Derivation of the Navier–Stokes equations

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The derivation of the Navier–Stokes equations as well as their application and formulation for different families of fluids, is an important exercise in fluid dynamics with applications in mechanical engineering, physics, chemistry, heat transfer, and electrical engineering. A proof explaining the properties and bounds of the equations, such as Navier–Stokes existence and smoothness, is one of the important unsolved problems in mathematics.

Basic assumptions

The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum – a continuous substance rather than discrete particles. Another necessary assumption is that all the fields of interest including pressure, flow velocity, density, and temperature are at least weakly differentiable.

The equations are derived from the basic principles of continuity of mass, conservation of momentum, and conservation of energy. Sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be applied. This finite volume is denoted by Ω and its bounding surface ∂Ω. The control volume can remain fixed in space or can move with the fluid.

The material derivative

Main article: material 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. 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 advective or material (or Lagrangian) derivative.

The material derivative is defined as the linear operator:

${\displaystyle \frac{D}{Dt}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }}$

where u is the flow velocity. The first term on the right-hand side of the equation is the ordinary Eulerian derivative (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). 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 through application of the chain rule in which all independent variables are checked for change along the path (which is to say, the total derivative).

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 observing the movement of a weather balloon. 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 flow.

Continuity equations

The Navier–Stokes equation is a special continuity equation. A continuity equation may be derived from conservation principles of:

• mass,
• momentum,
• energy.

A continuity equation (or conservation law) is an integral relation stating that the rate of change of some integrated property φ defined over a control volume Ω must be equal to the rate at which it is lost or gained through the boundaries Γ of the volume plus the rate at which it is created or consumed by sources and sinks inside the volume. This is expressed by the following integral continuity equation:

${\displaystyle \frac{d}{dt}{\int }_{\mathrm{\Omega }}\phi d\mathrm{\Omega }=-{\int }_{\mathrm{\Gamma }}\phi \mathbf{u}\cdot \mathbf{n}d\mathrm{\Gamma }-{\int }_{\mathrm{\Omega }}sd\mathrm{\Omega }}$

where u is the flow velocity of the fluid, n is the outward-pointing unit normal vector, and s represents the sources and sinks in the flow, taking the sinks as positive.

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

${\displaystyle \frac{d}{dt}{\int }_{\mathrm{\Omega }}\phi d\mathrm{\Omega }=-{\int }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot (\phi \mathbf{u})d\mathrm{\Omega }-{\int }_{\mathrm{\Omega }}sd\mathrm{\Omega }.}$

Applying the Reynolds transport theorem to the integral on the left and then combining all of the integrals:

${\displaystyle {\int }_{\mathrm{\Omega }}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}d\mathrm{\Omega }=-{\int }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot (\phi \mathbf{u})d\mathrm{\Omega }-{\int }_{\mathrm{\Omega }}sd\mathrm{\Omega }\Rightarrow {\int }_{\mathrm{\Omega }}\left(\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (\phi \mathbf{u})+s\right)d\mathrm{\Omega }=0.}$

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

${\displaystyle \frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (\phi \mathbf{u})+s=0.}$

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. Validity is retained if φ is a vector, in which case the vector-vector product in the second term will be a dyad.

Conservation of mass

Mass may be considered also. When the intensive property φ is considered as the mass, by substitution into the general continuity equation, and taking s = 0 (no sources or sinks of mass):

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

where ρ is the mass density (mass per unit volume), and u is the flow velocity. 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, ⁠Dρ/Dt⁠ = 0 (the density following the path of a fluid element is constant) and the equation reduces to:

${\displaystyle \mathrm{\nabla }\cdot \mathbf{u}=0}$

which is in fact a statement of the conservation of volume.

Conservation of momentum

A general momentum equation is obtained when the conservation relation is applied to momentum. When the intensive property φ is considered as the mass flux (also momentum density), that is, the product of mass density and flow velocity ρu, by substitution into the general continuity equation:

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}(\rho \mathbf{u})+\mathrm{\nabla }\cdot (\rho \mathbf{u}\otimes \mathbf{u})=\mathbf{s}}$

where u ⊗ u is a dyad, 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).

Using the formula for the divergence of a dyad,

${\displaystyle \mathrm{\nabla }\cdot (\mathbf{a}\otimes \mathbf{b})=(\mathrm{\nabla }\cdot \mathbf{a})\mathbf{b}+\mathbf{a}\cdot \mathrm{\nabla }\mathbf{b}}$

we then have

${\displaystyle \mathbf{u}\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\rho \frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\mathrm{\nabla }\cdot (\rho \mathbf{u})+\rho \mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}=\mathbf{s}}$

Note that the gradient of a vector is a special case of the covariant derivative, the operation results in second rank tensors; except in Cartesian coordinates, it is important to understand that this is not simply an element by element gradient. Rearranging :

${\displaystyle \mathbf{u}\left(\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (\rho \mathbf{u})\right)+\rho \left(\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}\right)=\mathbf{s}}$

The leftmost expression enclosed in parentheses is, by mass continuity (shown before), equal to zero. Noting that what remains on the left side of the equation is the material derivative of flow velocity:

${\displaystyle \rho \frac{D\mathbf{u}}{Dt}=\rho \left(\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}\right)=\mathbf{s}}$

This appears to simply be an expression of Newton's second law (F = ma) in terms of body forces instead of point forces. 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:

${\displaystyle \begin{array}{rlrl}\rho \frac{d}{dt}(\mathbf{u}(x,y,z,t))=\mathbf{s}& \Rightarrow & \rho \left(\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }x}\frac{dx}{dt}+\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }y}\frac{dy}{dt}+\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }z}\frac{dz}{dt}\right)& =\mathbf{s}\\ & \Rightarrow & \rho \left(\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }y}+w\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }z}\right)& =\mathbf{s}\\ & \Rightarrow & \rho \left(\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}\right)& =\mathbf{s}\end{array}}$

where u = (u, v, w). The reason why this is "less rigorous" is that we haven't shown that the choice of

${\displaystyle \mathbf{u}=\left(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right)}$

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. For this reason the convective derivative is also known as the particle derivative.

Cauchy momentum equation

See also: Cauchy momentum equation

The generic density of the momentum source s seen previously is made specific first by breaking it up into two new terms, one to describe internal stresses and one for external forces, such as gravity. By examining the forces acting on a small cube in a fluid, it may be shown that

${\displaystyle \rho \frac{D\mathbf{u}}{Dt}=\mathrm{\nabla }\cdot \mathit{\sigma }+\rho f}$

where σ is the Cauchy stress tensor, and f accounts for body forces present. This equation is called the Cauchy momentum equation and describes the non-relativistic momentum conservation of any continuum that conserves mass. σ is a rank two symmetric tensor given by its covariant components. In orthogonal coordinates in three dimensions it is represented as the 3 × 3 matrix:

${\displaystyle {\sigma }_{ij}=\left(\begin{array}{ccc}{\sigma }_{xx}& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{yx}& {\sigma }_{yy}& {\tau }_{yz}\\ {\tau }_{zx}& {\tau }_{zy}& {\sigma }_{zz}\end{array}\right)}$

where the σ are normal stresses and τ shear stresses. This matrix is split up into two terms:

${\displaystyle {\sigma }_{ij}=\left(\begin{array}{ccc}{\sigma }_{xx}& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{yx}& {\sigma }_{yy}& {\tau }_{yz}\\ {\tau }_{zx}& {\tau }_{zy}& {\sigma }_{zz}\end{array}\right)=-\left(\begin{array}{ccc}p& 0& 0\\ 0& p& 0\\ 0& 0& p\end{array}\right)+\left(\begin{array}{ccc}{\sigma }_{xx}+p& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{yx}& {\sigma }_{yy}+p& {\tau }_{yz}\\ {\tau }_{zx}& {\tau }_{zy}& {\sigma }_{zz}+p\end{array}\right)=-p\mathbf{I}+\mathit{\tau }}$

where I is the 3 × 3 identity matrix and τ is the deviatoric stress tensor. Note that the mechanical pressure p is equal to the negative of the mean normal stress:

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

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 τ in the equation above must be zero for a fluid at rest. Note that τ is traceless. The Cauchy equation may now be written in another more explicit form:

${\displaystyle \rho \frac{D\mathbf{u}}{Dt}=-\mathrm{\nabla }p+\mathrm{\nabla }\cdot \mathit{\tau }+\rho f}$

This equation is still incomplete. For completion, one must make hypotheses on the forms of τ and p, that is, one needs a constitutive law for the stress tensor which can be obtained for specific fluid families and on the pressure. Some of these hypotheses lead to the Euler equations (fluid dynamics), other ones lead to the Navier–Stokes equations. 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 general form of the equations of motion is not "ready for use", the stress tensor is still unknown so that more information is needed; this information is normally some knowledge of the viscous behavior of the fluid. For different types of fluid flow this results in specific forms of the Navier–Stokes equations.

Newtonian fluid

Main article: Newtonian fluid

Compressible Newtonian fluid

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

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

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 rate tensor or equivalently the velocity gradient.
• The fluid is isotropic.
• For a fluid at rest, ∇ ⋅ τ must be zero (so that hydrostatic pressure results).

The above list states the classic argument that the shear strain rate tensor (the (symmetric) shear part of the velocity gradient) is a pure shear tensor and does not include any inflow/outflow part (any compression/expansion part). This means that its trace is zero, and this is achieved by subtracting ∇ ⋅ u in a symmetric way from the diagonal elements of the tensor. The compressional contribution to viscous stress is added as a separate diagonal tensor.

Applying these assumptions will lead to :

${\displaystyle \mathit{\tau }=\mu \left(\mathrm{\nabla }\mathbf{u}+{\left(\mathrm{\nabla }\mathbf{u}\right)}^{\mathsf{T}}\right)+\lambda \left(\mathrm{\nabla }\cdot \mathbf{u}\right)\mathbf{I}}$

or in tensor form

${\displaystyle {\tau }_{ij}=\mu \left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)+{\delta }_{ij}\lambda \frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_k}}$

That is, the deviatoric of the deformation rate tensor is identified to the deviatoric of the stress tensor, up to a factor μ.

δ_ij is the Kronecker delta. μ and λ are proportionality constants associated with the assumption that stress depends on strain linearly; μ is called the first coefficient of viscosity or shear viscosity (usually just called "viscosity") and λ is the second coefficient of viscosity or volume viscosity (and it is related to bulk viscosity). 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 ≈ −⁠2/3⁠μ.

A straightforward substitution of τ_ij into the momentum conservation equation will yield the Navier–Stokes equations, describing a compressible Newtonian fluid:

${\displaystyle \rho \left(\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}\right)=-\mathrm{\nabla }p+\mathrm{\nabla }\cdot \left[\mu \left(\mathrm{\nabla }\mathbf{u}+{\left(\mathrm{\nabla }\mathbf{u}\right)}^{\mathsf{T}}\right)\right]+\mathrm{\nabla }\cdot \left[\lambda \left(\mathrm{\nabla }\cdot \mathbf{u}\right)\mathbf{I}\right]+\rho \mathbf{g}}$

The body force has been decomposed into density and external acceleration, that is, f = ρg. The associated mass continuity equation is:

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

In addition to this equation, an equation of state and an equation for the conservation of energy is needed. The equation of state to use depends on context (often the ideal gas law), the conservation of energy will read:

${\displaystyle \rho \frac{Dh}{Dt}=\frac{Dp}{Dt}+\mathrm{\nabla }\cdot (k\mathrm{\nabla }T)+\mathrm{\Phi }}$

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

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

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.

Incompressible Newtonian fluid

For the special (but very common) case of incompressible flow, the momentum equations simplify significantly. Using the following assumptions:

• Viscosity μ will now be a constant
• The second viscosity effect λ = 0
• The simplified mass continuity equation ∇ ⋅ u = 0

This gives incompressible Navier-Stokes equations, describing incompressible Newtonian fluid:

${\displaystyle \rho \left(\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}\right)=-\mathrm{\nabla }p+\mathrm{\nabla }\cdot \left[\mu \left(\mathrm{\nabla }\mathbf{u}+{\left(\mathrm{\nabla }\mathbf{u}\right)}^{\mathsf{T}}\right)\right]+\rho \mathbf{g}}$

then looking at the viscous terms of the x momentum equation for example we have:

${\displaystyle \begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(2\mu \frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(\mu \left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\right)\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(\mu \left(\frac{\mathrm{\partial }u}{\mathrm{\partial }z}+\frac{\mathrm{\partial }w}{\mathrm{\partial }x}\right)\right)\\ & =2\mu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\mu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\mu \frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }y\mathrm{\partial }x}+\mu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}+\mu \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }z\mathrm{\partial }x}\\ & =\mu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\mu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\mu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}+\mu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\mu \frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }y\mathrm{\partial }x}+\mu \frac{{\mathrm{\partial }}^{2}w}{\mathrm{\partial }z\mathrm{\partial }x}\\ & =\mu {\mathrm{\nabla }}^{2}u+\mu \frac{\mathrm{\partial }}{\mathrm{\partial }x}{\cancel{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}+\frac{\mathrm{\partial }w}{\mathrm{\partial }z}\right)}}^0\\ & =\mu {\mathrm{\nabla }}^{2}u\end{array}}$

Similarly for the y and z momentum directions we have μ∇²v and μ∇²w.

The above solution is key to deriving Navier–Stokes equations from the equation of motion in fluid dynamics when density and viscosity are constant.

Non-Newtonian fluids

Main article: Non-Newtonian fluid

A non-Newtonian fluid is a fluid whose flow properties differ in any way from those of Newtonian fluids. Most commonly the viscosity of non-Newtonian fluids is a function of shear rate or shear rate history. However, there are some non-Newtonian fluids with shear-independent viscosity, that nonetheless exhibit normal stress-differences or other non-Newtonian behaviour. Many salt solutions and molten polymers are non-Newtonian fluids, as are many commonly found substances such as ketchup, custard, toothpaste, starch suspensions, paint, blood, and shampoo. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin, the constant of proportionality being the coefficient of viscosity. In a non-Newtonian fluid, the relation between the shear stress and the shear rate is different, and can even be time-dependent. The study of the non-Newtonian fluids is usually called rheology. A few examples are given here.

Bingham fluid

Main article: Bingham plastic

In Bingham fluids, the situation is slightly different:

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}=\{\begin{array}{ll}0,& \tau <{\tau }_0\\ {\displaystyle \frac{\tau -{\tau }_0}{\mu }},& \tau \ge {\tau }_0\end{array}}$

These are fluids capable of bearing some stress before they start flowing. Some common examples are toothpaste and clay.

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

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

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).

Stream function formulation

In the analysis of a flow, it is often desirable to reduce the number of equations and/or the number of variables. The incompressible Navier–Stokes equation with mass continuity (four equations in four unknowns) can 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:

${\displaystyle \begin{array}{rl}\mathrm{\nabla }\times (\mathrm{\nabla }\varphi )& =0\\ \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{A})& =0\end{array}}$

for any differentiable scalar φ and vector A. The first identity implies that any term in the Navier–Stokes equation that may be represented as the gradient of a scalar will disappear when the curl of the equation is taken. Commonly, pressure p and external acceleration g will be eliminated, resulting in (this is true in 2D as well as 3D):

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

where it is assumed that all body forces are describable as gradients (for example it is true for gravity), and density has been divided so that viscosity becomes kinematic viscosity.

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 flow velocity being zero, we can replace the flow velocity with the curl of some vector ψ so that mass continuity is always satisfied:

${\displaystyle \mathrm{\nabla }\cdot \mathbf{u}=0\Rightarrow \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathit{\psi })=0\Rightarrow 0=0}$

So, as long as flow velocity is represented through u = ∇ × ψ, 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 unknown pressure variable and no longer dependent on a separate mass continuity equation:

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

Apart from containing fourth order derivatives, this equation is fairly complicated, and is thus uncommon. Note that if the cross differentiation is left out, the result is a third order vector equation containing an unknown vector field (the gradient of pressure) that may be determined from the same boundary conditions that one would apply to the fourth order equation above.

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. 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.

The 3D flow velocity is expressed as (note that the discussion not used coordinates so far):

${\displaystyle \mathbf{u}=u_1{\mathbf{e}}_1+u_2{\mathbf{e}}_2+u_3{\mathbf{e}}_3}$

where e_i are basis vectors, not necessarily constant and not necessarily normalized, and u_i are flow velocity components; let also the coordinates of space be (x₁, x₂, x₃).

Now suppose that the flow is 2D. This does not mean the flow is in a plane, rather it means that the component of flow 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):

${\displaystyle \mathbf{u}=u_1{\mathbf{e}}_1+u_2{\mathbf{e}}_2;\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }{x}_3}=\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{x}_3}=0}$

The vector function ψ is still defined via:

${\displaystyle \mathbf{u}=\mathrm{\nabla }\times \mathit{\psi }}$

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:

${\displaystyle u_1{\mathbf{e}}_1+u_2{\mathbf{e}}_2=\frac{{\mathbf{e}}_1}{h_2{h}_3}\left[\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}\left(h_3{\psi }_3\right)-\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_3}\left(h_2{\psi }_2\right)\right]+}$

${\displaystyle +\frac{{\mathbf{e}}_2}{h_3{h}_1}\left[\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_3}\left(h_1{\psi }_1\right)-\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}\left(h_3{\psi }_3\right)\right]+\frac{{\mathbf{e}}_3}{h_1{h}_2}\left[\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}\left(h_2{\psi }_2\right)-\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}\left(h_1{\psi }_1\right)\right]}$

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

${\displaystyle u_1{\mathbf{e}}_1+u_2{\mathbf{e}}_2=\frac{{\mathbf{e}}_1}{h_2{h}_3}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}\left(h_3{\psi }_3\right)-\frac{{\mathbf{e}}_2}{h_3{h}_1}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}\left(h_3{\psi }_3\right)}$

the significance here is that only one component of ψ remains, so that 2D flow becomes a problem with only one dependent variable. 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 equation for ψ can simplify since a variety of quantities will now equal zero, for example:

${\displaystyle \mathrm{\nabla }\cdot \mathit{\psi }=\frac{1}{h_1{h}_2{h}_3}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_3}\left(\psi h_1{h}_2\right)=0}$

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

${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }\times \mathit{\psi })=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathit{\psi })-{\mathrm{\nabla }}^2\mathit{\psi }=-{\mathrm{\nabla }}^2\mathit{\psi }}$

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

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left({\mathrm{\nabla }}^2\psi \right)+(\mathrm{\nabla }\times \mathit{\psi })\cdot \mathrm{\nabla }\left({\mathrm{\nabla }}^2\psi \right)=\nu {\mathrm{\nabla }}^4\psi }$

where ∇⁴ is the biharmonic operator. 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.

Derivation of the scalar stream function equation

The assumptions for the stream function equation are:

• The flow is incompressible and Newtonian.
• Coordinates are orthogonal.
• Flow is 2D: u₃ = ⁠∂u₁/∂x₃⁠ = ⁠∂u₂/∂x₃⁠ = 0
• The first two scale factors of the coordinate system are independent of the last coordinate: ⁠∂h₁/∂x₃⁠ = ⁠∂h₂/∂x₃⁠ = 0, otherwise extra terms appear.

The stream function has some useful properties:

• Since −∇²ψ = ∇ × (∇ × ψ) = ∇ × u, the vorticity of the flow is just the negative of the Laplacian of the stream function.
• The level curves of the stream function are streamlines.

The stress tensor

The derivation of the Navier–Stokes equation involves the consideration of forces acting on fluid elements, so that a quantity called the stress tensor appears naturally in the Cauchy momentum equation. 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. Recalling that σ = −pI + τ, for a Newtonian fluid the stress tensor is:

${\displaystyle {\sigma }_{ij}=-p{\delta }_{ij}+\mu \left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)+{\delta }_{ij}\lambda \mathrm{\nabla }\cdot \mathbf{u}.}$

If the fluid is assumed to be incompressible, the tensor simplifies significantly. In 3D cartesian coordinates for example:

${\displaystyle \begin{array}{rl}\mathit{\sigma }& =-\left(\begin{array}{ccc}p& 0& 0\\ 0& p& 0\\ 0& 0& p\end{array}\right)+\mu \left(\begin{array}{ccc}2{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}& {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x}}& {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }z}+\frac{\mathrm{\partial }w}{\mathrm{\partial }x}}\\ {\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }y}}& 2{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }y}}& {\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }z}+\frac{\mathrm{\partial }w}{\mathrm{\partial }y}}\\ {\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }z}}& {\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }z}}& 2{\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }z}}\end{array}\right)\\ & =-p\mathbf{I}+\mu \left(\mathrm{\nabla }\mathbf{u}+{\left(\mathrm{\nabla }\mathbf{u}\right)}^{\mathsf{T}}\right)\\ & =-p\mathbf{I}+2\mu \mathbf{e}\end{array}}$

e is the strain rate tensor, by definition:

${\displaystyle e_{ij}=\frac{1}{2}\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right).}$

 

Unfunded mandate

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Unfunded_mandate 

An unfunded mandate is a statute or regulation that requires any entity to perform certain actions, with no money provided for fulfilling the requirements. This can be imposed on state or local government, as well as private individuals or organizations. The key distinction is that the statute or regulation is not accompanied by funding to fulfill the requirement.

An example in the United States, would be those federal mandates that induce "responsibility, action, procedure or anything else that is imposed by constitutional, administrative, executive, or judicial action" for state and local governments and/or the private sector.

As of 1992, 172 federal mandates obliged state or local governments to fund programs to some extent. Beginning with the Civil Rights Act of 1957 and the Civil Rights Act of 1964, as well as the Voting Rights Act of 1965, the United States federal government has designed laws that require state and local government spending to promote national goals. During the 1970s, the national government promoted education, mental health, and environmental programs by implementing grant projects at a state and local level; the grants were so common that the federal assistance for these programs made up over a quarter of state and local budgets. The rise in federal mandates led to more mandate regulation. During the Reagan Administration, Executive Order 12291 and the State and Local Cost Estimate Act of 1981 were passed, which implemented a careful examination of the true costs of federal unfunded mandates. More reform for federal mandates came in 1995 with the Unfunded Mandates Reform Act (UMRA), which promoted a Congressional focus on the costs imposed onto intergovernmental entities and the private sector because of federal mandates. Familiar examples of Federal Unfunded Mandates in the United States include the Americans with Disabilities Act and Medicaid.

Background

An "intergovernmental mandate" generally refers to the responsibilities or activities that one level of government imposes on another by legislative, executive or judicial action. According to the Unfunded Mandates Reform Act of 1995 (UMRA), an intergovernmental mandate can take various forms:

• An enforceable duty – this refers to any type of legislation, statute or regulation that either requires or proscribes an action of state or local governments, excluding actions imposed as conditions of receiving federal aid.
• Certain changes in large entitlement programs – this refers to instances when new conditions or reductions in large entitlement programs, providing $5 billion or more annually to state or local governments, are imposed by the federal government.
• A reduction in federal funding for an existing mandate – this refers to a reduction or elimination of federal funding authorized to cover the costs of an existing mandate.

A 1993 study conducted by Price Waterhouse, sponsored by the National Association of Counties, determined that in fiscal year 1993 counties in the US spent $4.8 billion for twelve unfunded federal mandates. Medicaid was one of these twelve unfunded mandates, and comprised the second largest item in state budgets, accounting for almost 13 percent of state general revenues in 1993.

Mandates can be applied either vertically or horizontally. Vertically applied mandates are directed by a level of government at a single department or program. Conversely, horizontally applied, or "crosscutting", mandates refer to mandates that affect various departments or programs. For example, a mandate requiring county health departments to provide outpatient mental health programs would be considered a vertically applied mandate, whereas a requirement that all offices in a given jurisdiction to become handicap-accessible would be considered a horizontally applied mandate.

History

Federal unfunded mandates can be traced back to the post-World War II years, when the federal government initiated national programs in education, mental health services, and environmental protection. The method for implementing these projects at the state and local level was to involve state and local governments. In the 1970s, the federal government utilized grants as a way to increase state and local participation, which resulted in federal assistance constituting over 25 percent of state and local budgets.

The first wave of major mandates occurred in the 1960s and 1970s, concerning civil rights, education, and the environment. The arrival of the Reagan administration ostensibly undermined various federal mandate efforts, as the executive branch promised to decrease federal regulatory efforts. For example, the passage of Executive Order 12291 required a cost-benefit analysis and an Office of Management and Budget clearance on proposed agency regulations, and the State and Local Cost Estimate Act of 1981 required the Congressional Budget Office to determine the state and local cost effects of proposed federal legislation moving through the Legislative Branch. However, the U.S. Advisory Commission on Intergovernmental Relations (ACIR) reported that, during the 1980s, more major intergovernmental regulatory programs were enacted than during the 1970s.

According to a 1995 Brookings Institution report, in 1980 there were 36 laws that qualified as unfunded mandates. Despite opposition from the Reagan administration and George H. W. Bush administration, an additional 27 laws that could be categorized as unfunded mandates went into effect between 1982 and 1991.

The U.S. Supreme Court has been involved in deciding the federal government's role in the U.S. governmental system based on constitutionality. During the period between the New Deal era and the mid-1980s the court generally utilized an expansive interpretation of the interstate commerce clause and the 14th Amendment to validate the growth of the federal government's involvement in domestic policymaking. For example, the 1985 Supreme Court case Garcia v. San Antonio Metropolitan Transit Authority affirmed the ability for the federal government to directly regulate state and local governmental affairs.

The increase of mandates in the 1980s and 1990s incited state and local protest. In October 1993, state and local interest groups sponsored a National Unfunded Mandates Day, which involved press conferences and appeals to congressional delegations about mandate relief. In early 1995, Congress passed unfunded mandate reform legislation.

In 1992 the court determined in various cases that the US Constitution provides state and locality protections concerning unfunded mandate enactments. For example, in the 1992 case New York v. United States, the Court struck down a federal law that regulated the disposal of low-level radioactive waste, which utilized the Tenth Amendment to the United States Constitution to require states to dispose of the radioactive material.

Examples

Unfunded mandates are most commonly utilized in regulation of civil rights, anti-poverty programs and environmental protection programs.

Clean Air Act

The Clean Air Act was passed in 1963 to support the United States Environmental Protection Agency (EPA), established on December 2, 1970, in developing research programs looking into air pollution problems and solutions. The EPA received authority to research air quality. The 1970 Amendments to the Clean Air Act established the National Ambient Air Quality Standards, authorized requirements for control of motor vehicle emissions, increased the federal enforcement authority but required states to implement plans to adhere to these standards. The 1990 Amendments to the Clean Air Act of 1970 expanded and modified the National Ambient Air Quality Standards and expanded and modified enforcement authority. The amendments increased the mandates on states to comply with the federal standards for air quality. States have had to write up State Implementation Plans, have them approved by the EPA and must also fund the implementation.

The Americans with Disabilities Act of 1990

The Americans with Disabilities Act of 1990 prohibits discrimination based on disability, requires existing public facilities to be made accessible, requires new facilities to comply with accessibility expectations, and requires that employers provide anything a disabled employee might need, such as a sign language interpreter. Tax incentives encourage employers to hire people with disabilities. State institutions and local employers are expected to pay for changes made to existing facilities and are responsible for making sure that new facilities are in compliance with the federal requirements under the ADA.

Medicaid

Medicaid is a health program for low-income families and people with certain medical needs in the United States. It is funded jointly by the federal and state governments, but implemented by states. Federal funding covers a variable portion of at least half of Medicaid costs, and states are expected to cover the remainder. This means that any federally mandated increase in Medicaid spending forces states to spend more. However, as state participation in Medicaid is voluntary, it is not technically an unfunded mandate.

EMTALA

The Emergency Medical Treatment and Active Labor Act (EMTALA) was passed by the United States Congress in 1986 to halt certain practices of patient dumping. The act requires hospitals accepting payment from Medicare to provide emergency treatment to any patient coming to their emergency department, regardless of their insurance coverage or ability to pay. Though hospitals could theoretically choose to not participate in Medicare placing them outside of EMTALA's scope, very few do not accept payments from Medicare causing EMTALA to apply to nearly all US hospitals. Though EMTALA infers an obligation to provide certain emergency care, the statute does not contain any provision regarding funding or financing of said emergency care. EMTALA could therefore be characterized as an unfunded mandate.

The No Child Left Behind Act of 2001

The 2001 No Child Left Behind Act was passed in response to widespread concern about the quality of public education in America. The act was meant to decrease the gap between students who were performing very well and students who were performing poorly. The act required schools receiving federal funding to administer statewide standardized tests to students at the end of each year. If students did not show improvement from year to year on these tests, their schools were asked to work to improve the quality of the education by hiring highly qualified teachers and by tutoring struggling students. To continue receiving Federal grants, states had to develop plans that demonstrated their steps to improve the quality of education in their schools. The No Child Left Behind Act mandated that states fund the improvements in their schools and provide the appropriate training for less qualified teachers. Federally mandated K-12 education is also a (mostly) unfunded mandate.

Criticism

Critics argue that unfunded mandates are inefficient and are an unfair imposition of the national government on the smaller governments. While many scholars do not object to the goals of the mandates, the way they are enforced and written are criticized for their ineffectiveness. State and local governments do not always disagree with the spirit of the mandate, but they sometimes object to the high costs they must bear to carry out the objectives.

The debate on unfunded federal mandates is visible in cases such as New York v. United States, mentioned above. In School District of Pontiac, Michigan v. Duncan, the plaintiffs alleged that the school district need not comply with the No Child Left Behind Act of 2001 because the federal government did not provide them sufficient funding; the court concluded that insufficient federal funds were not a valid reason to not comply with a federal mandate.

Unfunded Mandates Reform Act

Main article: Unfunded Mandates Reform Act of 1995

Purpose

The Unfunded Mandates Reform Act (UMRA) was approved by the 104th Congress on March 22, 1995, and became effective October 5, 1995, during the Clinton administration. It is public law 104-4. The official legislation summarizes the bill as being: "An Act: To curb the practice of imposing unfunded Federal mandates on States and local governments; [...] and to ensure that the Federal Government pays the costs incurred by those governments in complying with certain requirements under Federal statutes and regulations, and for other purposes."

UMRA was enacted to avoid imposing mandates, when said mandates did not include federal funding to help the SLTG (State, Local, and Tribal Governments) carry out the goals of the mandate. It also allowed the Congressional Budget Office to estimate the cost of mandates to SLTGs and to the private sector, and allows federal agencies issuing mandates to estimate the costs of mandates to the entities that said mandates regulate.

Application

Most of the act's provisions apply to proposed and final rules for which a notice of the proposed rule was published, and that include a Federal mandate that could result in the expenditure of funds by SLTGs or the private sector of or in excess of $100 million in any given year. If a mandate meets these conditions, a written statement must be provided that includes the legal authority for the rule, a cost-benefit assessment, a description of the macroeconomic effects that the mandate will likely have, and a summary of concerns from the SLTG and how they were addressed. An agency enforcing the mandate must also choose the least-costly option that still achieves the goals of the mandate, as well as consult with elected officials of the SLTG to allow for their input on the implementation of the mandate and its goals. Section 203 of UMRA is a bit more extensive in that it applies to all regulatory requirements that significantly affect small governments, and requires federal agencies to provide notice of the requirements to the government(s), enable the officials of the government(s) to provide their input on the mandate, and inform and educate the government(s) on the requirements for implementation of the mandate.

UMRA allows the United States Congress to decline unfunded federal mandates within legislation if such mandates are estimated to cost more than the threshold amounts estimated by the Congressional Budget Office. UMRA does not apply to "conditions of federal assistance; duties stemming from participation in voluntary federal programs; rules issued by independent regulatory agencies; rules issued without a general notice of proposed rulemaking; and rules and legislative provisions that cover individual constitutional rights, discrimination, emergency assistance, grant accounting and auditing procedures, national security, treaty obligations, and certain elements of Social Security".

Effectiveness

Ever since UMRA was proposed, it has remained unclear, how effective the legislation actually is at limiting the burdens imposed by unfunded mandates on SLTGs, and whether or not unfunded mandates need to be limited so strictly. Proponents of the Act argue that UMRA is needed to limit legislation that imposes obligations on SLTGs and that creates higher costs and less efficiency, while opponents argue that sometimes federal unfunded mandates are necessary to achieve a national goal that state and local governments don't fund voluntarily. Opponents also question the effectiveness of the bill due to the aforementioned restrictions.

2015 Unfunded Mandates and Information Transparency Act

The Act was written to amend UMRA by having the CBO compare the authorized level of funding in legislation to the costs of carrying out any changes. It was done by also amending the Congressional Budget Act of 1974. The bill was introduced by Republican North Carolina Representative Virginia Foxx and passed by the House on February 4, 2015.

Foxx had authored a previous version of this bill, which also passed the house, as H.R. 899 (113th Congress) in February 2014. The bill would allow private companies and trade associations to look at proposed rules before they are announced to the public. The concern is that private companies could weaken upgrades to public protections.