Convected base vector

The stress tensor of the fluid particle (X,t) at time t, x(X, can be expanded either in terms of unit vectors at the present time t and Cartesian components Tij or in terms of the convected base vectors at position X and time t and contravariant components as shown in the following ... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Fig. 4.59 illustrates the flow pattern determined by Eqs (4.45) and (4.47). The distribution of the velocity vectors is shown for the case when the axial velocity component equals (ux - 1). Eqs. (4.45) and (4.47) allow us to find temperature and degree of conversion distributions in the frontal zone based on the fundamental balance equations. These equations differ from Eqs. (4.36) and (4.47) because they take into account convective heat transfer along the z-direction. The dimensionless forms of the main determining equations are as follows energy balance... 

This is a surface-related phenomenon based on the mass flux vector of component i and the surface area across which this flux acts. Relative to a stationary reference frame, p, v, is the mass flux vector of component i with units of mass of species i per area per time. It is extremely important to emphasize that p, v, contains contributions from convective mass transfer and molecular mass transfer. The latter process is due to diffusion. When one considers the mass of component i that crosses the surface of the control volume due to mass flux, the species velocity and the surface velocity must be considered. For example. Pi (Vr — Vsurface) is the mass flux vector of component i with respect to the surface... 

Answer The mass transfer calculation is based on the normal component of the total molar flux of species A, evaluated at the solid-liquid interface. Convection and diffusion contribute to the total molar flux of species A. For thermal energy transfer in a pure fluid, one must consider contributions from convection, conduction, a reversible pressure work term, and an irreversible viscous work term. Complete expressions for the total flux of speeies mass and energy are provided in Table 19.2-2 of Bird et al. (2002, p. 588). When the normal component of these fluxes is evaluated at the solid-liquid interface, where the normal component of the mass-averaged velocity vector vanishes, the mass and heat transfer problems require evaluations of Pick s law and Fourier s law, respectively. The coefficients of proportionality between flux and gradient in these molecular transport laws represent molecular transport properties (i.e., a, mix and kxc). In terms of the mass transfer problem, one focuses on the solid-liquid interface for x > 0 ... 

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) ... 

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