Cauchy momentum equation

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

These forms of the equation of motion are commonly called the Cauchy momentum equations. For generalized Newtonian fluids we can define the terms of the deviatoric stress tensor as a function of a generalized Newtonian viscosity, p, and the components of the rate of deformation tensor, as described in Table 5.3. 

Equations 2.6a-c are known collectively as the Cauchy momentum equation. Before we discuss the meanings of the terms in the equations, it is useful to note that the y-direction equation can be obtained from the x-direction equation simply by permuting indices x, y, z y, z, x. Similarly, the z direction is obtained from y, z, x ... 

The terms on the left side of the Cauchy momentum equation sum to the rate of change of momentum, or inertia, on a unit volume basis. There are four terms because momentum in a given direction changes as the velocity changes with time and as a fluid element changes direction. The first term on the right is the rate of... 

The Cauchy momentum equation can be transformed to other coordinate systems that might be more useful for particular problems. Flow in a capillary, for example, will be described most naturally in cylindrical (r, 9, z) coordinates. The momentum equation is shown in Table 2.2 in the three commonly used coordinate systems. The equation is often written in the shorthand vector form... 

The terms on the left side of the Cauchy momentum equation, which we have written symbolically as p(DylDt), represent the contribution of inertial effects to the momentum balance. The inertial effects are negligible relative to the stresses generated within the fluid in most polymer processing operations, and to a very good approximation the inertial terms can usually be dropped. (Commercial fiber spinning is an exception.) For a Newtonian fluid the relative contribution of inertial and viscous terms is expressed as a dimensionless group known as the Reynolds number,... 

With the assumed velocity field, all terms on the left side of the three components of the Cauchy momentum equation. Equations 2.6a-c and Table 2.2, are zero. The equations in terms of the equivalent pressure [Pg.38]

We assume axisymmetry and a steady state. Commercial fiber spinning takes place at speeds on the order of 4,000 m/min (240 km/hr) and greater, so in this case inertia is important, as is aerodynamic drag this is perhaps the only polymer melt process where inertia must be considered. We therefore use the full Cauchy momentum equations, and we write the basic equations in cylindrical coordinates (Tables 2.1,2.2, and 2.5) as follows ... 

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