The terminology of flow

For the mathematical description and understanding of transport processes, it is advantageous for their descriptions to have several common characteristics, regardless of the nature of the transport quantity, to allow them to be treated in a similar manner. Without knowledge of their fundamental causes at the molecular level, which corresponds to their historical development, transport processes can be described with help from quantities that can be quantitatively measured on a macroscopic level. One such quantity is that of flux. 

The flux J is understood to be the amount of a quantity transported per unit time through a unit surface area. 

Flux is a vector for which a direction must be specified in addition to the quantity or contribution J. This is accomplished with the help of the unit vector e giving  

Jz are the vector components in the x, y and z axis directions of the coordinate system, Jx, Jy, Jz are their contributions and i,jand k are the corresponding unit vectors. Given a mass quantity m that is transported during time t through an area A, then let/ represent the contribution of the mass flux. For energy transport, then J is the contribution of the energy flux with the dimensions J/m2s (where J = Joule). 

In a very general sense, the flux of a quantity G is proportional at a given location to the gradient of the scalar field produced by the flux, a(x, y, z). Mathematically, one obtains the contributions of the three components with the gradient of a, grad a, from the partial derivative of a at the coordinates x, y, z which for the flux G results in  

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