Kinematics

Kinematics is the branch of mechanics that explores the motion of material bodies from the standpoint of their space-time relationships, disregarding their masses and the forces acting on them. 

The term rheology dates back to 1929 (Tanner and Walters 1998) and is used to describe the mechanical response of materials. Polymeric materials generally show a more complex response than classical Newtonian fluids or linear viscoelastic bodies. Nevertheless, the kinematics and the conservation laws are the same for all bodies. The presentation here is condensed one may consult other books for amplification (Bird et al. 1987a Huilgol and Phan-Thien 1997 Tanner 2000). We begin with kinematics. 

There are two viewpoints that one may take to describe the fluid motion the Lagrangian and the Eulerian viewpoints. 

In the Lagrangian viewpoint, one keeps track of a fluid particle, which, at time t, is located at a position x, i — 1, 2, 3) in Cartesian coordinates. The trajectory of the particle would be given by x, (r), where x, is a dependent variable and t is an independent variable. The velocity of the fluid particle is therefore given by 

In the Eulerian viewpoint, the quantities in the flow field are described as a function of fixed position x,- and time t. Specifically, the flow velocity is described 

Zheng et al.. Injection Molding, DOI 10.1007/978-3-642-21263-5 2, Springer-Verlag Berlin Heidelberg 2011 

There are two basic flows used to characterize polymers shear and shear-free flows. (It so happens that processes are usually a combination of these flows or sometimes are dominated by one type or the other.) The velocity field for rectilinear shear flow is given below  

FIGURE 3.2 The deformation of (a) a unit cube of material from time ti to f2 (t2 fi) in (b) steady simple shear flow and (c) three kinds of shear-free flow. The volume of material is preserved in all of these flows. (Reprinted by permission of the publisher from Bird et al., 1987a.) 

There are significant differences in the behavior of polymeric fluids in these two types of deformation, and each type of deformation has a different effect on the orientation of macromolecules. For example, uniaxial and planar extensional flows impart significant molecular orientation in polymers during flow compared to shear flows. On the other hand, biaxial extensional flow is a weak flow and does not lead to a strong degree of molecular orientation. Furthermore, the rheological response can be significantly different for a polymer in extensional flow versus shear flow. We demonstrate these differences later in this chapter. 

For these two types of flows, the components of the rate deformation tensor (Table 2.8) take on a distinct form. For shear flow the components of the rate of deformation tensor are 

Kij is interpreted as the yth component of the total stress tensor while Xy is the yth component of the extra stress tensor. Sii is the Kronecker delta and is defined as 

The physical nature of the turbulent component of a spectral line in a molecular cloud is currently a source of considerable debate. Physical processes that have been suggested as sources of the turbulence in molecular clouds are expanding HII regions, supernova remnants, cloud-cloud collisions, galactic differential rotation, and stellar winds. Unfortunately, for all of these processes there are theoretical problems with couphng the energy produced into turbulence. 

---

The kinematic viscosity is defined as the ratio between the absolute viscosity and the density. It is expressed in m /s. The most commonly used unit is mm /s formerly called centistoke, cSt. 

They can also be calculated from tbe kinematic viscosities at 100 and 210°F. After having calculated the densities, pj and p, one obtains ( S )... 

Coefficients a and b can also be determined from the kinematic viscosities at 100 and 210°F. 

ASTM proposes representing the kinematic viscosity of hydrocarbons by a straight line on graph paper, called viscometric, for which the scales are such that ... 

T = temperature a, b = straight line coefficients c = kinematic viscosity log = common logarithm (base 10)... 

The exact calculation of the index is given in the ASTM D 2270 standard. The kinematic viscosity at 40°C (f/) of an oil whose viscosity index (V7) is being calculated is compared with those of two reference oils for which the viscosity indices are 0 and 100 respectively, and which have at 100°C the same kinematic viscosity as that of the oil being examined )... 

M = molecular weight j/joo = kinematic viscosity at lOOT Tff = pour point temperature... 

For optimum combustion, the fuel should vaporize rapidly and mix intimately with the air. Even though the design of the injection system and combustion chamber play a very important role, properties such as volatility, surface tension, and fuel viscosity also affect the quality of atomization and penetration of the fuel. These considerations justify setting specifications for the density (between 0.775 and 0.840 kg/1), the distillation curve (greater than 10% distilled at 204°C, end point less than 288°C) and the kinematic viscosity (less than 8 mm /s at -20°C). 

The heavy fuel should be heated systematically before use to improve its operation and atomization in the burner. The change in kinematic viscosity with temperature is indispensable information for calculating pressure drop and setting tbe preheating temperature. Table 5.20 gives examples of viscosity required for burners as a function of their technical design. 

Average kinematic viscosity required for the atomization of heavy fuels. 

For the refiner, the main problem is to meet the specifications for kinematic viscosity and sulfur content. Dilution by light streams such as home-heating oil and LCO, and selection of feedstocks coming from low-sulfur crude oils give him a measure of flexibility that will nevertheless lead gradually to future restrictions, most notably the new more severe antipollution rules imposing lower limits on sulfur and nitrogen contents. 

Cranking Simulator), by a pumpability temperature limit measured by a rotating mini viscometer, and by the minimum kinematic viscosity at 100°C. The five summer grades are defined by bracketing kinematic viscosities at 100°C. 

A chart by J. Groff relates the SAE grades, the kinematic viscosity, and the viscosity indices. This correlation is given in Figure 6.1 in the 1994 edition. 

This is the essential characteristic for every lubricant. The kinematic viscosity is most often measured by recording the time needed for the oil to flow down a calibrated capillary tube. The viscosity varies with the pressure but the influence of temperature is much greater it decreases rapidly with an increase in temperature and there is abundant literature concerning the equations and graphs relating these two parameters. One can cite in particular the ASTM D 341 standard. 

Other characteristics often required Kinematic viscosities at 40°C and 100°C NFT 60-100... 

The viscosity is determined by measuring the time it takes for a crude to flow through a capillary tube of a given length at a precise temperature. This is called the kinematic viscosity, expressed in mm /s. It is defined by the standards, NF T 60-100 or ASTM D 445. Viscosity can also be determined by measuring the time it takes for the oil to flow through a calibrated orifice standard ASTM D 88. It is expressed in Saybolt seconds (SSU). 

Kinematic viscosity NFT 60-100 ISO 3104 and 3105 ASTM D 445 and D 446 Measurement of time required to flow between 2 marks in a tube... 

Viscosity index NFT 60-136 ISO 2909 ASTM D 2270 Calculation based on kinematic viscosity... 

The same definition of viscosity applies to oil as gas (see Section 5.2.6), but sometimes the kinematic viscosity is quoted. This is the viscosity divided by the density (u = i7p), and has a straight line relationship with temperature. 

McCaffery A J 1999 Quasiresonant vibration-rotation transfer a kinematic interpretation J. Chem. Phys. Ill 7697... 

The discussion of Rutherford backscattering spectrometry starts with an overview of the experimental target chamber, proceeds to the particle kinematics that detennine mass identification and depth resolution, and then provides an example of the analysis of a silicide. 

---