Everything about The Navier-stokes Equations totally explained
The
Navier-Stokes equations, named after
Claude-Louis Navier and
George Gabriel Stokes, describe the motion of
fluid substances such as
liquids and
gases. These equations establish that changes in
momentum in infinitesimal volumes of fluid are simply the sum of dissipative viscous forces (similar to
friction), changes in
pressure, gravity, and other forces acting inside the fluid: an application of
Newton's second law to
fluid.
They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They may be used to
model weather,
ocean currents, water flow in a pipe, flow around an
airfoil (wing), and motion of
stars inside a
galaxy. As such, these equations in both full and simplified forms, are used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with
Maxwell's equations they can be used to model and study
magnetohydrodynamics.
The Navier-Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have yet to prove that in three dimensions solutions always exist (
existence), or that if they do exist they don't contain any infinities, singularities or discontinuities (smoothness). These are called the
Navier-Stokes existence and smoothness problems. The
Clay Mathematics Institute has called this one of the
seven most important open problems in mathematics, and offered a $1,000,000 prize for a solution or a counter-example.
The Navier-Stokes equations are
differential equations which, unlike
algebraic equations, don't explicitly establish a relation among the variables of interest (for example
velocity and
pressure). Rather, they establish relations among the
rates of change. For example, the Navier-Stokes equations for simple case of an
ideal fluid (inviscid) can state that acceleration (the
rate of change of
velocity) is proportional to the
gradient (a type of multivariate derivative) of pressure.
Contrary to what is normally seen in
solid mechanics, the Navier-Stokes equations dictate not
position but rather
velocity. A solution of the Navier-Stokes equations is called a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest (such as flow rate, drag force, or the path a "particle" of fluid will take) may be found.
Properties
Nonlinearity
The Navier-Stokes equations are
nonlinear partial differential equations in almost every real situation (exceptions include one dimensional flow and
creeping flow). The nonlinearity makes most problems difficult or impossible to solve and is part of the cause of
turbulence.
The nonlinearity is due to
convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity, an example of convective but
laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging
nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.
Turbulence
Turbulence is the time dependent
chaotic behavior seen in many fluid flows. It is generally believed that it's due to the
inertia of the fluid as a whole: the culmination of time dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the
Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier-Stokes equations model turbulence properly.
Even though turbulence is an everyday experience, it's extremely difficult to find solutions, quantify, or in general characterize. A $1,000,000
prize was offered in May 2000 by the
Clay Mathematics Institute to whoever makes
preliminary progress toward a mathematical theory which will help in the understanding of this phenomenon.
The numerical solution of the Navier-Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation (see
Direct numerical simulation). Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, several approximations such as the Reynolds averaged Navier-Stokes equations (RANS), supplemented with turbulence models (such as the k-ε model), are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Another technique for solving numerically the Navier-Stokes equation is the Large-eddy simulation (LES). This approach is computationally more expensive than the RANS method (in time and computer memory), but produces better results, since part of the turbulent characteristic scales are explicitly resolved.
Applicability
Together with supplemental equations (for example, conservation of mass) and well formulated boundary conditions, the Navier-Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations.
The Navier-Stokes equations assume that the fluid being studied is a
continuum. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier-Stokes equations. Depending on the
Knudsen number of the problem,
statistical mechanics or possibly even
molecular dynamics may be a more appropriate approach.
Another limitation is very simply the complicated nature of the equations. Time tested formulations exist for common fluid families, but the application of the Navier-Stokes equations to less common families tends to result in very complicated formulations which are an area of current research. For this reason, the Navier-Stokes equations are usually written for
Newtonian fluids.
Derivation and description
inertial frame of reference, the most general form of the Navier-Stokes equations ends up being:
»
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
nonlinear term makes this a very difficult problem to solve analytically (a lengthy
implicit solution may be found which involves
elliptic integrals and
roots of cubic polynomials). Issues with the actual existence of solutions arise for R > 22.609 (approximately), the parameter R being the
Reynolds number with appropriately chosen scales. This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.
Further Information
Get more info on 'Navier-stokes Equations'.
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