The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as the application and formulation for different families of fluids. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such

## Basic assumptions

The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum, in other words is not made up of discrete particles but rather a continuous substance. Another necessary assumption is that all the of fields of interest like pressure, velocity, density, temperature and so on are differentiable, weakly at least. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface In Physics, velocity is defined as the rate of change of Position. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a weak derivative is a generalization of the concept of the Derivative of a function ( strong derivative) for functions not assumed

The equations are derived from the basic principles of conservation of mass, momentum, and energy. The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov - Lavoisier law says that the Mass of In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός For that matter, sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be applied. In Fluid mechanics and Thermodynamics, a control volume is a mathematical abstraction employed in the process of creating Mathematical models This finite volume is denoted by Ω and its bounding surface $\partial \Omega$. The control volume can remain fixed in space or can move with the fluid.

## The convective derivative

Main article: convective derivative

Changes in properties of a moving fluid can be measured in two different ways. One can measure a given property by either carrying out the measurement on a fixed point in space as particles of the fluid pass by, or by following a parcel of fluid along its streamline. Fluid flow is described in general by a Vector field in three (for steady flows or four (for non-steady flows including time dimensions The derivative of a field with respect to a fixed position in space is called the Eulerian derivative while the derivative following a moving parcel is called the convective derivative.

The convective derivative is defined as the operator:

$\frac{D}{Dt}(\star) \ \stackrel{\mathrm{def}}{=}\ \frac{\partial}{\partial t}(\star) + \mathbf{v}\cdot\nabla (\star)$

where $\mathbf{v}$ is the velocity of the fluid. The first term on the right-hand side of the equation is the ordinary Eulerian derivative (i. e. the derivative on a fixed reference frame, representing changes at a point with respect to time) whereas the second term represents changes of a quantity with respect to position (see advection). Advection, in mechanical and chemical engineering is a transport mechanism of a substance or a conserved property with a moving Fluid. This "special" derivative is in fact the ordinary derivative of a function of many variables along a path following the fluid motion; it may be derived easily through application of the chain rule. In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions.

For example, the measurement of changes in wind velocity in the atmosphere can be obtained with the help of an anemometer in a weather station or by mounting it on a weather balloon. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five An anemometer is a device for measuring wind speed and is one instrument used in a Weather station. The anemometer in the first case is measuring the velocity of all the moving particles passing through a fixed point in space, whereas in the second case the instrument is measuring changes in velocity as it moves with the fluid.

## Conservation laws

The Navier–Stokes equation is a special case of the (general) continuity equation. A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity It, and associated equations such as mass continuity, may be derived from conservation principles of:

This is done via the Reynolds transport theorem, an integral relation stating that the changes of some intensive property (call it L) defined over a control volume Ω must be equal to what is lost (or gained) through the boundaries of the volume plus what is created/consumed by sources and sinks inside the control volume. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Reynolds transport theorem is a fundamental theorem used in formulating the basic conservation laws of Fluid dynamics. In the Physical sciences an intensive property (also called a bulk property) is a Physical property of a system that does not depend on the This is expressed by the following integral equation:

$\frac{d}{dt}\int_{\Omega} L \ dV = -\int_{\partial\Omega} L\mathbf{v\cdot n} \ dA + \int_{\Omega} Q \ dV$

where v is the velocity of the fluid and Q represents the sources and sinks in the fluid. Recall that Ω represents the control volume and $\partial \Omega$ its bounding surface.

The divergence theorem may be applied to the surface integral, changing it into a volume integral:

$\frac{d}{d t} \int_{\Omega} L \ dV = -\int_{\Omega} \nabla \cdot ( L\mathbf{v}) \ dV + \int_{\Omega} Q \ dV$

Applying Leibniz's rule to the integral on the left and then combining all of the integrals:

$\int_{\Omega} \frac{\partial L}{\partial t} \ dV = - \int_{\Omega}\nabla \cdot (L\mathbf{v}) \ dV + \int_{\Omega} Q \ dV\qquad \Rightarrow \qquad\int_{\Omega} \left( \frac{\partial L}{\partial t} + \nabla \cdot (L\mathbf{v}) + Q\ \right) dV = 0$

The integral must be zero for any control volume; this can only be true if the integrand itself is zero, so that:

$\frac{\partial L}{\partial t} + \nabla \cdot (L\mathbf{v}) + Q = 0$

From this valuable relation (a very generic continuity equation), three important concepts may be concisely written: conservation of mass, conservation of momentum, and conservation of energy. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought In Mathematics &mdash in particular in Multivariable calculus &mdash a volume integral refers to an Integral over a 3- Dimensional domain In Mathematics, Leibniz's rule for Differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an Integral A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity Validity is retained if L is a vector, in which case the vector-vector product in the second term will be a dyad. In Mathematics, in particular Multilinear algebra, the dyadic product \mathbb{P} = \mathbf{u}\otimes\mathbf{v} of two

### Conservation of momentum

The most elemental form of the Navier–Stokes equations is obtained when the conservation relation is applied to momentum. Writing momentum as $\rho \mathbf{v}$ gives:

$\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) + \mathbf{Q} = 0$

where $\mathbf{v} \mathbf{v}$ is an outer product, a special case of tensor product, which results in a second rank tensor; the divergence of a second rank tensor is again a vector (a first rank tensor)[1]. In Linear algebra, the outer product typically refers to the tensor product of two vectors. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the Noting that a body force (notated $\mathbf{b}$) is a source or sink of momentum (per volume) and expanding the derivatives completely:

$\frac{\partial \rho}{\partial t} \mathbf{v} + \rho \frac{\partial \mathbf{v}}{\partial t} + \nabla(\rho \mathbf{v}) \cdot \mathbf{v} + \rho \mathbf{v} \nabla \cdot \mathbf{v} = \mathbf{b}$
$\frac{\partial \rho}{\partial t} \mathbf{v} + \rho \frac{\partial \mathbf{v}}{\partial t} + \nabla(\rho) \mathbf{v} \cdot \mathbf{v} + \rho \nabla(\mathbf{v}) \cdot \mathbf{v} + \mathbf{v} \rho \nabla \cdot \mathbf{v} = \mathbf{b}$
$\mathbf{v} \frac{\partial \rho}{\partial t} + \rho \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \mathbf{v} \cdot \nabla \rho + \rho \mathbf{v} \cdot \nabla \mathbf{v} + \rho \mathbf{v} \nabla \cdot \mathbf{v} = \mathbf{b}$

Note that the gradient of a vector is a special case of the covariant derivative, the operation results in second rank tensors[1]; except in Cartesian coordinates, it's important to understand that this isn't simply an element by element gradient. A body force is a force that acts on the volume of a body and also can be defined as an external force acting throughout the mass of a body In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. Rearranging and recognizing that $\mathbf{v} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{v} = \nabla \cdot (\rho \mathbf{v})$:

$\mathbf{v} \left(\frac{\partial \rho}{\partial t} + \mathbf{v} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{v}\right) + \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = \mathbf{b}$
$\mathbf{v} \left(\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v})\right) + \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = \mathbf{b}$

The leftmost expression enclosed in parentheses is, by mass continuity (shown in a moment), equal to zero. Noting that what remains on the left side of the equation is the convective derivative and writing the equation as a vector equation yields:

$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = \mathbf{b}\qquad \Rightarrow \qquad\rho\frac{D \mathbf{v}}{D t} = \mathbf{b}$

This appears to simply be an expression of Newton's second law (F = ma) in terms of body forces instead of point forces. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the A body force is a force that acts on the volume of a body and also can be defined as an external force acting throughout the mass of a body Each term in any case of the Navier–Stokes equations is a body force. A shorter though less rigorous way to arrive at this result would be the application of the chain rule to acceleration:

$\rho \frac{d}{d t}(\mathbf{v}(x, y, z, t)) = \mathbf{b}\qquad \Rightarrow \qquad\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \frac{\partial \mathbf{v}}{\partial x}\frac{d x}{d t} + \frac{\partial \mathbf{v}}{\partial y}\frac{d y}{d t} + \frac{\partial \mathbf{v}}{\partial z}\frac{d z}{d t} \right) = \mathbf{b} \qquad \Rightarrow$
$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + u \frac{\partial \mathbf{v}}{\partial x} + v \frac{\partial \mathbf{v}}{\partial y} + w \frac{\partial \mathbf{v}}{\partial z} \right) = \mathbf{b}\qquad \Rightarrow \qquad\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = \mathbf{b}$

where $\mathbf{v} = (u, v, w)$. In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. The reason why this is "less rigorous" is that we haven't shown that picking $\mathbf{v} = \left(\frac{d x}{d t}, \frac{d y}{d t}, \frac{d z}{d t}\right)$ is correct; however it does make sense since with that choice of path the derivative is "following" a fluid "particle", and in order for Newton's second law to work, forces must be summed following a particle. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the For this reason the convective derivative is also known as the particle derivative.

### Conservation of mass

Mass may be considered also. Taking Q = 0 (no sources or sinks of mass) and putting in density:

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$

where ρ is the mass density (mass per unit volume), and $\mathbf{v}$ is the velocity of the fluid. This equation is called the mass continuity equation, or simply "the" continuity equation. This equation generally accompanies the Navier–Stokes equation.

In the case of an incompressible fluid, ρ is a constant and the equation reduces to:

$\nabla\cdot\mathbf{v} = 0$

which is in fact a statement of the conservation of volume. Fluid mechanics is the study of how Fluids move and the Forces on them

## General form of the Navier–Stokes equations

The generic body force $\mathbf{b}$ seen previously is made specific first by breaking it up into two new terms, one to describe forces resulting from stresses and one for "other" forces such as gravity. By examining the forces acting on a small cube in a fluid, it may be shown that

$\rho\frac{D\mathbf{v}}{D t} = \nabla \cdot \sigma_{ij} + \mathbf{f}$

where σij is the stress tensor, and $\mathbf{f}$ accounts for other body forces present. Stress is a measure of the average amount of Force exerted per unit Area. This equation is called the Cauchy momentum equation and describes the non-relativistic momentum conservation of any continuum that conserves mass. The Cauchy momentum equation is a vector Partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any Continuum σij is a symmetric tensor given by:

$\sigma_{ij} = \begin{pmatrix}\sigma_{xx} & \tau_{xy} & \tau_{xz} \\\tau_{yx} & \sigma_{yy} & \tau_{yz} \\\tau_{zx} & \tau_{zy} & \sigma_{zz}\end{pmatrix}$

where the σ are normal stresses and τ shear stresses. Stress is a measure of the average amount of Force exerted per unit Area. A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material This tensor is split up into two terms:

$\sigma_{ij} = \begin{pmatrix}\sigma_{xx} & \tau_{xy} & \tau_{xz} \\\tau_{yx} & \sigma_{yy} & \tau_{yz} \\\tau_{zx} & \tau_{zy} & \sigma_{zz}\end{pmatrix}=-\begin{pmatrix}p&0&0\\0&p&0\\0&0&p\end{pmatrix}+ \begin{pmatrix}\sigma_{xx}+p & \tau_{xy} & \tau_{xz} \\\tau_{yx} & \sigma_{yy}+p & \tau_{yz} \\\tau_{zx} & \tau_{zy} & \sigma_{zz}+p\end{pmatrix}= -p I + \mathbb{T}$

where I is the 3 x 3 identity matrix and the pressure p is minus the mean normal stress:[2]

$p = - \frac{1}{3} \left( \sigma_{xx} + \sigma_{yy} + \sigma_{zz} \right).$

The motivation for doing this is that pressure is typically a variable of interest, and also this simplifies application to specific fluid families later on since the rightmost tensor $\mathbb{T}$ in the equation above must be zero for a fluid at rest. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface Note that $\mathbb{T}$, called the Deviatoric stress tensor, is traceless. Stress is a measure of the average amount of Force exerted per unit Area. In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal The Navier–Stokes equation may now be written in the most general form:

$\rho\frac{D\mathbf{v}}{D t} = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f}$

This equation is still incomplete. For completion, one must make hypotheses on the form of $\mathbb{T}$, that is, one needs a constitutive law for the stress tensor which can be obtained for specific fluid families; additionally, if the flow is assumed compressible an equation of state will be required, which will likely further require a conservation of energy formulation.

## Application to different fluids

The most general form of the Navier–Stokes equations is not "ready for use", the stress tensor contains too many unknowns so that more information is needed; this information is normally some knowledge of the viscous behavior of the fluid.

### Newtonian fluid

Main article: Newtonian fluid

The formulation for Newtonian fluids stems from an observation made by Newton that, for most fluids,

$\tau \propto \frac{\partial u}{\partial y}$

In order to apply this to the Navier–Stokes equations, three assumptions were made by Stokes:

• The stress tensor is a linear function of the strain rates. A Newtonian fluid (named for Isaac Newton) is a Fluid whose stress versus rate of strain curve is linear and passes through the origin Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements
• The fluid is isotropic.
• For a fluid at rest, $\nabla \cdot \mathbb{T}$ must be zero (so that hydrostatic pressure results). Fluid statics (also called hydrostatics) is the Science of Fluids at rest and is a sub-field within Fluid mechanics.

Applying these assumptions will lead to:

$\mathbb{T}_{ij} = \mu\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) + \delta_{ij} \lambda \nabla \cdot \mathbf{v}$

δij is the Kronecker delta. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two μ and λ are proportionality constants associated with the assumption that stress depends on strain linearly; μ is called the first coefficient of viscosity (usually just called "viscosity") and λ is the second coefficient of viscosity (related to bulk viscosity). Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. Volume viscosity (also called bulk viscosity or second viscosity) appears in the Navier-Stokes equation if it is written for Compressible fluid The value of λ, which produces a viscous effect associated with volume change, is very difficult to determine, not even its sign is known with absolute certainty. Even in compressible flows, the term involving λ is often negligible; however it can occasionally be important even in nearly incompressible flows and is a matter of controversy. When taken nonzero, the most common approximation is λ ≈ - ⅔ μ. [3]

A straightforward substitution of $\mathbb{T}_{ij}$ into the momentum conservation equation will yield the Navier–Stokes equations for a compressible Newtonian fluid:

$\rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right) = -\frac{\partial p}{\partial x} + \frac{\partial}{\partial x}\left(2 \mu \frac{\partial u}{\partial x} + \lambda \nabla \cdot \mathbf{v}\right) + \frac{\partial}{\partial y}\left(\mu\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)\right) + \frac{\partial}{\partial z}\left(\mu\left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right)\right) + \rho g_x$
$\rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}+ w \frac{\partial v}{\partial z}\right) = -\frac{\partial p}{\partial y} + \frac{\partial}{\partial x}\left(\mu\left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right)\right) + \frac{\partial}{\partial y}\left(2 \mu \frac{\partial v}{\partial y} + \lambda \nabla \cdot \mathbf{v}\right) + \frac{\partial}{\partial z}\left(\mu\left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}\right)\right) + \rho g_y$
$\rho \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y}+ w \frac{\partial w}{\partial z}\right) = -\frac{\partial p}{\partial z} + \frac{\partial}{\partial x}\left(\mu\left(\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}\right)\right) + \frac{\partial}{\partial y}\left(\mu\left(\frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}\right)\right) + \frac{\partial}{\partial z}\left(2 \mu \frac{\partial w}{\partial z} + \lambda \nabla \cdot \mathbf{v}\right) + \rho g_z$

or, more compactly in vector form,

$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \nabla \cdot (\mu \cdot (\nabla \mathbf{v} + (\nabla \mathbf{v})^T)) + \nabla (\lambda \nabla \cdot \mathbf{v})+ \rho \mathbf{g}$

where the transpose has been used. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a Gravity has been accounted for as "the" body force, ie $\mathbf{f} = \rho \mathbf{g}$. The associated mass continuity equation is:

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$

In addition to this equation, an equation of state and an equation for the conservation of energy is needed. In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic The equation of state to use depends on context (often the ideal gas law), the conservation of energy will read:

$\rho \frac{D h}{D t} = \frac{D p}{D t} + \nabla \cdot (k \nabla T) + \Phi$

Here, h is the enthalpy, T is the temperature, and Φ is a function representing the dissipation of energy due to viscous effects:

$\Phi = \mu \left(2\left(\frac{\partial u}{\partial x}\right)^2 + 2\left(\frac{\partial v}{\partial y}\right)^2 + 2\left(\frac{\partial w}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right)^2 + \left(\frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}\right)^2 + \left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right)^2\right) + \lambda (\nabla \cdot \mathbf{v})^2$

With a good equation of state and good functions for the dependence of parameters (such as viscosity) on the variables, this system of equations seems to properly model the dynamics of all known gases and most liquids. The ideal gas law is the Equation of state of a hypothetical Ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834 In Thermodynamics and molecular chemistry, the enthalpy (denoted as H, h, or rarely as χ) is a quotient or description of Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature

For the special but very common case of incompressible flow, the momentum equations simplify significantly. For example, looking at the viscous terms of the x momentum equation (note that viscosity will now be a constant and the second viscosity effect will be zero):

\begin{align} &\frac{\partial}{\partial x}\left(2 \mu \frac{\partial u}{\partial x} + \lambda \nabla \cdot \mathbf{v}\right) + \frac{\partial}{\partial y}\left(\mu\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)\right) + \frac{\partial}{\partial z}\left(\mu\left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right)\right) \\ \\ & = 2 \mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + \mu \frac{\partial^2 v}{\partial y \, \partial x} + \mu \frac{\partial^2 u}{\partial z^2} + \mu \frac{\partial^2 w}{\partial z \, \partial x} \\ \\ & = \mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + \mu \frac{\partial^2 u}{\partial z^2} + \mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 v}{\partial y \, \partial x} + \mu \frac{\partial^2 w}{\partial z \, \partial x} \\ \\ & = \mu \nabla^2 u + \mu \frac{\partial}{\partial x} \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) = \mu \nabla^2 u\end{align}

The Navier–Stokes equations are almost universally dealt with for Newtonian fluids. Part of this is because, as of 2007, good models for non-Newtonian flow simply do not exist. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. As with Newtonian flow, formulations are inspired by examining specific cases, but unlike Newtonian flow there are no models that will work beyond such special cases. Development and implementation of good non-Newtonian models is an area of ongoing research.

### Bingham fluid

Main article: Bingham plastic

In Bingham fluids, the situation is slightly different:

$\frac {\partial u} {\partial y} = \left\{\begin{matrix} 0 &, \quad \tau < \tau_0 \\ (\tau - \tau_0)/ {\mu} &, \quad \tau \ge \tau_0 \end{matrix}\right.$

These are fluids capable of bearing some shear before they start flowing. A Bingham plastic is a Viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous Fluid at high stress Some common examples are toothpaste and clay. Toothpaste is a Paste or Gel Dentifrice used to clean and maintain the aesthetics and health of Teeth. Clay is a naturally occurring material composed primarily of fine-grained Minerals which show plasticity through a variable range of Water content, and

### Power-law fluid

Main article: Power-law fluid

A power law fluid is an idealised fluid for which the shear stress, τ, is given by

$\tau = K \left(\frac{\partial u}{\partial y}\right)^n$

This form is useful for approximating all sorts of general fluids, including shear thinning (such as latex paint) and shear thickening (such as corn starch water mixture). A Power-law fluid is a type of Generalized Newtonian fluid for which the Shear stress, &tau, is given by \tau = K \left( \frac {\partial FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material

## Stream function formulation

In the analysis of a flow, it is often desirable to reduce the number of equations or the number of variables being dealt with, or both. The incompressible Navier-Stokes equation with mass continuity (three equations in three unknowns) can, in fact, be reduced to a single equation with a single dependent variable in 2D, or one vector equation in 3D. This is enabled by two vector calculus identities:

$\nabla \times (\nabla \phi) = 0$
$\nabla \cdot (\nabla \times \mathbf{A}) = 0$

for any differentiable scalar φ and vector $\mathbf{A}$. The following identities are important in Vector calculus: Single operators (summary This section explicitly lists what some symbols mean for clarity The first identity implies that any term in the Navier-Stokes equation represented by a gradient will disappear when the curl of the equation is taken. cURL is a Command line tool for transferring files with URL syntax. Commonly, pressure and gravity are what eliminate, resulting in (this is true in 2D as well as 3D):

$\nabla \times \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = \nu \nabla \times (\nabla^2 \mathbf{v})$

where it's assumed that all body forces are describable as gradients (true for gravity), and density has been divided so that viscosity becomes kinematic viscosity. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress.

The second vector calculus identity above states that the divergence of the curl of a vector field is zero. Since the (incompressible) mass continuity equation specifies the divergence of velocity being zero, we can replace the velocity with the curl of some vector $\vec \psi$ so that mass continuity is always satisfied:

$\nabla \cdot \mathbf v = 0 \quad \Rightarrow \quad \nabla \cdot (\nabla \times \vec \psi) = 0 \quad \Rightarrow \quad 0 = 0$

So, as long as velocity is represented through $\mathbf v = \nabla \times \vec \psi$, mass continuity is unconditionally satisfied. With this new dependent vector variable, the Navier-Stokes equation (with curl taken as above) becomes a single fourth order vector equation, no longer containing the unkown pressure variable and no longer dependent on a separate mass continuity equation:

$\nabla \times \left(\frac{\partial}{\partial t}(\nabla \times \vec \psi) + (\nabla \times \vec \psi) \cdot \nabla (\nabla \times \vec \psi)\right) = \nu \nabla \times (\nabla^2 (\nabla \times \vec \psi))$

Apart from containing fourth order derivatives, this equation is fairly complicated, and is thus uncommon.

### 2D flow in orthogonal coordinates

The true utility of this formulation is seen when the flow is two dimensional in nature and the equation is written in a general orthogonal coordinate system, in other words a system where the basis vectors are orthogonal. In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2. Note that this by no means limits application to Cartesian coordinates, in fact most of the common coordinates systems are orthogonal, including familiar ones like cylindrical and obscure ones like toroidal. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually Toroidal coordinates are a three-dimensional orthogonal Coordinate system that results from rotating the two-dimensional bipolar coordinate system

The 3D velocity is expressed as (note that the discussion has been coordinate free up till now):

$\mathbf v = v_1 \mathbf e_1 + v_2 \mathbf e_2 + v_3 \mathbf e_3$

where $\mathbf e_i$ are basis vectors, not necessarily constant and not necessarily normalized, and vi are velocity components; let also the coordinates of space be

$\mathbf x = x_1 \mathbf e_1 + x_2 \mathbf e_2 + x_3 \mathbf e_3$.

Now suppose that the flow is 2D. This doesn't mean the flow is in a plane, rather it means that the component of velocity in one direction is zero and the remaining components are independent of the same direction. In that case (take component 3 to be zero):

$\mathbf v = v_1 \mathbf e_1 + v_2 \mathbf e_2$
$\frac{\partial v_1}{\partial x_3} = \frac{\partial v_2}{\partial x_3} = 0$

The vector function $\vec \psi$ is still defined via:

$\mathbf v = \nabla \times \vec \psi$

but this must simplify in some way also since the flow is assumed 2D. If orthogonal coordinates are assumed, the curl takes on a fairly simple form, and the equation above expanded becomes:

$v_1 \mathbf e_1 + v_2 \mathbf e_2 = \frac{\mathbf{e}_{1}}{h_{2} h_{3}} \left[\frac{\partial}{\partial x_{2}} \left( h_{3} \psi_{3} \right) - \frac{\partial}{\partial x_{3}} \left( h_{2} \psi_{2} \right)\right] + \frac{\mathbf{e}_{2}}{h_{3} h_{1}} \left[\frac{\partial}{\partial x_{3}} \left( h_{1} \psi_{1} \right) - \frac{\partial}{\partial x_{1}} \left( h_{3} \psi_{3} \right)\right] + \frac{\mathbf{e}_{3}}{h_{1} h_{2}} \left[\frac{\partial}{\partial x_{1}} \left( h_{2} \psi_{2} \right) - \frac{\partial}{\partial x_{2}} \left( h_{1} \psi_{1} \right)\right]$

Examining this equation shows that we can set ψ1 = ψ2 = 0 and retain equality with no loss of generality, so that:

$v_1 \mathbf e_1 + v_2 \mathbf e_2 = \frac{\mathbf{e}_{1}}{h_{2} h_{3}} \frac{\partial}{\partial x_{2}} \left( h_{3} \psi_{3} \right)- \frac{\mathbf{e}_{2}}{h_{3} h_{1}} \frac{\partial}{\partial x_{1}} \left( h_{3} \psi_{3} \right)$

the significance here is that only one component of $\vec \psi$ remains, so that 2D flow becomes a problem with only one dependent variable. cURL is a Command line tool for transferring files with URL syntax. The cross differentiated Navier–Stokes equation becomes two 0 = 0 equations and one meaningful equation.

The remaining component ψ3 = ψ is called the stream function. The stream function is defined for two-dimensional flows of various kinds The equation for ψ can simplify since a variety of quantities will now equal zero, for example:

$\nabla \cdot \vec \psi = \frac{1}{h_{1} h_{2} h_{3}} \frac{\partial}{\partial x_3} \left(\psi h_1 h_2\right) = 0$

if the scale factors h1 and h2 also are independent of x3. Also, from the definition of the vector Laplacian

$\nabla \times (\nabla \times \vec \psi) = \nabla(\nabla \cdot \vec \psi) - \nabla^2 \vec \psi = -\nabla^2 \vec \psi$

Manipulating the cross differentiated Navier–Stokes equation using the above two equations and a variety of identities[4] will eventually yield the 1D scalar equation for the stream function:

$\frac{\partial}{\partial t}(\nabla^2 \psi) + (\nabla \times \vec \psi) \cdot \nabla(\nabla^2 \psi) = \nu \nabla^4 \psi$

where $\nabla^4$ is the biharmonic operator. In Mathematics and Physics, the vector Laplace operator, denoted by \scriptstyle \nabla^2 named after Pierre-Simon Laplace, is a differential In Mathematics, the biharmonic equation is a fourth-order Partial differential equation which arises in areas of Continuum mechanics, including Linear This is very useful because it is a single self contained scalar equation that describes both momentum and mass conservation in 2D. The only other equations that this partial differential equation needs are initial and boundary conditions. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i

The assumptions for the stream function equation are listed below:

• The flow is incompressible and Newtonian.
• Coordinates are orthogonal. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i
• Flow is 2D: $v_3 = \frac{\partial v_1}{\partial x_3} = \frac{\partial v_2}{\partial x_3} = 0$
• The first two scale factors of the coordinate system are independent of the last coordinate: $\frac{\partial h_1}{\partial x_3} = \frac{\partial h_2}{\partial x_3} = 0$

The stream function has some useful properties:

• Since $-\nabla^2 \vec \psi = \nabla \times (\nabla \times \vec \psi) = \nabla \times \mathbf v$, the vorticity of the flow is just the negative of the Laplacian of the stream function. The stream function is defined for two-dimensional flows of various kinds Vorticity is a mathematical concept used in Fluid dynamics. It can be related to the amount of " circulation " or "rotation" (or more strictly the
• The level curves of the stream function are streamlines. In Mathematics, a level set of a real -valued function f of n variables is a set of the form { ( x 1 Fluid flow is described in general by a Vector field in three (for steady flows or four (for non-steady flows including time dimensions

## The stress tensor

The derivation of the equations involves the consideration of forces acting on fluid elements, so that a quantity called the stress tensor appears naturally in the Cauchy momentum equation. Stress is a measure of the average amount of Force exerted per unit Area. The Cauchy momentum equation is a vector Partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any Continuum Since the divergence of this tensor is taken, it is customary to write out the equation fully simplified, so that the original appearance of the stress tensor is lost.

However, the stress tensor still has some important uses, especially in formulating boundary conditions at fluid interfaces. In Fluid mechanics and Mathematics, a capillary surface is a Surface that represents the interface between two different Fluids As a consequence Recalling that $\sigma_{ij} = -p I + \mathbb{T}$, for a Newtonian fluid the stress tensor is:

$\sigma_{ij} = -\begin{pmatrix}p&0&0\\0&p&0\\0&0&p\end{pmatrix} + \mu\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) + \delta_{ij} \lambda \nabla \cdot \mathbf{v}$

If the fluid is assumed to be incompressible, the tensor simplifies significantly:

\begin{align}\sigma_{ij} &= -\begin{pmatrix}p&0&0\\0&p&0\\0&0&p\end{pmatrix} + \mu \begin{pmatrix}2 \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} & \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \\\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} & 2 \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \\\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} & \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} & 2\frac{\partial w}{\partial z}\end{pmatrix} \\&= -p I + \mu (\nabla \mathbf{v} + (\nabla \mathbf{v})^T) = -p I + 2 \mu e\\\end{align}

e is the strain rate tensor, by definition:

$e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)$

## References

• Batchelor, G.K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0521663962
• White, Frank M. Strain rate, with regards to Materials science, is the change in strain over the change in time and is denoted as έ. George Keith Batchelor ( March 8 1920 - March 30 2000) was an Australian Applied mathematician and Fluid dynamicist (2006). Viscous Fluid Flow. New York, NY: McGraw Hill.
• Surface Tension Module, by John W. M. Bush, at MIT OCW. MIT OpenCourseWare (MIT OCW is an initiative of the Massachusetts Institute of Technology (MIT to put all of the educational materials from its undergraduate - and

## Notes

1. ^ a b Lebedev, Leonid P. (2003). Tensor Analysis. World Scientific. ISBN 9812383603.
2. ^ See Batchelor (1967), §3. 3, p. 141.
3. ^ See Batchelor (1967), §3. 3, p. 144.
4. ^ Eric W. Weisstein. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld Vector Derivative. MathWorld. MathWorld is an online Mathematics reference work created and largely written by Eric W Retrieved on 2008-06-7.

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