Navier-Stokes equation definition

Conservation Law for a System Conservation laws (e.g., Newton s second law or the conservation of energy) are most conveniently written for a system, which, by definition, is an identified mass of material. In fluid mechanics, however, since the fluid is free to deform and mix as it moves, a specific system is difficult to follow. The conservation of momentum, leading to the Navier-Stokes equations, is stated generally as... 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

In Chapter 2 considerable effort is devoted to establishing the relationship between the stress tensor and the strain-rate tensor. The normal and shear stresses that act on the surfaces of a fluid particle are found to depend on the velocity field in a definite, but relatively complex, manner (Eqs. 2.140 and 2.180). Therefore, when these expressions for the forces are substituted into the momentum equation, Eq. 3.53, an equation emerges that has velocities (and pressure) as the dependent variables. This is a very important result. If the forces were not explicit functions of the velocity field, then more dependent variables would likely be needed and a larger, more complex system of equations would emerge. In terms of the velocity field, the Navier-Stokes equations are stated as... 

Taking the vector curl of the right-hand side causes the first and last terms to drop out, since the curl of the gradient vanishes. However, for variable density, the left-hand side expands to long, complex, and not-too-useful expression (see Section A.14). Therefore let us restrict attention to incompressible flows, namely constant density. The curl of the incompressible Navier-Stokes equation, incorporating the definition of vorticity u = VxV, yields... 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters De defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, De = 2 Q. L/ nAPY . Equivalent mameters are not the same as hydraulic diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/IkDeH). Equivalent diameter De is not to be used in the friction factor and Reynolds number / 16/Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydrauhc diameter Dh used for turbulent flow. 

The latter comes from the matching requirement that the magnitude of the pressure in the end regions must be comparable with that in the core. The definitions (6-130) lead to the following nondimensionalized form of the Navier-Stokes equations for the end regions ... 

The membranous SCC duct is modeled as a section of a rigid torus filled with an incompressible Newtonian fluid. The governing equations of motion for the fluid are developed from the Navier-Stokes equation. Refer to the nomenclature section for definition of aU variables, and Figure 64.5 for a cross section of the SCC and membranous utricle sack. 

Thus, the complete definition of a Newtonian fluid is that it not only possesses a constant viscosity but it also satisfies the condition of equation (1.9), or simply that it satisfies the complete Navier-Stokes equations. Thus, for instance, the so-called constant viscosity Boger fluids [Boger, 1976 Prilutski et al., 1983] which display constant shear viscosity but do not conform to equation (1.9) must be classed as non-Newtonian fluids. 

These definitions of Vj and Vy can then be used in the x and y components of the differential equation of motion, Eqs. (3.7-36) and (3.7-37), with Vj = 0 to obtain a differential equation for ip that is equivalent to the Navier-Stokes equation. Details are given elsewhere (B2). 

Re is a dimensionless parameter. Its definition constitutes a powerful tool for the transfer of information from experiments performed at the laboratory scale on various hydrodynamic phenomena to very large (e.g. airplanes) or very small (particles) scales. Via eq 1.36 Re also provides the possibilty of further simplification of the Navier-Stokes equation if the value of Re is very large or very small. 

In contrast, for high Reynolds numbers, the pressure term in the Navier-Stokes equation is of the same order of magnitude as the nonlinear term (which itself, by definition, is large compared to the viscous term), and the viscous force is negligible. We therefore have successively ... 

In the two-fluid formulation, the velocity field of each of these two continuous phases is described by its own continuity equation and Navier—Stokes equation (e.g., Anderson and Jackson, 1967 Rietema and Van den Akker, 1983 Sokolichin et al, 2004 Tabib et al, 2008). Each of these Navier-Stokes equations comprises a mutual phase interaction force. Only in the size, or numerical value, of this phase interaction force, particle size may come out. However, in selecting the computational grid or the numerical technique for solving the flow fields of the two continuous phases, particle size is completely irrelevant—due to the definition of mutually interpenetrating continua. 

Equation 1 is the Poisson equation. This equation should be solved in order to obtain the electric potential distribution in the computational domain. On the right hand of this equation, the term F Z] iZ,c, shows the gradient influence of the co-ions and counterions on the electric potential inside the domain. The electric field is the gradient of the electric potential (Eq. 2). Equation 3 is the Nemst-Planck equation, where the definition of ionic flux is given by Eq. 4. On the right-hand side of this equation, (m c,), (D, Vc,), and (z,/t,c,V( ) represent flow field (the electroosmosis), diffusion, and electric field (the electrophoresis), respectively, which contribute to the ionic mass transfer. The ionic concentrations of each species can be found by solving these two equations. Equations 5 and 6 are the Navier-Stokes and the continuity equations, respectively, which describe the velocity field and the pressure gradient in the computational domain. 

To estimate (ys), one may note that the velocity profile in the liquid is influenced by the soUd zones only in a region of their size, a, in all directions (see Fig. 1) this behavior actually reflects the Laplacian character of the Stokes equation obeyed by the fluid velocity. One, therefore, expects (ys) U/a, where U is the slip velocity of the fluid on the shear-free zones, so that we eventually obtain Ff = A(f>sr ilJ/a. Now if one recalls the definition of the effective slip length, as given by the Navier BC, Ff also reads Ff = ArnV/bee, with F [/ the averaged slip velocity over the superhydrophobic surface. Combining the two independent estimates, one deduces ... 

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