User:Petwil/worksinprogress

Engineers, physicists, geophysical and atmospherical fluid dynamicists, and astrophysicists, not to mention rheologists, have a slew of different terms for idealized fluids and flows. Idealized fluids in turn may be split into thermodynamic idealizations and constitutive idealizations.

This page attempts to collect in one point various idealized fluids and flows in the subject of fluid dynamics, modulo non-Newtonian models. There are so many non-Newtonian constitutive models that it is best to leave those to another page so as not to obfuscate the more common idealizations in the rest of (Newtonian) fluid mechanics.

Note that the words "fluid" and "gas" are largely interchangeable here. In particular fluid does not imply liquid. The choice of nomenclature for idealized fluids - be they referred to as fluids or gases - is for the most part simply a matter of convention and should not be taken to imply anything in particular.

=IDEALIZED FLUIDS: Thermodynamic=

Incompressible fluid
For a homogenous incompressible fluid, $$ \rho $$ is a constant. For an inhomogenous mixture of incompressible fluids, such as oil and water, $$ (\partial_t + v \cdot \nabla) \rho = - \rho (\nabla \cdot v) = 0 $$. Note that incompressibility must be relaxed to obtain sound waves.

Ideal gas
An ideal gas has the equation of state, $$ P V = n R T = N k T $$. Here are a few fundamental relations for an ideal gas that are handy to know. Note that $$ \gamma $$ here is the ratio of specific heats, $$ \gamma = c_P / c_V $$. Also, $$ c_P - c_V = R / \mu $$, where $$ R = N_A k $$ and $$ \mu = $$ average mass per particle (right? ugh). For fully ionized material of solar abundances, $$ \mu = 0.613\,{\rm amu} $$.


 * sound speed:

$$ c_s^2 = \gamma \frac{P}{\rho} = \gamma \frac{R}{\mu} T = \gamma (c_P - c_V) T = \gamma (\gamma - 1)c_V T = \gamma(\gamma - 1)\epsilon $$


 * heat capacity:

$$ c_V = \frac{R}{\mu} \frac{1}{\gamma - 1} $$


 * energy density (up to arbitrary additive constant, chosen by convention to be zero):

$$ \epsilon = c_V T = \frac{P}{(\gamma - 1)\rho} = \frac{c_s^2}{\gamma(\gamma - 1)} $$


 * enthalpy:

$$ h = \epsilon + Pv = \left(\frac{\gamma}{\gamma-1}\right) \frac{P}{\rho} = \frac{c_s^2}{\gamma - 1} $$


 * entropy:

Most places you look it up, the entropy of an ideal gas is given in the form of the Sackur-Tetrode equation. For a monatomic ideal gas,

$$ \frac{S}{Nk} = \ln\left( \frac{V}{N} \left( \frac{4 \pi m U}{3 N h^2}\right)^{3/2}\right) + \frac{5}{2} $$

Barotropic fluid
A "barotropic" fluid is a fluid with an extremely simple equation of state: The pressure is a function of the density only, $$ P = P(\rho) $$. Examples: An isothermal ideal gas, an isentropic ideal gas.

baroclinic fluid
A fluid which is not barotropic. Thus, the baroclinic vector $$ (\nabla \rho \times \nabla P) / \rho^2 $$ need not be zero.

perfect gas
A gas in which the specific heat is independent of temperature.

Polytropic gas
=IDEALIZED FLUIDS: constitutive/transport idealizations=

Ideal fluid
A type of invisicd or Euler fluid: A fluid in which thermal conductivity and viscosity are zero. See Landau & Lifshitz volume 6. The momentum transport equation (aka the fluid equation) for an ideal fluid is the Euler equation(s), rather than the Navier-Stokes equations. The energy transport equation is also, of course, greatly simplified.

Inviscid or Euler fluid
A fluid with zero viscosity, so that the fluid momentum transport equation is the Euler equation(s), rather than the Navier-Stokes equation(s).

Newtonian fluid
Basically your garden-variety viscous fluid. A fluid in which the internal stress is assumed to be linear in the rate-of-strain tensor, plus an overall isotropic stress (thermal pressure) which is independent of the rate-of strain tensor. There are a few further assumptions. (1) The stress should not depend on any overall rotational motion of the fluid, so that the stress tensor is independent of the antisymmetric part of the rate-of-strain tensor. (2) The fluid does not have any internal coordinate system, so that there are only two constants of proportionality needed, namely the bulk viscosity (not much used) and the shear viscosity (typically called simply the viscosity), which respectively determine the stress due to the trace of the rate-of-strain tensor and the traceless symmetric part of the rate-of-strain tensor.

=IDEALIZED FLOWS=

Potential flow
Potential flow is flow which is irrotational, so that the velocity at any point may be expressed as the gradient of a potential.

=RATE-OF-STRAIN TENSOR= The rate-of-strain tensor in Eulerian terms is $$ L_{ij} = \partial_j v_i $$ It is convenient to break this into its trace, an antisymmetric part, and a symmetric traceless part:

$$ D_{ij} = \frac{1}{3} \left( \partial_k v_k \right) \delta_{ij} $$

$$ \Omega_{ji} = \frac{1}{2} \left[ \partial_i v_j - \partial_j v_i \right] $$

$$ \Sigma_{ij} = \frac{1}{2}\left[ \partial_i v_j + \partial_j v_i \right] - \frac{1}{3} \left(\partial_k v _k\right) \delta_{ij} $$

Note then that

$$ \partial_j v_i = D_{ij} + \Omega_{ij} + \Sigma_{ij} $$