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For example, the Navier–Stokes equations are often used to model fluid flows that are turbulent, which means that the fluid is highly chaotic and unpredictable. Turbulence is a difficult phenomenon to model and understand, and it adds another layer of complexity to the problem of solving the Navier–Stokes equations.
However, theoretical understanding of their solutions is incomplete, despite its importance in science and engineering. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proven that smooth solutions always exist. This is called the Navier–Stokes existence and smoothness problem.
Examples of degenerate cases—with the non-linear terms in the Navier–Stokes equations equal to zero—are Poiseuille flow, Couette flow and the oscillatory Stokes boundary layer. But also, more interesting examples, solutions to the full non-linear equations, exist, such as Jeffery–Hamel flow , Von Kármán swirling flow , stagnation ...
In fluid dynamics, Rayleigh problem also known as Stokes first problem is a problem of determining the flow created by a sudden movement of an infinitely long plate from rest, named after Lord Rayleigh and Sir George Stokes. This is considered as one of the simplest unsteady problems that have an exact solution for the Navier-Stokes equations.
The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier–Stokes equations.The inertial forces are assumed to be negligible in comparison to the viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations: [1]
The cross differentiated Navier–Stokes equation becomes two 0 = 0 equations and one meaningful equation. The remaining component ψ 3 = ψ is called the stream function . The equation for ψ can simplify since a variety of quantities will now equal zero, for example:
In computational fluid dynamics (CFD), the SIMPLE algorithm is a widely used numerical procedure to solve the Navier–Stokes equations. SIMPLE is an acronym for Semi-Implicit Method for Pressure Linked Equations. The SIMPLE algorithm was developed by Prof. Brian Spalding and his student Suhas Patankar at Imperial College London in the early ...
This is considered one of the simplest unsteady problems that has an exact solution for the Navier–Stokes equations. [1] [2] In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.