Navier-Stokes equations

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 liquids and gases ( fluids ). The equations are an extension of the Euler equations in fluid mechanics to include terms describing viscosity .

In the narrower sense, especially in physics, the Navier-Stokes equation is the momentum equation for flows. In a broader sense, especially in numerical fluid mechanics , this momentum equation is extended by the continuity equation and the energy equation and then forms a system of non-linear partial differential equations of the second order. This is the basic mathematical model of fluid mechanics. In particular, the equations depict turbulence and boundary layers . A de-dimensionalization of the Navier-Stokes equations provides various dimensionless parameters such as the Reynolds number or the Prandtl number .

The Navier-Stokes equations depict the behavior of water, air and oils and are therefore used in a discretized form in the development of vehicles such as cars and airplanes . This is done as an approximation, since no exact analytical solutions are known for these complicated applications. The existence and uniqueness of a solution of the equations has also not yet been proven in the general case, which is one of the most important unsolved mathematical problems, the Millennium Problems .

history

Isaac Newton published his three-volume Principia with the Laws of Motion in 1686 and also defined the viscosity of a linearly viscous (today: Newtonian ) liquid in the second book . In 1755 Leonhard Euler derived the Euler equations from the laws of motion , with which the behavior of viscosity-free fluids (liquids and gases) can be calculated. The prerequisite for this was his definition of pressure in a fluid, which is still valid today. Jean-Baptiste le Rond d'Alembert (1717–1783) introduced Euler's approach , derived the local mass balance and formulated the d'Alembert's paradox , according to which the flow of viscous fluids exerted no force on a body in the direction of the flow becomes (which Euler already proved). 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 formulated the momentum law for Newtonian fluids in differential form independently of one another in the first half of the 19th century. Navier (1827) and Poisson (1831) set up the momentum equations after considering the effects 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 this (1845). However, the name Navier-Stokes equations for the momentum equations prevailed.

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, numerical fluid mechanics developed to such an extent that with its help for practical problems solutions of the Navier-Stokes equations can be found, which - as it turns out - agree well with the real flow processes.

formulation

Momentum equation

The Navier-Stokes equation in the narrower sense is the law of momentum as the application of Newton's axioms to a continuum . One form used for compressible fluids is:

${\displaystyle \rho {\dot {\vec {v}}}=\rho \left({\frac {\partial {\vec {v}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {v}}\right)=-\nabla p+\mu \Delta {\vec {v}}+(\lambda +\mu )\nabla (\nabla \cdot {\vec {v}})+{\vec {f}}.}$

Here is the density , the (static) pressure , the speed of a particle in the flow, the superpoint as well as the substantial time derivative below , the partial derivative according to time with the fluid element fixed, " " the (formal) scalar product with the nabla operator and the Laplace operator . To the left of the equals sign is the substantial acceleration of the fluid elements and the term formed with the Nabla operator represents their convective part. The vector stands for a volume force density such as gravitation or the Coriolis force, each related to the unit volume and has the unit Newton / cubic meter . The parameters and are the dynamic viscosity and the first Lamé constant . In the literature they are also referred to as Lame viscosity constants. ${\displaystyle \rho }$ ${\displaystyle p}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\tfrac {\mathrm {D} }{\mathrm {D} t}}}$ ${\displaystyle \partial /\partial t}$ ${\displaystyle \cdot }$ ${\displaystyle \nabla }$ ${\displaystyle \Delta }$ ${\displaystyle {\vec {f}}}$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$

Another notation for the form used in literature is:

${\displaystyle \rho {\dot {\vec {v}}}=\rho \left({\frac {\partial {\vec {v}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {v}}\right)=-\nabla p+\mu \Delta {\vec {v}}+\left(\zeta +{\frac {\mu }{3}}\right)\nabla (\nabla \cdot {\vec {v}})+{\vec {f}}.}$

Where ζ is the bulk viscosity . With the continuity equation and application of Stokes' hypothesis ζ = 0, the equation for the momentum density becomes : ${\displaystyle {\vec {m}}=\rho {\vec {v}}}$

${\displaystyle {\frac {\partial {\vec {m}}}{\partial t}}+\nabla \cdot ({\vec {v}}\otimes {\vec {m}})=-\nabla p+\mu \Delta {\vec {v}}+{\frac {\mu }{3}}\nabla (\nabla \cdot {\vec {v}})+{\vec {f}}.}$

The arithmetic symbol forms the dyadic product . To complete the equations, the mass balance or continuity equation ( conservation of mass ) and, for gases, the energy balance ( conservation of energy ) must be added. Depending on the further assumptions that are made of the fluid, the complete system results in different forms. The most commonly used form is the Navier-Stokes equations for incompressible fluids, because they are well suited for subsonic flows and their calculation is easier than those for compressible fluids. ${\displaystyle \otimes }$

Momentum equation in components

The vector form of the equations apply in every coordinate system . Here the component equations of the momentum equation are to be given specifically for Cartesian coordinates .

${\displaystyle {\begin{aligned}&{\frac {\partial (\rho v_{x})}{\partial t}}+{\frac {\partial (\rho v_{x}^{2})}{\partial x}}+{\frac {\partial (\rho v_{x}v_{y})}{\partial y}}+{\frac {\partial (\rho v_{x}v_{z})}{\partial z}}=\dotsm \\&\qquad \dotsm =-{\frac {\partial p}{\partial x}}+{\frac {\partial }{\partial x}}\left[\mu \left(2{\frac {\partial v_{x}}{\partial x}}-{\frac {2}{3}}(\nabla \cdot {\vec {v}})\right)\right]+{\frac {\partial }{\partial y}}\left[\mu \left({\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}}\right)\right]+{\frac {\partial }{\partial z}}\left[\mu \left({\frac {\partial v_{x}}{\partial z}}+{\frac {\partial v_{z}}{\partial x}}\right)\right]+f_{x}\\&{\frac {\partial (\rho v_{y})}{\partial t}}+{\frac {\partial (\rho v_{x}v_{y})}{\partial x}}+{\frac {\partial (\rho v_{y}^{2})}{\partial y}}+{\frac {\partial (\rho v_{y}v_{z})}{\partial z}}=\dotsm \\&\qquad \dotsm =-{\frac {\partial p}{\partial y}}+{\frac {\partial }{\partial x}}\left[\mu \left({\frac {\partial v_{y}}{\partial x}}+{\frac {\partial v_{x}}{\partial y}}\right)\right]+{\frac {\partial }{\partial y}}\left[\mu \left(2{\frac {\partial v_{y}}{\partial y}}-{\frac {2}{3}}(\nabla \cdot {\vec {v}})\right)\right]+{\frac {\partial }{\partial z}}\left[\mu \left({\frac {\partial v_{y}}{\partial z}}+{\frac {\partial v_{z}}{\partial y}}\right)\right]+f_{y}\\&{\frac {\partial (\rho v_{z})}{\partial t}}+{\frac {\partial (\rho v_{x}v_{z})}{\partial x}}+{\frac {\partial (\rho v_{y}v_{z})}{\partial y}}+{\frac {\partial (\rho v_{z}^{2})}{\partial z}}=\dotsm \\&\qquad \dotsm =-{\frac {\partial p}{\partial z}}+{\frac {\partial }{\partial x}}\left[\mu \left({\frac {\partial v_{z}}{\partial x}}+{\frac {\partial v_{x}}{\partial z}}\right)\right]+{\frac {\partial }{\partial y}}\left[\mu \left({\frac {\partial v_{z}}{\partial y}}+{\frac {\partial v_{y}}{\partial z}}\right)\right]+{\frac {\partial }{\partial z}}\left[\mu \left(2{\frac {\partial v_{z}}{\partial z}}-{\frac {2}{3}}(\nabla \cdot {\vec {v}})\right)\right]+f_{z}\end{aligned}}}$

In it are and the vector components in the spatial -, - and - directions. In this form, a possible location dependence of the shear viscosity due to its temperature dependence and temperature fluctuations in the fluid can be taken into account. ${\displaystyle v_{x,y,z}}$ ${\displaystyle f_{x,y,z}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

De-dimensionalization

The Navier-Stokes equations can be de-dimensionalized with characteristic measures of the entire flow area for length , velocity and density . This creates the dimensionless quantities ${\displaystyle L}$ ${\displaystyle v_{\infty }}$ ${\displaystyle \rho _{\infty }}$

${\displaystyle {\begin{aligned}{\vec {x}}^{\ast }:={\frac {\vec {x}}{L}},\quad \nabla ^{\ast }:=L\nabla ,\quad \Delta ^{\ast }:=L^{2}\Delta ,\quad {\vec {v}}^{\ast }:={\frac {\vec {v}}{v_{\infty }}},\quad \\t^{\ast }:={\frac {v_{\infty }t}{L}},\quad \rho ^{\ast }:={\frac {\rho }{\rho _{\infty }}},\quad p^{\ast }:={\frac {p}{\rho _{\infty }v_{\infty }^{2}}},\quad {\vec {f}}^{\ast }:={\frac {L{\vec {f}}}{\rho _{\infty }v_{\infty }^{2}}}\end{aligned}}}$

which lead to the dimensionless momentum equation:

${\displaystyle \rho ^{\ast }\left({\frac {\partial {\vec {v}}^{\ast }}{\partial t^{\ast }}}+({\vec {v}}^{\ast }\cdot \nabla ^{\ast }){\vec {v}}^{\ast }\right)=-\nabla ^{\ast }p^{\ast }+{\frac {1}{\mathrm {Re} }}\Delta ^{\ast }{\vec {v}}^{\ast }+{\frac {1}{3\mathrm {Re} }}\nabla ^{\ast }(\nabla ^{\ast }\cdot {\vec {v}}^{\ast })+{\vec {f}}^{\ast }}$

This characterizes the dimensionless Reynolds number

${\displaystyle \mathrm {Re} ={\frac {L\rho _{\infty }v_{\infty }}{\mu }}}$

the flow in terms of the ratio of inertia to shear forces.

For flows with a free surface, the dimensionless force density contains the Froude number , which characterizes the ratio of inertia to gravitational forces. ${\displaystyle {\vec {f}}^{\ast }}$

Derivation of the momentum equation

The Chapman-Enskog development of Boltzmann equations of the kinetic theory leads to the Navier-Stokes equations with vanishing bulk viscosity so . This development is based on a distribution function that depends only on the speed of the particles, i.e. neglects their angular momentum. This is a good assumption in monatomic gases at low to medium pressure, but does not apply to polyatomic gases. The Chapman-Enskog development is so mathematically demanding that it cannot be presented here. ${\displaystyle \zeta =0}$

In the phenomenological approach of continuum mechanics, the Navier-Stokes equations with volume viscosity result as follows from Newton's assumption of linear viscosity. The viscosity is based on the experiment, according to which a force is required to maintain a shear flow , which, based on its effective area, corresponds to a shear stress . The pressure, which represents a uniform normal stress in all spatial directions, also acts in the fluid . Cauchy's stress tensor summarizes the state of stress in a fluid element into a mathematical object and embodies its divergence accordingly ${\displaystyle {\boldsymbol {\sigma }}}$

${\displaystyle {\vec {F}}=\int _{A}{\vec {s}}\,\mathrm {d} A=\int _{V}\operatorname {div} {\boldsymbol {\sigma }}\,\mathrm {d} V}$

the flow of force in the fluid. The force that acts with area-distributed forces on the surface of the volume is the volume integral over the divergence of the stress tensor. This therefore contributes to the substantial acceleration ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {s}}}$ ${\displaystyle A}$ ${\displaystyle V}$

${\displaystyle {\dot {\vec {v}}}:={\frac {\partial {\vec {v}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {v}}={\frac {\partial {\vec {v}}}{\partial t}}+{\vec {v}}\cdot (\nabla \otimes {\vec {v}})={\frac {\partial {\vec {v}}}{\partial t}}+(\nabla \otimes {\vec {v}})^{\top }\cdot {\vec {v}}={\frac {\partial {\vec {v}}}{\partial t}}+\operatorname {grad} ({\vec {v}})\cdot {\vec {v}}}$

of the fluid elements. In addition to the divergence of the stress tensor, a volume-distributed force such as gravity can act on a fluid element, and so the first Cauchy-Euler law of motion results with the density : ${\displaystyle {\vec {f}}}$ ${\displaystyle \rho }$

${\displaystyle \rho {\dot {\vec {v}}}=\operatorname {div} {\boldsymbol {\sigma }}+{\vec {f}}}$

A Newtonian fluid is able to transfer forces via the pressure in the fluid and via tensions, which depend on the spatial change in the flow velocity and which are macroscopically noticeable as viscosity . The spatial change in the flow velocity is summarized in the velocity gradient . However, there are no stresses in a rigid rotation, which is measured by the asymmetrical part of the velocity gradient, see kinematics in fluid mechanics . Accordingly, only the symmetrical part of the velocity gradient, the distortion velocity tensor, carries ${\displaystyle \operatorname {grad} {\vec {v}}}$ ${\displaystyle \mathbf {d} }$

${\displaystyle \mathbf {d} :={\frac {1}{2}}[\nabla \otimes {\vec {v}}+(\nabla \otimes {\vec {v}})^{\top }]={\frac {1}{2}}{\begin{pmatrix}2{\frac {\partial v_{x}}{\partial x}}&{\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{x}}{\partial z}}+{\frac {\partial v_{z}}{\partial x}}\\&2{\frac {\partial v_{y}}{\partial y}}&{\frac {\partial v_{y}}{\partial z}}+{\frac {\partial v_{z}}{\partial y}}\\{\text{sym.}}&&2{\frac {\partial v_{z}}{\partial z}}\end{pmatrix}}}$

contributes to viscosity. In a reference system invariant material model of the linear viscosity, the stress tensor can only depend on and its main linear invariant . The material model of the classical material theory for the linearly viscous, isotropic fluid reads accordingly ${\displaystyle \mathbf {d} }$ ${\displaystyle \operatorname {Sp} (\mathbf {d} )}$

${\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {1} +\lambda \operatorname {Sp} (\mathbf {d} )\mathbf {1} +2\mu \mathbf {d} =-p\mathbf {1} +\zeta \operatorname {Sp} (\mathbf {d} )\mathbf {1} +2\mu \mathbf {d} ^{\rm {D}}}$

It denotes the (static) pressure, the unit tensor , the trace , the superscript the deviator , the shear viscosity , the first Lamé constant and the volume viscosity . ${\displaystyle p}$ ${\displaystyle \mathbf {1} }$ ${\displaystyle \operatorname {Sp} }$ ${\displaystyle {\rm {D}}}$ ${\displaystyle \mu }$ ${\displaystyle \lambda }$ ${\displaystyle \zeta =\lambda +2\mu /3}$

Inserting the divergence of the stress tensor into the first Cauchy-Euler law of motion yields the Navier-Stokes equations.

proof

For Cauchy-Euler's law of motion, the divergence of the stress tensor is calculated using ${\displaystyle {\begin{aligned}2\mathbf {d} =&\nabla \otimes {\vec {v}}+(\nabla \otimes {\vec {v}})^{\top }\\\operatorname {Sp} \mathbf {d} =&\nabla \cdot {\vec {v}}\end{aligned}}}$ and the derivation rules ${\displaystyle {\begin{aligned}\nabla \cdot (f\mathbf {1} )=&\nabla f\\\nabla \cdot (\nabla \otimes {\vec {f}})=&(\nabla \cdot \nabla ){\vec {f}}=\Delta {\vec {f}}\\\nabla \cdot (\nabla \otimes {\vec {f}})^{\top }=&\nabla (\nabla \cdot {\vec {f}})\end{aligned}}}$ see formula collection Tensoranalysis , provided: ${\displaystyle {\begin{aligned}\nabla \cdot {\boldsymbol {\sigma }}=&\nabla \cdot \left[-p\mathbf {1} +\lambda \operatorname {Sp} (\mathbf {d} )\mathbf {1} +2\mu \mathbf {d} \right]\\=&-\nabla p+\lambda \nabla (\nabla \cdot {\vec {v}})+\mu \nabla \cdot (\nabla \otimes {\vec {v}})+\mu \nabla \cdot (\nabla \otimes {\vec {v}})^{\top }\\=&-\nabla p+\mu \Delta {\vec {v}}+(\lambda +\mu )\nabla (\nabla \cdot {\vec {v}})\end{aligned}}}$ Therein is the Laplace operator . The viscosity parameters are temperature-dependent and the temperature is locally variable, especially in gases, which should be taken into account when diverging. That was neglected here (as usual). This is how the Navier-Stokes equations arise ${\displaystyle \Delta }$ ${\displaystyle {\begin{aligned}\rho {\dot {\vec {v}}}=&-\nabla p+\mu \Delta {\vec {v}}+(\lambda +\mu )\nabla (\nabla \cdot {\vec {v}})+{\vec {f}}\\\rho {\dot {\vec {v}}}=&-\nabla p+\mu \Delta {\vec {v}}+{\vec {f}}\end{aligned}}}$ where the equation below assumes incompressibility . The product rule is used to calculate the momentum density ${\displaystyle \nabla \cdot {\vec {v}}=0}$ ${\displaystyle {\vec {m}}=\rho {\vec {v}}}$ ${\displaystyle {\begin{aligned}{\frac {\partial {\vec {m}}}{\partial t}}={\frac {\partial (\rho {\vec {v}})}{\partial t}}=&{\frac {\partial \rho }{\partial t}}{\vec {v}}+\rho {\frac {\partial {\vec {v}}}{\partial t}}\\\nabla \cdot ({\vec {v}}\otimes {\vec {m}})=&\nabla \cdot ({\vec {m}}\otimes {\vec {v}})=(\nabla \cdot {\vec {m}}){\vec {v}}+({\vec {m}}\cdot \nabla ){\vec {v}}\\\rightarrow {\frac {\partial {\vec {m}}}{\partial t}}+\nabla \cdot ({\vec {v}}\otimes {\vec {m}})=&{\underline {{\frac {\partial \rho }{\partial t}}{\vec {v}}}}+\rho {\frac {\partial {\vec {v}}}{\partial t}}+{\underline {(\nabla \cdot {\vec {m}}){\vec {v}}}}+\rho ({\vec {v}}\cdot \nabla )\cdot {\vec {v}}=\rho {\dot {\vec {v}}}\end{aligned}}}$ The underlined terms are omitted because of the continuity equation and the equation for the momentum density is created: ${\displaystyle {\tfrac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\vec {v}})=0}$ ${\displaystyle {\frac {\partial {\vec {m}}}{\partial t}}+\nabla \cdot ({\vec {v}}\otimes {\vec {m}})=-\nabla p+\mu \Delta {\vec {v}}+(\lambda +\mu )\nabla (\nabla \cdot {\vec {v}})+{\vec {f}}.}$

The pressure, the density and the rate of distortion tensor are objective, see Euclidean transformation , so they are perceived by different observers in the same way. Therefore the Navier-Stokes equations are invariant to a Galilei transformation . ${\displaystyle \mathbf {d} }$

Navier-Stokes equations for incompressible fluids

Liquids can be regarded as incompressible to a good approximation.

If the density does not change along particle trajectories, the flow is called incompressible . This is, for example, a reasonable assumption for water or gases well below the speed of sound ( Mach number <0.3). The continuity equation is simplified so that the velocity field is free of divergence

${\displaystyle \nabla \cdot {\vec {v}}=0,}$

The momentum equation simplifies to:

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

Here stands for the physical pressure, is a volume force based on the unit volume and is the dynamic viscosity. An incompressible flow is completely described by a partial differential equation system with two equations for the two quantities velocity and pressure as a function of location and time. Conservation of energy is not required to close the system. This set of equations is also known as the incompressible Navier-Stokes variable density equations . Application examples for this equation are problems in oceanography, when water of different salinity is incompressible but does not have a constant density. ${\displaystyle p}$ ${\displaystyle {\vec {f}}}$ ${\displaystyle \mu }$ ${\displaystyle {\vec {v}}}$ ${\displaystyle p}$

In many practical problems, the flow is not only incompressible, but even has a constant density. Here you can divide by the density and include it in the differential operators:

${\displaystyle {\frac {\partial {\vec {v}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {v}}=-\nabla {\overline {p}}+\nu \Delta {\vec {v}}+{\overline {\vec {f}}}.}$

In this equation stands for the quotient of physical pressure and density and is an acceleration due to gravity . These variables thus represent the pressure or the volume force related to the unit mass. The variable is the kinematic viscosity and measures the diffusive impulse transport. ${\displaystyle {\overline {p}}=p/\rho }$ ${\displaystyle {\overline {\vec {f}}}={\vec {f}}/\rho }$ ${\displaystyle \nu =\mu /\rho }$

The latter equations are also referred to in the literature as incompressible Navier-Stokes equations, or simply the Navier-Stokes equations, because they are the best studied and most frequently used in practice. They are also easier to solve than the equations for compressible fluids. The equations can be used for many important flow problems, for example with air currents far below the speed of sound ( Mach number <0.3), for water currents and for liquid metals. However, as soon as the densities of the fluids under consideration change significantly, such as in supersonic flows or in meteorology, the Navier-Stokes equations for incompressible fluids no longer represent a suitable model of reality and must be replaced by the complete Navier-Stokes equations for compressible fluids Fluids are replaced.

Momentum equation for incompressibility in components

The vector form of the equations apply in every coordinate system . Here the component equations of the momentum equation for incompressibility should be given in Cartesian, cylindrical and spherical coordinates.

In a Cartesian system, the momentum balance is written: ${\displaystyle xyz}$

${\displaystyle {\begin{aligned}\rho {\frac {\mathrm {D} v_{x}}{\mathrm {D} t}}=&-{\frac {\partial p}{\partial x}}+\mu \Delta v_{x}+f_{x}\\\rho {\frac {\mathrm {D} v_{y}}{\mathrm {D} t}}=&-{\frac {\partial p}{\partial y}}+\mu \Delta v_{y}+f_{y}\\\rho {\frac {\mathrm {D} v_{z}}{\mathrm {D} t}}=&-{\frac {\partial p}{\partial z}}+\mu \Delta v_{z}+f_{z}\\{\frac {\mathrm {D} }{\mathrm {D} t}}=&{\frac {\partial }{\partial t}}+v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}\end{aligned}}}$

The operator stands for the substantial derivative . ${\displaystyle {\tfrac {D}{\mathrm {D} t}}}$

In cylindrical coordinates ( ) are the equations ${\displaystyle R,\varphi ,z}$

${\displaystyle {\begin{aligned}\rho \left({\frac {\mathrm {D} v_{R}}{\mathrm {D} t}}-{\frac {v_{\varphi }^{2}}{R}}\right)=&-{\frac {\partial p}{\partial R}}+\mu \left(\Delta v_{R}-{\frac {v_{R}}{R^{2}}}-{\frac {2}{R^{2}}}{\frac {\partial v_{\varphi }}{\partial \varphi }}\right)+f_{R}\\\rho \left({\frac {\mathrm {D} v_{\varphi }}{\mathrm {D} t}}+{\frac {v_{R}v_{\varphi }}{R}}\right)=&-{\frac {1}{R}}{\frac {\partial p}{\partial \varphi }}+\mu \left(\Delta v_{\varphi }-{\frac {v_{\varphi }}{R^{2}}}+{\frac {2}{R^{2}}}{\frac {\partial v_{R}}{\partial \varphi }}\right)+f_{\varphi }\\\rho {\frac {\mathrm {D} v_{z}}{\mathrm {D} t}}=&-{\frac {\partial p}{\partial z}}+\mu \Delta v_{z}+f_{z}\\{\frac {\mathrm {D} }{\mathrm {D} t}}=&{\frac {\partial }{\partial t}}+v_{R}{\frac {\partial }{\partial R}}+{\frac {1}{R}}v_{\varphi }{\frac {\partial }{\partial \varphi }}+v_{z}{\frac {\partial }{\partial z}}\end{aligned}}}$

The equations are in spherical coordinates ( ) ${\displaystyle r,\varphi ,\theta }$

${\displaystyle {\begin{aligned}\rho \left({\frac {\mathrm {D} v_{r}}{\mathrm {D} t}}-{\frac {v_{\varphi }^{2}+v_{\theta }^{2}}{r}}\right)=&-{\frac {\partial p}{\partial r}}+\mu \left[\Delta v_{r}-{\frac {2}{r^{2}}}\left(v_{r}+{\frac {\partial v_{\theta }}{\partial \theta }}+v_{\theta }\cot \theta +{\frac {1}{\sin \theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}\right)\right]+f_{r}\\\rho \left({\frac {\mathrm {D} v_{\varphi }}{\mathrm {D} t}}+{\frac {v_{r}v_{\varphi }+v_{\varphi }v_{\theta }\cot \theta }{r}}\right)=&-{\frac {1}{r\sin \theta }}{\frac {\partial p}{\partial \varphi }}+\mu \left[\Delta v_{\varphi }+{\frac {1}{r^{2}\sin ^{2}\theta }}\left(-v_{\varphi }+2{\frac {\partial v_{r}}{\partial \varphi }}+2{\frac {\partial v_{\theta }}{\partial \varphi }}\cos \theta \right)\right]+f_{\varphi }\\\rho \left({\frac {\mathrm {D} v_{\theta }}{\mathrm {D} t}}+{\frac {v_{r}v_{\theta }-v_{\varphi }^{2}\cot \theta }{r}}\right)=&-{\frac {1}{r}}{\frac {\partial p}{\partial \theta }}+\mu \left[\Delta v_{\theta }+{\frac {2}{r^{2}}}\left({\frac {\partial v_{r}}{\partial \theta }}-{\frac {v_{\theta }}{\sin ^{2}\theta }}-{\frac {\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}\right)\right]+f_{\theta }\\{\frac {\mathrm {D} }{\mathrm {D} t}}=&{\frac {\partial }{\partial t}}+v_{r}{\frac {\partial }{\partial r}}+{\frac {v_{\varphi }}{r\sin \theta }}{\frac {\partial }{\partial \varphi }}+{\frac {v_{\theta }}{r}}{\frac {\partial }{\partial \theta }}\end{aligned}}}$

Navier-Stokes equations for compressible fluids

Gases are compressible fluids, which are used for technical purposes in internal combustion engines, for example . The graphic shows the working process of a two-stroke engine

For compressible gases, the above momentum equations are expanded to include the energy balance and the equation of state of an ideal gas . The complete set of equations therefore consists of the continuity equation ( conservation of mass ), momentum balance ( conservation of momentum ), energy balance ( conservation of energy ) and an equation of state. The laws given in brackets apply in closed systems, but the incoming and outgoing flows must be balanced on a fluid particle, which leads to balance equations that can be looked up under fluid mechanics . Assuming that the density is constant along the particle trajectories, the equations for incompressible fluids arise again.

In the following, the derivative of a variable means according to time and is the Nabla operator , which forms the derivative according to location, i.e. the divergence or the gradient, depending on the link, and are the three location coordinates in a Cartesian coordinate system. The specified balance equations lead to conservation equations in closed systems. ${\displaystyle \partial _{t}}$ ${\displaystyle \nabla }$ ${\displaystyle x_{i}~(i=1,2,3)}$

Mass conservation

The continuity equation corresponds to the conservation of mass and is formulated here with the momentum density : ${\displaystyle {\vec {m}}=\rho {\vec {v}}}$

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot {\vec {m}}=0.}$

Conservation of momentum

The momentum balance corresponds to the conservation of momentum and is in index notation

${\displaystyle \rho {\dot {v}}_{i}:=\partial _{t}m_{i}+\sum _{j=1}^{3}\partial _{x_{j}}m_{i}v_{j}=-\partial _{x_{i}}p+\sum _{j=1}^{3}\partial _{x_{j}}S_{ij}+f_{i}\qquad (i=1,2,3),}$

where the Kronecker delta and ${\displaystyle \delta _{ij}}$

${\displaystyle S_{ij}=\mu (\partial _{x_{j}}v_{i}+\partial _{x_{i}}v_{j})+\lambda \delta _{ij}\sum _{k=1}^{3}\partial _{x_{k}}v_{k}\qquad (i,j=1,2,3)}$

are the friction tensor or viscous stress tensor . The material parameter is the dynamic viscosity, the first Lamé constant and is the -th component of the volume force vector. In the alternative, coordinate-free notation, the momentum balance is ${\displaystyle \mu }$ ${\displaystyle \lambda }$ ${\displaystyle f_{i}}$ ${\displaystyle i}$

${\displaystyle \rho {\dot {\vec {v}}}={\frac {\partial {\vec {m}}}{\partial t}}+\nabla \cdot ({\vec {v}}\otimes {\vec {m}})=\nabla \cdot \left(-p\mathbf {1} +\mathbf {S} \right)+{\vec {f}},}$

in which

${\displaystyle \mathbf {S} =\mu \left[(\nabla \otimes {\vec {v}})^{\top }+\nabla \otimes {\vec {v}}\right]+\lambda (\nabla \cdot {\vec {v}})\mathbf {1} =2\mu \mathbf {d} +\lambda \operatorname {Sp} (\mathbf {d} )\mathbf {1} }$

the viscous stress tensor, d is the strain rate tensor, which is the symmetrical part of the velocity gradient and has the trace , the stress tensor, 1 is the unit tensor and the dyadic product , see # derivation of the momentum equation above. ${\displaystyle (\nabla \otimes {\vec {v}})^{\top }}$ ${\displaystyle \operatorname {Sp} (\mathbf {d} )=\nabla \cdot {\vec {v}}}$ ${\displaystyle -p\mathbf {1} +\mathbf {S} ={\boldsymbol {\sigma }}}$ ${\displaystyle \otimes }$

Energy conservation

The energy balance on the fluid particle in the earth's gravitational field reads

${\displaystyle \partial _{t}\rho E+\nabla \cdot (H{\vec {m}})=\nabla \cdot (\mathbf {S} \cdot {\vec {v}}-{\vec {W}})+q-\rho {\vec {v}}\cdot {\vec {g}},}$

where the gravitational acceleration and ${\displaystyle {\vec {g}}}$

${\displaystyle H=E+{\frac {p}{\rho }}}$

is the enthalpy per unit mass. The negative sign in front of the gravitational acceleration results from the downward directed vector , so that potential energy is gained in an upward flowing current . The heat flux can by means of the coefficient of thermal conductivity as a ${\displaystyle {\vec {g}}}$ ${\displaystyle {\vec {W}}}$ ${\displaystyle \kappa }$

${\displaystyle {\vec {W}}=-\kappa \nabla T}$

to be written. The source term can be used, for example, to describe the absorption and emission of heat from greenhouse gases as a result of radiation . The total energy per unit mass is the sum of internal ( ), kinetic and potential energy, so it can be written (with the height ) as ${\displaystyle q}$ ${\displaystyle E}$ ${\displaystyle e}$ ${\displaystyle h}$

${\displaystyle E=e+{\frac {1}{2}}|{\vec {v}}|^{2}+h|{\vec {g}}|.}$

Equation of state

Now there are four equations for five variables and the system is completed by the following equation of state :

${\displaystyle p=(\gamma -1)\rho \left(E-{\frac {1}{2}}|{\textbf {v}}|^{2}-h|{\vec {g}}|\right).}$

The thermodynamic quantities density, pressure and temperature are linked by the ideal gas law :

${\displaystyle T={\frac {p}{\rho R}}\qquad {\text{und}}\qquad e=\int _{T_{0}}^{\top }c_{v}(\tau )\,\mathrm {d} \tau .}$

Often one also assumes a perfect gas with constant specific heat capacity . Then the integral is simplified and the following applies: ${\displaystyle c_{v}}$

${\displaystyle e=c_{v}T={\frac {RT}{\gamma -1}}={\frac {p}{\rho \cdot (\gamma -1)}}}$

In both cases the hanging adiabatic exponent and the gas constant by the specific heat coefficient for constant pressure respectively constant volume through and together. ${\displaystyle \gamma }$ ${\displaystyle R}$ ${\displaystyle c_{p}}$ ${\displaystyle c_{v}}$ ${\displaystyle \gamma ={\frac {c_{p}}{c_{v}}}}$ ${\displaystyle R=c_{p}-c_{v}}$

boundary conditions

An essential point in the Navier-Stokes equations is the experimentally very well proven slip condition ( no-slip condition ), in which zero is specified as the relative speed on a wall both in the normal direction and especially in the tangential direction . The fluid particles stick to the wall. This leads to the formation of a boundary layer , which is responsible for essential phenomena only modeled by the Navier-Stokes equations. Only if the free path of moving molecules is large compared to the characteristic length of the geometry (e.g. for gases with extremely low densities or flows in extremely narrow gaps) this condition is no longer useful.

Due to dynamic (i.e. force) boundary conditions on a surface, the surface is generally deformed and the flow follows it. The problem then includes determining the area. It results from the specification of the surface force or stress vector for all points on the surface and the fact that the surface is a material surface, because surface forces can only be applied to fluid particles. The following applies to the surface , where is the normal unit vector of the surface and the stress tensor is calculated from the material equation. Mostly, especially in the technical field such as B. at the outlet of a pipe through which there is a flow, the area is known, which considerably simplifies the task. ${\displaystyle {\vec {s}}_{0}}$ ${\displaystyle {\vec {s}}_{0}={\boldsymbol {\sigma }}\cdot {\hat {n}}}$ ${\displaystyle {\hat {n}}}$ ${\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {1} +\lambda \operatorname {Sp} (\mathbf {d} )\mathbf {1} +2\mu \mathbf {d} }$

With correspondingly small-scale flows, the surface tension must be taken into account, which, according to the Young-Laplace equation, depends on the curvature of the surface. If the curvature is weak, the equation for the pressure on the surface is obtained

${\displaystyle p(x,y,z=h)=p_{0}-\gamma \left({\frac {\partial ^{2}h(x,y)}{\partial x^{2}}}+{\frac {\partial ^{2}h(x,y)}{\partial y^{2}}}\right)}$

Here is the specified pressure on the surface , which here has the surface parameters and , and is a parameter that scales the strength of the surface tension. ${\displaystyle p_{0}}$ ${\displaystyle h}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \gamma }$

In addition, if necessary, either a temperature or a heat flow must be specified at the edge.

Possible solutions

Theoretical solution

To date it has not been possible to prove the existence of global solutions. Mathematicians like P.-L. Lions (see literature list) essentially consider the important special case of the incompressible Navier-Stokes equations. While Olga Alexandrovna Ladyschenskaja , Roger Temam and Ciprian Foias, among others , have already been able to prove extensive statements about existence, uniqueness and regularity for the two-dimensional case, so far there are no results for the general three-dimensional case, as there are some fundamental embedding theorems for so-called Sobolev spaces can no longer be used. However, there are existence and uniqueness statements for finite times or special, in particular small, initial data also in the three-dimensional case - especially for weak solutions . The case of weak solutions of the Navier-Stokes equations also in three dimensions was dealt with by Jean Leray in 1934. He showed that the weak solutions he introduced show no pathological behavior in two dimensions (no divergence (blow up) in finite time) and thus global exist in time. However, research by Tristan Buckmaster and Vlad Vicol showed that for another type of weak solution (weaker than Leray's definition), the Navier-Stokes equations show pathological behavior (ambiguity) in three dimensions.

According to the Clay Mathematics Institute , the problem of the general, incompressible proof of existence in three dimensions is one of the most important unsolved mathematical problems at the turn of the millennium.

In practice, analytical solutions are obtained by simplifying the physical models / boundary conditions (special cases). The non-linearity of the convective acceleration presents a particular problem here . The representation with the help of the vorticity is useful here : ${\displaystyle ({\vec {v}}\cdot \nabla ){\vec {v}}}$ ${\displaystyle {\vec {\omega }}=\nabla \times {\vec {v}}=\operatorname {rot} \;{\vec {v}}}$

${\displaystyle ({\vec {v}}\cdot \nabla ){\vec {v}}={\frac {1}{2}}\nabla (\|{\vec {v}}\|)^{2}-{\vec {v}}\times {\vec {\omega }}}$.

Closed analytical solutions exist almost only for cases in which the second term vanishes. This is the case with the assumption that with 3-dimensional flows the eddies always form along the streamline (i.e. the Helmholtz vortex law ) . However, this assumption does not apply to all real currents. An analytical solution with is in the Hamel-Oseenschen vortex . ${\displaystyle {\vec {\omega }}\|{\vec {v}}}$ ${\displaystyle {\vec {\omega }}\bot {\vec {v}}}$

The Navier-Stokes equations are an important field of application in numerical mathematics (theory deals with the existence and uniqueness of solutions; however, there are generally no closed solution formulas). The sub-area that deals with the construction of numerical approximation methods for the Navier-Stokes equations is numerical fluid mechanics or Computational Fluid Dynamics (CFD).

Numerical solution

Visualization of the numerical calculation of the wind flow around a house

For the numerical solution of the Navier-Stokes equations, methods of numerical fluid mechanics are used. Finite-difference , finite-element and finite-volume methods are used as discretizations , as well as spectral methods and other techniques for special tasks . In order to be able to resolve the boundary layer correctly, the grids must be extremely finely resolved in the normal direction near the wall. This is not done in the tangential direction, so that the cells on the wall have extremely large aspect ratios.

The fine resolution forces extremely small time steps because of the compliance with the CFL condition with explicit time integration. Therefore, implicit procedures are usually used. Because of the non-linearity of the system of equations , the system has to be solved iteratively (e.g. using the multi-grid or Newton method ). The combination of momentum and continuity equations in the incompressible equations has a saddle point structure that can be used here.

A simple model for simulating fluids that satisfies the Navier-Stokes equation within the hydrodynamic limit is the FHP model . Its further development leads to the Lattice-Boltzmann methods , which are particularly attractive in the context of parallelization for execution on supercomputers .

In the field of computer graphics , several numerical solution methods were used in which a real-time representation can be achieved through certain assumptions, although in some cases the physical correctness is not always guaranteed. One example of this is the "stable fluids" process developed by Jos Stam . Here the Chorin projection method was used for the field of computer graphics.

Calculation of turbulent flows

Visualization of the large eddy simulation of a Kármán vortex street

In order to calculate turbulent flows , the Navier-Stokes equations can be calculated numerically directly . However, the resolution of the individual turbulence forces a very fine grid, so that this is only economical in research with the aid of supercomputers and with small Reynolds numbers .

In practice, the solution of the Reynolds equations has prevailed. Here, however, a turbulence model is necessary to close the system of equations.

The middle way is the large eddy simulation , which calculates at least the large eddies numerically and only simulates the small scales using a turbulence model.

A much studied convection, which can be described with the Navier-Stokes equation, is Rayleigh-Bénard convection . It is an important example of self-organizing structures and chaos theory .

Simplifications

Due to the difficult solvability properties of the Navier-Stokes equations, attempts will be made in the applications (insofar as this is physically sensible) to consider simplified versions of the Navier-Stokes equations.

Euler equations

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

If the viscosity is neglected ( ), the Euler equations are obtained (here for the compressible case) ${\displaystyle \eta =\lambda =0}$

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

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

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

If the Helmholtz projection is applied to the equation, the pressure disappears in the equation: ${\displaystyle P}$

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

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

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

${\displaystyle P\Delta }$is also called the Stokes operator .

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

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

Boussinesq approximation

→ Main article : Boussinesq approximation

The Boussinesq approximation is often used for gravitation-dependent flows with small density variations and temperature fluctuations that are not too large.

Stochastic Navier-Stokes equations

Since there is still no proof of existence for solutions of the general Navier-Stokes equations, it is also not certain that they reflect turbulence in fluids and, if so, how realistic. Furthermore, random external disturbances can influence the flow ( butterfly effect ) and it is known that fluid elements carry out a random Brownian movement . Such random fluctuations can be captured with a stochastic approach . It becomes a stochastic differential equation in differential notation

${\displaystyle \mathrm {d} {\vec {v}}_{t}=[-\nabla p_{t}+\mu \Delta {\vec {v}}_{t}-({\vec {v}}_{t}\cdot \nabla ){\vec {v}}_{t}+{\vec {f}}_{t}]\mathrm {d} t+b({\vec {v}}_{t},\operatorname {grad} {\vec {v}}_{t})\mathrm {d} W_{t}}$

considered. The term in square brackets represents the Navier-Stokes equations for incompressibility and the following term a stochastic influence such as Brownian motion. This approach is the subject of brisk research activity at the turn of the millennium.

literature

• H. Oertel (ed.): Prandtl guide through fluid mechanics. Fundamentals and phenomena . 13th edition. Springer Vieweg, 2012, ISBN 978-3-8348-1918-5 .
• GK Batchelor : An introduction to Fluid Dynamics. Cambridge University Press, Cambridge u. a. 2000, ISBN 0-521-66396-2 ( Cambridge mathematical library ).
• Alexandre Chorin , Jerrold Marsden : A Mathematical Introduction to Fluid Mechanics. 3rd edition corrected, 3rd printing. Springer, New York NY a. a. 1998, ISBN 3-540-97918-2 ( Texts in Applied Mathematics 4).
• Robert Kerr, Marcel Oliver: Regular or not regular? - Tracking down flow singularities. In: Dierk Schleicher, Malte Lackmann: An Invitation to Mathematics: Insights into Current Research . Springer Spektrum Verlag, 2013. ISBN 978-3-642-25797-1 .
• LD Landau, EM Lifschitz: Textbook of theoretical physics , Volume VI: Hydrodynamics . Akademie Verlag, Berlin 1991, ISBN 3-05-500070-6 .
• Pierre-Louis Lions : Mathematical Topics in Fluid Mechanics. Volume 1: Incompressible Models. Clarendon Press, Oxford et al. a. 1996, ISBN 0-19-851487-5 ( Oxford lecture series in mathematics and its applications 3).
• Pierre-Louis Lions: Mathematical Topics in Fluid Mechanics. Volume 2: Compressible Models. Clarendon Press, Oxford et al. a. 1998, ISBN 0-19-851488-3 ( Oxford lecture series in mathematics and its applications 10).
• Thomas Sonar : Turbulence around fluid mechanics . Spectrum of Science Dossier 6/2009: “The greatest riddles in mathematics”, ISBN 978-3-941205-34-5 , pp. 64–73.
• Karl Wieghardt : Theoretical fluid mechanics. 2nd revised and expanded edition. Teubner, Stuttgart 1974, ISBN 3-519-12034-8 Guidelines for applied mathematics and mechanics. Teubner study books; (Reprint: Universitäts-Verlag Göttingen, Göttingen 2005, ISBN 3-938616-33-4 ( Göttingen Classics of Fluid Mechanics 2)).
• Lars Davidson: Fluid mechanics, turbulent flow and turbulence modeling . (PDF) Lecture notes, Chalmers University of Technology, Gothenburg, Sweden

Web links

• Description of the Millennium Problem Navier-Stokes Equations (PDF; 98 kB)
• Navier-Stokes First Exact Transformation. (PDF; 185 kB)
• Video: Navier-Stokes equation . Institute for Scientific Film (IWF) 2007, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-13096 .

Individual evidence

1. ↑ LD Landau, EM Lifshitz: Fluid Mechanics - Course of Theoretical Physics , Institute of Physical Problems, Pergamon Press, 1966, pp. 47-53
2. ↑ A. Chorin, J.-E. Marsden: A Mathematical Introduction to Fluid Mechanics . Springer Verlag, 2000
3. ^ T. Sonar: Turbulence around fluid mechanics . Spektrum der Wissenschaft Verlag, April 2009, pp. 78–87
4. ^ ^{A b} G. G. Stokes: On the Theories of Internal Friction of Fluids in Motion . In: Transactions of the Cambridge Philosophical Society . tape 8 , 1845, p. 287-305 ( archive.org [accessed April 7, 2017]).
5. ↑ H. Schlichting, Klaus Gersten: boundary layer theory . Springer-Verlag, 1997, ISBN 978-3-662-07554-8 , pp. 73 ( books.google.de ).
6. ↑ F. Durst: Fundamentals of fluid mechanics . Springer, 2006, ISBN 3-540-31323-0 , pp. 10-16 .
7. ↑ An Introduction to Continuum Mechanics, J.-N. Reddy, Cambridge 2008, pp. 212-214
8. ↑ LD Landau, EM Lifshitz: Fluid Mechanics - Course of Theoretical Physics , Institute of Physical Problems, Pergamon Press, 1966, pp. 47-53
9. ↑ Oertel (2012), p. 252.
10. ↑ Oertel (2012), p. 267ff.
11. ^ Sydney Chapman, TG Cowling: The Mathematical Theory of Non-uniform Gases . An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge University Press, 1970, ISBN 978-0-521-40844-8 .
12. ^ Bergmann, Schaefer: Textbook of Experimental Physics . Gases nanosystems liquids. Ed .: Thomas Dorfmüller, Karl Kleinermanns. 2nd edition volume 5 . Walter de Gruyter, Berlin 2006, ISBN 978-3-11-017484-7 , p. 45 f . ( google.de [accessed on April 2, 2017]).
13. ↑ A multi-page summary can be found in Jonas Toelke: Lattice-Boltzmann method for simulating two-phase flows . Ed .: Faculty of Civil Engineering and Surveying at the Technical University of Munich. 2001, p. 11–15 ( tu-braunschweig.de [PDF]).
14. ^ M. Bestehorn: hydrodynamics and structure formation . Springer, 2006, ISBN 978-3-540-33796-6 .
15. ↑ LD Landau, EM Lifshitz: Fluid Mechanics - Course of Theoretical Physics , Volume 6, Institute of Physical Problems, Pergamon Press, 1966
16. ↑ P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2002, ISBN 3-540-43111-X , pp. 182 ff .
17. ^ M. Bestehorn: hydrodynamics and structure formation . Springer, 2006, ISBN 978-3-540-33796-6 , pp. 64 .
18. ^ Tristan Buckmaster, Vlad Vicol: Nonuniqueness of weak solutions to the Navier-Stokes equation, Annals of Mathematics, Volume 189, 2019, pp. 101-144, Arxiv
19. ↑ Hannelore Inge Breckner: Approximation and optimal control of the stochastic navier-stokes equation . Ed .: Mathematical-Natural-Scientific-Technical Faculty of the Martin Luther University Halle-Wittenberg. 1999, p. 1 (English, uni-halle.de [accessed April 10, 2017]).