I am reading the article by Zu and He (Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts, Physical review E, 2013). There the authors mention that for incompressible flow, the fluctuations of pressure $p$ are of the order $\mathcal{O}(Ma^2)$ where Ma is Mach number. i.e., \begin{align} \partial_t p = \mathcal{O}(Ma^2) \end{align} As far as I understand, the references cited in this article also assume the same instead of showing that $\partial_t p = \mathcal{O}(Ma^2)$. I am interested to know, how this estimate is derived and how the incompressible fluid assumption is used to get the estimate. I will be greateful, if somebody points out a useful reference or explain it here.
1 Answer
I am interested to know, how this estimate is derived and how the incompressible fluid assumption is used to get the estimate.
The incompressible approximation in a fluids is simply shown by the expression that $\nabla \cdot \mathbf{u} = 0$, i.e., the bulk flow of any parcel of fluid is divergenceless. One can assume incompressible flow in the limit where the speeds are much less than the relevant communication speed of the medium (e.g., speed of sound in Earth's atmosphere).
One can derive an expression from the incompressible Euler equations that can be expressed as something like: $$ \frac{ 1 }{ 2 } u^{2} + \phi + \frac{ P }{ \rho } = constant \tag{0} $$ where $u$ is the bulk flow speed, $\phi$ is some external field acting on a fluid parcel, $P$ is the pressure, and $\rho$ is the mass density. If we ignore the external fields for the moment and solve for pressure, we find: $$ P \simeq \rho \left( C_{o} - \frac{ 1 }{ 2 } u^{2} \right) \tag{1} $$ where $C_{o}$ is the constant from Equation 0 above.
Suppose the relevant communication speed is the speed of sound, $C_{s}$, and the Mach number is defined as $M = \tfrac{ u }{ C_{s} }$, then we can show that: $$ \partial_{t} P \simeq \partial_{t} \rho \left( C_{o} - \frac{ 1 }{ 2 } C_{s}^{2} M^{2} \right) \tag{2} $$ where $\partial_{t} = \tfrac{ \partial }{ \partial t }$ and $C_{s}$ is constant for all intents and purposes.
We know how to determine $\partial_{t} \rho$ from the continuity equation for incompressible flow, if we so wish be we can see from Equation 2 already that $\partial_{t} P \propto M^{2}$. In the limit that $u \ll C_{s}$ and for small disturbances, the Mach number will be very small such that $M^{2} \sim 0$ can be used/assumed.
