Equations Stokes

The viscous drag force is considered to be proportional to the velocity of the particle  

The proportionality factor, /, is called the ffiction(al) coefficient and has dimensions of mass/time (e.g. kg s ). Note that in some areas of the technical sciences a dimensionless friction coefficient can be used, e.g. in courses on unit operations dealing with pipe flow. Equation 8.2 does not 

However, in one specific case (spherical particles), an equation for/can be obtained. This is the equation derived by G. G. Stokes in 1850 for spherical particles in very dilute dispersions. Both the viscosity of the dispersion medium and the radius of the particle enter this eqpiation  

A can of recently stirred paint is placed on a shelf during the winter season. The paint contains only one pigment type, TiOa, with an average particle size of 0.2 pm and a density of 4230 kg m . The density 

It is assumed that Stokes law is valid. The sedimentation velocity is then given by Equation 8.4  

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Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

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. 

Let Ap, Au and AT denote the deviations of the mass density, p, the velocity field, u, and the temperature, T, fiom their fiill equilibrium values. The fluctuating, linearized Navier-Stokes equations are... 

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 ... 

Quian Y H, D Humieres D and Lallemand P 1992 Lattice BGK models for Navier-Stokes equation Euro. Phys. Lett. 17 479... 

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... 

At Che opposite limit, where Che density Is high enough for mean free paths to be short con ared with pore diameters, the problem can be treated by continuum mechanics. In the simplest ease of a straight tube of circular cross-section, the fluid velocity field can easily be obtained by Integrating Che Nsvler-Stokes equations If an appropriate boundary condition at Che... 

Navler-Stokes equation, 26 Non-isothermal catalyst pellet, 156-... 

Equations (1.6) and (1.7) are used to formulate explicit relationships between the extra stress components and the velocity gradients. Using these relationships the extra stress, t, can be eliminated from the governing equations. This is the basis for the derivation of the well-known Navier-Stokes equations which represent the Newtonian flow (Aris, 1989). 

Brooks, A. N, and Hughes, T. J.R., 1982. Streamline-upwind/Petrov Galerldn formulations for convection dominated hows with particular emphasis on the incompressible Navier -Stokes equations. Cornput. Methods Appl Meek Eng. 32, 199-259. 

Crouzeix, M. and Raviart, P. A., 1973. Conforming and non-conforming finite elements for solving the stationary Navier-Stokes equations. RAIRO, Seric Rouge 3, 33 -76. 

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

Papanastasiou et al. (1992) suggested that in order to generate realistic solutions for Navier-Stokes equations the exit conditions should be kept free (i.e. no outflow conditions should be imposed). In this approach application of Green s theorem to the equations corresponding to the exit boundary nodes is avoided. This is eqvrivalent to imposing no exit conditions if elements with... 

Gresho, P. M., Lee, R. L. and Sani, R. L., 1980. On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in fluids, Ch. 2, Pineridge Press, Swansea, pp. 27-75. 

As already mentioned, the present code corresponds to the solution of steady-state non-isothennal Navier-Stokes equations in two-dimensional Cartesian domains by the continuous penalty method. As an example, we consider modifications required to extend the program to the solution of creeping (Stokes) non-isothermal flow in axisymmetric domains ... 

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. 

We shall mate further use of the Stokes equation later in this chapter for the... 

Temam R. (1979) Navier-Stokes equations. Theory and numerical analysis. North-Holland, Amsterdam, New-York, Oxford. 

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 ... 

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