# Scalar Flow Across a Small Area Element

I've just started reading the text "Vectors, Tensors, and the Basic Equations of Fluid Mechanics" by Rutherford Aris and I came across the following problem.

If $\rho$ is any scalar property per unit volume of a fluid in motion, show how to define a flux vector $\vec{f}$ such that $f_i$ is the rate of flow of $\rho$ per unit area across a small element perpendicular to the axis $0i$.

My instincts tell me it will be related to the velocity of the fluid in motion. Any help would be appreciated.

• Have you tried some form of dimensional analysis? – nluigi Aug 2 '18 at 8:19
• @nluigi $\vec{f}=\rho\vec{v}$? – Alex S Aug 2 '18 at 8:25
• Maybe :), does that make sense to you physically? – nluigi Aug 2 '18 at 8:26
• @nluigi It seems to make sense; the flux vector will be flowing in the same direction as the velocity of the fluid, and it also relates the density of the fluid with the flux vector. – Alex S Aug 2 '18 at 8:30
• You need to use the component of the velocity in the 0i direction. – Farcher Aug 2 '18 at 8:38

Let's consider a one-dimensional control volume, $V=DHdx$ where $dx$, $D$ and $H$ are the length, depth and height of the control volume respectively.

The change of a scalar quantity per volume $\rho$ in the control volume by purely convective transport is then: $$\frac{\partial(\rho V)}{\partial t} = \rho v DH|_x - \rho v DH|_{x+dx}$$

We identify $F=\rho v DH$ to be the total amount of the scalar quantity flowing in and out of the system.

Using the definition of the gradient: $$\frac{\partial\rho}{\partial t}DHdx = -\frac{\partial(\rho v)}{\partial x} DHdx$$

and simplifying: $$\frac{\partial\rho}{\partial t} = -\frac{\partial(\rho v)}{\partial x}$$

We can identify $f=\rho v$ as the total amount of the scalar quantity per unit area flowing in and out of the system.

This analysis is easily extended to multiple dimensions using vector analysis.

• What does the quantity $B$ represent? Or was that a typo – Alex S Aug 2 '18 at 8:53
• Very nicely done by the way; this is a bit beyond my skill level at the moment, but hopefully with some practice I'll be able to derive these kinds of results. – Alex S Aug 2 '18 at 8:56
• Also, what does the subscript in the first align environment represent? – Alex S Aug 2 '18 at 9:04
• @AlexS - it was a typo. The control volume extends from position $x$ to $x+dx$, to relate the ingoing and outgoing fluxes to the change in the controle volume you need to evaluate the fluxes at the faces of the control volume, i.e. at $x$ and $x+dx$. – nluigi Aug 2 '18 at 9:17