Convective time derivative

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

Quoting Cornille [87, p. 168] The calculation concerning the electromagnetic conservation laws given in most textbooks, for example, in Jackson [7, p. 239] is not correct, as noted by Selak et al. [88], because it is not permissible to substitute a convective time derivative for an Eulerian time derivative even when we have a constant volume of integration. ... 

The relatively simple concept represented by the sky-diver example is easily generalized to provide a relationship between the convected derivative of any scalar quantity B associated with a fixed material point and the partial derivatives of B with respect to time and spatial position in a fixed (inertial) reference frame. Specifically, B changes for a moving material point both because B may vary with respect to time at each fixed point at a rate dB/dt and because the material point moves through space and B may be a function of spatial position in the direction of motion. The rate of change of B with respect to spatial position is just V/i. The rate at which B changes with time for a material point with velocity u is then just the projection of VB onto the direction of motion multiplied by the speed, which is u VB. It follows that the convected time derivative of any scalar B can be expressed in terms of the partial derivatives of B with respect to time and spatial position as... 

Derive the appropriate form of the Navier-Stokes equation for an electrically conducting Newtonian fluid with constant density and viscosity. This equation is called the magnetohydrodynamic momentum equation. Your derivation should begin from first principles. However, you may assume - but should state in appropriate places - the necessary properties of pressure, viscous stress, and convected time derivatives. 

With the homogeneous flow assumption, whereby V(RR) = 0, the definition of (RR)(i) as given by Eq. (6.25) is the same as the codeformational time derivative or convected time derivative shown in Appendix 6.A. 

Ata = -l,b = c = 0, equations [7.2.24] correspond to the Maxwell liquid with a discrete spectrum of relaxation times and the upper convective time derivative. For solution of polymer in a pure viscous liquid, it is convenient to represent this model in such a form that the solvent contribution into total stress tensor will be explicit ... 

The upper-convected time derivative is a time derivative in a special coordinate system whose base coordinate vectors stretch and rotate with material lines. With this definition of the upper-convected time derivative, stresses are produced only when material elements are deformed mere rotation produces no stress (see Section 1.4). Because of the way it is defined, the upper-convected time (teriva-tive of the Finger tensor is identically zero (see eqs. 2.2.3S and 1.4.13) ... 

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) ... 

Since K is homogeneous, vK = 0. Thus, the left-hand side is the upper-convected time derivative referred to in Chapter 4 (eq. 4.3.2), which is often designated by v over the tensor... 

Just as there are various possible finite strain tensors, there are various time derivatives that can be used in place of the ordinary derivative of stress in Eq. 10.21 to satisfy the continuum mechanics requirements for a model to be able to describe large, rapid deformations in arbitrary coordinate systems. The derivative that yields a differential model equivalent to Lodge s Eq. 10.6 is the upper convected time derivative (defined in Eq. 11.19), and the resulting model is called the upper-convected Maxwell model. Other possibilities include the lower-convected derivative and the corotational derivative. Furthermore, a weighted-sum of two of these derivatives can be used to formulate a differential constitutive equation for polymeric liquids. In particular, the Gordon-Schowalter convected derivative [7] is defined in this manner. 

---