# Newtonian approach

Ultimately, the compromise between the realized speedup and the accuracy obtained for the governing dynamic model should depend on the applications for which the dynamic simulations are used for. For very detailed dynamic pathways, only the Newtonian approach is probably adequate. For general conformational sampling questions, many other simulation methodologies can work well. In particular, if a weak coupling to a phenomenological heat bath, as in the LN method, is tolerated, the general efficiency of force splitting methods can be combined with the long-timestep stability of methods that resolve harmonic and anharmonic motions separately (such as LIN) to alleviate severe resonances and yield speedup. The speedup achieved in LN might be exploited in general thermodynamic studies of macromolecules, with possible extensions into enhanced sampling methods envisioned. [c.257]

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as [c.14]

In a significant number of polymer processes the influence of fluid elasticity on the flow behaviour is small and hence it is reasonable to use the generalized Newtonian approach to analyse the flow regime. In generalized Newtonian fluids tire extra stress is explicitly expressed in terms of velocity gradients and viscosity and can be eliminated from the equation of motion. This results in the derivation of Navier-Stokes equations with velocity and pressure as tire only prime field unknowns. Solution of Navier-Stokes equations (or Stokes equation for creeping flow) by the finite element schemes is the basis of computer modelling of non-elastic polymer flow regimes. In contrast, in viscoelastic flow models the extra stress can only be given through implicit relationships with the rate of strain, and hence remains as a prime field unknown in the governing equations. In this case therefore, in conjunction with the governing equations of continuity and momentum (generally given as Cauchy s equation of motion) an appropriate constitutive equation must be solved. Numerical solution of viscoelastic constitutive equations has been the subject of a considerable amount of research in the last two decades. This has given rise to a plethora of methods [c.79]

As discussed in the previous chapters, utilization of viscoelastic constitutive equations in the finite element schemes requires a significantly higher computational effort than the generalized Newtonian approach. Therefore an important simplification in the model development is achieved if the elastic effects in a flow system can be ignored. However, almost all types of polymeric fluids exhibit some degree of viscoelastic behaviour during their flow and deformation. Hence the neglect of these effects, without a sound evaluation of the flow regime characteristics, which may not allow such a simplification, can yield inaccurate results. [c.150]

The Newtonian approach gives the equation of motion as follows [c.183]

Nowadays the position is changing because, as ever increasing demands are being put on materials and moulding machines it is becoming essential to be able to make reliable quantitative predictions about performance. In Chapter 4 it was shown that a simple Newtonian approach gives a useful first approximation to many of the processes but unfortunately the assumption of constant viscosity can lead to serious errors in some cases. For this reason a more detailed analysis using a Non-Newtonian model is often necessary and this will now be illustrated. [c.343]

Newtonian Approach Let Nb(t) and Nyj(t) represent the number of black and white balls at time t, respectively. Let Tib(t) and n (f) be the number of black and white balls having a marked site directly ahead of them at time t. The Newtonian equations of motion are then given by [c.460]

The Newton-Raphson approach, being essentially a point-slope method, converges most rapidly for near linear objective functions. Thus it is helpful to note that tends to vary as 1/P and as exp(l/T). For bubble-point-temperature calculation, we can define an objective function [c.118]

A novel optimization approach based on the Newton-Kantorovich iterative scheme applied to the Riccati equation describing the reflection from the inhomogeneous half-space was proposed recently [7]. The method works well with complicated highly contrasted dielectric profiles and retains stability with respect to the noise in the input data. However, this algorithm like others needs the measurement data to be given in a broad frequency band. In this work, the method is improved to be valid for the input data obtained in an essentially restricted frequency band, i.e. when both low and high frequency data are not available. This [c.127]

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One [c.2339]

Various other ways to incorporate the out-of-plane bending contribution are possible. For e3

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. [c.287]

The plane of the rotor blade cross-section representing the flow field configuration at the start of mixing in a partially filled single-blade mixer is shown in Figure 5.1. Initial distribution of the compound inside the mixer chamber corresponds to a fill factor of 71 per cent and is chosen arbitrarily. It is evident that the flow field within this domain should be modelled as a free surface regime with random moving boundaries. Available options for the modelling of such a flow regime are explained in Chapter 3, Section 5. In this example, utilization of the volume of fluid (VOF) approach based on an Eulerian framework is described. To maintain simplicity we neglect elastic effects and the variations of compound viscosity with mixing, and focus on the simulation of the flow corresponding to a generalized Newtonian fluid. In the VOF approach [c.142]

The Indian-born physicist Subramanyan Chandrasekhar (Nobel Prize 1983) had a personal style of research, which 1 learned about only recently, that seems to parallel mine. He intensely studied a selected subject for years. At the end of this period he generally summarized his work and thoughts in a book or a comprehensive review and then moved on to something else. He also refuted Huxley s claim that scientists over 60 do more harm than good by sharing the response of Rayleigh (who was 67 at the time) that this may be the case if they only undertake to criticize the work of younger men (women were not yet mentioned) but not when they stick to the things they are competent in. He manifested this belief in his seminal work on black holes, on which he published a fundamental book when he was 72, and his detailed analysis of Newton s famous Principia published when he was 84, shortly before his death. I have also written books and reviews whenever I felt that I had sufficiently explored a field in my research and it was time to move on this indeed closely resembles Chandrasekhar s approach (vide infra). [c.227]

The molecular orbital approach to chemical bonding rests on the notion that as elec trons m atoms occupy atomic orbitals electrons m molecules occupy molecular orbitals Just as our first task m writing the electron configuration of an atom is to identify the atomic orbitals that are available to it so too must we first describe the orbitals avail able to a molecule In the molecular orbital method this is done by representing molec ular orbitals as combinations of atomic orbitals the linear combination of atomic orbitals molecular orbital (LCAO MO) method [c.61]

The synchronous transit method is also combined with quasi-Newton methods to find transition states. Quasi-Newton methods are very robust and efficient in finding energy minima. Based solely on local information, there is no unique way of moving uphill from either reactants or products to reach a specific reaction state, since all directions away from a minimum go uphill. HyperChem has two synchronous transit methods implemented. The linear synchronous transit method (LST) searches for a maximum along a linear path between reactants and products. It may happen that this method will end up with a structure having two or more negative eigenvalues. The quadratic synchronous transit method (QST) is an improvement of the LST approach and searches for a maximum along a parabola connecting reactants and products, and for a minimum in all directions perpendicular to the parabola. [c.67]

Let X, R and P be the coordinates of the current point, the reactants, and products. (Note that it is necessary to evaluate the geometries of the reactants and products using the same method or the same basis set as the one used here to calculate the transition structure). In the linear synchronous transit approach, X is on a path that is a linear interpolation between Rand P. The quadratic synchronous transit method uses a curved path through X, Rand P. In both cases, a maximum is found along the path. The tangent to the synchronous transit path (LST or QST) is used to guide the optimization to the quadratic region of the transition state. Then the tangent to the path is used to choose the best eigenvector for the ascent direction and the quasi-Newton method is used to complete the optimization. It is also possible to use an eigenvector-following step to complete optimization. [c.309]

In the next few sections we shall examine some of the theoretical effort which has been directed toward an understanding of these exponents. The principal difference in the theoretical approach to the two regimes of Fig. 2.10 involves the absence or presence of chain entanglements. In discussing the Eyring theory, we saw that different relaxation times might be associated with the relatively loose associations arising from London forces, while the knotting together of chains would result in more sluggish motion. A little reflection will convince us that polymers are not essentially different from any other class of compounds with respect to the first of these interaction modes All molecules attract one another. The knotting mechanism, by comparison, is unique to long-chain structures, implying an ability of molecules to become entangled. Thus we expect that some sort of critical chain length must be exceeded before the mechanism of entanglement becomes effective. Although precise incorporation of this notion into a theoretical treatment of the phenomena is still incomplete, there is no doubt that this is the source of the breaks in the curves in Fig. 2.10. The subscript c which we have used to designate the degree of polymerization and molecular weight at the break point indicates a critical molecular weight above which entanglement effects make significant contributions to the observed viscosity. [c.105]

The increase in fuel viscosity with temperature decrease is shown for several fuels in Figure 9. The departure from linearity as temperatures approach the pour point illustrates the non-Newtonian behavior created by wax matrices. The freezing point appears before the curves depart from linearity. It is apparent that the low temperature properties of fuel are closely related to its distillation range as well as to hydrocarbon composition. Wide-cut fuels have lower viscosities and freezing points than kerosenes, whereas heavier fuels used in ground turbines exhibit much higher viscosities and freezing points. [c.415]

Water-soluble EHEC is a moderate ethyl DS (- 1.0) modification of high hydroxyethyl MS (>2.0) HEC. Ethyl groups lower the surface and interfacial tensions, thereby increasing surface activity. This group also modifies adsorption properties of the polymer to particulates found in many formulations such as clays, pigments, and latices. Aqueous solutions have pseudoplastic rheology. High viscosity grades are more pseudoplastic than low viscosity materials, which approach Newtonian flow behavior. Viscosities decrease reversibly with increasing temperature. Above 65°C, EHEC precipitates from solution. Salts lower the temperature at which precipitation occurs. Solution viscosities are insensitive to pH between about 3 to 11. Aqueous solutions are miscible with lower alcohols, glycols, and ketones up to equal proportions. Water-soluble EHECs are used to thicken and stabilize a variety of materials, including water-borne paints, plasters, detergents, cosmetics, and pharmaceuticals (49). [c.276]

Solutions of methylceUuloses are pseudoplastic below the gel point and approach Newtonian flow behavior at low shear rates. Above the gel point, solutions are very thixotropic because of the formation of three-dimensional gel stmcture. Solutions are stable between pH 3 and 11 pH extremes wiU cause irreversible degradation. The high substitution levels of most methylceUuloses result in relatively good resistance to enzymatic degradation (16). [c.276]

In general, the model-based reasoning approach is best appHed, not as a method in itself, but as an add-on to a knowledge-based system. The main reason is that modeling is hard, and problem-solving based solely on fundamental models is computationally complex (41). Using the hybrid approach can take advantage of the efficiency of compiled knowledge in rapidly focusing on the solution, while retaining the robustness of models when confronted with the need for behavioral detail. Several recent research papers in the chemical engineering Hterature have explored this hybrid problem-solving notion (42,43). For more information on model-based reasoning in general see References 44 and 45. [c.536]

The advantage of this approach is that it is easier to program than a full Newton-Raphson method. If the transport coefficients do not vary radically, then the method converges. If the method does not converge, then it maybe necessary to use the full Newton-Raphson method. [c.476]

From the above list of rate-based model equations, it is seen that they total 5C -t- 6 for each tray, compared to 2C -t-1 or 2C -t- 3 (depending on whether mole fractious or component flow rates are used for composition variables) for each stage in the equihbrium-stage model. Therefore, more computer time is required to solve the rate-based model, which is generally converged by an SC approach of the Newton type. [c.1292]

If the objective function is considered two-dimensional, consisting of Equations (7-13) and (7-14) and the vector X includes only T and a, then the only change in the iteration is that the derivatives of with respect to composition are ignored in establishing the Newton-Raphson corrections to T and a. The new compositions can then be determined from Equations (7-8) and (7-9). Such a simplified procedure sacrifices little in convergence rate for vapor-liquid systems, where the contributions of compfosition-derivatives to changes in T and a are almost always smad 1. This approach requires only two evaluations of per iteration and still avoids creeping since it is essentially second-order in the limit as convergence is approached. [c.117]

The question stated above was fomuilated in two ways, each using an exact result from classical mechanics. One way, associated witii the physicist Losclnnidt, is fairly obvious. If classical mechanics provides a correct description of the gas, then associated with any physical motion of a gas, there is a time-reversed motion, which is also a solution of Newton s equations. Therefore if //decreases in one of these motions, tliere ought to be a physical motion of the gas where H increases. This is contrary to the //-tlieorem. The other objection is based on the recurrence theorem of Poincare [H], and is associated with the mathematician Zemielo. Poincare s theorem states that in a bounded mechanical system with finite energy, any mitial state of the gas will eventually recur as a state of the gas, to within any preassigned accuracy. Thus, if //decreases during part of the motion, it must eventually increase so as to approach, arbitrarily closely, its initial value. [c.686]

Particle models offer a simple means for easily and efficiently incorporating the symplectic structure. In some sense, the particle description is exceedingly natural the standard definition of a rigid body is a relatively rigid collection of massive point particles (see e.g. [1]). The particles need not have direct physical significance given any rigid body whose inertial tensor, center of mass, and total mass are provided, one can develop an equivalent representation in terms of point masses subject to rigid rod constraints. An important benefit of this choice of integration variables is that the equations of holonomically constrained particle motion are easily solved by use of the SHAKE (or RATTLE) discretization [30, 3], a generalization of Verlet which has been shown to be symplectic [22]. This approach was used by Ciccotti et al [9] to treat small rigid polyatoms, albeit without recognition of the symplectic character of the algorithm. More recently [5], the technique was generalized and applied to treat chains of rigid bodies, with the assistance of special SHAKE-SOR and sparse Newton methods for treating the nonlinear equations arising at each step of integration. [c.351]

Equation (5.49) derived for isothermal Newtonian flow in thin cavities, is called the pressure potential or Hele-Shaw equation. Analogous equations in terms of pressure gradients can be obtained using other types of boundary conditions in the integration of components of the equation of motion given as Equation (5.41). The lubrication approximation approach has also been generalized to obtain solutions for non-isothernial generalized Newtonian flow in thin layers. The generalized Hele-Shaw equation for non-isotherraal generalized Newtonian fluids is used extensively to model narrow gap flow regimes in injection and compression moulding (Hieber and Shen, 1980 Lee et al, 1984). Other generalized equations, derived on the basis of the lubrication approximation, are used to model laminar flow in calendering, coating and other processes where the domain geometry allows utilization of this approach (Soh and Chang, 1986 Hannart and Hopfinger, 1989). [c.173]

As the table indicates C—H bond dissociation energies m alkanes are approxi mately 375 to 435 kJ/mol (90-105 kcal/mol) Homolysis of the H—CH3 bond m methane gives methyl radical and requires 435 kJ/mol (104 kcal/mol) The dissociation energy of the H—CH2CH3 bond m ethane which gives a primary radical is somewhat less (410 kJ/mol or 98 kcal/mol) and is consistent with the notion that ethyl radical (primary) is more stable than methyl [c.169]

We shall follow the same approach as the last section, starting with an examination of the predicted behavior of a Voigt model in a creep experiment. We should not be surprised to discover that the model oversimplifies the behavior of actual polymeric materials. We shall continue to use a shear experiment as the basis for discussion, although a creep experiment could be carried out in either a tension or shear mode. Again we begin by assuming that the Hookean spring in the model is characterized by a modulus G, and the Newtonian dash-pot by a viscosity 77. [c.168]

The principle approach to immunoassay is illustrated in Figure 1, which shows a basic sandwich immunoassay. In this type of assay, an antibody to the analyte to be measured is immobilized onto a soHd surface, such as a bead or a plastic (microtiter) plate. The test sample suspected of containing the analyte is mixed with the antibody beads or placed in the plastic plate, resulting in the formation of the antibody—analyte complex. A second antibody which carries an indicator reagent is then added to the mixture. This indicator maybe a radioisotope, for RIA an enzyme, for EIA or a fluorophore, for fluorescence immunoassay (FIA). The antibody-indicator binds to the first antibody—analyte complex, free second antibody-indicator is washed away, and the two-antibody—analyte complex is quantified using a method compatible with the indicator reagent, such as quantifying radioactivity or enzyme-mediated color formation (see Automated insteut ntation, clinical chemistey). [c.22]

The mechanism by which silver sulfide enhances sensitivity of silver haUde grains is related to improved efficiency during latent-image formation. It has been suggested that sulfur sensitization increases the depth of individual electron traps (139,191,192) however, sulfur may reduce the repulsive potential energy associated with the surface space charge and thereby faciUtate the approach of a photoelectron to the surface for subsequent latent-image formation (193). These models share the common notion that sulfur sensitization enhances electron-trapping propensities at sites where latent image formation can occur. On the other hand, other experiments have suggested that under certain conditions sulfur sensitization can enhance hole-trapping probabihties (19,156,194—196) or can have a dual role, trapping either of the photocarriers depending on the size and/or location of the silver sulfide cluster (197-199). [c.448]

Water. Eor a long time in the United States, the approach to water pollution control was through the estabUshment of water quaUty standards for receiving bodies of water, ie, rivers, streams, or lakes, with most limits estabUshed on a state-by-state basis. There was no effective, national, legal authority to limit the discharge of pollutants. In the late 1960s, the U.S. government revived an old law, the Rivers and Harbor Act of 1899 (the Refuse Act) (1). The law prohibited the discharge of anything into navigable waters unless a permit was obtained from the Corps of Engineers, thus providing a first step toward control of industrial discharges. This was followed by additional legislation, culminating in the passage of the Eederal Water Pollution Control Act Amendments (FWPCA) of 1972 and the Clean Water Act (CWA) of 1977 (2). The objective of the FWPCA was to restore and maintain the chemical, physical, and biological integrity of the nation s waters. [c.76]

An appropriate set of iadependent reference dimensions may be chosen so that the dimensions of each of the variables iavolved ia a physical phenomenon can be expressed ia terms of these reference dimensions. In order to utilize the algebraic approach to dimensional analysis, it is convenient to display the dimensions of the variables by a matrix. The matrix is referred to as the dimensional matrix of the variables and is denoted by the symbol D. Each column of D represents a variable under consideration, and each tow of D represents a reference dimension. The /th tow andyth column element of D denotes the exponent of the reference dimension corresponding to the /th tow of D ia the dimensional formula of the variable corresponding to theyth column. As an iEustration, consider Newton s law of motion, which relates force E, mass Af, and acceleration by (eq. 2) [c.104]

See pages that mention the term

**Newtonian approach**:

**[c.186] [c.451] [c.696] [c.726] [c.74] [c.139] [c.238] [c.251] [c.90] [c.286] [c.641] [c.126] [c.47] [c.408] [c.317] [c.1034]**

Gas turbine engineering handbook (2002) -- [ c.183 , c.186 ]