Shear stress tensor

The shear stress tensor t for a Newtonian fluid is the product of the deformation rate tensor and the viscosity. 

For the momentum conservation of a single-phase fluid, the momentum per unit volume / is equal to the mass flux pU. The momentum flux is thus expressed by the stress tensor i/r = (pi — t). Here p is the static pressure or equilibrium pressure / is a unit tensor and r is the shear stress tensor. Since <1> = —pf where / is the field force per unit mass, Eq. (5.12) gives rise to the momentum equation as... 

The Ellis model (41), is a three-parameter model, in which the non-Newtonian viscosity is a function of the absolute value of the shear stress tensor, x,... 

For steady-state shearing flows, the relationship between the shear-stress tensor and the shear-rate tensor is given by Criminale-Ericksen-... 

Two traditional approaches to the closure of the Reynolds equation are outlined below. These approaches are based on Boussinesq s model of turbulent viscosity completed by Prandtl s or von Karman s hypotheses [276, 427]. For simplicity, we confine our consideration to the case of simple shear flow, where the transverse coordinate Y = Xi is measured from the wall (the results are also applicable to turbulent boundary layers). According to Boussinesq s model, the only nonzero component of the Reynolds turbulent shear stress tensor and the divergence of this tensor are defined as... 

Figure 1.1 illustrates schematically a shear strain, y, produced by applying a force F to the upper face of a cubic volume of a fluid. Relative to the three axes shown in Figure 1.1 (i.e., 1 = direction of flow, 2 = shear gradient, 3 = vorticity axis), one may define three materials parameters relating the shear stress tensor, a, to the shear rate, Y [Macosko, 1994] ... 

The heat transfer rate from a flush-mounted shear stress tensor depends on the near-wall flow, i.e., the magnitude of the velocity gradient. For a laminar two-dimensional thermal boundary layer developing over the heated sensor with an approaching linear velocity profile (Fig. 2) and negligible free convectimi effect, the heat loss from the thermal element can be derived from the thermal boundary layer equation as... 

The solid phase shear stress tensor r is defined by ... 

Here, n is the unit normal vector, f is the shear stress tensor, and p is the pressure. The first term in the right-hand side is the tangential component, and the second term is the normal component of the force. 

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