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5.7: The Linear Transport Model

Steady-State Transport Phenomena

There are many phenomena that involve the motion or transport of some quantity that behave similarly to the way fluids and electric charge flow in those sections of a circuit without sources of energy density (pumps or batteries). In steady-state, i.e., after all transients have settled out, we can model these phenomena the same way we model fluids and electric charge flow. These are such common and general phenomena that it is useful to collect them together under their own specially named model—the linear transport model. First, we recast the energy-density equations we had for fluids and charge in a way that emphasizes the flow, i.e., the current. Then, we follow standard practice and write the current as a flux.

The transport equations for fluids and current electricity (without pumps or batteries) and using the subscripts “F” to denote fluids and “E” to denote electric charge become:

\[ \underset{\text{fluids}}{\Delta (total~ head) = – I_F R_F}\]

\[ \underset{ \text{current electricity}} {\Delta (V) = – I_E R_E} \]

Solving for current we have:

\[ \underset{\text{fluids}}{I_F = – \dfrac{1}{R_F} \Delta (total ~head)}\]

\[\underset{\text{ current electricity }}{I_E = – \dfrac{1}{R_E} \Delta V}\]

Next we write the resistance in terms of the conductivity k, the cross sectional area of the pipe, the wire, or whatever is constraining the flow, A, and the length of the circuit under consideration L (across which \(\Delta\)(total head) or \(\Delta \)V is measured).

\[R = \dfrac{1}{k} \left(\dfrac{L}{A}\right)\]

This relation says that the resistance increases with the length of the pipe or wire, is inversely proportional to the cross-sectional area, and is inversely proportional to the conductivity of the pipe or wire.

Figure 5.7.1: Flow through a pipe of length \(L\) and cross section \(A\)

This should make sense, we recall that the resistance is what causes fluid or electrical energy density to be converted to thermal energy. Rearranging, we get

\[\dfrac{1}{R} = k \dfrac{A}{L} \label{Eq4}\]

which we can substitute into the expressions we previously had for current. We now drop the subscripts and use the general symbol  \(\phi\) for the voltage \(V\) and total head. These last two expressions are often referred to as potentials, so, \(\phi\) is referred to as a potential.

\[I = – k \dfrac{A}{L} \Delta \phi ~~~~(fluids~ or~ current ~electric)\]

We can generalize these relations further by defining the flux \(j\) of whatever it is that flows. Flux is the amount of what is flowing that passes a unit area per unit time.

\[j = \dfrac{I}{A}\]

Then the expression for j becomes

\[j = –  \dfrac{k}{L} \Delta \phi \label{Eq5}\]

Our final generalization is to imagine the length, L, shrinking down to infinitesimal size, so that \(\Delta \phi\)/L becomes simply the derivative of the potential d\(\phi/\)dx in the direction of the flow, called the gradient of the potential. Hence

\[\dfrac{\Delta \phi}{L} = \dfrac{d\phi}{dx} \label{Eq6}\]

and Equation \(\ref{Eq5}\) becomes

\[j = – k \dfrac{d \phi }{dx} \label{genlinEQ}\]

this is the generalized linear transport equation, where

\[I = jA\]

In words this relation says that the flux of whatever it is that flows is proportional to a constant that depends of the characteristics of the physical system, which is the conductivity for fluids and electric current, and to the gradient of the potential. The faster the potential changes in the direction of the flow, the higher the flow. The gradient of the potential is the “driving force” that causes the fluid or electric charge to flow in the presence of the resistance. Thinking back to our energy-density model, the gradient of the potential is simply the change (loss) in fluid or electric energy density as you move along with the flow. The minus sign is consistent with the convention that the flux is positive in the direction of flow. Because the gradient of the potential is negative in the direction of flow, the extra minus sign in the relation, makes the flux positive.

The units of the flux and current, conductivity, and potential depend on whatever it is that is flowing. We now look at several specific cases. To get an expression in terms of resistance, we use the relation in Equations \(\ref{Eq4}\) and \(\ref{Eq6}\).

We have imagined steady state conditions in the discussion up to this point. Among other things, this means that the gradient of the potential remains constant in time. This will typically be the case when there is a large reservoir of whatever it is that is being transported on both sides of the region through which the transport is occurring. Then, if the reservoirs remain at constant potential, one higher than the other, the potential gradient remains constant. In the more general case, the potential (and the potential gradient) is a function of both time and spatial dimensions. By combining the transport equation with the condition of continuity, a differential equation is obtained that can be solved for the particular boundary conditions and initial conditions that obtain of the phenomenon in question. We won’t pursue this non-steady state further here.

Ways of Expressing the Resistance for Fluid and Electrical Charge Flow

Resistance in Fluid Flow

We consider a current flowing through a round pipe of radius r, cross sectional area A and length L. If the flow is in the low-velocity regime (streamline or laminar flow) the expression for resistance is simply the product of two factors. One factor incorporates pipe geometry and the other incorporates fluid properties. We can intuitively see how geometry, the length and area of the pipe (or other similar structure) influences resistance. If fluid flows through the pipes that differ only in length, which one will have more loss to the thermal energy system? The longer one will lose more. If two pipes differ in cross-sectional area, which pipe will have a greater proportion of fluid interacting with the pipe walls? The one with smaller area will, and thus it will lose more energy to the thermal energy system.

Incorporating pipe geometry properties with a fluid property, viscosity, we have the following relations:

\[R = (fluid~ properties) \times (geometry~ properties) \]

\[= 8 \eta \times \dfrac{L}{Ar^2} \]

\[= \dfrac{8 \eta L}{  \pi  r^4 }\]

where \(\eta\) is the viscosity of the fluid. Note the very strong dependence of flow resistance on the radius of the pipe or tube—inversely proportional to the fourth power of the radius. This has interesting consequences in many common situations, including partially clogged water pipes and arteries. Viscosity has units of (J\(\times\)s)/m3or (N\(\times\)s)/m2. A table of viscosities for several fluids is given in Figure 5.7.1.

Figure 5.7.1: Viscosities of select fluids at select temperatures
Fluid at specific temperature (20° C) viscosity \(\eta\)
air  (20° C) \(1.8 \times 10^{-5}\)
water (20° C) \(1.8 \times 10^{-3}\)
water (20° C) \(1.0 \times 10^{-3}\)
water (90° C) \(0.32 \times 10^{-3}\)
blood (37° C) \(4 \times 10^{-3}\)
light motor oil (20° C) \(0.03\)
glycerin (20° C) \(1.5\)

When the velocity of the fluid is high, the flow is turbulent, a chaotic and irregular flow. The value of the resistance when flow is turbulent is always greater than for smooth streamline (laminar) flow. Because it is complicated to model and not independent of velocity, we won’t discuss turbulent flow quantitatively in this course. It is useful to know, however, in case you encounter fluid flow in your future studies, that it is possible to calculate a particular parameter, which is a function of current, fluid properties as well as dimensions and shape of the pipe, that allows one to roughly predict whether laminar or turbulent flow will occur. This dimensionless parameter is called the Reynolds number. The flow is laminar for values of the Reynolds number less than about 2000 and turbulent for values greater than about 3000. The flow is unstable for intermediate values. For now, the important point to remember is simply that as flow velocity increases, the flow will eventually become turbulent and the resistance will increase significantly. This phenomenon of increasing resistance occurs in many common fluid systems including air moving in the ductwork in forced-air heating and air conditioning systems, water flowing in typical household water distribution systems, and blood flowing in arteries.

Resistance in Electric Charge Flow

As with the resistance to fluid flow through pipes, the resistance to the movement of electric charge can be separated into a factor dependent solely on the geometry of the material through which charge moves and another factor independent of geometry, but dependent on the details of the charge carriers and how they interact with the material through which they move.

Focusing first on the geometry factors, the resistance will increase proportionally with the length, \(L\), and the inverse of the cross sectional area, \(A\). In fluid flow, the geometry independent factor is a property of the fluid itself, viscosity; in charge flow it is a property of the material through which the charge flows. This geometry independent factor is called resistivity, which is the reciprocal of the conductivity. The symbol for resistivity is a Greek rho (sometimes with a subscript \(r\) to distinguish it from the mass density). Combining the two factors, the resistance is:

\[R ~=~ \dfrac{ \rho L}{A}~ =~ \dfrac{1 L}{ kA}\]

Note the extremely large range or resistivities for common materials. In most materials there is a small dependence of resistivity on temperature. In semiconductors the dependence of resistivity on temperature is typically larger than for insulators or metals. A strong current dependence arises if the thermal heating, due to the resistance, causes the temperature of the material to increase substantially. This is the case with ordinary tungsten light bulbs–the kind that get hot. The resistance of the filament of these light bulbs increases many times as they go from room temperature to their typical operating temperature, which is several thousand degrees.

Often, instead of electrical resistivity, its reciprocal, conductivity is tabulated. Substances that have high conductivities (low resistivities) are good electrical conductors. Metals are usually good electrical conductors. Substances that have low conductivities (high resistivities) are said to be good insulators. Glass and most plastics are good insulators.

Heat Flow (Thermal Energy Flow)

In heat conduction, the quantity that flows is thermal energy. We are imagining a non equilibrium situation in which the temperature is hotter at one end of a rod, for example, than at the other end. The “driving potential” is the gradient of the temperature in the material. The “current” is the thermal energy per time, which is power. Power in heat flow is analogous to the current in fluid or charge flow. Here, power is the amount of thermal energy that is transferred per second past a plane which is oriented perpendicular to the gradient of the temperature. The thermal flux is the power per unit area. Thermal energy (heat) flows due to a temperature gradient within a solid, liquid, or gas.

For heat flow we can re-write the general linear transport equation

\[j = – k \dfrac{d \phi}{dx}\]

where \(k\) is the thermal conductivity in units of Watts per (meter kelvin) and d\(\phi\)/dx becomes the temperature gradient in units of kelvin per meter.

Then the linear transport equation (Equation \(\ref{genlinEQ}\)) gives the heat flux in units of Watts/square meter

\[\dfrac{P}{A} = – k \dfrac{d \phi}{dx}\]

For an object (window pane perhaps) with cross sectional area A, thickness L, and temperature difference across the surfaces\( \Delta \)T, the total heat transported across the object would be

\[P = – k A \dfrac{ \Delta T}{L}.\]

Metals that are good electrical conductors are usually also good thermal conductors. This is because the movement of “free” electrons in the metal is primarily responsible for both electrical and thermal conduction. However, the vibrations of atoms can also transport thermal energy with little resistance, provided the material is very pure. Diamond is an electrical insulator; its electrons are strongly bound to the carbon atoms and are not free to transport electric charge. However, diamond has a thermal conductivity considerably larger (thermal resistivity smaller) than any metal at room temperature!

We see from the table that air is an extremely poor conductor of heat (as long as there is no convection; i.e., macroscopic movement of the air), which makes it an excellent insulator. When you bundle up on those ski trips in the mountains, it is most effective to trap air between you and your outerwear with a woolly sweater and/or long johns, and wear a down or Holofil™ jacket with many microscopic pockets of trapped air. The walls and roofs of our houses are layered with fiberglass mats or similar material that are designed to trap air in order to be effective insulators. Double-pane windows also trap "dead air" between the outside environment and the inner window.



There are many phenomena in various branches of science that are generally referred to as diffusion. In its basic form, diffusion is simply the net motion (net transport) of a particular species of particles that is due to (1) the presence of random thermal motion that naturally exists at all temperatures above \(0 K\) and (2) to the presence of a concentration gradient of that species of particle. Diffusion of particles occurs in solids, liquids, and gases. Another common example occurs when permeable or semipermeable membranes are present. If a concentration difference of some particle species exists across a membrane that is permeable to those particles, then there will be a flow of those particles across the membrane from the region of high concentration to the region of low concentration. If the concentration gradient is removed, transport ceases. The greater the concentration gradient, the greater the number of particles transported per unit time.

Steady-state diffusion is described by the linear transport model. We re-write the general linear transport equation

\[j = – k \dfrac{d \phi }{dx} \]


\[j = – D \dfrac{dc}{dx} ~~~~ (historically ~known~ as ~Fick’s~ law) \]

\(where~ D~ is ~the ~diffusion~ constant ~in~ SI ~units ~of ~square~ meters ~per ~sec,~ \dfrac{m^2}{s} \)

\(c~ is~ the~ particle~ concentration ~gradient~ in~ SI~ units ~of ~number~ per~ cubic~ meter, ~m^{-3} \)

\(and~ \dfrac{dc}{dx} ~is ~the~ particle~ concentration~ gradient~ in~ SI ~units ~of ~number~ per~ m^{-4}.\)

The particle flux, \(j\), has units of number per (square meter second). The units of \(j\) and D are often expressed in terms of moles rather than number of particles. In this case, the units of D are mol m2/s.

In steady state conditions, the linear transport model using the constructs of concentration and concentration gradient works well. However, in transient conditions, a better approach is to use the mobility, rather than concentration, and the free energy gradient, rather than the concentration gradient in the linear transport equation. This latter approach avoids having to let the diffusion constant be a function of the concentration.

Values of diffusion constants range over many orders of magnitude. The diffusion of nuclear magnetization (due to the very weak magnetic moments of nuclei) in pure insulating crystals can be as small as \(1.0 \times 10^{-17} m^2s^{-1}\) while the diffusion of hydrogen in air at STP is nearly \(1.0 \times 10^{-4} m^2s^{-1}.\) The table lists some representative values of solute-fluid combinations at 20°C and 1 atm pressure.

Other Examples of Linear Transport

As previously indicated, there are many transport phenomena that can be understood with the linear transport model. Many of these phenomena have been given special names (laws). It is easy to not realize that they are all understandable from the general perspective of the linear transport model. For example in studying subsurface water flow Henry Darcy established in the 1850’s with a series of sand column experiments that, for a given type of sand, the volume flow rate, i.e., the current I, was proportional to the head difference and cross-sectional area of the column and inversely proportional to the column length. From what you know about linear transport, you should be able to predict the law that bears his name, Darcy’s law, simply by assuming that there is a soil-dependent conductivity (known today as the hydraulic conductivity of the soil) and writing down the general linear transport equation:

\[\dfrac{I}{A} = k_{soil} \dfrac{d(head)}{dx}. \]

A summary of the algebraic relationships and relationships in Linear Transport Model as it applies to fluid flow, current electricity, heat conduction and diffusion is given on the next page.

Algebraic Representations

\[Flux:~~ j = \dfrac{I}{A} \]

\[Generic~ Transport Eq.~~ j = – k_{generic ~conductivity} \dfrac{d \phi }{dx}\]

Relationship of Resistance and Conductivity to Resistivity, \(\rho\) :

\[Resistance: ~~R = \rho \dfrac{L}{A} ~~ Conductivity:~~ k = \dfrac{1}{\rho} \]

Summary and Comparison of Several Steady-State Transport Systems

  Laminar of fluid flow current electrcity Heat conduction Diffusion
transported quantity

volume of fluid,



charge q

[coulombs, C]

thermal energy


particles[no units]

(can be expressend in [mol])



flux,j: volume per time per area


flux, j:charge per time per area


thermal flux, j:thermal nergy per time per area,


flux, j: number of particles per tiem per area,




energy per unit 


[J/m3] or [N/m2] or [Pa]

voltage, V

energy per unit charge

[J/C] or [volts, V]

temperature, T

[kelvin, K]

Particle concentration, c


Potential gradient, 


change in total head along flow,



change in voltaage with distance along circuit,



charge in temp with distance along path,


change in particle conc with distance along path,



(inverse of resistivity)

fluid conductivity, k


electricity conductivity, k


thermal conductivity, k


diffusion constant, D


Transport Equation j=-kd(head)/dx j=-kdV/dx j=-kdT/dx j=-kdD/dx


(I=jA) or (P=jA)

flow, I: volume per time


current, I: charge per time 

[C/s] or [amps, A]

Power, P;thermal energy per time

particle current, I: number per time



(inverse of conductivity)

fluid resistivity, \(\rho\)



electrical resistivity, \(\rho\)


thermal resistivity, \(\rho\)



fluid resistivity, R



electrical resistivity, R


[V/A] or [ohms,\(\omega\)]

thermal resistivity, R




power dissipated

or transferred










no power dissipated in thermal currents


no power dissipated in diffusion currents