I wanted to model a real life problem using the Navier-Stokes equations and was wondering what the assumptions made by the same are so that I could better relate my entities with a 'fluid' and make or set assumptions on them likewise. For example one of the assumptions of a Newtonian fluid is that the viscosity does not depend on the shear rate. Similarly what are the assumptions that are made on a fluid or how does the Navier-Stokes equations define a fluid for which the equation is applicable?
[Physics] What are the assumptions of the Navier-Stokes equations
fluid dynamicsnavier-stokes;
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Firstly, i want to set the coordinate system so that there is no confusion:
- I assume the x-coordinate is along the incline wall and the y-coordinate is perpendicular to the incline wall.
In such a coordinate system, the acceleration due to gravity $\boldsymbol{g}$ is expressed as:
$$\boldsymbol{g}=\left(g_x,g_y\right)=\left(g\sin\theta,g\cos\theta\right)$$
where $g$ is the gravitational acceleration constant and $\theta$ is the angle of the incline.
Secondly, these typical text-book questions on velocity profiles for inclined planes always have a number of implicit assumptions:
- The flow is fully developed - This means that any transient effects have vanished and we are dealing with a steady-state system.
- The flow is laminar - This means that the fluid flow parallel to the incline wall in organized layers such that the y-component of the velocity is zero and the x-component of the velocity only depends on the y-coordinate.
- Entrance/exit effects are negligible - This means that any open boundaries to the flow are very far away from the section of the incline we are dealing with such that they do not interfere with the flow.
With these assumptions the Navier-Stokes equations reduce to:
$$0=-\frac{1}{\rho}\partial_xp + \nu\partial_y^2u_x + g_x \qquad 0=-\frac{1}{\rho}\partial_yp + g_y$$
Note that the continuity equation is satisfied identically with the above assumptions.
I do not understand why it is reasonable to assume that there is no pressure gradient along the incline wall, i.e. $\partial_xp=0$?
Consider the ways in which a pressure gradient may form:
- By an applied pressure at the in/out-flow boundaries
- By a hydrostatic contribution due to fluid resting on a wall.
Neither of these cases apply here as no applied pressure is specified and the in/out-flow boundaries are open (otherwise no flow) such that a hydrostatic pressure cannot be generated. Conclusion the pressure along the incline wall must vanish.
In addition, there is no pressure change out of the page and perpendicular to the incline axis (duh)
Actually, since $g_y\neq0$, there exists a pressure gradient in the y-direction and it is purely hydrostatic:
$$\partial_yp = \rho g_y = \rho g \cos\theta$$
If this may seem unintuitive, consider the case where the incline is perfectly horizontal ($\theta=0$). In that case there is no flow ($g_x=0$) and the fluid is simply resting on the wall exerting a hydrostatic pressure due to gravity ($g_y=g$).
Best Answer
Most importantly, the Navier-Stokes equations are based on a continuum assumption. This means that you should be able to view the fluid as having properties like density and velocity at infinitely small points. If you look at e.g. liquid flows in nanochannels or gas flows in microchannels you could be in a regime where this assumption breaks down. As far as I know there is no hard limit for the continuum assumption, but the Knudsen number is a useful indicator.
Additionally there is, as @ShuchangZhang mentioned, an assumption on the nature of the stress in the fluid. Although I am not sure whether you could really call this an assumption or whether it should be considered a theory (like the NS equations itself).
The strongest assumptions are typically not in the Navier-Stokes equations themselves, but rather in the boundary conditions that should be applied in order to solve them. To give an example, whether the no-slip boundary condition (fluid velocity at the wall equals wall velocity) or the navier slip boundary condition (fluid velocity equals a scaled velocity gradient at the wall) has been a much debated subject, in particular for hydrophobic surfaces (see e.g. Phys. Rev. Lett. 94, 056102 (2005) and references therein and thereto)