Material time derivative

Therein, ( ) a denotes the material time derivative with respect to pa. 

In our development and notation we mainly follow Truesdell [10] and use a subscript to indicate a constituent and a prime to denote material time derivative following the motion of that constituent thus, vt = x[ and at = x ( are the 2th peculiar velocity and acceleration, respectively. / ), Li = grad vt and I), = Li — Lf) are the peculiar gradient of deformation, velocity gradient and rate of deformation, respectively. 

It is important to emphasize that Q. itself is time-dependent since the material volume element of interest is undergoing deformation. Note that in our statement of the balance of linear momentum we have introduced notation for our description of the time derivative which differs from the conventional time derivative, indicating that we are evaluating the material time derivative. The material time derivative evaluates the time rate of change of a quantity for a given material particle. Explicitly, we write... 

Here, the symbol D/Dt stands for the convected or material time derivative, which we shall subsequently discuss in some detail. In the context of (2-6) it is clear that D/Dt represents the time derivative of the total mass of material in the material volume Vm(t). Alternatively, we could say that it is the time derivative of the total mass associated with the fixed set of material points that comprise Vm(t). We shall see shortly that Eq. (2 6), which derives directly from the definition of a material volume for a fluid that conserves mass, is entirely equivalent to (2-3) or (2 4) and leads precisely to the pointwise continuity equation, (2-5). However, this cannot be seen easily without further discussion of the convected or material time derivative. 

It may now be evident why the convected derivative D/Dt is also known as the material time derivative. DB/Dt is, in fact, the time derivative of the quantity B for a fixed material point. Material points are often specified by the position vector xo corresponding to their position at t = 0. The position vector of the material point xo at an arbitrary time / > 0 is thus... 

The material time derivative of a quantity ( ) following the motion of the i -constituent is... 

We begin by writing down the conservation or balance laws (Ericksen ). We shall employ the cartesian tensor notation, repeated tensor indices being subject to the usual summation convention. The comma denotes partial differentiation with respect to spatial coordinates and the superposed dot a material time derivative. For example,... 

Although the spatial (Euler) description x, t is simpler in fluids, the material (time) derivative, expressed by (3.8), is preferred below because it gives more concise results. 

From (10.55) we see that the tensors gij,g and behave as though they were constants in material differentiation with respect to t. However, this is not the case with surface metric tensor ttap as can be seen from (10.56), and it is the consequence of its explicit dependence of time. Then the operation of raising and lowering of indices of tensor fields with respect to aap is not, generally, commutative with material time derivative. Particularly, this is tme for Indeed, it is easy to show that... 

The acceleration vector is given by a material time derivative of (2.26) ... 

Recall the definition of the deformation gradient dx = F dX its material time-derivative defines the following velocity gradient ... 

Under this relation the material time derivative of the integral of an arbitrary function d> can be calculated as... 

The material time derivative of a vector-valued or tensor-valued function is not always objective as described above even if the original function is objective. It can be said that the convected derivative and corotational derivative are introduced to ensure objectivity of the time-derivative. For example we have... 

We here denote the material time-derivative d a)4 /dt of a function [Pg.124]

The material time derivative of a function at a space-time (at, t) with reference to the mean velocity v is given by... 

For a vector field function y/, the material time derivative is defined by... 

A superposed dot denotes the material time derivative. The Leslie coefficients a,- are assumed to be independent on velocity gradient and time. They depend on temperature and pressure. Four of the coefficients are related one with another via the Parodi equation [14, 24]... 

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