Momentum and Navier-Stokes Equations

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 fluid mechanics texts (e.g., Slattery [ibid.] Denn Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) leads to the Navier-Stokes equations, whose three Cartesian components are 

Fluid f statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, Vp = pg. Letting z be directed vertically upward, so thatp = —g where g is the gravitational acceleration (9.806 m2/s), the pressure field is given by 

This equation applies to any incompressible or compressible static fluid. For an incompressible liquid, pressure varies linearly with depth. For compressible gases, p is obtained by integration accounting for the variation of p with z. 

The force exerted on a submerged planar surface of area A is given by F = p, A where pc is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example, Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. 

Examples Four examples follow, illustrating the application of the conservation equations to obtain useful information about fluid flows. 

---

Note that these equations are obtained formally from the Cauchy momentum and Navier-Stokes equations in Tables 2.3 and 2.4, respectively, by setting p = 0. This is consistent with the notion that mass (density) enters the momentum balance only through the inertial terms. 

Isotropy of the Momentum Flux Density Tensor If we trace back our derivation of the macroscopic LG Euler s and Navier-Stokes equations, we see that the only place where the geometry of the underlying lattice really enters is through the form for the momentum flux density tensor, fwhere cp = x ) + y ), k = 1,..., V... 

As a solution to momentum transport (Navier-Stokes equation) is usually not available for bioreactor velocity field, we are left with mass and heat balances. A local balance describing mixing, flow and interface transfer, including reaction is known as the dispersed model (3) ... 

The key of the simulation is to solve the continuity and navier-stokes equations in an Eulerian Cartesian coordinate system. Driving forces from the fluid flow are applied to the particles as body forces. These forces are also added to the fluid equations and cause change in momentum, as reflected by the change in the pressure gradient in the flow direction. 

To preview the results somewhat, it will be shown that the general form of the transport equations contains expressions for the property flux variables (momentum flux P, energy flux q, and entropy flux s) involving integrals over lower-order density functions. In this form, the transport equations are referred to as general equations of change since virtually no assumptions are made in their derivation. In order to finally resolve the transport equations, expressions for specific lower-ordered distribution functions must be determined. These are, in turn, obtained from solutions to reduced forms of the Liouville equation, and this is where critical approximations are usually made. For example, the Euler and Navier-Stokes equations of motion derived in the next chapter have flux expressions based on certain approximate solutions to reduced forms of the Liouville equation. Let s first look, however, at the most general forms of the transport equations. 

If we substitute Equations 6.33 through 6.38 into Equations 6.30 through 6.32, we get the complete momentum or Navier-Stokes equations written in the conservation form of the Cartesian coordinate system for time-dependent, compressible, and viscous flow in terms of velocities. The x component of the momentum equations is... 

Since the computational model is time independent, a stationary solver can be used. The continuity equation together with the momentum equations is solved separately from the species equations to smoothly achieve converged solution. Returned values from the continuity and Navier-Stokes equations are then automatically called by the chosen solver and used in the convection-diffusion equation. 

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. 

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... 

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. 

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) ... 

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 ... 

Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations or continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conseiwation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

General equations of momentum and energy balance for dispersed two-phase flow were derived by Van Deemter and Van Der Laan (V2) by integration over a volume containing a large number of elements of the dispersed phase. A complete system of solutions of linearized Navier-Stokes equations... 

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. ... 

A standard approach to modeling transport phenomena in the field of chemical engineering is based on convection-diffusion equations. Equations of that type describe the transport of a certain field quantity, for example momentum or enthalpy, as the sum of a convective and a diffusive term. A well-known example is the Navier-Stokes equation, which in the case of compressible media is given as... 

---