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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.

J A = D A C A MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahQeadaWgaaWcbaGaamyqaaqabaGccqGH9aqpcqGHsislcaWHebWaaSbaaSqaaiaadgeaaeqaaOGaeyOiGCRaey4bIeTaam4qamaaBaaaleaacaWGbbaabeaaaaa@4037@

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,

q = k T MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahghacqGH9aqpcqGHsislcaWHRbGaeyOiGCRaey4bIeTaamivaaaa@3DAC@

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.

u = k μ ( p ρ g ) = ϕ v

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.

ρ a = ρ d v d t = ρ f + T

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.

V ( t ) ( ρ a ρ f ) d V = V ( t ) T d V = S ( t ) T n d S MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaqqabaWaa8Wuaeaadaqadaqaaiabeg8aYjaaykW7caWHHbGaeyOeI0IaeqyWdiNaaGPaVlaahAgaaiaawIcacaGLPaaaaSqaaiaadAfacaGGOaGaamiDaiaacMcaaeqaniabgUIiYlabgUIiYlabgUIiYdGccaaMc8UaamizaiaadAfacqGH9aqpdaWdtbqaaiabgEGirlabgkci3kaahsfaaSqaaiaadAfacaGGOaGaamiDaiaacMcaaeqaniabgUIiYlabgUIiYlabgUIiYdGccaaMc8UaamizaiaadAfaaeaacqGH9aqpdaWdsbqaaiaahsfacqGHIaYTcaWHUbaaleaacaWGtbGaaiikaiaadshacaGGPaaabeqdcqGHRiI8cqGHRiI8aOGaaGPaVlaadsgacaWGtbaaaaa@6D89@

The stress tensor for a Newtonian fluid is as follows.

T = ( p + λ Θ ) I + 2 μ e e = 1 2 ( v + v t ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaCivaiabg2da9maabmaabaGaeyOeI0IaamiCaiabgUcaRiabeU7aSjaaykW7cqqHyoquaiaawIcacaGLPaaacaaMc8UaaCysaiabgUcaRiaaikdacqaH8oqBcaaMc8UaaCyzaaqaaiaahwgacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaGaey4bIeTaaCODaiabgUcaRiabgEGirlaahAhadaahaaWcbeqaaiaadshaaaaakiaawIcacaGLPaaaaaaa@551E@

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.

× H = J + D t B = 0 × E = B t D = ρ Constitutive equations: B = μ H D = ε E J = σ E MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7AA2@

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.

E = V B = × A

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.

Reading assignment

Read Chapter 1 and Appendix A and B of Aris.

Questions & Answers

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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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