Time Derivatives of Second-Order Tensors

Now we seek the rule of transformation of the material derivative of tensor which is a type of time derivative following the motion of the material element. It is important to realize, however, that all tensor quantities described in reference to the convected coordinate system f must be transformed into tensor quantities fixed in space (i.e fixed coordinate system x ) because all physical measurements are made relative to the fixed coordinafe sysfem. 

The derivative of each term on the right-hand side of Eq. (2B.11) is given by da- 

Equation (2B.15) may be used for any fixed coordinate system (i.e., for Cartesian coordinates as well as for any curvilinear coordinates). A similar procedure can be followed for a contravariant tensor a, yielding (Oldroyd 1950) 

The physical significance of each of the terms in Eq. (2B.15) (also in Eq. (2B.16)) is as follows. The first two terms ((9a y/9r)+n (9a y/9x )) describe the rate of time variation while following the motion (translation) of a material element. The third and fourth terms describe the rotational motion as well as the deformation of the element, as the quantity dv /dxi (and 9u /9jc ) is the velocity gradient that can be represented by the rate-of-deformation tensor d/ j and the vorticity tensor that is, 

The covariant and contravariant forms of a given tensor involving convected 

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