Variable-mass system events. The following derivation is for a body that is gaining mass (accretion). A body of time-varying mass "m" moves at a velocity v at an initial time "t". In the same instant, a particle of mass dm moves with velocity u. The initial momentum can be written as Now at a time "t" + d"t", let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as Since d"m"dv is the product of two small values, it can be ignored, meaning during d"t"
Velocity gradient analysis, the dimensions of velocity gradient can be determined. The dimensions of velocity are formula_4 , and the dimensions of distance are formula_5. Since the velocity gradient can be expressed as formula_6. Therefore, the velocity gradient has the same dimensions as this ratio, i.e. formula_7. In 3 dimensions, the gradient, formula_8 , of the velocity formula_9 is a second-order tensor which can be expressed as the matrix L: formula_11 can be decomposed into the sum of a symmetric matrix formula_12 and a skew-symmetric matrix formula_13 as follows formula_12 is called the strain rate tensor and describes the rate of stretching