Where } } is the material derivative operator, also denoted by in capital notation as } }, is the flow velocity, is the local fluid density, is the local pressure, is the viscous stress tensor and represents the sum of the external body forces.
22.
Where is the velocity vector, is the density of the fluid, D / Dt is the material derivative, is the viscosity of the fluid, is the electric charge density, is the applied electric field, and is the electric field due to the zeta potential at the walls.
23.
The momentum theorem states that the integrated force exerted at the boundaries of the control volume ( a surface integral ), is equal to the integrated time rate of change ( material derivative ) of the momentum of fluid parcels passing through the interior of the control volume ( a volume integral ).
24.
The material derivative finally is obtained when the frame of reference path "'x "'( " t " ) is solidal with the local stream in the continuum ( Lagrangian reference system ), so the reference velocity is equal to the macroscopic velocity in the continuum:
25.
Where bold symbols denote vectors, \ mathbf { u } is the fluid flow velocity, \ mathbf { v } is the spherical particle velocity, \ rho _ \ mathrm { c } is the mass density of the fluid ( continuous phase ), V _ \ mathrm { p } is the volume of the particle, and D / D " t " denotes the material derivative.
26.
Here \ boldsymbol { U } is the velocity field of the air, \ boldsymbol { \ Omega } is the angular velocity vector of the planet, \ rho is the density of the air, p is the air pressure, \ boldsymbol { F } _ r is the friction, \ boldsymbol { g } is the acceleration vector due to gravity and { D \; \ over Dt } is the material derivative.
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