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The incompressible Navier-Stokes equations widely used in hydrodynamics don't have the gravitational acceleration. $$ \begin{align} \frac{\partial u_i}{\partial x_i} & = 0, \\ \frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}&=-\frac{1}{\rho}\frac{\partial p}{\partial x_i}+\nu\frac{\partial^2u_i}{\partial x_j \partial x_j}. \end{align} $$

One explanation is that the physical magnitude of $g$ is in a far small level than other forces, e.g., pressure and viscous forces, but how this term is eliminated?

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If you go through the process of non-dimensionalizing the equations, the math becomes more clear. If you start with the momentum equation (ignoring viscous forces because they aren't important for the analysis): $$ \frac{\partial u_i}{\partial t} + \frac{\partial u_i u_j}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + g $$

Then introduce relevant scales to non-dimensionalize things: $\bar{u}_i = u_i/u_0$, $\bar{x}_i = x_i/L$, $\bar{\rho} = \rho/\rho_0$, $\bar{g} = g/g_0$, $\tau = u_0/L t$ and $\bar{p} = p/p_0$, you get:

$$ \frac{\partial \bar{u}_i}{\partial \tau} + \frac{\partial \bar{u}_i \bar{u}_j}{\partial \bar{x_j}} = -\frac{\text{Eu}}{\bar{\rho}} \frac{\partial \bar{p}}{\partial \bar{x}_i} + \frac{1}{\text{Fr}^2}\bar{g} $$

$\text{Eu} = \frac{p_0}{\rho_0 u_0^2}$ is the Euler number and $\text{Fr} = \frac{u_0}{\sqrt{g_0 L}}$ is the Froude number.

The Froude number is the ratio of convective forces to gravity forces. When convective forces are much, much larger than gravity forces, the Froude number is large and so $\frac{1}{\text{Fr}^2} \ll 1$, and the gravity term can be neglected relative to the convective terms. This is how we can mathematically justify dropping the gravity term when the convective forces are large.

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The NS equations do include the gravity term, see the Wikipedia entry where it's included as a body force term.

Under some conditions, it can be neglected: large Froude number (as tpg2114 showed) or hydrostatics (in which the gravity force is balanced by the pressure gradient) or horizontal flows ($g$ in $z$-direction but flow is in $x,\,y$ plane). It can also absorbed into other terms: gravity being conservative, it can be absorbed into the pressure gradient term for incompressible flows, $$-\nabla w+\mathbf g=-\nabla(w+\phi)=-\nabla w'.$$ where $w=p/\rho$, but this really would not be 'ignored' as you've suggested.

A pedagogical reason might be that the solution would end up being more complex than reasonably expected in a test or homework assignment.

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protected by Qmechanic May 18 '18 at 18:23

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