## D t

In these equations, n is the molecular flux of momentum and g and F are gravitational acceleration and external body forces, respectively. The physical interpretation of the various terms appearing in these equations again follows similar lines: the first term is the rate of increase in momentum per unit volume; the second term represents change in momentum per unit volume, caused by convection; the fourth and fifth terms represent the gravitational force per unit volume and any other external force, if present, respectively. The third term represents molecular contributions, which include pressure and viscous force per unit volume and is given as where p is pressure and t is the viscous stress tensor. It may be noted that the second term of the right-hand side is not simple divergence because of the tensorial nature of t .

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton's law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written:

where Sij is the Kronecker delta (Sj = 1 if i = j and Sij = 0 if i = j) function, ^ is the coefficient of viscosity and k is the coefficient of bulk viscosity. The superscript 'T' denotes the transpose of a tensor quantity. In general, it is believed that, except in the study of shock waves and in the absorption and attenuation of acoustic waves, it is convenient to ignore the coefficient of bulk viscosity. Substitution of Eq. (2.7) into Eq. (2.6) and Eq. (2.5) results in the complete momentum conservation equation. A special case of these momentum conservation equations for constant density and constant viscosity fluids is the famous Navier-Stokes equation, which provides the usual starting point for the analysis of flow processes (Bird et al., 1960; Deen, 1998).

Any fluid, which does not obey Newton's law of viscosity, is called a non-Newtonian fluid. This class covers widely different materials/fluids, varying from those exhibiting slight deviation from Newtonian behavior to almost elastic solids. Fluids exhibiting slight deviations from Newtonian behavior, such as power law fluids (which exhibit a power law relationship between stress and strain) or Bingham plastic fluids (which require finite yield stress for flowing), can be readily modeled using the same framework. The molecular viscosity term appearing in Eq. (2.7) is replaced by an effective viscosity term, which may be a function of local stress and strain values. More complex behavior, e.g. viscoelastic behavior, requires a completely different framework to develop satisfactory constitutive equations. The subject of developing suitable constitutive equations for viscoelastic fluids, is extremely complex and outside the scope of this book. As stated earlier, the focus in this book is on simulating turbulent, multiphase and reactive flows. Detailed discussion about the rheology and motion of complex fluids can be found in Tanner (1985), Bird et al. (1987) (constitutive equations, models) and Crochet et al. (1984) (numerical simulation).

2.1.3. Conservation of Energy

Application of the law of conservation of energy can be used to derive transport equations for total energy. In order to derive an equation for internal energy, it is first necessary to derive a transport equation for mechanical energy, which can then be subtracted from the equation for total energy to obtain the governing equation for internal energy (Bird et al., 1960; Bird, 1998). Without going into details, the equation for internal energy written in terms of static enthalpy is given below:

dt (ph) + V-(p Uh) = -V-(q) + Dp - (t : V U) -V ■ ^ hk jk j + Sh (2.8) Here h is an enthalpy, which is defined as

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