# Introduction  (Page 5/5)

 Page 5 / 5

## Densities, potential gradients, and fluxes

Velocity: and flux by convection . Transport or flux of the various quantities discussed in this course will be due to convection (or advection) or due to the gradient of a potential. Common to all of these transport process is the convective transport resulting from the net or average motion of the molecules or the velocity field, v . The convective flux of a quantity is equal to the product of the density of that quantity and the velocity. In this sense, the velocity vector can be interpreted as a “volumetric flux” as it has the units of the flow of volume across a unit area of surface per unit of time. Because the flux by convection is common to all forms of transport, the integral and differential calculus that follow the convective motion of the fluid will be defined. These will be known as the Reynolds’ transport theorem and the convective or material derivative.

Mass density and mass flux . If ρ is the mass density, the mass flux is ρ v .

Species concentration . Suppose the concentration of species A in a mixture is denoted by C A . The convective flux of species A is C A v . Fick’s law of diffusion gives the diffusive flux of A.

${\mathbf{J}}_{A}=-{\mathbf{D}}_{A}•\nabla {C}_{A}$

The diffusivity, D A , is in general a tensor but in an isotropic medium, it is usually expressed as a scalar.

Internal energy (heat). The density of internal energy is the product of density and specific internal energy, ρ E . The convective flux is ρ E v . For an incompressible fluid, the convective flux becomes ρ C p (T-T o ) v . The conductive heat flux, q , is given by Fourier’s law for conduction of heat,

$\mathbf{q}=-\mathbf{k}•\nabla T$

where k is the thermal conductivity tensor (note: same symbol as for permeability).

Porous media . The density of a single fluid phase per unit bulk volume of porous media is ϕ ρ , where ϕ is the porosity. Darcy’s law gives the volumetric flux, superficial velocity, or Darcy’s velocity as a function of a potential gradient.

$\begin{array}{c}\mathbf{u}=-\frac{\mathbf{k}}{\mu }•\left(\nabla p-\rho \text{\hspace{0.17em}}\mathbf{g}\right)\\ =\varphi \text{\hspace{0.17em}}\mathbf{v}\end{array}$

where k is the permeability tensor and v is the interstitial velocity or the average velocity of the fluid in the pore space. Darcy’s law is the momentum balance for a fluid in porous media at low Reynolds number.

Momentum balance . Newton’s law of motion for an element of fluid is described by Cauchy’s equation of motion.

$\begin{array}{c}\rho \text{\hspace{0.17em}}a=\rho \text{\hspace{0.17em}}\frac{dv}{dt}\\ =\rho \text{\hspace{0.17em}}f+\nabla •T\end{array}$

where f is the sum of body forces and T is the stress tensor. The stress tensor can be interpreted as the flux of force acting on the bounding surface of an element of fluid.

$\begin{array}{ccc}\underset{V\left(t\right)}{\iiint }\left(\rho \text{\hspace{0.17em}}\mathbf{a}-\rho \text{\hspace{0.17em}}\mathbf{f}\right)\text{\hspace{0.17em}}dV\hfill & =\hfill & \underset{V\left(t\right)}{\iiint }\nabla •\mathbf{T}\text{\hspace{0.17em}}dV\hfill \\ & =\hfill & \underset{S\left(t\right)}{\iint }\mathbf{T}•\mathbf{n}\text{\hspace{0.17em}}dS\hfill \end{array}$

The stress tensor for a Newtonian fluid is as follows.

$\begin{array}{l}T=\left(-p+\lambda \text{\hspace{0.17em}}\Theta \right)\text{\hspace{0.17em}}I+2\mu \text{\hspace{0.17em}}e\\ e=\frac{1}{2}\left(\nabla v+\nabla {v}^{t}\right)\end{array}$

where p is the thermodynamic pressure, Θ is the divergence of velocity, μ is the coefficient of shear viscosity, (λ+2/3μ) coefficient of bulk viscosity, and e is the rate of deformation tensor. Thus the anisotropic part (not identical in all directions) of the stress tensor is proportional to the symmetric part of the velocity gradient tensor and the constant of proportionality is the shear viscosity.

Electricity and Magnetism . We will not be solving problems in electricity and magnetism but the fundamental equations are presented here to illustrate the similarity between the field theory of transport phenomena and the classical field theory of electricity and magnetism. The Maxwell’s equations and the constitutive equations are as follows.

$\begin{array}{l}\nabla ×\mathbf{H}=\mathbf{J}+\frac{\partial \mathbf{D}}{\partial t}\\ \nabla •\mathbf{B}=0\\ \nabla ×\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}\\ \nabla •\mathbf{D}=\rho \\ \text{Constitutive equations:}\\ \mathbf{B}=\mathbf{\mu }\mathbf{H}\\ \mathbf{D}=\mathbf{\epsilon }\mathbf{E}\\ \mathbf{J}=\mathbf{\sigma }\mathbf{E}\end{array}$

where

• electric field intensity
• electric flux density or electric induction
• magnetic field intensity
• magnetic flux density or magnetic induction
• electric current density
• charge density
• magnetic permeability (tensor if anisotropic)
• electric permittivity (tensor if anisotropic)
• electric conductivity (tensor if anisotropic)

When the fields are quasi-static, the coupling between the electric and magnetic fields simplify and the fields can be represented by potentials.

$\begin{array}{l}\mathbf{E}=-\nabla V\\ \mathbf{B}=\nabla ×\mathbf{A}\end{array}$

where V is the electric potential and A is the vector potential. The electric potential is analogous to the flow potential for invicid, irrotational flow and the vector potential is analogous to the stream function in two-dimensional, incompressible flow.

Read Chapter 1 and Appendix A and B of Aris.

where we get a research paper on Nano chemistry....?
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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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
how nano science is used for hydrophobicity
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
what is Nano technology ?
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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