Effective mass (solid-state physics) from both sides gives where is the th element of , is the th element of , and are the th and th elements of , respectively, and is the total energy of the particle according to the Planck–Einstein relation. The index is contracted by the use of Einstein notation (there is an implicit summation over ). Since Newton's second law uses the inertial mass (not the gravitational mass), we can identify the inverse of this mass in the equation above as the tensor This tensor expresses the change in group velocity due to a change in crystal momentum. Its
Superficial velocity and subject to assumptions. Superficial velocity can be expressed as: where: Using the concept of porosity, the dependence between the advection velocity and the superficial velocity can be expressed as (for one-dimensional flow): where: The local physical velocity can still be different than the average fluid velocity because the vector of the local fluid flow does not have to be parallel to that of average flow. Also, there may be local constriction in the flow channel. Superficial velocity Superficial velocity (or superficial flow velocity), in engineering of multiphase flows and flows in porous media, is a hypothetical (artificial) flow velocity