Editing Navier–Stokes equations (section)

==Application to specific problems==
The Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as [[multiphase flow]] driven by [[surface tension]].

Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by [[Scale analysis (mathematics)|scale analysis]] to further simplify the problem.

[[Image:NSConvection vectorial.svg|thumb|400px|Visualization of '''(a)''' parallel flow and '''(b)''' radial flow]]

===Parallel flow===
Assume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) [[boundary value problem]] is:
<math display="block">\frac{\mathrm{d}^2 u}{\mathrm{d} y^2} = -1; \quad u(0) = u(1) = 0.</math>

The boundary condition is the [[no slip condition]]. This problem is easily solved for the flow field:
<math display="block">u(y) = \frac{y - y^2}{2}.</math>

From this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.

===Radial flow===
Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the ''radial'' flow between parallel plates; this involves convection and thus non-linearity. The velocity field may be represented by a function {{math|''f''(''z'')}} that must satisfy:
<math display="block">\frac{\mathrm{d}^2 f}{\mathrm{d} z^2} + R f^2 = -1; \quad f(-1) = f(1) = 0.</math>

This [[ordinary differential equation]] is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The [[Nonlinearity|nonlinear]] term makes this a very difficult problem to solve analytically (a lengthy [[Implicit function|implicit]] solution may be found which involves [[elliptic integral]]s and [[Cubic formula|roots of cubic polynomials]]). Issues with the actual existence of solutions arise for <math display="inline">R > 1.41
</math> (approximately; this is not [[square root of 2|{{sqrt|2}}]]), the parameter <math display="inline">R
</math> being the Reynolds number with appropriately chosen scales.<ref name="TM Shah">{{Cite journal|last=Shah |first=Tasneem Mohammad|title=Analysis of the multigrid method|journal=NASA Sti/Recon Technical Report N|volume=91|page=23418|bibcode=1989STIN...9123418S|year=1972}}</ref> This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.<ref name="TM Shah"/>

===Convection===
A type of natural convection that can be described by the Navier–Stokes equation is the [[Rayleigh–Bénard convection]]. It is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility.