The U-V-P scheme

72 FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES 3.1.1 The U-V-P scheme 

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as 

Further details of the BB, sometimes referred to as Ladyzhenskaya-Babuska-Brezi (LBB) condition and its importance in the numerical solution of incompressible flow equations can be found in textbooks dealing with the theoretical aspects of the finite element method (e.g. see Reddy, 1986), In practice, the instability (or checker-boarding) of pressure in the U-V-P method can be avoided using a variety of strategies. 

The main strategies for obtaining stable results by the U -V -P scheme for incompressible flow are as follows  

Triangular Taylor- Hood Quadratic Linear Vertices and mid-sides Vertices 

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The U-V-P scheme based on the slightly compressible continuity equation... 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

Working equations of the U-V-P scheme in Cartesian coordinate systems... 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

Using a procedure similar to the derivation of Equation (4.13) the working equations of the U-V-P scheme for steady-state Stokes flow in a polar (r, 6) coordinate system are obtained on the basis of Equations (4.5) and (4.6) as... 

Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

As it can be seen the working equations of the penalty scheme are more compact than their counterparts obtained for the U-V-P method. 

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

As explained in Chapter 3, it is possible to use equal order interpolation models for the spatial discretization of velocity and pressure in a U-V-P scheme based on Equations (4.127) and (4.128) without violating the BB stability condition. 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

The described application of Green s theorem which results in the derivation of the weak statements is an essential step in the formulation of robu.st U-V-P and penalty schemes for non-Newtonian flow problems. 

The Petrolite method for calixarene synthesis also leads to dihomooxacalix[4]arenes, as represented by the p-tert-butyl compound 9. Proton-NMR spectra and TLC data indicate that the mixture obtained from p-terf-butylphenol and paraformaldehyde contains appreciable amounts of 9, but it is difficult to obtain a pure sample of 9 from this mixture 23,2, 28>. Compound 9 is much more readily accessible by the thermally-induced dehydration of the (u. v(hydroxymethyl)tetramer 46 which can be readily prepared by the convergent synthesis outlined in Scheme 5 followed by bis hydroxymethylation 62), as shown in Scheme 6. Other oxacalixarenes can also be obtained by thermally-induced dehydration. For example, 2,6-6 (hydroxymethyl)-phenols (42) yield hexahomotrioxacalix[3]arenes (43)28,62,101 > j cfroxymethyl)... 

It can be shown by induction an y, that far input (e,y v) in D the computation must halt with output (ay,v). Thus during the computation in P or in any strongly equivalent single variable scheme the value of the single register x must at all times be of the form (u, y v) where u is an initial substring of ay and y is a terminal substring of y. If the tail of val(x) at some point in the computation starts with a, then the second symbol added to the head afterwards must be a similarly if the tail starts with b, the next symbol but one added to the head must be b. ... 

The program scheme PCS) we shall construct in Example VIII-6 has an extra test T which does not appear in S and which we assume to be an n+2-way test with possible outcomes 0,l,...,n, for some new symbol such a test could be simulated by binary tests in the standard way. Scheme P(S) has variables x, u, v, and z. Register z holds the eventual output and register x is input and program variable. The registers u and v are special program variables which simulate the pushdown store of a pushdown store machine implementing the computations of S. ... 

Bradshaw et al. (B3) use Eqs. (40) to derive a differential equation for the turbulent shear stress t. The transport velocity Qa is taken as (Tmei/p), where Tm x is the maximum value of riy) in the boundary layer. G and I are prescribed as functions of the position across the boundary layer, and o is essentially taken as constant. Together with Eqs. (10a,b), Eq. (36) gives a closed set of equations for U, V, and t this system is of hyperbolic type, with three real characteristic lines. Bradshaw et al. construct a numerical solution using the method of characteristics it can also be done using small streamwise steps with an explicit difference scheme (Nl A. J. Wheeler and J. P. Johnston, private communications). There is a great physical appeal to the characteristics, especially since it is found that the solutions along the outward-going characteristic dominates the total solution. This... 

Six different weighting schemes are proposed (1) the unweighted case u (yvi = 1 i —, n, where A is the number of atoms for each compound), (2) atomic mass m, (3) the - van der Waals volume v, (4) the Sanderson - atomic electronegativity e, (5) the - atomic polarizability p and (6) the - electrotopological state indices of Kier and Hall 5. All the weights are also scaled with respect to the carbon atom, and their values are shown in Table W-5 moreover, as all the weights must be positive, the electrotopological indices are scaled thus ... 

Here it is the vector of the horizontal velocity, U the vertically integrated horizontal velocity, w is the vertical velocity, r is the sea-level elevation, V/, the horizontal nabla operator, q a source term of water flux, T the temperature, S the salinity, p the pressure, and p is the density. Moreover,/is the inertial frequency,/= 2 1 sin 0, where 1 Zn (1 I 1/365.2425)724 h is the earth s angular velocity and 0 is the latitude. Turbulent viscosity is indicated by the term D, . Wind forcing enters the scheme as a vertical boundary condition. The equations are solved usually in spherical coordinates, but are written here for simplicity in Cartesian form. 

Using the notation of Section 3.3, we recall that the codewords of an SBE scheme are typical sequences of the r.v. U = aS + W. We assume that these codewords, when considered as random samples u(M, S) (u is the modulating mapping of the SBE scheme), can be well approximated by n i.i.d. realizations of U. Such an approximation will be faithful, in cases where the allowed distortion level P and the noise power N are significantly smaller than the covertext signal power Q. This situation is typical to certain transform domain image watermarking schemes where [Pg.30]

By going from X to X we see immediately that the conditions and of SGA 1 V 4 are fulfilled. As to G, i.e., P(u) F(jf) F( l ) an isomorphism implies u an isomorphism, we proceed as follows First F(u) an isomorphism implies that the corresponding morphism of the usual schemes over V is an isomorphism (2.4.2 ), and hence u V is an isomorphism. From the connectedness of follows then that all fibre functors are isomorphic and hence u is an isomorphism. 

When irradiated with u.v. light, through a Corex filter, cocaine and its p-methoxy and p-methyl derivatives, in solution in methanol, produced one mole of formaldehyde besides the norcocaines nor-(8) after a tt-tt transition had occurred. However, phenylacetylecgonine methyl ester was not N-demethylated. The piperidine analogues did not respond to the irradiation either. The mechanism for cocaine is envisaged as being that shown in Scheme 2. 

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