1) Knudsen number
The Knudsen number (Kn)
is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length
scale.
It is defined as:
where
Nusselt number
The Nusselt
number is a dimensionless number which measures the enhancement of heat transfer
from a surface which occurs in a 'real' situation, compared to the heat
transfer that would be measured if only conduction could occur. Typically it is
used to measure the enhancement of heat transfer when convection takes place.
where
The Nusselt
number can also be viewed as being a dimensionless temperature gradient at the
surface.
Prandtl number
Prandtl Number is a dimensionless number approximating the ratio of momentum
diffusivity and thermal diffusivity,
It is defined as:
where ν is the kinematic viscosity and α
is the thermal diffusivity.
Typical values for Pr are:
For mercury, conduction is very effective compared to convection: thermal diffusivity is dominant. For engine oil,
convection is very effective in transferring energy from an area, compared to pure conduction: momentum
diffusivity is dominant.
Schmidt number is proportional
to { (kinetic viscosity) / (molecular diffusivity) }
and is used in mass transfer in general and diffusion in flowing systems
calculations in particular. It is normally defined in the following form :
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2)
fluid dynamics
This article or section should be merged with Fluid mechanics
Fluid
dynamics is the study of fluids (liquids and gases) in motion, and the effect of the fluid motion on fluid
boundaries, such as solid containers or other fluids. Fluid dynamics is a
branch of fluid mechanics, and has a number of subdisciplines,
including aerodynamics (the study of gases in motion) and hydrodynamics (liquids in motion). These fields are used in
such wide-ranging fields as calculating forces and moments on aircraft, the mass flow of petroleum through pipelines, prediction of weather patterns, and even traffic engineering, where traffic is treated as a continuous
flowing fluid.
Gases are composed of molecules which collide with one another and solid objects.
The continuity assumption, however, considers fluids to be continuous. That is, properties such as density, pressure,
temperature, and velocity are taken to be well-defined at infinitely small
points, and are assumed to vary continuously from one point to another. The
discrete, molecular nature of a fluid is ignored.
Those problems for which the continuity
assumption does not give answers of desired accuracy are solved using statistical mechanics. In order to determine whether to use
conventional fluid dynamics (a subdiscipline of continuum mechanics) or statistical mechanics, the Knudsen number is evaluated for the problem. Problems with
Knudsen numbers at or above unity must be evaluated using statistical mechanics
for reliable solutions.
The foundational axioms of fluid dynamics
are the conservation laws, specifically, conservation of mass, conservation of momentum (also known as
Newton's second law or the balance law), and conservation of energy. These are based on classical mechanics and are modified in relativistic mechanics.
The central equations for fluid dynamics
are the Navier-Stokes equations, which are non-linear differential equations that describe the flow of a fluid
whose stress depends linearly on velocity and on pressure. The unsimplified equations do not have a general closed-form
solution, so they are only of use in computational fluid dynamics. The equations can be simplified
in a number of ways. All of the simplifications make the equations easier to
solve. Some of them allow appropriate fluid dynamics problems to be solved in
closed form.
Compressible vs
incompressible flow
A fluid problem is called compressible if changes in the density of the fluid have significant
effects on the solution. If the density changes have negligible effects on the
solution, the fluid is called incompressible and the changes in density are ignored.
In order to determine whether to use
compressible or incompressible fluid dynamics, the Mach number of the problem is evaluated. As a rough guide,
compressible effects can be ignored at Mach numbers below approximately 0.3.
Nearly all problems involving liquids are in this regime and modeled as
incompressible.
The incompressible Navier-Stokes
equations are simplifications of the Navier-Stokes
equations in which the density has been assumed to be constant. These can be
used to solve incompressible problems.
Viscous problems are those in which fluid friction have
significant effects on the solution. Problems for which friction can safely be
neglected are called inviscid.
The Reynolds number can be used to evaluate whether viscous or inviscid equations are appropriate
to the problem. High Reynolds numbers indicate that the inertial forces are
more significant than the viscous forces. However, even in high Reynolds number
regimes certain problems require that viscosity be
included. In particular, problems calculating net forces on bodies (such as
wings) should use viscous equations. As illustrated by d'Alembert's paradox, a body in an inviscid fluid will experience no force.
The standard equations of inviscid flow are the Euler equations. Another often used model, especially in
computational fluid dynamics, is to use the Euler equations far from the body and the boundary layer equations close to the body.
The Euler equations can be integrated along a streamline to get Bernoulli's equation. When the flow is everywhere irrotational as well as inviscid,
Bernoulli's equation can be used to solve the problem.
Another simplification of fluid dynamics
equations is to set all changes of fluid properties with time to zero. This is called steady flow, and is applicable to a large class of
problems, such as lift and drag on a wing or flow through a pipe. Both
the Navier-Stokes equations and the Euler equations
become simpler when their steady forms are used.
Whether a problem is steady or unsteady
depends on the frame of reference. For instance, the flow around a ship in a
uniform channel is steady from the point of view of the passengers on the ship,
but unsteady to an observer on the shore. Fluid dynamicists
often transform problems to frames of reference in which the flow is steady in
order to simplify the problem.
If a problem is incompressible, irrotational, inviscid, and
steady, it can be solved using potential flow, governed by Laplace's equation. Problems in
this class have elegant solutions which are linear combinations of well-studied
elementary flows.
Turbulence is flow dominated by recirculation, eddies, and
apparent randomness. Flow in which turbulence is not exhibited is called laminar. Mathematically, turbulent flow is often represented
via Reynolds decomposition where the flow is broken down into the
sum of a steady component and a perturbation component.
It is believed that turbulent flows obey
the Navier-Stokes equations. Direct Numerical
Simulation (DNS), based on the Navier-Stokes and
incompressibility equations, makes it possible to simulate turbulent flows with
moderate Reynolds numbers (restrictions depend on the power of computer). The
results of DNS agree with the experimental data.
There are a large number of other possible
approximations to fluid dynamic problems. Stokes flow is flow at very low Reynold's numbers, such that inertial forces can be
neglected compared to viscous forces. The Boussinesq approximation neglects
variations in density except to calculate buoyancy forces.
Mathematical equations and objects