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What are the Navier-Stokes Equations?

The movement of fluid in the physical domain is driven by various properties. For the purpose of bringing the behavior of fluid flow to light and developing a mathematical model, those properties have to be defined precisely as to provide transition between the physical and the numerical domain. Velocity, pressure, temperature, density, and viscosity are the main properties that should be considered simultaneously when conducting a fluid flow examination. In accordance with the physical incidents such as combustion, multiphase flow, turbulent, mass transport, etc., those properties diversify enormously, which can be categorized into kinematic, transport, thermodynamic, and other miscellaneous properties\(^1\).

Thermo-fluid incidents directed by governing equations are based on the laws of conservation. The Navier-Stokes equations are the broadly applied mathematical model to examine changes on those properties during dynamic and/or thermal interactions. The equations are adjustable regarding the content of the problem and are expressed based on the principles of conservation of mass, momentum, and energy\(^1\):

  • Conservation of Mass: Continuity Equation
  • Conservation of Momentum: Momentum Equation of Newton’s Second Law
  • Conservation of Energy: First Law of Thermodynamics or Energy Equation
Conservation equations

Figure 1: Conservation Equations

Although some sources specify the expression of Navier-Stokes equations merely for conservation of momentum, some of them also use all equations of conservation of the physical properties. Regarding the flow conditions, the Navier-Stokes equations are rearranged to provide affirmative solutions in which the complexity of the problem either increases or decreases. For instance, having a numerical case of turbulence according to the pre-calculated Reynolds number, an appropriate turbulent model has to be applied to obtain credible results.

History

Despite the fact that motion of fluid is an exploratory topic for human beings, the evolution of mathematical models emerged at the end of 19\(^{th}\) century after the industrial revolution. The initial appropriate description of the viscous fluid motion had been indicated in the paper “Principia” by Sir Isaac Newton (1687) in which dynamic behavior of fluids under constant viscosity was investigated\(^1\). Later, Daniel Bernoulli (1738) and Leonhard Euler (1755) subsequently derived the equation of inviscid flow which is now expressed as Euler’s inviscid equations. Even though Claude-Louis Navier (1827), Augustin-Louis Cauchy (1828), Siméon Denis Poisson (1829), and Adhémar St.Venant (1843) had carried out studies to explore the mathematical model of fluid flow, they had overlooked the viscous (frictional) force. In 1845, Sir George Stokes had derived the equation of motion of a viscous flow by adding Newtonian viscous terms, thereby the Navier-Stokes Equations had been brought to their final form which has been used to generate numerical solutions for fluid flow ever since\(^{1,2}\).

The matematicians Newton, Navier and Stokes

Figure 2: Newton\(^{9}\), Navier\(^{10}\), and Stokes\(^{11}\), the mathematicians behind the Navier-Stokes equations

The Navier-Stokes Equations

The observation method of fluid flow based on kinematic properties is a fundamental issue for generating a convenient mathematical model. Movement of fluid can be investigated with either Lagrangian or Eulerian methods\(^3\). Lagrangian description of fluid motion is based on monitoring a fluid particle which is large enough to detect properties. Between the initial coordinates at time \(t_0\) and coordinates of the same particle at time \(t_1\) millions of separate particles have to be examined through the path that is almost impossible to follow. In the Eulerian method, any specific particle across the path is not followed; instead, the velocity field as a function of time and position is examined. The missile example (Figure 3) precisely fits to emphasize these methods.

Conversion equations

Figure 3: Observation of fluid motion with the methods Lagrangian and Euler

The Lagrangian formulation of motion is always time dependent. As \(a\), \(b\), and \(c\) are the initial coordinates of a particle; \(x\), \(y\), and \(z\) are coordinates of the same particle at time \(t\). Description of motion for Lagrangian:

\[x=x(a,b,c,\,t),y=y(a,b,c,\,t),z=z(a,b,c,\,t) \tag1\]

In the Eulerian method, \(u\), \(v\) and \(w\) are the components of velocity at the point \((x,y,z)\) while \(t\) is the time. The velocity components \(u\), \(v\) and \(w\) are the unknowns which are functions of the independent variables \(x\), \(y\), \(z\) and \(t\). The description of motion with the Eulerian method for any particular value of \(t\) is:

\[u=u(x,y,z,\,t),v=v(x,y,z,\,t),w=w(x,y,z,\,t) \tag2\]

The equations of conservation in the Eulerian system in which fluid motion is described are expressed as Continuity Equation for mass, Navier-Stokes Equations for momentum and Energy Equation for the first law of Thermodynamics. The equations are all considered simultaneously to examine fluid and flow fields.

Conservation of Mass

The mass in the control volume can be neither created nor destroyed in accordance with physical laws. The conservation of mass, also expressed as Continuity Equation, states that the mass flow difference throughout system between inlet- and outlet-section is zero:

\[\frac{D_\rho}{D_t}+\rho\left(\nabla\cdot\vec{V}\right)=0 \tag3\]

where \(\rho\) is density, \(V\) is velocity and gradient operator \(\nabla\);

\[\vec{\nabla}=\vec{i}\frac{\partial}{\partial x}+\vec{j}\frac{\partial}{\partial y}+\vec{k}\frac{\partial}{\partial z} \tag4\]

While the density is constant, the flow is assumed incompressible and then continuity is simplified as below, which indicates a steady-state process:

\[\frac{D_\rho}{D_t}=0 \longrightarrow \nabla\cdot\vec{V}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} =0 \tag5\]

Conservation of Momentum

The momentum in a control volume is kept constant, which implies conservation of momentum that we call ‘The Navier-Stokes Equations’. The description is set up in accordance with the expression of Newton’s Second Law of Motion:

\[F=m\cdot a \tag6\]

where \(F\) is the net force applied to any particle, \(a\) is the acceleration, and \(m\) is the mass. In case the particle is a fluid, it is convenient to divide the equation to volume of particle to generate a derivation in terms of density as follows:

\[\rho\frac{DV}{D_t}=f=f_{body}+f_{surface} \tag7\]

in which \(f\) is the force exerted on the fluid particle per unit volume, and \(f_{body}\) is the applied force on the whole mass of fluid particles as below:

\[f_{body}=p\cdot g \tag8\]

where \(\rho\) is the density of fluid and \(g\) is the gravitational acceleration. External forces which are deployed through the surface of fluid particles, \(f_{surface}\) is expressed by pressure and viscous forces as shown below:

\[f_{surface}=\nabla\cdot\tau_{ij}=\frac{\partial\tau_{ij}}{\partial x_i}=f_{pressure}+f_{viscous} \tag9\]

where \(\tau_{ij}\) is expressed as stress tensor. According to the general deformation law of Newtonian viscous fluid given by Stokes, \(\tau_{ij}\) is expressed as\(^2\):

\[\tau_{ij}=-p\delta_{ij}+\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)+\delta_{ij}\lambda\nabla\cdot V \tag{10}\]

Hence, Newton’s equation of motion can be specified in the form as follows:

\[\rho\frac{DV}{D_t}=\rho\cdot g+\nabla\cdot\tau_{ij} \tag{11}\]

Substitution of equation (10) into equation (11) results in the Navier-Stokes equations of Newtonian viscous fluid in one equation:

\[\underbrace{\rho\frac{DV}{D_t}}_{I} = \underbrace{\rho\cdot g} _ {II} - \underbrace{\nabla p} _ {III}+\underbrace{\frac{\partial}{\partial x _ i}\left[\mu\left(\frac{\partial v _ i}{\partial x _ j} + \frac{\partial v _ j}{\partial x _ i}\right)+\delta _ {ij} \lambda\nabla\cdot V\right]} \tag{12} _ {IV}\]

\(I\): Momentum convection

\(II\): Mass force

\(III\): Surface force

\(IV\): Viscous force

where static pressure \(\rho\) and gravitational force \(\rho\vec{g}\). The equation (12) is convenient for fluid and flow fields both transient and compressible. \(D/D_t\) indicates the substantial derivative as follows:

\[\frac{D(\,)}{D_t}=\frac{\partial(\,)}{\partial t}+u\frac{\partial(\,)}{\partial x}+v\frac{\partial(\,)}{\partial y}+w\frac{\partial(\,)}{\partial z}=\frac{\partial(\,)}{\partial t}+V\cdot\nabla(\,) \tag{13}\]

If the density of fluid is accepted to be constant, the equations are greatly simplified in which the viscosity coefficient \(\mu\) is assumed constant and \(\nabla\cdot V=0\) in equation (12). Thus, the Navier-Stokes equations for an incompressible three-dimensional flow can be expressed as follows:

\[\rho\frac{DV}{Dt}=\rho g-\nabla p + \mu\nabla^2V \tag{14}\]

For each dimension when the velocity is \(V(u,v,w)\):

\[\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=\rho g_x-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right) \tag{15}\]
\[\rho\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)=\rho g_y-\frac{\partial p}{\partial y}+\mu\left(\frac{\partial^2v}{\partial x^2}+\frac{\partial^2v}{\partial y^2}+\frac{\partial^2v}{\partial z^2}\right) \tag{16}\]
\[\rho\left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)=\rho g_z-\frac{\partial p}{\partial z}+\mu\left(\frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}+\frac{\partial^2w}{\partial z^2}\right) \tag{17}\]

\(p\), \(u\), \(v\) and \(w\) are unknowns where a solution is sought by application of both continuity equation and boundary conditions. Besides, the energy equation has to be considered if any thermal interaction is available in the problem.

Conservation of Energy

The Conservation of Energy is the first law of thermodynamics which states that the sum of the work and heat added to the system will result in the increase of energy of the system:

\[dE_t=dQ+dW \tag{18}\]

where \(dQ\) is the heat added to the system, \(dW\) is the work done on the system, and \(dE_t\) is the increment in the total energy of the system. One of the common types of energy equation is:

\[\rho\left[\underbrace{\frac{\partial h}{\partial t}} _ {I} + \underbrace{\nabla\cdot(hV)} _ {II}\right]=\underbrace{-\frac{\partial p}{\partial t}} _ {III} + \underbrace{\nabla\cdot(k\nabla T)} _ {IV} + \underbrace{\phi} _ {V} \tag{19}\]

\(I\): Local change with time

\(II\): Convective term

\(III\): Pressure work

\(IV\): Heat flux

\(V\): Heat dissipation term

Variations of the Navier-Stokes Equations

The Navier-Stokes equations are extremely thorough in order to simulate physical incidents. Nonetheless, having a non-linear structure and various complexities, it is hardly possible to conduct an exact solution of those equations. Thus, with regard to the physical domain, both approaches and assumptions are partially applied to grind the equations. Beyond this, though, some assumptions have to be applied as to provide a reliable model in which the equation is carried out, up to a further step in terms of complexity such as turbulence.

The mathematical model merely gives ties among parameters which are part of the whole process. Hence, the solution of the Navier-Stokes equations can be realized with either analytical or numerical methods. The analytical method is the process that only compensates solutions in which non-linear and complex structures in the Navier-Stokes equations are ignored within several assumptions. It is only valid for simple / fundamental cases such as Couette flow, Poisellie flow, etc\(^3\). On the other hand, almost every case in fluid dynamics comprises non-linear and complex structures in the mathematical model which cannot be ignored to sustain reliability. Hence, the solution of the Navier-Stokes equations are carried out within several numerical methods, the omnipresence of Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). A computational analysis of fluid flow can be described as shown in Figure 4.

Analytical and Numerical stages of a project

Figure 4: The panoramic structure of a CFD project and its stages

Time Domain

The analysis of fluid flow can be conducted in either steady (time-independent) or unsteady (time-dependent) condition depending on the physical incident. In case the fluid flow is steady, it means the motion of fluid and parameters do not rely on change in time, the term \(\frac{\partial()}{\partial t}=0\) where the continuity and momentum equations are re-derived as follows:

Continuity equation:

\[\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0 \tag{20}\]

The Navier-Stokes equation in \(x\) direction:

\[\rho\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right) = \rho g_x-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right) \tag{21}\]

While the steady flow assumption negates the effect of some non-linear terms and provides a convenient solution, variation of density is a hurdle that keeps the equation in a complex formation.

Compressibility

Due to the malleable structure of fluids, the compressibility of particles is a significant issue. Despite the fact that all types of fluid flow are compressible in a various range regarding molecular structure, most of them can be assumed to be incompressible in which the density changes are negligible. Thus, the term \(\frac{\partial\rho}{\partial t}=0\) is thrown away regardless of whether the flow is steady or not, as below:

Continuity equation:

\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0 \tag{22}\]

The Navier-Stokes equation in \(x\) direction:

\[\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=\rho g_x-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right) \tag{23}\]

As incompressible flow assumption provides reasonable equations, the application of steady flow assumption concurrently enables us to ignore non-linear terms where \(\frac{\partial()}{\partial t}=0\). Moreover, the density of fluid in high speed cannot be accepted as incompressible in which the density changes are important. “The Mach Number” is a dimensionless number that is convenient to investigate fluid flow, whether incompressible or compressible\(^3\):

\[Ma=\frac{V}{a}\leq0.3 \tag{24}\]

where \(Ma\) is the Mach number, \(V\) is the velocity of flow, and \(a\) is the speed of sound at \(340.29 \frac{m}{s}\) at sea level. As in equation (24), when the Mach number is lower than 0.3, the assumption of incompressibility is acceptable. On the contrary, the change in density cannot be negligible in which density should be considered as a significant parameter. For instance, if the velocity of a car is higher than 100\(\frac{m}{s}\), the suitable approach to conduct credible numerical analysis is the compressible flow. Apart from velocity, the effect of thermal properties on the density changes has to be considered in geophysical flows\(^3\).

Low and High Reynolds Numbers:

The Reynolds number, the ratio of inertial and viscous effects, is also effective on Navier-Stokes equations to truncate the mathematical model. While \(Re\longrightarrow\infty\), the viscous effects are presumed negligible where viscous terms in Navier-Stokes equations are thrown away. The simplified form of Navier-Stokes equation, described as Euler equation, can be specified as follows\(^8\):

The Navier-Stokes equation in \(x\) direction:

\[\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=\rho g_x-\frac{\partial p}{\partial x} \tag{25}\]

Even though viscous effects are relatively important for fluids, the inviscid flow model partially provides a reliable mathematical model as to predict real process for some specific cases. For instance, high-speed external flow over bodies is a broadly used approximation where inviscid approach reasonably fits. While \(Re\ll1\), the inertial effects are assumed negligible where related terms in Navier-Stokes equations drop out. The simplified form of Navier-Stokes equations is called either creeping flow or Stokes flow\(^8\):

The Navier-Stokes equation in \(x\) direction:

\[\rho g_x-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}+\right)=0 \tag{26}\]

Having tangible viscous effects, creeping flow is a suitable approach to investigate the flow of lava, swimming of microorganisms, flow of polymers, lubrication, etc.

Turbulence

The behavior of the fluid under dynamic conditions is a challenging issue that is compartmentalized as laminar and turbulent. The laminar flow is orderly at which motion of fluid can be predicted precisely. Except that, the turbulent flow has various hindrances, therefore it is hard to predict the fluid flow which shows a chaotic behavior. The Reynolds number, the ratio of inertial forces to viscous forces, predicts the behavior of fluid flow whether laminar or turbulent regarding several properties such as velocity, length, viscosity, and also type of flow. Whilst the flow is turbulent, a proper mathematical model is selected to carry out numerical solutions. Various turbulent models are available in literature and each of them has a slightly different structure to examine chaotic fluid flow.

Turbulent flow can be applied to the Navier-Stokes equations in order to conduct solutions to chaotic behavior of fluid flow. Apart from the laminar, transport quantities of the turbulent flow, it is driven by instantaneous values. Direct numerical simulation (DNS) is the approach to solving the Navier-Stokes equation with instantaneous values. Having district fluctuations varies in a broad range, DNS needs enormous effort and expensive computational facilities. To avoid those hurdles, the instantaneous quantities are reinstated by the sum of their mean and fluctuating parts as follows:

\[instantaneous\,value = \overline{mean\,value}+fluctuating\,value' \tag{27}\]
\[u = \overline{u}+u'\]
\[v = \overline{v}+v'\]
\[w = \overline{w}+w'\]
\[T= \overline{T}+T'\]

where \(u\), \(v\), and \(w\) are velocity components and \(T\) is temperature. The differences among values are shown in Figure 5 both for steady and unsteady conditions:

Instant, mean and fluctuating velocities; a) steady process, b) unsteady process

Figure 5: Instant, mean and fluctuating velocities; a) steady process, b) unsteady process\(^8\)

Instead of instantaneous values which cause non-linearity, carrying out a numerical solution with mean values provides an appropriate mathematical model which is named “The Reynolds-averaged Navier-Stokes (RANS) equation”\(^4\). The fluctuations can be negligible for most engineering cases which cause a complex mathematical model. Thus, RANS turbulence model is a procedure to close the system of mean flow equations. The general form of The Reynolds-averaged Navier-Stokes (RANS) equation can be specified as follows:

Continuity equation:

\[\frac{\partial\overline{u}}{\partial x}+\frac{\partial\overline{v}}{\partial y}+\frac{\partial\overline{w}}{\partial z}=0 \tag{28}\]

The Navier-Stokes equation in \(x\) direction:

\[\rho\left(\frac{\partial\overline{u}}{\partial t}+\overline{u}\frac{\partial\overline{u}}{\partial x}+\overline{v}\frac{\partial\overline{u}}{\partial y}+\overline{w}\frac{\partial\overline{u}}{\partial z}\right)=\rho g_x-\frac{\partial\overline{p}}{\partial x}+\mu\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right) \tag{29}\]

The turbulence model of RANS can also vary regarding methods such as k-omega, k-epsilon, k-omega-SST, and Spalart-Allmaras which have been used to seek a solution for different types of turbulent flow.

Likewise, large eddy simulation (LES) is another mathematical method for turbulent flow which is also comprehensively applied for several cases. Tough LES ensures more accurate results than RANS, it requires much more time and computer memory. As in DNS, LES considers to solve the instantaneous Navier-Stokes equations in time and three-dimensional space\(^4\).

Application of the Navier-Stokes Equations

The simple form of the Navier-Stokes equations only encompasses the change in properties such as velocity, pressure, and density under dynamic conditions for one phase laminar flow. Most engineering applications require further mathematical models to simulate physical incidents with the aim of obtaining affirmative results in the numerical domain. Some of the most common engineering problems and relevant mathematical models as to carry out numerical simulations can be given as below:

  • Air flow in a duck: Single phase flow, laminar / turbulent, steady / unsteady
  • Water flow in an open channel: Multiphase flow, laminar / turbulent, steady / unsteady
  • Combustion in cylinder: Multiphase flow, laminar / turbulent, unsteady, chemical reaction, heat transfer, mass transfer
  • Condensation/evaporation of water: Multiphase flow, laminar / turbulent, unsteady, heat transfer, mass transfer

Microfluidics

The Navier-Stokes equations cannot compensate the physical model of the flow at very small scales such as the motion of single bacteria — also called microfluidics. Thereby, it is convenient to either change or reinstate the Navier-Stokes equations with a suitable mathematical model. The Knudsen number (Kn) is a dimensionless number that is the ratio of the mean free path of molecular structure to observation scale. The preferred model in accordance with the Knudsen number is shown in Figure 6:

Preferred mathematical models for motion of fluid in accordance with the Knudsen number

Figure 6: Preferred mathematical models for motion of fluid in accordance with the Knudsen number\(^7\)

Real-time simulation:

The Navier-Stokes equations have been considered an important issue by animation / video game companies\(^5\). For the purpose of generating a real-time animation, the application of Navier-Stokes equations provides stunning results when it comes to enhancing reality\(^6\). They are also part of the reason that modern video games appear to be realistic in many more ways than several years ago. Just think about the movement of flags in the wind: doesn’t this look absolutely realistic in modern games?!

Resources

\(^1\): White, Frank (1991).Viscous Fluid Flow. 3rd Edition. McGraw-Hill Mechanical Engineering. ISBN-10: 0072402318.

\(^2\): Stokes, George (1851). “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums”. Transactions of the Cambridge Philosophical Society. 9: 8–106.

\(^3\): White, Frank (2002). Fluid Mechanics. 4th edition. McGraw-Hill Higher Education. ISBN: 0-07-228192-8.

\(^4\): Cebeci, T., Shao, J.P., Kafyeke, F., Laurendeau, E (2005). Computational Fluid Dynamics for Engineers. Horizon Publishing Inc. ISBN: 0-9766545-0-4.

\(^5\): http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf

\(^6\): https://software.intel.com/en-us/articles/fluid-simulation-for-video-games-part-1

\(^7\): http://www.mathcces.rwth-aachen.de/2research/0mms/0gases/start

\(^8\): Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2001). Transport Phenomena, 2th edition. John Wiley & Sons. ISBN 0-471-41077-2.

\(^9\): https://en.wikipedia.org/wiki/Isaac_Newton

\(^{10}\): https://en.wikipedia.org/wiki/Claude-Louis_Navier#/media/File:Claude-Louis_Navier.jpg

\(^{11}\): https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet#/media/File:Ggstokes.jpg