Mathematical modeling Stokes equation

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. 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

Mathematically, the PPDF method is based on the Finite Volume Method of solving full Favre averaged Navier-Stokes equations with the k-e model as a closure for the Reynolds stresses and a presumed PDF closure for the mean reaction rate. 

Our intent here is not to suggest a solution method but rather to use the stream-function-vorticity formulation to comment further on the mathematical characteristics of the Navier-Stokes equations. In this form the hyperbolic behavior of the pressure has been lost from the system. For low-speed flow the pressure gradients are so small that they do not measurably affect the net pressure from a thermodynamic point of view. Therefore the pressure of the system can simply be provided as a fixed parameter that enters the equation of state. Thus pressure influences density, still accommodating variations in temperature and composition. Since the pressure or the pressure gradients simply do not appear anywhere else in the system, pressure-wave behavior has been effectively filtered out of the system. Consequently acoustic behavior or high-speed flow cannot be modeled using this approach. 

P 7] The topic has only been treated theoretically so far [28], A mathematical model was set up slip boundary conditions were used and the Navier-Stokes equation was solved to obtain two-dimensional electroosmotic flows for various distributions of the C, potential. The flow field was determined analytically using a Fourier series to allow one tracking of passive tracer particles for flow visualization. It was chosen to study the asymptotic behavior of the series components to overcome the limits of Fourier series with regard to slow convergence. In this way, with only a few terms highly accurate solutions are yielded. Then, alternation between two flow fields is used to induce chaotic advection. This is achieved by periodic alteration of the electrodes potentials. 

Mathematical modeling and computer simulation have been applied for various flow studies in rectangular microchannels (see Table 3.1). An equation to describe the flow in a rectangular channel has been given [124]. Simulation of fluid flow can be conducted by solving the coupled Poisson and Navier-Stokes equation for fluid velocity [532]. However, this complicated computation has been simplified by solving the Laplace equation for the electric field because it is proportional to fluid velocity [321]. 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

The modelling example of the previous section shows that to simplify the general mathematical model of the studied process, the real flow in the filter unit has been considered in terms of its own simplified model. Indeed, it is difficult to understand why we have used a flow model, when in fact, for the flow characterization, we already have the Navier-Stokes equations and their expression for the computational fluid dynamics. To answer this question some precisions about the general aspects of the computational fluids dynamics have to be given. 

Solving the previous set of equations, especially with realistic boundary conditions, is a formidable task and a lot of issues are still unanswered. This is not surprising because of the complexity of the equations, and because of their recent derivation, around 1950 for the first nonlinear models, the Oldroyd models. On the other hand, the mathematical theory for the Euler and the Navier-Stokes equations for incompressible Newtonian fluids is still not complete though these equations were derived in 1755 md 1821 respectively ... 

Each of these different types of flows is governed by a set of equations having special features. It is essential to understand these features to select an appropriate numerical method for each of these types of equations. It must be remembered that the results of the CFD simulations can only be as good as the underlying mathematical model. Navier-Stokes equations rigorously represent the behavior of an incompressible Newtonian fluid as long as the continuum assumption is valid. As the complexity increases (such as turbulence or the existence of additional phases), the number of phenomena in a flow problem and the possible number of interactions between them increases at least quadratically. Each of these interactions needs to be represented and resolved numerically, which may put strain on (or may exceed) the available computational resources. One way to deal with the resolution limits and... 

Antille et al. [79, 80] have shown how measurements of the anodic current fluctuations that arise from oscillation of the electrolyte-metal interface beneath the anode, can be used, in conjunction with their mathematical model, to deduce the steady velocity fields in a cell. The method has the advantage that there is no need to determine the electromagnetic forces, or to solve the Navier-Stokes equations... 

Polezhaev, V. I., Bune, A. V., Verezub, N. A., et al., Mathematical Modeling of Convective Heat and Mass Transfer on the Base of the Navier-Stokes Equations, Nauka, Moscow, 1987 [in Russian],... 

At the lower temperature (783 K open symbols in Fig. 70) a substantially different behavior is observed. The imide band (A in Fig. 69 bottom) decreases quasi-linearly with the elapsed time (see Eq. 24). The aromatic band (V in Fig. 70 top) is complex, revealing two distinct decomposition patterns. At the beginning (first half) of the normalized time a slow linear decrease is observed, followed by a fast decrease. The decrease of the imide band and the change of the aromatic band in the second part of the curve are typical for a film diffusion-controlled reaction of shrinking particles in a gas flow in the Stokes regime. To confirm this observation a new mathematical model is used to fit the curves [321]. Starting from Eq. 20, the reaction velocity ks is substituted with kg=D Rf1 [321]. D is the diffusion velocity and kg the mass transfer coefficient between fluid and particle. The differential equation is solved and the time necessary to reduce a particle from a starting radius R0 to Rt is obtained [see Eq. (22)] [321],... 

The starting point is a Mathematical Model, i.e. the set of equations and boundary conditions, which covers the physics of the flow most suitable. For some problems the governing equations are known accurately (e.g. the Navier-Stokes equations for incompressible Newtonian fluids). However for many phenomena (e.g. turbulence or multiphase flow) and especially for the description of ceramic materials or wall slip phenomena the exact equations are either not available or a numerical solution is not feasible. 

So far, all governing equations are presented. It can be seen that a closed mathematical model of the combined pressure-driven flow and electroosmotic flow in microfluidic channels should include the continuity equarirm, the Stokes equation, the Poisson equation, the Boltzmann distribution, and the Laplace equation. A set of the governing equations for such combined pressure-driven flow and electroosmotic flow can be rewritten in dimensional form as... 

Mathematical formalism has been developed using semi-empirical considerations [36, 37]. Computer simulation smdies show that resulting equation predicts oscillations. Attempt has been made to provide justification on the bases of Navier-Stokes equation but it is open to question. Dimensional analysis has recently been employed for investigating the phenomena [31]. Flow dynamics and stability in a density oscillator have been examined by Steinbock and co-workers [38], They have related it to Rayleigh-Taylor instability of two different dense viscous liquids. A theoretical description has been presented which is based on a one-fluid model and a steady state approximation for a two-dimensional flow using Navier-Stokes equation. However, the treatment is quite complex and cannot explain the generation of electric potential oscillations. 

Continuum models (i.e. Navier-Stokes equation) should be used as long as possible in the applicable range. The mathematical handling is easier compared with molecular models, which have to be used in the non[Pg.257]

Computational tools analyzing the elementary chemical processes which take place at the gas-surface interface and couple them to the surrounding gas phase have been recently developed. The mathematical models are based on the numerical solution of the Navier-Stokes equations. 

A number of commercially available computational fluid dynamics (CFD) models could be used for the prediction of squat. At the core of any CFD problem is a computational grid or mesh where the solution is divided into thousands of elements. These elements are usually 2D quadrilaterals or triangles and three-dimensional (3D) hexahedral, tetrahedral, or prisms. Mathematical equations are solved for each element by the numerical model. For hydrodynamics the Navier-Stokes equations (NSEs) can be solved to include viscosity and turbulence. The NSEs provide detailed prediction (vortices) of the flow field, but require very thin meshes, high central processing unit (CPU) time, and memory storage. Its resolution is also quite difficult with numerical instabilities. Examples of commercial CFD models include Fluent and Fidap. 

In the foregoing chapters, the simulation is based on the macroscopic point of view that the fluid is continuous medium and its physical properties, such as density, velocity, and pressure, are functions of time and space. Thus, the Navier—Stokes equation can be employed as modeling equation in the mathematical simulation. In this chapter, we turn to the mesoscopic point of view and use the lattice Boltzmann... 

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