Navier–Stokes equations 48395 224384361 2008-07-08T16:49:07Z 18.125.6.145 /* Compressible flow of Newtonian fluids */ coefficient of \mu should be 1/3 not 4/3 {{Continuum mechanics}} The '''Navier–Stokes equations''', named after [[Claude-Louis Navier]] and [[George Gabriel Stokes]], describe the motion of viscous [[fluid]] substances such as [[liquid]]s and [[gas]]es. These equations arise from applying [[Newton's second law]] to [[fluid]] motion, together with the assumption that the fluid [[stress (physics)|stress]] is the sum of a [[diffusion|diffusing]] [[viscosity|viscous]] term (proportional to the gradient of velocity), plus a [[pressure]] term. They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They may be used to [[model (abstract)|model]] [[weather]], [[ocean current]]s, water flow in a pipe, flow around an [[airfoil]] (wing), and motion of [[star]]s inside a [[galaxy]]. As such, these equations in both full and simplified forms, are used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with [[Maxwell's equations]] they can be used to model and study [[magnetohydrodynamics]]. The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist ([[Existence theorem|existence]]), or that if they do exist they do not contain any infinities, singularities or discontinuities (smoothness). These are called the [[Navier–Stokes existence and smoothness]] problems. The [[Clay Mathematics Institute]] has called this one of the [[Millennium Prize Problems|seven most important open problems in mathematics]], and offered a $1,000,000 prize for a solution or a counter-example. The Navier–Stokes equations are [[differential equation]]s which, unlike [[algebraic equations]], do not explicitly establish a relation among the variables of interest (e.g. [[velocity]] and [[pressure]]). Rather, they establish relations among the [[derivative|rates of change]]. For example, the Navier–Stokes equations for simple case of an [[ideal fluid]] (inviscid) can state that acceleration (the [[derivative|rate of change]] of [[velocity]]) is proportional to the [[gradient]] (a type of multivariate derivative) of pressure. Contrary to what is normally seen in [[solid mechanics]], the Navier–Stokes equations dictate not [[position]] but rather [[velocity]]. A solution of the Navier–Stokes equations is called a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest (such as flow rate, drag force, or the path a "particle" of fluid will take) may be found. ==Properties== ===Nonlinearity=== The Navier–Stokes equations are [[nonlinear]] [[partial differential equations]] in almost every real situation (exceptions include one dimensional flow and [[creeping flow]]). The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the [[turbulence]] that the equations model. The nonlinearity is due to [[convective]] acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity, an example of convective but [[laminar]] (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging [[nozzle]]. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood. ===Turbulence=== [[Turbulence]] is the time dependent [[Chaos theory|chaotic]] behavior seen in many fluid flows. It is generally believed that it is due to the [[inertia]] of the fluid as a whole: the culmination of time dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the [[Reynolds number]] quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly. Even though turbulence is an everyday experience, it is extremely difficult to find solutions, quantify, or in general characterize. A $1,000,000 [[Navier–Stokes existence and smoothness|prize]] was offered in May 2000 by the [[Clay Mathematics Institute]] to whoever makes [[Navier–Stokes existence and smoothness|preliminary progress]] toward a mathematical theory which will help in the understanding of this phenomenon. The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation (see [[Direct numerical simulation]]). Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the [[Reynolds-averaged Navier-Stokes equations]] (RANS), supplemented with turbulence models (such as the k-ε model), are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Another technique for solving numerically the Navier–Stokes equation is the Large-eddy simulation (LES). This approach is computationally more expensive than the RANS method (in time and computer memory), but produces better results since the larger turbulent scales are explicitly resolved. ===Applicability=== Together with supplemental equations (for example, conservation of mass) and well formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations. The Navier–Stokes equations assume that the fluid being studied is a [[continuum]]. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. Depending on the [[Knudsen number]] of the problem, [[statistical mechanics]] or possibly even [[molecular dynamics]] may be a more appropriate approach. Another limitation is very simply the complicated nature of the equations. Time tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations which are an area of current research. For this reason, the Navier–Stokes equations are usually written for [[Newtonian fluid]]s. ==Derivation and description== {{main|Derivation of the Navier–Stokes equations}} The derivation of the Navier–Stokes equations begins with the conservation of mass, momentum, and energy being written for an arbitrary control volume. In an [[inertial frame of reference]], the most general form of the Navier–Stokes equations ends up being: :<math>\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f},</math> where <math>\mathbf{v}</math> is the flow velocity, <math>\rho</math> is the fluid density, ''p'' is the pressure, <math>\mathbb{T}</math> is the ([[Stress (physics)#Stress deviator tensor|deviatoric]]) stress [[tensor field|tensor]], and <math>\mathbf{f} </math> represents [[body force]]s (per unit volume) acting on the fluid and <math>\nabla</math> is the [[del]] operator. This is a statement of the conservation of momentum in a fluid and it is an application of [[Newton's second law]] to a [[continuum]]. This equation is often written using the [[substantive derivative]], making it more apparent that this is a statement of Newton's law: :<math>\rho \frac{D \mathbf{v}}{D t} = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f}.</math> The left side of the equation describes acceleration, and may be composed of time dependent or convective effects (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of [[body force]]s (such as gravity) and divergence of [[Stress (physics)|stress]] (pressure and stress). ===Convective acceleration=== [[Image:ConvectiveAcceleration.png|thumb|An example of convection. Though the flow is steady (time independent), the fluid decelerates as it moves down the diverging duct (when the flow is subsonic), hence there is acceleration.]] A very significant feature of the Navier–Stokes equations is the presence of [[Convection|convective acceleration]]: the effect of time independent acceleration of a fluid with respect to space, represented by the quantity: :<math>\mathbf{v} \cdot \nabla \mathbf{v}</math> where <math>\nabla \mathbf{v}</math> is the [[tensor derivative]] of the velocity vector, equal in Cartesian coordinates to the component by component gradient. This term may, by a [[vector calculus identities#Vector_dot_product|vector calculus identity]], be expressed without a tensor derivative:<ref>See Batchelor (1967), §3.5, p. 160.</ref><ref>{{cite web | url = http://mathworld.wolfram.com/ConvectiveDerivative.html | title = Convective Derivative | author = [[Eric W. Weisstein]] | publisher = [[MathWorld]] | accessdate = 2008-05-20 }}</ref> :<math>\mathbf{v} \cdot \nabla \mathbf{v} = \nabla \left( \frac{|\mathbf{v}|^2}{2} \right) + \left( \nabla \times \mathbf{v} \right) \times \mathbf{v}.</math> The form has use in irrotational flow, where the [[Curl (mathematics)|curl]] of the velocity (called vorticity) <math>\nabla \times \mathbf{v}</math> is equal to zero. Convection is also often written as :<math>(\mathbf{v} \cdot \nabla) \mathbf{v}</math> where the [[advection|advection operator]] <math>\mathbf{v} \cdot \nabla</math> is used; however this will yield the wrong expression outside of [[Cartesian coordinates]] since the tensor derivative is lost. This is because the advection operator is applicable only to scalar physical quantities, the vectorial nature of momentum requires a tensor derivative. Regardless of what kind of fluid is being dealt with, convective acceleration is a [[nonlinear]] effect. Convection is present in most flows, exceptions include [[creeping flow]] and incompressible flow in one dimension. ===Stresses=== The effect of [[Stress (physics)|stress]] in the fluid is represented by the <math>\scriptstyle \nabla p</math> and <math>\scriptstyle \nabla \cdot\mathbb{T}</math> terms, these are gradients of surface forces, analogous to stresses in a solid. <math>\scriptstyle \nabla p</math> is called the pressure gradient and arises from [[normal stress]]es that turn up in almost all situations, dynamic or not. <math>\scriptstyle \nabla \cdot\mathbb{T}</math> conventionally describes viscous forces; for incompressible flow, this is only a shear effect. <math>\scriptstyle \mathbb{T}</math> is the [[deviatoric stress tensor]], and the stress tensor is equal to: :<math>\sigma_{ij} = -pI + \mathbb{T}</math> where <math>I</math> is the 3x3 [[identity matrix]]. Interestingly, only the ''gradient'' of pressure matters, not the pressure itself. The effect of the pressure gradient is that fluid flows from high pressure to low pressure. The stress derivative term <math>\scriptstyle \nabla \cdot\mathbb{T}</math> contains too many unknowns to be immediately usable, so the general form of the Navier–Stokes equation is not immediately usable to solve problems. For this reason, assumptions on the specific viscous behavior of a fluid are made (based on natural observations) and applied in order to specify this quantity in terms of familiar variables, such as velocity. For example, this term becomes the useful quantity <math>\scriptstyle \mu \nabla^2 \mathbf{v}</math> when the fluid is assumed incompressible and [[Newtonian fluid|Newtonian]]. ===Other forces=== <math>\scriptstyle \mathbf{f}</math> represents "other" ([[body force]]) forces. Typically this is only [[gravity]], but may include other fields (such as electromagnetic). In a non-inertial coordinate system, other "forces" such as that associated with [[Fictitious force|rotating coordinates]] may be inserted. Often, these forces may be represented as the gradient of some scalar quantity. Gravity in the <math>z</math> direction, for example, is the gradient of <math>-\rho g z</math>. Since pressure shows up only as a gradient, this implies that solving a problem without any such body force can be mended to include the body force by modifying pressure. ===Other equations=== The Navier–Stokes equations are strictly a statement of the conservation of momentum. In order to fully describe fluid flow, more information is needed (how much depends on the assumptions made), this may include boundary data ([[no-slip condition|no-slip]], [[capillary surface]], etc), the conservation of mass, the conservation of energy, and/or an [[equation of state]]. Regardless of the flow assumptions, a statement of the [[conservation of mass]] is generally necessary. This is achieved through the mass [[continuity equation]], given in its most general form as: :<math>\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0</math> or, using the [[substantive derivative]]: :<math>\frac{D\rho}{Dt} + \rho (\nabla \cdot \mathbf{v}) = 0.</math> ==Incompressible flow of Newtonian fluids== The vast majority of work on the Navier–Stokes equations is done under an [[incompressible flow]] assumption for [[Newtonian fluid]]s. The incompressible flow assumption typically holds well even when dealing with a "compressible" fluid, such as air at room temperature (even when flowing up to about [[Mach number|Mach]] 0.3). Taking the incompressible flow assumption into account and assuming constant viscosity, the Navier–Stokes equations will read<ref name="Ach">See Acheson (1990).</ref> (in vector form): :<math>\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}</math> '''f''' represents "other" [[body force]]s (forces per unit volume), such as [[gravity]] or [[centrifugal force]]. It's well worth observing the meaning of each term: :<math> \overbrace{\rho \Big( \underbrace{\frac{\partial \mathbf{v}}{\partial t}}_{ \begin{smallmatrix} \text{Unsteady}\\ \text{acceleration} \end{smallmatrix}} + \underbrace{\mathbf{v} \cdot \nabla \mathbf{v}}_{ \begin{smallmatrix} \text{Convective} \\ \text{acceleration} \end{smallmatrix}}\Big)}^{\text{Inertia}} = \underbrace{-\nabla p}_{ \begin{smallmatrix} \text{Pressure} \\ \text{gradient} \end{smallmatrix}} + \underbrace{\mu \nabla^2 \mathbf{v}}_{\text{Viscosity}} + \underbrace{\mathbf{f}}_{ \begin{smallmatrix} \text{Other} \\ \text{forces} \end{smallmatrix}} </math> Note that only the convective terms are nonlinear for incompressible Newtonian flow. The convective acceleration is an acceleration caused by a (possibly steady) change in velocity over ''position'', for example the speeding up of fluid entering a converging [[nozzle]]. Though individual fluid particles are being accelerated and thus are under unsteady motion, the flow field (a velocity distribution) will not necessarily be time dependent. Another important observation is that the viscosity is represented by the [[vector Laplacian]] of the velocity field. This implies that Newtonian viscosity is '''diffusion of momentum''', this works in much the same way as the [[diffusion]] of heat seen in the [[heat equation]] (which also involves the Laplacian). If temperature effects are also neglected, the only "other" equation (apart from initial/boundary conditions) needed is the mass continuity equation. Under the incompressible assumption, density is a constant and it follows that the equation will simplify to: :<math>\nabla \cdot \mathbf{v} = 0</math> This is more specifically a statement of the conservation of volume (see [[divergence]]). These equations are commonly used in 3 coordinates systems: [[Cartesian]], [[Cylindrical coordinate system|cylindrical]], and [[Spherical coordinate system|spherical]]. The Cartesian equations follow directly from the vector equation above, obtaining equations in other coordinate systems will require a [[Del in cylindrical and spherical coordinates|change of variables]]. ===Cartesian coordinates=== Writing the vector equation explicitly, :<math> \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} + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) + \rho g_x</math> :<math> \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} + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) + \rho g_y</math> :<math> \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} + \mu \left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) + \rho g_z</math> Note that gravity has been accounted for as a body force, and the values of <math>g_x, g_y, g_z</math> will depend on the orientation of gravity with respect to the chosen set of coordinates. The continuity equation reads: :<math>{\partial u \over \partial x} + {\partial v \over \partial y} + {\partial w \over \partial z} = 0</math> Note that the velocity components (the dependent variables to be solved for) are <math>u</math>, <math>v</math>, <math>w</math>. This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other representations, this is a [[nonlinear]] system of [[partial differential equations]] for which solutions are difficult to obtain. ===Cylindrical coordinates=== A change of variables on the Cartesian equations will yield<ref name="Ach"/> the following momentum equations for ''r'', θ, and ''z'': :<math> \rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_r}{\partial \theta} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\theta}^2}{r}\right) = -\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_r}{\partial \theta^2} + \frac{\partial^2 u_r}{\partial z^2} - \frac{u_r}{r^2} - \frac{2}{r^2}\frac{\partial u_{\theta}}{\partial \theta}\right] + \rho g_r</math> :<math> \rho \left(\frac{\partial u_{\theta}}{\partial t} + u_r \frac{\partial u_{\theta}}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_{\theta}}{\partial \theta} + u_z \frac{\partial u_{\theta}}{\partial z} + \frac{u_r u_{\theta}}{r}\right) = -\frac{1}{r}\frac{\partial p}{\partial \theta} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\theta}}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_{\theta}}{\partial \theta^2} + \frac{\partial^2 u_{\theta}}{\partial z^2} + \frac{2}{r^2}\frac{\partial u_r}{\partial \theta} - \frac{u_{\theta}}{r^2}\right] + \rho g_{\theta}</math> :<math> \rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\theta}}{r} \frac{\partial u_z}{\partial \theta} + u_z \frac{\partial u_z}{\partial z}\right) = -\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u_z}{\partial \theta^2} + \frac{\partial^2 u_z}{\partial z^2}\right] + \rho g_z</math> The gravity components will generally not be constants, however for most applications either the coordinates are chosen so that the gravity components are constant or else it is assumed that gravity is counteracted by a pressure field (for example, flow in horizontal pipe is treated normally without gravity and without a vertical pressure gradient). The continuity equation is: :<math> \frac{1}{r}\frac{\partial}{\partial r}\left(r u_r\right) + \frac{1}{r}\frac{\partial u_\theta}{\partial \theta} + \frac{\partial u_z}{\partial z} = 0.</math> This cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above). Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. A very common case is axisymmetric flow, where there is no tangential velocity (<math>u_{\theta} = 0</math>) and the remaining quantities are independent of <math>\theta</math>: :<math> \rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + u_z \frac{\partial u_r}{\partial z}\right) = -\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) + \frac{\partial^2 u_r}{\partial z^2} - \frac{u_r}{r^2}\right] + \rho g_r</math> :<math> \rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + u_z \frac{\partial u_z}{\partial z}\right) = -\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) + \frac{\partial^2 u_z}{\partial z^2}\right] + \rho g_z</math> :<math> \frac{1}{r}\frac{\partial}{\partial r}\left(r u_r\right) + \frac{\partial u_z}{\partial z} = 0.</math> ===Spherical coordinates=== In [[spherical coordinates]], the <math>r</math>, <math>\theta</math>, and <math>\phi</math> momentum equations are<ref name="Ach"/> (note the convention used: <math>\phi</math> is [[colatitude]]<ref>{{cite web | url = http://mathworld.wolfram.com/SphericalCoordinates.html | title = Spherical Coordinates | author = [[Eric W. Weisstein]] | publisher = [[MathWorld]] | date = [[2005-10-26]] | accessdate = 2008-01-22 }}</ref>): :<math> \rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\theta}}{r \sin(\phi)} \frac{\partial u_r}{\partial \theta} + \frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} - \frac{u_{\theta}^2 + u_{\phi}^2}{r}\right) = -\frac{\partial p}{\partial r} + \rho g_r</math> ::<math> \mu \left[ \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_r}{\partial r}\right) + \frac{1}{r^2 \sin(\phi)^2} \frac{\partial^2 u_r}{\partial \theta^2} + \frac{1}{r^2 \sin(\phi)} \frac{\partial}{\partial \phi}\left(\sin(\phi) \frac{\partial u_r}{\partial \phi}\right) - 2 \frac{u_r + \frac{\partial u_{\phi}}{\partial \phi} + u_{\phi} \cot(\phi)}{r^2} + \frac{2}{r^2 \sin(\phi)} \frac{\partial u_{\theta}}{\partial \theta} \right] </math> :<math> \rho \left(\frac{\partial u_{\theta}}{\partial t} + u_r \frac{\partial u_{\theta}}{\partial r} + \frac{u_{\theta}}{r \sin(\phi)} \frac{\partial u_{\theta}}{\partial \theta} + \frac{u_{\phi}}{r} \frac{\partial u_{\theta}}{\partial \phi} + \frac{u_r u_{\theta} + u_{\theta} u_{\phi} \cot(\phi)}{r}\right) = -\frac{1}{r \sin(\phi)} \frac{\partial p}{\partial \theta} + \rho g_{\theta}</math> ::<math> \mu \left[ \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_{\theta}}{\partial r}\right) + \frac{1}{r^2 \sin(\phi)^2} \frac{\partial^2 u_{\theta}}{\partial \theta^2} + \frac{1}{r^2 \sin(\phi)} \frac{\partial}{\partial \phi}\left(\sin(\phi) \frac{\partial u_{\theta}}{\partial \phi}\right) + \frac{2 \frac{\partial u_r}{\partial \theta} + 2 \cos(\phi) \frac{\partial u_{\phi}}{\partial \theta} - u_{\theta}}{r^2 \sin(\phi)^2} \right] </math> :<math> \rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} + \frac{u_{\theta}}{r \sin(\phi)} \frac{\partial u_{\phi}}{\partial \theta} + \frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + \frac{u_r u_{\phi} - u_{\theta}^2 \cot(\phi)}{r}\right) = -\frac{1}{r} \frac{\partial p}{\partial \phi} + \rho g_{\phi}</math> ::<math> \mu \left[ \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_{\phi}}{\partial r}\right) + \frac{1}{r^2 \sin(\phi)^2} \frac{\partial^2 u_{\phi}}{\partial \theta^2} + \frac{1}{r^2 \sin(\phi)} \frac{\partial}{\partial \phi}\left(\sin(\phi) \frac{\partial u_{\phi}}{\partial \phi}\right) + \frac{2}{r^2} \frac{\partial u_r}{\partial \phi} - \frac{u_{\phi} + 2 \cos(\phi) \frac{\partial u_{\theta}}{\partial \theta}}{r^2 \sin(\phi)^2} \right] </math> Mass continuity will read: :<math> \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 u_r\right) + \frac{1}{r \sin(\phi)}\frac{\partial u_\theta}{\partial \theta} + \frac{1}{r \sin(\phi)}\frac{\partial}{\partial \phi}\left(\sin(\phi) u_\phi\right) = 0</math> These equations could be (slightly) simplified by, for example, factoring <math>1/r^2</math> from the viscous terms. This isn't done to preserve the structure of the Laplacian and other quantities. ===Stream function formulation=== Taking the [[Curl (mathematics)|curl]] of the Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (<math>w = 0</math> and no dependence of anything on <math>z</math>), where the equations reduce to: :<math> \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}\right) = -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + \rho g_x</math> :<math> \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}\right) = -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}\right) + \rho g_y</math> Differentiating the first with respect to <math>y</math>, the second with respect to <math>x</math> and subtracting the resulting equations will eliminate pressure and any [[potential]] force. Defining the [[stream function]] <math>\psi</math> through :<math>u = \frac{\partial \psi}{\partial y} \quad ; \quad v = -\frac{\partial \psi}{\partial x}</math> results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation degrade into one equation: :<math>\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial \psi}{\partial y} \frac{\partial}{\partial x}\left(\nabla^2 \psi\right) - \frac{\partial \psi}{\partial x} \frac{\partial}{\partial y}\left(\nabla^2 \psi\right) = \nu \nabla^4 \psi</math> where <math>\nabla^4</math> is the (2D) [[biharmonic operator]] and <math>\nu</math> is the [[kinematic viscosity]]. This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for [[creeping flow]] results when the left side is assumed zero. In [[axisymmetric]] flow another stream function formulation, called the [[Stokes stream function]], can be used to describe the velocity components of an incompressible flow with one [[scalar]] function. ==Compressible flow of Newtonian fluids== There are some exceptional phenomena that are closely linked with fluid [[compressibility]]. One of the obvious examples is [[sound]]. Description of such phenomena requires more general presentation of the Navier–Stokes equation that takes into account fluid compressibility. If viscosity is assumed a constant, one additional term appears, as shown here: :<math>\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f} +(\mu /3 + \mu^v) \nabla (\nabla \cdot \mathbf{v}) </math> where <math>\mu^v</math> is second viscosity coefficient. It is related to [[volume viscosity]] or [[bulk viscosity]]. This additional term disappears for ''incompressible fluid'', when the [[divergence]] of the flow equals 0. ==Application to specific problems== The Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as multiphase flow driven by [[surface tension]]. Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by [[Scale analysis (mathematics)|scale analysis]] to further simplify the problem. For example, after assuming steady, parallel, one dimensional, nonconvective pressure driven flow between parallel plates, the resulting scaled (dimensionless) [[boundary value problem]] is: [[Image:NSConvection.png|thumb|400px|Visualization of a) parallel flow and b) radial flow.]] :<math>\frac{d^2 u}{d y^2} = -1 \quad ; \quad u(0) = u(1) = 0</math> The boundary condition is the [[no slip condition]]. This problem is easily solved for the flow field: :<math>u(y) = \frac{y - y^2}{2}</math> From this point onward more quantities of interest can be easily obtained, such as viscous drag force or net flow rate. Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the ''radial'' flow between parallel plates; this involves convection and thus nonlinearity. The velocity field may be represented by a function <math>f(z)</math> that must satisfy: :<math>\frac{d^2 f}{d z^2} + R f^2 = -1 \quad ; \quad f(0) = f(1) = 0</math> This [[ordinary differential equation]] is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The [[nonlinear]] term makes this a very difficult problem to solve analytically (a lengthy [[Implicit function|implicit]] solution may be found which involves [[elliptic integrals]] and [[Cubic formula|roots of cubic polynomials]]). Issues with the actual existence of solutions arise for R > 22.609 (approximately), the parameter R being the [[Reynolds number]] with appropriately chosen scales. This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows. == See also == *[[Computational fluid dynamics]] *[[Reynolds transport theorem]] *[[Reynolds number]] *[[Mach number]] *[[Multiphase flow]] *[[Adhémar Jean Claude Barré de Saint-Venant]] *[[Navier–Stokes existence and smoothness|Millennium prize problem details]] *[[Churchill-Bernstein Equation]] *[[Reynolds-averaged Navier-Stokes equations]] *[[Coanda Effect]] *[[Fokker-Planck equation]] *[[Boltzmann equation]] *[[Double_layer_(plasma)#The_Vlasov-Poisson_equation|Vlasov equation]] == References == *{{cite book | last = Acheson | first = D. J. | title = Elementary Fluid Dynamics | publisher = [[Oxford University Press]] | series = Oxford Applied Mathematics and Computing Science Series | year = 1990 | isbn = 0198596790 }} *{{citation | first=G.K. | last=Batchelor | authorlink=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0521663962 }} *{{citation | first=Inge L. | last=Rhyming | title=Dynamique des fluides | year=1991 | publisher=Presses Polytechniques et Universitaires Romandes, Lausanne }} *{{citation | first1=A.D. | last1=Polyanin | first2=A.M. | last2=Kutepov | first3=A.V. | last3=Vyazmin | first4=D.A. | last4=Kazenin | title=Hydrodynamics, Mass and Heat Transfer in Chemical Engineering | publisher=Taylor & Francis, London | year=2002 | isbn=0-415-27237-8 }} ==Notes== {{ref-list}} == External links == * [http://www.allstar.fiu.edu/aero/Flow2.htm Simplified derivation of the Navier–Stokes equations] * [http://www.qeden.com/wiki/Navier-Stokes_Existence_and_Smoothness QEDen] Millennium Prize Problems Wiki * [http://www.cfd-online.com/Resources/soft.html CFD online software list] A compilation of codes, including Navier–Stokes solvers. 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