The Navier-Stokes equation

To use Eq. 7.8, it is normally necessary to replace the normal-stress and shear-stress terms with terms involving measurable properties, such as viscosities and velocities. No one has yet found a way to do this without introducing very severe restrictive assumptions. The set of assumptions commonly used is as follows  

The three-dimensional stresses in a flowing, constant-density newtonian fluid have the same form as the three-dimensional stress in a solid body that obeys Hooke s j law (i.e., a perfectly elastic, isotropic solid). 

If we make these assumption, then it can be shown that 

The derivation ofjthis equation is shown in numerous texts [6, p. 66 7]. The intuitive meaning of three of the terms on the right is obvious. The fourth is a bit harder to see. The dPIdx term is the result of pressure force on the infinitesimal cube The fi dW ldy ) and jx d VJdz ) terms represent the net shear force on the cube due to changes of the x velocity in the y and z directions. One can see how these arise by assuming that the shear forces are independent of each other and by substituting Newton s law of viscos ity, Eq. 1.5, in the dr ldy and dr Jdz terms in Eq. 7.79. 

This is the differential momentum balance for the x direction, subject to the list of assumptions given above. Analogous balances can be made up for the y and z directions the three together make up the Navier-Stokes equations. 

We will presume a Newtonian fluid and neglect the normally very small bulk viscosity. The statement (3.56) for the stress tensor is then transformed into Stokes formulation, the so-called Stokes hypothesis  

3 Convective heat and mass transfer. Single phase flow 

Putting this expression into Cauchy s equation of motion, (3.48), yields the so-called Navier-Stokes equation 

For an incompressible fluid we have dwk/dxk = dwi/dxi = 0. Furthermore, if we presume a constant viscosity, the equation is simplified to 

The equation clearly shows that the forces acting on a fluid element are made up of body, pressure and viscosity forces. A momentum balance exists for each of the three coordinate directions j = 1, 2, 3, so that (3.58) and (3.59) each represent three equations independent of each other. (3.59) can be seen written out in Appendix A3. Its formulation for cylindrical coordinates is also given there. 

Having developed the relations between the components of stress and the components of velocity at a given point within the body of a simple viscous fluid, let us consider the dynamical equations of motion. Let the center of the rectangular element Sx y6z be at the point x, y, z. Resolution of the internal forces parallel to the x-direction gives a traction of (30j /3x) 6x 5y6z due to normal stress on the yz-plane and traction of (3T /3y)6y6x6z and Sy x due to tangential 

With the aid of Eqns 3-15, Eqns 3-17 can be reduced to a set of relations involving only the internal pressure p and the three components of velocity, u, v, and w. 

These are the general Navier-Stokes equations in Cartesian coordinates. There are only three equations for the four variables p, u, v, and w, but a fourth relation is supplied by the continuity equation  

Equations 3-18 can be simplified considerably by applying some restrictions derived from the physical properties of the fluid and the geometry of a lubricating film. These reduce the Navier-Stokes equations to a set of two differential equations from which a generalized version of the Reynolds equation is obtained. 

The height i/ of the fluid film is very small compared to the 

Rather than setting up a force-momentum balance for a particular flow problem as was done in Chapter 1, general equations, known as the Navier-Stokes equations, may be formulated. Before discussing the Navier-Stokes equations, it is necessary to consider some related matters. 

Let / represent some property of the flow, for example the velocity, temperature or density of the fluid. In general / is a function of the time t and the spatial coordinates x, y, z. Then the total derivative of f with respect to r is given by 

Denoting these derivatives by wx, wy, wz, respectively, equation A.l can be written as 

In the special case when the velocity components are those of the fluid, the total rate of change of 4 is denoted by D /Dr, which is known as the substantive derivative  

In equation A.3, vx, vy, vz are the velocity components of the fluid. Thus, /Dt gives the rate of change of tb for a material element as it flows along. This is known as differentiation following the flow. 

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It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

These are the two components of the Navier-Stokes equation including fluctuations s., which obey the fluctuation dissipation theorem, valid for incompressible, classical fluids ... 

Ocily n. - 1 of the n equations (4.1) are independent, since both sides vanish on suinming over r, so a further relation between the velocity vectors V is required. It is provided by the overall momentum balance for the mixture, and a well known result of dilute gas kinetic theory shows that this takes the form of the Navier-Stokes equation... 

Clearly then, the continuum approach as outlined above is faulty. Furthermore, since our erroneous result depends only on the non-slip boundary condition for V together with the Navier-Stokes equation (4.3), one or... 

Taylor, C. and Hood, P., 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73-100. 

Lee, R. L., Gresho, P. M. and Sani, R. L., 1979. Smoothing techniques for certain primitive variable solutions of the Navier-Stokes equations. Int. J. Numer. Methods Eng. 14, 1785-1804. 

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

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. 

Exact Solutions to the Navier-Stokes Equations. As was tme for the inviscid flow equations, exact solutions to the Navier-Stokes equations are limited to fairly simple configurations that aHow for considerable simplification both in the equation and in the boundary conditions. For the important situation of steady, fully developed, laminar, Newtonian flow in a circular tube, for example, the Navier-Stokes equations reduce to... 

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part ... 

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... 

The principle can be illustrated by examining the Navier-Stokes equation for two-dimensional incompressible flow. The x-component of the equation is... 

Dynamic meteorological models, much like air pollution models, strive to describe the physics and thermodynamics of atmospheric motions as accurately as is feasible. Besides being used in conjunction with air quaHty models, they ate also used for weather forecasting. Like air quaHty models, dynamic meteorological models solve a set of partial differential equations (also called primitive equations). This set of equations, which ate fundamental to the fluid mechanics of the atmosphere, ate referred to as the Navier-Stokes equations, and describe the conservation of mass and momentum. They ate combined with equations describing energy conservation and thermodynamics in a moving fluid (72) ... 

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... 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

The Navier-Stokes equations are greatly simplified when it is noted that = = 0 and dv /dx = three components are written in terms of the equivalent pressure P ... 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

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 Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie 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/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

When the continmty equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = X, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side ... 

Porous media is typically characterized as an ensemble of channels of var ious cross sections of the same length. The Navier-Stokes equations for all chaimels passing a cross section normal to the flow can be solved to give ... 

The Navier-Stokes equation with viscous stresses for the x, y, and z directions are given by ... 

Assuming laminar flow for a linear momentum equation in the a direction (an approximation from the Navier-Stokes equations) gives... 

The full concentration equation for the contaminant may be simplified in the same manner as the Navier-Stokes equations to derive a boundary-layer approximation for the concentration, namely,... 

The incompressible, time-averaged continuity and the Navier-Stokes equations can be written as... 

The Navier-Stokes equation in the direction of gravity (> -direction) is given by the expression... 

The dimensionless numbers are important elements in the performance of model experiments, and they are determined by the normalizing procedure ot the independent variables. If, for example, free convection is considered in a room without ventilation, it is not possible to normalize the velocities by a supply velocity Uq. The normalized velocity can be defined by m u f po //ao where f, is the height of a cold or a hot surface. The Grashof number, Gr, will then appear in the buoyancy term in the Navier-Stokes equation (AT is the temperature difference between the hot and the cold surface) ... 

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