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    Topics covered in this chapter

  • Particle paths and material derivatives
  • Streamlines
  • Streaklines
  • Dilatation
  • Reynolds' transport theorem
  • Conservation of mass and the equation of continuity
  • Deformation and rate of strain
  • Physical interpretation of the deformation tensor
  • Principal axis of deformation
  • Vorticity, vortex lines, and tubes

Reading assignment: Chapter 4 of Aris

Kinematics is the study of motion without regard to the forces that bring about the motion. Already, we have described how rigid body motion is described by its translation and rotation. Also, the divergence and curl of the field and values on boundaries can describe a vector field. Here we will consider the motion of a fluid as microscopic or macroscopic bodies that translate, rotate, and deform with time. We treat fluids as a continuum such that the fluid identified to be at a specific point in space at one time with neighboring fluid will be at another specific point in space at a later time with the same neighbors, with the exception of certain bifurcations. This identification of the fluid occupying a point in space requires that the motion is deterministic rather than stochastic, i.e., random motions such as diffusion and turbulence are not described. Central to the kinematics of fluid motion is the concept of convection or following the motion of a "particle" of fluid.

Particle paths and material derivatives

Fluid motion will be described as the motion of a "particle" that occupies a point in space. At some time, say t = 0 , a fluid particle is at a position ξ = ( ξ 1 , ξ 2 , ξ 3 ) and at a later time the same particle is at a position x . The motion of the particle that occupied this original position is described as follows.

x = x ( ξ , t ) or x i = x ( ξ 1 , ξ 2 , ξ 3 , t )

The initial coordinates ξ of a particle will be referred to as the material coordinates of the particles and, when convenient, the particle itself may be called the particle ξ . The terms convected and Lagrangian coordinates are also used. The spatial coordinates x of the particle may be referred to as its position or place . It will be assumed that the motion is continuous, single valued and the previous equation can be inverted to give the initial position or material coordinates of the particle which is at any position x at time t ; i.e.,

ξ = ξ ( x , t ) or ξ i = ξ i ( x 1 , x 2 , x 3 , t )

are also continuous and single valued. Physically this means that a continuous arc of particles does not break up during the motion or that the particles in the neighborhood of a given particle continue in its neighborhood during the motion. The single valuedness of the equations mean that a particle cannot split up and occupy two places nor can two distinct particles occupy the same place. Exceptions to these requirements may be allowed on a finite number of singular surfaces, lines or points, as for example a fluid divides around an obstacle. It is shown in Appendix B that a necessary and sufficient condition for the inverse functions to exist is that the Jacobian

J = ( x 1 , x 2 , x 3 ) ( ξ 1 , ξ 2 , ξ 3 )
should not vanish.

The transformation x = x ( ξ , t ) may be looked at as the parametric equation of a curve in space with t as the parameter. The curve goes through the point ξ , corresponding to the parameter t = 0 , and these curves are the particle paths . Any property of the fluid may be followed along the particle path. For example, we may be given the density in the neighborhood of a particle as a function ρ ( ξ , t ) , meaning that for any prescribed particle ξ we have the density as a function of time, that is, the density that an observer riding on the particle would see. (Position itself is a "property" in this general sense so that the equations of the particle path are of this form.) This material description of the change of some property, say ( ξ , t ) , can be changed to a spatial description ( x , t ) .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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