I don't know if this is a proper question. I am not so familiar with fluids. I am just curious about what Navier-Stokes equations for fluids will look like in zero gravity. Are they stay the same? If so, how can I tell the zero gravity condition? If not, please explain it to me.
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2$\begingroup$ Could one of the close-voters explain what is unclear about this question? $\endgroup$– Kyle KanosCommented Jan 22, 2016 at 11:38
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2 Answers
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Gravity acts as a source term in the equations, and it is a source term on the energy and momentum equations. The mass conservation equation is not modified by gravity. So, looking at the momentum equation with gravity, we have:
$$ \frac{\partial \rho u_i}{\partial t} + \frac{\partial \rho u_i u_j}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \mu \frac{\partial^2 u_j}{\partial x_i x_j} + \rho g_i$$
where $g_i$ is the component of the gravity acceleration vector in the $i-th$ direction. Under "normal" conditions people would use, the gravity acceleration vector would be $(0,-9.81,0)$ for example, and so only the y-momentum equation has any influence.
Without gravity, that term just goes away and your equation is:
$$ \frac{\partial \rho u_i}{\partial t} + \frac{\partial \rho u_i u_j}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \mu \frac{\partial^2 u_j}{\partial x_i x_j}$$
It's that simple -- there is just a source term that appears if there is gravity and goes away if there isn't. The same is true for the energy equation, there is a gravity source term that is there when there is gravity and gone where there isn't. So in the Wikipedia article, just set $\mathbf{g} = 0$.
For the vast majority of simulations that are not atmospheric-scale flows like weather, gravity is ignored anyway. You can decide whether gravity matters by looking at the Froude number. If you non-dimensionalize the Navier-Stokes equations including the gravity term, the Froude number will appear and based on the size of it you can tell if gravity will be important.
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$\begingroup$ Just curious for the gravity vector why do you have it in the y cor instead of z cor? $\endgroup$– VogtsterCommented Feb 13, 2019 at 15:05
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$\begingroup$ @Vogtster Aerospace engineering background -- usually 2D things (like looking at the cross section of a wing, or the centerplane of a body or engine) are in the X-Y plane and Z is for the out-of-page. Obviously it isn't important which coordinates are which, so long as the system remains right-handed, but that is the convention I was trained under. $\endgroup$– tpg2114Commented Feb 13, 2019 at 15:37
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$\begingroup$ There is much more to gravity than just the Froude number. The Rayleigh number, the Richardson number and more. I'd say the Froude number is the least interesting of them. $\endgroup$ Commented Nov 9, 2020 at 14:41
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Gravity is important for stratified flows. No matter if the stratification is caused by temperature differences, moisture differences (H2O is lighter than N2) or salinity differences. It is the gravity acceleration that is responsible for buoyancy. Without gravity we have no hydrostatic pressure and no buoyancy.
What changes in the equations? It is best visible in the compressible flow equations simplified for the Boussinesq approximation. You get $$ \frac{\mathrm{d}\boldsymbol{u}}{\mathrm{d}t} =-\frac{\nabla p'}{\rho_{0}}+\nu\nabla^{2}\boldsymbol{u}+\frac{\rho - \rho_{0}}{\rho_{0}}\boldsymbol{g}. $$ the last term is responsible for the force acting on lighter and heavier parts of the fluid. If $g=0$, this term is zero and we use many interesting effects. Suddenly, density changes in the fluid become much less important.
In gravity, when you heat the bottom of fluid container, the fluid at the bottom will get hot and light. The Rayleigh-Taylor instability will kick in and Rayleigh-Benard convection will start in the container. This will not happen in zero gravity. This container can be pretty small, very non-atmospheric scale.
Thermal convection is very important in real life. It is the process that allows the distribution of heat from heating radiators. Without convection, we would have a layer of very hot air at the radiators and the heat would diffuse very slowly. One would have to blow some air to create fluid flow mechanically to distribute the heat.
In gravity you have stable stratification suppressing turbulence and you get laminar flows if the Richardson number is too large. There is $g$ in the Richardson number. This is relevant even for laboratory-scale flows https://youtu.be/mf_143gkKSQ?t=120
In gravity we have large waves on the free surface of liquids, driven by the weight of the liquid. In zero gravity we will be left just with capillary waves driven by the surface tension. Those are also important, but they are not those larger waves you know from the seaside.
In gravity we have cold air coming down from the mountains at night, we have the sea breeze and many other effects. These do not exist in zero gravity.
TL;DR; Almost everything will change, except for the very limited and boring world of incompressible isothermal flows (or weakly compressible neutrally-stratified).
answered Nov 9, 2020 at 12:27