Navier–Stokes equations

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The Navier-Stokes equations [navˈjeː stəʊks] (after Claude Louis Marie Henri Navier and George Gabriel Stokes) are a mathematical model of the flow of linear-viscous Newtonian fluids. The equations are an extension of the Euler equations of fluid mechanics to include terms describing viscosity.

In a narrower sense, especially in physics, the Navier-Stokes equations mean the momentum equation for flows. In a broader sense, especially in computational fluid dynamics, this momentum equation is extended by the continuity equation and the energy equation and then forms a system of nonlinear partial differential equations of second order. This is the fundamental mathematical model of fluid mechanics. In particular, the equations represent turbulence and boundary layers. A de-dimensionalization of the Navier-Stokes equations yields various dimensionless ratios such as the Reynolds number or the Prandtl number.

The Navier-Stokes equations represent the behavior of water, air and oils and are therefore applied in discretized form in the development of vehicles such as cars and aircraft. This is done in approximate form since no exact analytical solutions are known for these complicated applications. Moreover, the existence and uniqueness of a solution to the equations has not yet been proven in the general case, which is one of the most important unsolved mathematical problems, the millennium problems.

History

In 1686 Isaac Newton published his three-volume Principia containing the laws of motion and also defined the viscosity of a linearly viscous (today: Newtonian) fluid in the second book. In 1755 Leonhard Euler derived the Euler equations from the laws of motion, with which the behaviour of viscosity-free fluids (liquids and gases) can be calculated. The prerequisite for this was his definition of the pressure in a fluid, which is still valid today. Jean-Baptiste le Rond d'Alembert (1717-1783) introduced the Eulerian approach, derived the local mass balance and formulated the d'Alembert paradox, according to which no force is exerted on a body in the direction of flow by the flow of viscosity-free fluids (which Euler had already proved earlier). Because of this and other paradoxes of viscosity-free flows, it was clear that Euler's equations of motion had to be supplemented.

Claude Louis Marie Henri Navier, Siméon Denis Poisson, Barré de Saint-Venant and George Gabriel Stokes independently formulated the momentum theorem for Newtonian fluids in differential form in the first half of the 19th century. Navier (1827) and Poisson (1831) established the momentum equations after considering the action of intermolecular forces. In 1843 Barré de Saint-Venant published a derivation of the momentum equations from Newton's linear viscosity approach, two years before Stokes did so (1845). However, the name Navier-Stokes equations prevailed for the momentum equations.

Ludwig Prandtl made a significant advance in the theoretical and practical understanding of viscous fluids in 1904 with his boundary layer theory. From the middle of the 20th century, computational fluid dynamics developed to such an extent that with its help solutions of the Navier-Stokes equations can be found for practical problems, which - as has been shown - agree well with the real flow processes.

Simplifications

Due to the difficult solvability properties of the Navier-Stokes equations, one will try to consider simplified versions of the Navier-Stokes equations in the applications (as far as this is physically reasonable).

Euler equations

→ Main article: Euler's equations (fluid mechanics)

If the viscosity is neglected ( $\eta =\lambda =0$ ), one obtains the Euler equations (here for the compressible case)

$\rho {\frac {\partial {\vec {v}}}{\partial t}}+\rho ({\vec {v}}\cdot \nabla ){\vec {v}}=-\nabla p+{\vec {f}}.$

The Euler equations for compressible fluids play a role especially in aerodynamics as an approximation of the full Navier-Stokes equations.

Stokes equation

Another type of simplification is common, for example, in geodynamics, where the mantle of the Earth (or other terrestrial planets) is treated as an extremely viscous fluid (creeping flow). In this approximation, the diffusivity of momentum, i.e., the kinematic viscosity, is many orders of magnitude higher than the thermal diffusivity, and the inertia term can be neglected. Introducing this simplification into the stationary Navier-Stokes momentum equation, we obtain the Stokes equation:

$-\nabla p+\mu \cdot \Delta {\vec {v}}+{\vec {f}}=0$

Applying the Helmholtz projection to the equation, the pressure in the equation vanishes:

$\mu \cdot P\Delta {\vec {v}}+{\tilde {\vec {f}}}=0$

where ${\tilde {\vec {f}}}=P{\vec {f}}$ . This has the advantage that the equation now depends only on . ${\vec {v}}$ The original equation is obtained with

$\nabla p=(\operatorname {Id} -P)(\Delta {\vec {v}}+f),$

$P\Delta$ is also called the Stokes operator.

On the other hand, geomaterials have a complicated rheology, which leads to the fact that the viscosity is not considered constant. For the incompressible case this results:

$-\nabla p+\nabla \cdot \{\mu [\nabla {\vec {v}}+(\nabla {\vec {v}})^{\mathrm {T} }]\}+{\vec {f}}=0$

Boussinesq approximation

→ Main article: Boussinesq approximation

For gravity-dependent flows with small density variations and not too large temperature variations, the Boussinesq approximation is often used.