How can mass flux be defined for a one-dimensional flow?

I am reading a textbook where an example of a one-dimensional binary diffusion is given. In a binary system composed of substances A and B, we write a relation for the substance A:

$$\dot{m}_A'' = Y_A(\dot{m}_A'' + \dot{m}_B'') - \rho \mathcal{D}_{AB} \frac{dY_A}{dx}$$

where the quantity $$\dot{m}_A''$$ is explained as the mass flow of substance A per unit area and its units are $$\frac{kg}{s m^2}$$. $$Y_A$$ is mass fraction of substance A, $$\rho$$ is density and $$\mathcal{D}_{AB}$$ is binary diffusivity. In words, the above equation states: mass flux of A = bulk transport of A + diffusion of A.

The problem seemed straightforward at first, but then I thought, what does it actually mean that the mass flow is per unit of area since the problem is assumed to be one-dimensional? The way I visualise this, all transport of mass in this example happens along direction $$x$$ only and there is no other direction for which it would make sense to express the area. When there is a mass flow from one point on the $$x$$ axis to another, this mass flows through a series of points on the $$x$$ axis, and not areas.

How should I treat this variable $$\dot{m}_A''$$ for a one-dimensional flow and how can I make sense of defining it as mass flow per unit of area?

• The flow area is perpendicular to the x direction, and $\dot{m}"$ is the rate of mass flow through a unit area perpendicular to the x direction. Jan 14 '19 at 23:37