Navier-Stokes equations constant viscosity

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... 

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... 

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 ... 

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

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. 

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)... 

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... 

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. 

The second is the law of conservation of momentum which, for a fluid of constant density and viscosity, is the Navier—Stokes equation... 

Let us deal with the equation of motion for turbulent flow. In the case of laminar flow under the condition of constant density and constant viscosity, the equation of motion is expressed by the Navier-Stokes equation as... 

Taylor bubbles (gas plugs) in a vertical tube move under the influence of surface tension, inertia, gravitation, and viscous effects. For a Newtonian fluid with constant viscosity and density these phenomena can be described by the Navier-Stokes equations for circular geometry using cylindrical coordinates ... 

A comparison of experimental CO2 viscosities obtained over a wide range of temperatures and pressures with that of previously reported viscosites (14) taken over that same range is illustrated in Figure 3. The three different experimental curves shown correspond to a particular calibration fluid, water or CO2, or the constant derived from the Navier-Stokes equation. 

The curve which corresponds to the calibration based on CO2 at 1000 psia was, as expected, the most accurate of the three with an absolute average percent deviation (AAPD) of 3.84. The constant derived from the Navier-Stokes equation was found to be extremely sensitive to the gap size between the aluminum cylinder and tube wall. For example, changing the tube diameter to values within the specified tolerance (- /- 0.0002 inches), changed viscosity calculations by as much as 17.60 percent. 

A.3 Navier-Stokes equations for an incompressible fluid of constant viscosity in cartesian coordinates... 

We limit our discussion here to laminar flows governed by the steady or unsteady, incompressible Navier-Stokes equations. In addition, we restrict ourselves to flows where the solution to the energy or the concentration equation does not influence the flow field, a circumstance not uncommon to isothermal constant viscosity liquid flows of relevance for many electrochemical systems. The incompressible, constant-property, Navier-Stokes equations are given below, with summation over repeated indices ... 

Derive the appropriate form of the Navier-Stokes equation for an electrically conducting Newtonian fluid with constant density and viscosity. This equation is called the magnetohydrodynamic momentum equation. Your derivation should begin from first principles. However, you may assume - but should state in appropriate places - the necessary properties of pressure, viscous stress, and convected time derivatives. 

Certain simple forms of the differential equations of fluid flow are given in later chapters. A complete treatment requires vector and tensor equations outside the scope of this text, the best known of which are the Navier-Stokes equations for fluids of constant density and viscosity. They underlie more advanced study of the phenomena and are discussed in many texts dealing with applied mechanics and transport processes. 

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