Transport fluxes

In a previous post, I discussed heat transfer and derived Fourier’s law of heat conduction. In one dimension, it is given by:

$$ q = -\kappa \frac{dT}{dx} $$

where \( q \) is the heat flux, \( \kappa \) is the thermal conductivity, and \( \frac{dT}{dx} \) is the temperature gradient. It can be interpreted as follows: the temperature gradient is the driving force; because of it, a heat flux results that minimizes the temperature gradient, hence the negative sign. The resulting heat flux is proportional to the temperature gradient, and the proportionality constant is the thermal conductivity, which characterizes the ability of the system under study to conduct heat. A higher thermal conductivity leads to a faster transfer of heat – keep in mind the heat flux is the amount of heat transferred per unit area per unit time and is thus the rate of heat transfer per unit area.

Following a similar argument to mass transfer, a concentration gradient, \( \frac{dC}{dx} \), where \( C \) is the mass concentration of a species, results in a mass diffusion flux, denoted by \( J \), that minimizes the concentration gradient, as shown below:

$$ J = -D \frac{dC}{dx} $$

where \( D \) is known as the diffusion coefficient that describes the ability of the species to diffuse within the system. A higher diffusion coefficient corresponds to a larger mass diffusion flux, or the mass transfer rate per unit area, which is the amount of mass transferred per unit area per unit time. This is Fick’s (first) law of diffusion. Note that \( C \) can be measured in different ways, typically mass per unit volume or number of moles per unit volume. Correspondingly, the resulting flux is termed mass diffusion flux or molar diffusion flux.

The flow of fluid is often considered as momentum transfer. Consider a fluid between two stationary plates as shown in Figure 1 below. If the top plate starts to move at a speed of \( V \) along the \( x \) direction, then the layer of fluid right below the plate will also start to move. Because of the intermolecular forces between the fluid molecules, we expect the next layer of fluid will pick up a velocity as well. Eventually all the fluid between the plates will flow, and the only exception is the layer right above the bottom plate, which cannot move due to the friction of the plate.

The whole process can be described as a transfer of velocity (and thus momentum) along the \( -y \) direction, driven by the velocity gradient along the \( +y \) direction, as given by Newton’s law of viscosity:

$$ \tau = -\mu \frac{dv}{dy} $$

Here \( \mu \) is the proportionality constant known as the viscosity of the fluid. \( \tau \) is the momentum flux, or the amount of momentum transferred per unit area per unit time. We can quickly show that the dimension of momentum flux is equivalent to that of stress, and Newton’s law of viscosity can also be interpreted as that the flow of fluid results in shear stress in the fluid. Also note that we assumed a model of layer-by-layer motion of the fluid, which is true only in the laminar flow regime.

Extending to three dimensions, both \( q \) and \( J \) become vectors, and the derivative becomes the \( \nabla \) operator:

$$ \vec{q} = -\kappa \nabla T $$

$$ \vec{J} = -D \nabla C $$

For momentum transfer, it becomes a little tricky because velocity, or momentum, is already a vector (unlike \( T \) or \( C \)). Indeed, in the one-dimensional example above, we are working with the transfer of the \( x \) component of the momentum along the \( y \) direction, and a better way to write Newton’s law of viscosity is then:

$$ \tau_{xy} = -\mu \frac{dv_x}{dy} $$

\( x \)-momentum can be transferred along \( x \), \( y \), and \( z \) directions, so are \( y \)-momentum and \( z \)-momentum. Thus, we can write nine momentum fluxes. In other words, the momentum flux has nine components in three dimensions, and such quantities are known as tensors. Thus, the momentum flux, or equivalently the stress, is a tensor quantity. The most general form of Newton’s law of viscosity is quite complex, but for incompressible flow it can be written in a simplified form:

$$ \tau = -[\nabla \vec{v} + (\nabla \vec{v})^T] $$

Our discussions on three types of transport fluxes are summarized in the table below.

Type of flux (as resulted by the driving force)	Driving force	Materials Parameter	One-dimensional equation	Three-dimensional equation
Heat flux	Temperature gradient	Thermal conductivity	$$ q = -\kappa \frac{dT}{dx} $$	$$ \vec{q} = -\kappa \nabla T $$
Mass flux	Concentration gradient	Mass diffusivity	$$ J = -D \frac{dC}{dx} $$	$$ \vec{J} = -D \nabla C $$
Momentum flux	Velocity gradient	Viscosity	$$ \tau_{xy} = -\mu \frac{dv_x}{dy} $$	$$ \tau = -[\nabla \vec{v} + (\nabla \vec{v})^T] $$