Continuity and Navier-Stokes Equations

This is the continuity equation for constant density flow. 

As was mentioned above, the Navier-Stokes equations are obtained by the appli-cation of the conservation of momentum principle to the fluid flow. The same control volume that was introduced above in the discussion of the continuity equation is considered and the conservation of momentum in each of the three coordinate directions is separately considered. The net force acting on the control volume in any of these directions is then set equal to the difference between the rate at which momentum leaves the control volume in this direction and the rate at which it enters in this direction. The net force arises from the pressure forces and the shearing forces acting on the faces of the control volume. The viscous shearing forces for two-dimensional flow (see later) are shown in Fig. 2.3. They are expressed in terms of the velocity field by assuming the fluid to be Newtonian and are then given by [4],[5]  

Using the expressions for the shearing stresses in the momentum balance for the control volume and taking the limit of the resulting equations as the size of the control volume tends to zero then gives the following set of equations [6],[7], 

These are the Navier-Stokes equations for steady constant fluid property flow. They are sometimes termed the x-, y-, and z-momentum equations, respectively. Basically these equations state that die net rate of change of momentum per unit mass in any direction (the left-hand side) is the sum of the net pressure force and the net viscous force in that direction. 

It should be clearly noted that since the fluid properties are being assumed constant, the set of eqs. (2.5) to (2.7) is independent of the temperature field and can, therefore, be solved independently of the energy equation to give the velocity and pressure distribution. 

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Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

Moreover the analysis retains both radial and circumferential pressure variations. Under these circumstances the continuity and Navier-Stokes equations reduce to... 

The temperature, composition, and density are presumed to have only radial variations. The pressure, however, is allowed to vary throughout the flow, but in a very special way as will be derived shortly. Also the magnitude of the pressure variations is assumed to be small compared to the mean thermodynamic pressure. Using these assumptions, and invoking the Stokes hypothesis to give X = —2p./3, we can reduce the mass-continuity and Navier-Stokes equations to the following ... 

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

In the microfluid dynamics approaches the continuity and Navier-Stokes equation coupled with methodologies for tracking the disperse/continuous interface are used to describe the droplet formation in quiescent and crossflow continuous conditions. Ohta et al. [54] used a computational fluid dynamics (CFD) approach to analyze the single-droplet-formation process at an orifice under pressure pulse conditions (pulsed sieve-plate column). Abrahamse et al. [55] simulated the process of the droplet break-up in crossflow membrane emulsification using an equal computational fluid dynamics procedure. They calculated the minimum distance between two membrane pores as a function of crossflow velocity and pore size. This minimum distance is important to optimize the space between two pores on the membrane... 

The computational fluid dynamics investigations listed here are all based on the so-called volume-of-fluid method (VOF) used to follow the dynamics of the disperse/ continuous phase interface. The VOF method is a technique that represents the interface between two fluids defining an F function. This function is chosen with a value of unity at any cell occupied by disperse phase and zero elsewhere. A unit value of F corresponds to a cell full of disperse phase, whereas a zero value indicates that the cell contains only continuous phase. Cells with F values between zero and one contain the liquid/liquid interface. In addition to the above continuity and Navier-Stokes equation solved by the finite-volume method, an equation governing the time dependence of the F function therefore has to be solved. A constant value of the interfacial tension is implemented in the summarized algorithm, however, the diffusion of emulsifier from continuous phase toward the droplet interface and its adsorption remains still an important issue and challenge in the computational fluid-dynamic framework. 

In terms of these dimensionless variables, the equations governing the velocity field, i.e., the continuity and Navier-Stokes equations, become ... 

Consider the velocity field first. It is governed by the continuity and Navier-Stokes equations which, subject to the assumptions introduced above, are, as previously presented ... 

Computational fluid dynamic models (CFD) Computational models of fluid flow based on numerical solution of the continuity and Navier-Stokes equations (in either instantaneous or, more commonly, some type of averaged form). 

Dandy and Dwyer [30] computed numerically the three-dimensional flow around a sphere in shear flow from the continuity and Navier-Stokes equations. The sphere was not allowed to move or rotate. The drag, lift, and heat flux of the sphere was determined. The drag and lift forces were computed over the surface of the sphere from (5.28) and (5.33), respectively. They examined the two contributions to the lift force, the pressure contribution and the viscous contribution. While the viscous contribution always was positive, the pressure contribution would change sign over the surface of the sphere. The pressure... 

The continuity and Navier-Stokes equations in cylindrical and spherical coordinate systems are given in Supplement 5. 

A pseudo-homogeneous, two-dimensional reactor model for membrane reactors consists of the total gas-phase continuity and Navier-Stokes equations augmented with gas-phase component mass balances and the overall energy balance. 

If the above two equations are substituted in the continuity and Navier-Stokes equations and inviscid flow is assumed, after several steps of derivations, time-averaged second-order variation is given as follows [1] ... 

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. 

For incompressible fluid with constant density, the continuous and navier-stokes equations are ... 

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. 

The governing equations for the discrete phase (particles) have been already discussed in Section 7.1 and the description of the CFD method is given in Chapter 6. In brief, in this method, the continuity and Navier-Stokes equations for the fluid phase in the fluid-solid two-phase model, for an incompressible fluid with constant density, are given by Equations 7.32 and 7.33, respectively ... 

The following characteristic scales, radius, R, velocity scale. Us, timescale, R/ Us, pressure scale, fiUs/R, and fiUsIR are used to make the continuity and Navier-Stokes equation dimensionless. The dimensionless density, viscosity, thermal conductivity, and heat capacity distributions are defined as... 

The continuity and Navier-Stokes equations, the standard k-e turbulence model, and the transport equations for species concentration and enthalpy are solved using a three-dimensional computational domain for the inlet region (Figure 9.30(a)). 

Any experimental validation of that assumption is difficult, but recently direct numerical simulation of turbulent reactive flows appeared possible, for very moderate Reynolds number, however. Direct numerical simulations use the primitive equations themselves, the continuity and Navier Stokes equations, jointly with one or two diffusion-reaction equations like (1) a very big computer is required for the integration... 

Following the approach of Chouly and co-workers (1), a simplified three-dimensional flow configuration representing the pharyngeal airway is illustrated in Fig. 1 with the tongue idealised as a pressurised shell. With Strouhal Numbers in the order of 10 (5), the flow is considered quasi-steady and is characterised by the standard continuity and Navier-Stokes equation. For laminar conditions, these reads ... 

Two-dimensional (2D) models represent a second tier of spatial complexity with respect to sediment transport models. Models in this class have become more prevalent during the last couple of decades due to advancements in computer hardware and software capabilities. Two-dimensional models typically solve the depth-averaged flow continuity and Navier-Stokes equations with respect to hydrodynamic behavior and mass balance equations with respect to sediment transport. Computational methods employed in 2D models include finite difference, finite element, and finite volume. Examples of 2D models include Environmental Eluid Dynamics Code (EFDC), SEDZLJ, SEDZL, USTARS, MIKE21, and Delft 2D. 

The continuum fluid field is calculated from the continuity and Navier— Stokes equations based on the local mean variables over a computational ceU, which can be written as (Zhou et al., 2010a) ... 

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