Stokes problem difference equations

An element for the stress components composed of 16 sub-elements (4x4) on which bilinear (continuous) polynomials are used, was introduced by Marchal and Crochet in [28]. This leads to a continuous C° approximation of the three variables. The velocity is approximated by biquadratic polynomials while the pressure is linear. Fortin and Pierre ([17]) made a mathematical analysis of the Stokes problem for this three-field formulation. They conclude that the polynomial approximations of the different variables should satisfy the generalized inf-sup (Brezzi-Babuska) condition introduced by Marchal and Crochet and they proved it was the case for the Marchal and Crochet element. In order to take into account the hyperbolic character of the constitutive equation, Marchal and Crochet have implemented and compared two different methods. The first is the Streamline-Upwind/Petrov-Galerkin (SUPG). Thus a so-called non-consistent Streamline-Upwind (SU) is also considered (already used in [13]). As a test problem, they selected the "stick-slip" flow. With SUPG method applied to this problem, wiggles in the stress and the velocity field were obtained. In the SU method, the modified weighting function only applies to the convective terms in the constitutive equations. 

Diffusion, as indicated above, is a related problem, differing however in the fact that molecules of different sizes are involved. Many years ago, Einstein (5), in considering the Brownian movement of colloids where the particles are very large in comparison to the molecules of the solvent and assuming that Stokes law described the motion of the particles, arrived at what is known as the Stokes-Einstein equation ... 

Deviations from the Carman-Kozeny equation (4.9) become more pronounced in these beds of fibres as the voidage increases, because the nature of the flow changes from one of channel flow to one in which the fibres behave as a series of obstacles in an otherwise unobstructed passage. The flow pattern is also different in expanded fluidised beds and the Carman-Kozeny equation does not apply there either. As fine spherical particles move far apart in a fluidised bed, Stokes law can be applied, whereas the Carman-Kozeny equation leads to no such limiting resistance. This problem is further discussed by Carman 141. 

A physicist would view the expression (10) as typical in quantum mechanics and as corresponding to the evolution operator. Equations (8) and (9) are, incidentally, very typical in gauge theory, such as in QCD. Thus, guided by our intuition, we can reformulate our chief problem as a quantum-mechanical one. In other words, the approaches to the l.h.s. of the non-Abelian Stokes theorem are analogous to the approaches to the evolution operator in quantum mechanics. There are the two main approaches to quantum mechanics, especially to the construction of the evolution operator opearator approach and path-integral approach. Both can be applied to the non-Abelian Stokes theorem successfully, and both provide two different formulations of the non-Abelian Stokes theorem. 

Stagnation flows can be viewed either as a similarity reduction of the flow equations in a boundary-layer region or as an exact reduction of the Navier-Stokes equations under certain simplifying assumptions. Depending on the circumstances of a particular problem of interest, one or the other view may be more natural. In either case, the same governing equations emerge, with the differences being in boundary conditions. The alternatives are explored in later sections, where particular problems and boundary conditions are discussed. 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

The difference between this equation for turbulent flow and the Navier-Stokes equation for laminar flow is the Reynolds stress/turbulent stress term —pujuj appears in the equation of motion for turbulent flow. This equation of motion for turbulent flow involves non-linear terms, and it is impossible to be solved analytically. In order to solve the equation in the same way as the Navier-Stokes equation, the Reynolds stress or fluctuating velocity must be known or calculated. Two methods have been adopted to avoid this problem—phenomenological method and statistical method. In the phenomenological method, the Reynolds stress is considered to be proportional to the average velocity gradient and the proportional coefficient is considered to be turbulent viscosity or mixing length ... 

This theory takes into account the micro-rotational effects due to rotation of molecules. This becomes important with polymers or polymeric suspensions. The physical model assigns a substructure to each continuum particle. Each material volume element contains microvolume elements which can translate, rotate, and deform independently of the motion of the microvolime. In the simplest case, these fluids are characterised by 22 viscosity coefficients and the problem is formulated in terms of a system of 19 equations with 19 unknowns. The equations for a 2-D case were solved numerically and compared to experimental results. It is concluded that the model based on the micropolar fluid theory gives a better fit than the Navier - Stokes equations. However, it seems that the difference is small. 

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... 

The problem considered here differs from the canonical problem by the presence of a source term (i.e. the force) on the right-hand side of the complete Navier—Stokes equations (3.29). This force vanishes outside the EPR, for z (h, 1 - h), is opposite to the local flow direction, and is proportional to some power of its velocity (here, we consider the linear or quadratic law). The boundary condition at the entrance x = 0 is evident, U = 1, V = 0 (homogeneous velocity distribution). There are non-slip conditions on the walls z = 0 and z = 1. The further formulation of the problem is somewhat different for linear and quadratic EPRs. 

It is possible to solve a flow problem in either dimensional or dimensionless form. The variables can be assigned values using a consistent set of dimensions, which must be the SI system for turbulent flow. The dimensional formula is convenient since the problem is usually specified in that way, but in some cases the iterations may not converge. Alternatively, the equations can be made dimensionless. The dimensionless formulations are good when you are having trouble getting the iterations to converge, since you have a better sense of the problem when you specify the Reynolds number. This section takes the dimensional Navier-Stokes equation, Eq. (10.40), and derives two different dimensionless versions ... 

The reader may find the result (7-16) surprising. As already noted, it is well known that a rotating and translating sphere in a stationary fluid will often experience a sideways force (that is, lift) that will cause it to travel in a curved path-think, for example, of a curve ball in baseball or an errant slice or hook in golf. The difference between these familiar examples and the problem previously analyzed is that the Reynolds numbers are not small and the governing equations are the full, nonlinear Navier-Stokes equations rather than the linear creeping-flow approximation. Thus the decomposition to a set of simpler component problems cannot be used, and it is not possible to deduce anything about the forces on the... 

In chemical technology one often meets the problem of a steady-state motion of a spherical particle, drop, or bubble with velocity U in a stagnant fluid. Since the Stokes equations are linear, the solution of this problem can be obtained from formulas (2.2.12) and (2.2.13) by adding the terms Vr = -U cos6 and V = U[ sin 6, which describe a translational flow with velocity U, in the direction opposite to the incoming flow. Although the dynamic characteristics of flow remain the same, the streamline pattern looks different in the reference frame fixed to the stagnant fluid. In particular, the streamlines inside the sphere are not closed. 

The main distinction of the theory of a dynamic adsorption layer formed under weak and strong retardation arises when formulating the convective diffusion equation. At weak retardation the Hadamard-Rybczynski hydrodynamic velocity field is used while at strong retardation the Stokes velocity field. Different formulas for the dependence of the diffusion layer thickness on Peclet numbers are obtained. The problem of convective diffusion in the neighbourhood of a spherical particle with an immobile surface at small Reynolds numbers and condition (8.74) is solved, so that the well-known expression for the density distribution of the diffusion flow along the surface can be used. As a result, Eq. (8.10) takes the form (Dukhin, 1982),... 

Several attempts have been made to modify Stokes equation so as to render it applicable to ionic solutions. Whether theoretical or semi-empirical, these efforts have not had the success at first hoped for and it has become fashionable to deride Stokes law as having reached a state of sterility. Such a view is overly pessimistic, and considerable insight into the problem will be gained by a careful examination of the different approaches that have been employed so far. 

The analogy of these results to that in fixed-bed simulations without sidestream is pronounced and can be explained with the predominance of the bed friction-force term in the extended Navier-Stokes equation (Eq. (5.6)). HydrodynamicaUy developed flow is achieved after a distance of just about one particle diameter in the axial direction. However, the developed profile in a PBMR is characterized by a radial velocity different from zero. One can prove analytically that for reactive flow problems with negligible change in the physical properties (density, viscosity) the superficial radial velocity decreases linearly towards the core (Kiirten, 2003). In Fig. 5.17b the superficial radial velocities are compared. Using the radial porosity profile, smaller absolute values of the local superficial velocity are calcu-... 

---