# 0.3 The kinematics of fluid motion

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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 $\xi =\left({\xi }_{1},{\xi }_{2},{\xi }_{3}\right)$ and at a later time the same particle is at a position $\mathbf{x}$ . The motion of the particle that occupied this original position is described as follows.

$\mathbf{x}=\mathbf{x}\left(\xi ,t\right)\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}{x}_{i}=x\left({\xi }_{1},{\xi }_{2},{\xi }_{3},t\right)$

The initial coordinates $\xi$ 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 $\xi$ . The terms convected and Lagrangian coordinates are also used. The spatial coordinates $\mathbf{x}$ 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 $\mathbf{x}$ at time $t$ ; i.e.,

$\xi =\xi \left(\mathbf{x},t\right)\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}{\xi }_{i}={\xi }_{i}\left({x}_{1},{x}_{2},{x}_{3},t\right)$

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

$J=\frac{\partial \left({x}_{1},{x}_{2},{x}_{3}\right)}{\partial \left({\xi }_{1},{\xi }_{2},{\xi }_{3}\right)}$
should not vanish.

The transformation $\mathbf{x}=\mathbf{x}\left(\xi ,t\right)$ may be looked at as the parametric equation of a curve in space with $t$ as the parameter. The curve goes through the point $\xi$ , 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 $\rho \left(\xi ,t\right)$ , meaning that for any prescribed particle $\xi$ 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 $\Im \left(\xi ,t\right)$ , can be changed to a spatial description $\Im \left(\mathbf{x},\mathbf{t}\right)$ .

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
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Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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