Substantial time derivative

The special operator, DT/Dt is the substantial time derivative, and represents the time rate of change if the observer moves with the substance. 

The local balance equations for an observer moving along with the fluid are expressed in substantial time derivative form. From Eq. (3.71), we can express the substantial time derivative of e by... 

From Eqs. (3.76) and (3.63), we can represent the mass balance in the substantial time derivative... 

Equation (3.133) can be applied to a fluid element moving with the mass average velocity v. After replacing the differential operators with substantial time derivative operators inEq. (3.133), we have... 

The dot in denotes a time derivative. In flows that are not spatially homogeneous, this time derivative is the substantial time derivative defined by... 

Here the partial time derivation, 8/8t, describes how the variable (concentration, velocity and temperature) changes with time at a fixed position. Substantial time derivation, D/Dt, is a special kind of total time derivation computed by an observer floating downstream with the fluid. In this form the equation of motion states that a... 

In Cartesian coordinates the substantial time derivative, D/Dt, is given by... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Re3molds stress tensor The first and second terms on the RHS denote the production... 

In this notation D/Dt represents a generalization of the substantial time derivative operator. The B — D terms are the net birth and death terms collectively representing the net rate of production of particles of state (x, r) at time t in an environment of state Y. 

Comparing (A.8) and (A.9) we note that to make these relations coincide the total time derivative must be specified equal to the substantial time derivative. In this way the substantial derivative may be considered a special kind of the total time derivative [2, 28], and thus the Reynolds transport theorem is a special kind of the Leibnitz theorem. 

Substantial time derivative DB/Dt. This derivative is a special kind of total time derivative where the velocity of the observer is just the same as the velocity of the stream, i.e., the observer drifts along with the current ... 

The interaction of G with the velocity field is concealed in the substantial time derivative of G. If the velocity is taken along the X direction and the velocity gradient along the z direction, one has v = ejid, where z and x are the components of r. 

Substantial time derivative. Another useful type of time derivative is obtained if the observer floats along with the velocity v of the flowing stream and notes the change in density with respect to time. This is called the derivative that follows the motion, or the substantial time derivative, Dp/Dt. 

Perhaps the simplest way to combine time-dependent phenomena and rheological nonlinearity is to incorporate nonlinearity into the simple Maxwell equation, eq. 3.2.18. This can be done by replacing the substantial time derivative in a tensor version of eq 3.2.18 with the upper-convected time derivative of r, using eq. 4.3.2 (Oldrpyd, 1950) ... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

The terms in brackets equal an operator analogous to the substantial time derivative known from the transport phenomena literature. The total differential of / (r, c, t) is given by ... 

Hypothetical substantial time derivative operator in phase space... 

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