# 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.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
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Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Uday
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