General Vector Form

The purpose of this appendix is to spell out explicitly the Navier-Stokes and mass-continuity equations in different coordinate systems. Although the equations can be expanded from the general vector forms, dealing with the stress tensor T usually makes the expansion tedious. Expansion of the scalar equations (e.g., species or energy) are much less trouble. 

The equations in this section retain some compact notation, including the substantial derivative operator D/Dt, the divergence of the velocity vector V-V, and the Laplacian operator V2. The expansion of these operations into the various coordinate systems may be found in Appendix A. 

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In a general vector form, the net force exerted on the surfaces of a control volume is related to the stresses at the surfaces as... 

Begin with the general vector form of the substantial derivative, as stated below ... 

The general vector form for incompressible, constant-viscosity equations is... 

In the general vector form, the thermal-energy equation is De ... 

Consider the behavior of the Navier-Stokes equations for the two-dimensional flow in a conical channel as illustrated by Fig. 3.17. Begin with the constant-viscosity Navier-Stokes equations written in the general vector form as... 

The objective is to derive a system of equations in general vector form that describes the overall gas-phase mass continuity and the species continuity equations for A and all other species k in the mixture. Assume that there is convective and diffusive transport of the species, but no chemical reaction. 

Considering a general differential control volume, use a conservation law and the Reynolds transport theorem to write a species conservation equation for gas-phase species A in general vector form. Considering that the system consists of the gas phase alone, the droplet evaporation represents a source of A into the system. 

Again, considering a general control volume, write the species conservation equations for all the other species k in a general vector form. 

Derive the general vector form of the overall mass-continuity equation, recognizing that the droplet evaporation represents a source of mass to the system. 

For an incompressible fluid, the continuity equation is written in general vector form as... 

If the system of equations is written in the general vector form... 

To have the free energy density in a more general form including all the nine elementary distortions tzi, 2... agWQ should add the terms dnjdz, dnjdx and dnj dy and rewrite the Eq. (8.15b) in the vector notations for arbitrary distortion of n with respect to the Cartesian frame. Then we obtain Frank formula for the density of elastic energy in the general vector form ... 

In (1.25), the terms inside the brackets can be reformulated by use of vector and tensor notations. By comparing the terms inside the brackets with the mathematical definitions of the nabla or del operator, the vector product between this nabla operator and the mass flux vector we recognize that these terms can be written as V. (pv). The mass balance equation, (1.25), written in Cartesian coordinates can thus be rewritten in the general vector form as ... 

The expressions for 11 and 11 in (6.73) and (6.74), which also appear in the equilibrium equations (6.79) and (6.80), can be reformulated in general vector form following the methodology of Nakagawa [210]. It is a straightforward exercise to find the general form when... 

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