Viscosity Navier-Stokes equation

Expanded into cylindrical coordinates, the constant-viscosity Navier-Stokes equations are given as... 

The pressure does not appear directly in the vorticity-transport equation. Thus, it is apparent that the convective and diffusive transport of vorticity throughout a flow cannot depend directly on the pressure field. Nevertheless, it is completely clear that pressure affects the velocity field, which, in turn, affects the vorticity. By taking the divergence of the incompressible, constant-viscosity Navier-Stokes equations, a relationship can be derived among the velocity, pressure, and vorticity fields. Beginning with the Navier-Stokes equations as... 

Consider the behavior of the Navier-Stokes equations for the two-dimensional flow in a conical channel as illustrated by Fig. 3.17. Begin with the constant-viscosity Navier-Stokes equations written in the general vector form as... 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

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... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

The Navier-Stokes equation and the enthalpy equation are coupled in a complex way even in the case of incompressible fluids, since in general the viscosity is a function of temperature. There are, however, many situations in which such interdependencies can be neglected. As an example, the temperature variation in a microfluidic system might be so small that the viscosity can be assumed to be constant. In such cases the velocity field can be determined independently from the temperature field. When inserting the computed velocity field into Eq. (77) and expressing the energy density e by the temperature T, a linear equahon in T is... 

We thus find that the lattice BGK model describes, to second order in St, the fluid according to the Navier-Stokes equation with a viscosity... 

When considering flow of a liquid in contact with a solid surface, a basic understanding of the hydrodynamic behavior at the interface is required. This begins with the Navier-Stokes equation for constant-viscosity, incompressible fluid flow, such that Sp/Sf = 0,... 

Runnels and Eyman [41] report a tribological analysis of CMP in which a fluid-flow-induced stress distribution across the entire wafer surface is examined. Fundamentally, the model seeks to determine if hydroplaning of the wafer occurs by consideration of the fluid film between wafer and pad, in this case on a wafer scale. The thickness of the (slurry) fluid film is a key parameter, and depends on wafer curvature, slurry viscosity, and rotation speed. The traditional Preston equation R = KPV, where R is removal rate, P is pressure, and V is relative velocity, is modified to R = k ar, where a and T are the magnitudes of normal and shear stress, respectively. Fluid mechanic calculations are undertaken to determine contributions to these stresses based on how the slurry flows macroscopically, and how pressure is distributed across the entire wafer. Navier-Stokes equations for incompressible Newtonian flow (constant viscosity) are solved on a three-dimensional mesh ... 

Application of Newton s second law of motion to an infinitesimal element of an incompressible Newtonian fluid of density p and constant viscosity p, acted upon by gravity as the only body force, leads to the Navier-Stokes equation of motion ... 

Ibrahin (II) has published an addition to the Navier-Stokes equations which was intended to modify them for use with non-Newtonian fluids. The modification was only for the purpose of taking fluid elasticity into account, a factor which does not appear to be necessary for the majority of materials showing non-Newtonian behavior. Instead, a redevelopment of the original equations to allow for variations in viscosity with shear rate is required, an approach that would appear to be very complex but perhaps rewarding. 

The components of this equation are sometimes referred to as the Navier-Stokes equations when the viscosity is set equal to zero (inviscid fluid), then these equations reduce to the Euler equations. 

At the phenomenological level, there are enough further relations between the 14 variables to reduce the number to 5 and make the problem determinate. These further relations are the thermodynamic ones and Stokes and Newton s laws of viscosity and heat flow. These lead from the transport equations to the Navier-Stokes equations. It is noted that these are irreversible. 

The viscosity of the medium is t, and 1 is the unit tensor. (The Oseen tensor is the Green s function for the Navier-Stokes equation under the conditions that the fluid is incompressible, convective effects can be neglected, and inertial effects coming from the time derivative can be neglected.)... 

Viscous Forces In the momentum equation (Navier-Stokes equation), forces F acting on the system result from viscous stresses. It is necessary to relate these stresses to the velocity field and the fluid s viscosity. This relationship follows from the stress and strain-rate tensors, using Stokes postulates. 

The final objective of this chapter was to develop quantitative relationships between a fluid s strain-rate and stress fields. Expressions for the strain rates were developed in terms of velocities and velocity gradients. Then, using Stokes s postulates, the stress field was found to be proportional to the strain rates and a physical property of the fluid called viscosity. The fact that the stress tensor and strain-rate tensor share the same principal coordinates is an important factor in applying Stokes s postulates. The stress-strain-rate relationships are fundamental to the Navier-Stokes equations, which describe conservation of momentum in fluids. 

In later chapters we discuss the evaluation of transport properties for multicomponent mixtures of gases. At this point, however, the intent is to provide only a brief introduction on viscosity, since it is the principal fluid property that appears in the Navier-Stokes equations. The discussion here is limited to single-component fluids. 

There are a very wide variety of theories and approaches to determine transport properties and to report their functional dependencies [178,332]. The brief discussion here serves only to establish the basic functional dependencies, and thus facilitate understanding the role of viscosity in the Navier-Stokes equations. 

As discussed in Section 3.3, viscosity varies as a function of temperature and pressure. For isothermal, uniform-composition flows, viscosity is a constant. For many situations of interest, in which temperature and composition vary over only relatively small ranges, it can be appropriate to consider constant properties. For gases, viscosity is roughly proportional to T0-645—a relatively weak dependence. Moreover there is essentially no pressure dependence. In any case it is instructive to see how the Navier-Stokes equations behave in the limiting case of constant viscosity. 

Incompressible Navier-Stokes Equations Given that the flow in gas-dynamically incompressible, V V 0, the Navier-Stokes equations reduce to the following. As stated here, the viscosity is not presumed to be a constant. 

For the purpose of understanding pressure filtering, attention may be restricted to the single-component, constant-property, nonreacting equations for a perfect gas. Introducing the nondimensional variables into the vector forms of the mass-continuity, constant-viscosity Navier-Stokes, and perfect-gas thermal-energy equations yields the following nondimensional system ... 

A vorticity-transport equation can be derived by taking the taking the vector curl of the full Navier-Stokes equations. For incompressible flows with constant viscosity, the vorticity-transport equation can be expressed in a form that is quite similar to the other transport equations. Begin with the full Navier-Stokes equations, which for constant viscosity can be written in compact vector form as (Eq. 3.61)... 

By substitution of the velocity field into the constant-viscosity, incompressible, Navier Stokes equations, determine an expression for the pressure field around the sphere (i.e p(z, r)). 

Beginning with the constant-viscosity, incompressible Navier Stokes equations, write a reduced form of the radial and circumferential momentum equations that is appropriate to represent the fully developed flow in the circular channel. 

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