When this operator is applied on a field, it results in the time partial derivative of the field plus the velocity of the fluid dotted to the gradient of the field.
Although not the time derivative, for Kelvin's theorem around a curve, this operator is applied to circulation and results in zero. When applied to total energy, this operator yields time change of heat and of work. The time rate of a property C within a volume fixed in space plus the net flux of the property C passing through a bounding surface is equal to this operator applied to the property C as seen in the Reynolds transport theorem. When this operator is applied on a field, it results in the time partial derivative of the field plus the velocity of the fluid dotted to the gradient of the field. For 10 points, name this operator denoted capital D divided by capital Dt with lowercase t that describes the time rate of change of a property of a fluid as it moves.
Although not the time derivative, for Kelvin's theorem around a curve, this operator is applied to circulation and results in zero.
{material} derivative [accept {convective derivative}; accept {advective derivative}; accept {substantive derivative}; accept {Lagrangian derivative}; accept {Stokes derivative}; accept {particle derivative}; accept {hydrodynamic derivative}; accept {total derivative}; accept {capital D divided} by {Dt} or equivalents before mention; prompt on {derivative}]