic overpressure is noted \(p(x,t)\) and the change in density with respect to the equilibrium value, denoted \(\rho(x,t)\).
The mass conservation equation linearized gives in one dimension :
\(\rho_0 \frac {\partial v(x,t)}{\partial x}+\frac {\partial \rho(x,t)}{\partial t}=0\)
Similarly, in connection with the acoustic hypothesis, the Navier-Stokes becomes :
\(\rho_0 \frac {\partial v(x,t)}{\partial t}=-\frac {\partial p(x,t)}{\partial x}+\eta \frac {\partial ^2 v(x,t)}{\partial x^2}\)
Furthermore, the isentropic compressibility coefficient is written as :
\(\chi_S=\frac {1}{\rho_0}\frac{\rho(x,t)}{p(x,t)}\)
The mass conservation equation can be written as :
\(\frac {\partial v(x,t)}{\partial x}+\chi_S\frac {\partial p(x,t)}{\partial t}=0\)
Is :
\(\frac {\partial^2 v(x,t)}{\partial x^2}=-\chi_S \frac {\partial^2 p(x,t)}{\partial x\partial t}\)
Referring to the Navier-Stokes :
\(\rho_0 \frac {\partial v(x,t)}{\partial t}=-\frac {\partial p(x,t)}{\partial x}-\eta \chi_S \frac {\partial^2 p(x,t)}{\partial x\partial t}\)
The above equation is derived with respect to space and, using the equality between the two derivatives :
\(\frac {\partial^2 v(x,t)}{\partial t \partial x}=\frac {\partial^2 v(x,t)}{\partial x \partial t}=-\chi_S\frac {\partial^2 p(x,t)}{\partial t^2}\)
Finally leads to the equation :
\(\frac {\partial ^2 p}{\partial x^2}+ \frac{\eta }{{\rho _0 c_s^2 }}\frac{{\partial^2 }}{{\partial x^2}}(\frac {\partial p}{\partial t}) - \frac{1}{{c_s^2 }}\frac{{\partial ^2 p}}{{\partial t^2 }} = 0\;\;\;\;\;\;\;\;\;\;(with\;:\;\rho _0 \chi _s c_s^2 = 1)\)
Question
We seek a solution in the form of a monochromatic progressive plane wave of the type :
\(\underline p = p_0 e^{j(\omega t - \underline k x)}\)
To determine the relationship between \(\underline k\) and \(\omega\).
We set \(\underline k = k_1-jk_2\). For a low viscosity fluid, \(k_2<<k_1\).
Give the expression of \(k_2\) in the first order of \(\eta\).
What is its physical meaning ?
Indice
In complex notation, a derivative with respect to time becomes multiplying by \(j\omega\) and derivate with respect to \(x\) becomes multiplying by \(-jk\).
Solution
The dispersion relation is obtained from the propagation equation, noting that a derivative with respect to time is equivalent to multiplying by \(j\omega\) and derivate with respect to \(x\) is equivalent to multiplying by \(-jk\).
So :
\(-\underline k^2\underline p(x,t) -\frac {1}{c^2}(-\omega^2)\underline p(x,t) +\frac{\eta}{\rho_0 c^2}(-\underline k^2)(j\omega)\underline p(x,t)=0\)
Whence :
\(\underline k^2=\frac{\omega^2}{c^2}\frac{1}{1+j\omega \tau}\)
With :
\(\tau = \frac{\eta}{\rho_0 c^2}\)
To a low viscosity fluid, \(\omega \tau <<1\). Therefore :
\(\underline k=\frac{\omega}{c}( 1+j\omega \tau)^{-1/2} \approx \frac{\omega }{c} ( 1-j \omega \tau/2) \)
For identification :
\(k_1=\frac {\omega}{c}\) and \(k_2=\frac{\omega^2 \tau}{2c}=\frac{\omega^2\eta}{2\rho_0c^2}\)
In real notation, the pressure is expressed in the form :
\(p(x,t)=p_0e^{-k_2x}cos\omega(t-\frac{x}{c})\)
A characteristic distance can be defined :
\(\delta=\frac {1}{k_2}\)
that represents the attenuation characteristic length of the pressure wave in the viscous medium.