Viscous stress tensor

The remaining six quantities are called shear stresses. They have two subscripts associated with the coordinates, and are referred to as the components of the molecular momentum flow tensor, or the components of the molecular stress tensor, as they are associated with molecular motion. Usually, the viscous stress tensor, t, and the molecular stress tensor, it, are simply referred to as stress tensors. For a Newtonian fluid, we may express the stresses in terms of velocity gradients and viscosities in Cartesian coordinates as follows ... 

These results require further that w, v, R, 0, and Yi be continuous in the first approximation and rely on the assumptions that fi2, and 6 are continuous. The conditions obtained from equations (89), (90), and (93) in effect involve only the longitudinal components of the stress tensor and of the heat-flux vector. The first of the conditions quoted from equations (89) and (90) expresses a pressure discontinuity that balances the discontinuity in the viscous stress tensor, and the second states that the streamwise gradients of the components of velocity tangent to the sheet are continuous. 

Each stress tensor can then always be expressed as the sum of a mean pressure tensor, a viscous stress tensor, and a viscous diffusion-stress tensor thus, ... 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

For each continuous phase k present in a multiphase system consisting of N phases, in principle the set of conservation equations formulated in the previous section can be applied. If one or more of the N phases consists of solid particles, the Newtonian conservation laws for linear and angular momentum should be used instead. The resulting formulation of a multiphase system will be termed the local instant formulation. Through the specification of the proper initial and boundary conditions and appropriate constitutive laws for the viscous stress tensor, the hydrodynamics of a multiphase system can in principle be obtained from the solution of the governing equations. 

In addition, a viscous stress tensor arises from viscous friction on the rods ... 

The constitutive equation for a Newtonian fluid in vector form is formulated by use of sophisticated tensor analysis (e.g., [11] [93] [184] [89] [2]). The following expression for the viscous stress tensor is applied ... 

The structure of the expression for totai is that of a bilinear form it consists of a sum of products of two factors. One of these factors in each term is a flow quantity (heat flux q, mass diffusion flux jc, momentum flux expressed by the viscous stress tensor o, and chemical reaction rate rr)- The other factor in each term is related to a gradient of an intensive state variable (gradients of temperature, chemical potential and velocity) and may contain the external force gc or a difference of thermodynamic state variables, viz. the chemical affinity A. These quantities which multiply the fluxes in the expression for the entropy production are called thermodynamic forces or affinities. 

Anyhow, (1.255) is not properly closed yet as the pressure drop variable is still undetermined. Therefore, before we can apply (1.255) we need to parameterize the losses in terms of known flow parameters in pipes, valves, fittings, and other internal flow devices. Assuming that the viscous stress tensor reduces to a single shear stress component per unit wall surface (e.g., [102] ]185]), Apf per unit cross sectional area can be related directly to the friction drag force on the tube wall surface —Cwt DL. That is, since the friction drag force in a horizontal tube with a constant rate of flow is given by Z p/(- )D, the wall shear stress 3uelds ... 

If we split the total stress tensor T into a pressure term pe and a viscous stress tensor o term, the vector equation can be expressed as ... 

However, without showing all the lengthy details of the method by which the two scalar functions are determined, we briefly sketch the problem definition in which the partial solution (2.247) is used to determine expressions for the viscous-stress tensor o and the heat flux vector q. 

Prom the previous section we recognize that term 1 on the RHS of (2.250) equals the pressure term, pe, as for equilibrium systems. It can be shown by kinetic theory that term 2 on the RHS corresponds to the viscous stress tensor O as defined by the pressure tensor (2.69). 

Note that the viscosity parameter p has been introduced as a prefactor in front of the tensor functions by substitution of the kinetic theory transport coefficient expression after comparing the kinetic theory result with the definition of the viscous stress tensor o, (2.69). In other words, this model inter-comparison defines the viscosity parameter in accordance with the Enskog theory. 

The nine components of the viscous stress tensor usigma are defined by ... 

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