query: What is viscosity?

title: Viscosity

text: Viscosity The viscosity of a fluid is the measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity is the property of a fluid which opposes the relative motion between two surfaces of the fluid that are moving at different velocities. In simple terms, viscosity means friction between the molecules of fluid. When the fluid is forced through a tube, the particles which compose the fluid generally move more quickly near the tube's axis and more slowly

title: Viscosity

text: is the local shear velocity. This expression is referred to as Newton's law of viscosity. In shearing flows with planar symmetry, it is what "defines" formula_10. It is a special case of the general definition of viscosity (see below), which can be expressed in coordinate-free form. Use of the Greek letter mu (formula_10) for the viscosity is common among mechanical and chemical engineers, as well as physicists. However, the Greek letter eta (formula_19) is also used by chemists, physicists, and the IUPAC. The viscosity formula_10 is sometimes also referred to as the "shear viscosity". However, at least one author discourages

title: Viscosity solution

text: "formula_92-weak solutions", referring to their stability properties, or "comparison solutions", referring to their most characteristic property. Viscosity solution In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in optimal control (the Hamilton–Jacobi equation), differential games (the Isaacs equation) or front evolution problems, as

title: Viscosity solution

text: Viscosity solution In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in optimal control (the Hamilton–Jacobi equation), differential games (the Isaacs equation) or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential

title: Viscosity

text: a relatively high viscosity, such as pitch, may appear to be a solid. The word "viscosity" is derived from the Latin """", meaning mistletoe and also a viscous glue made from mistletoe berries. In materials science and engineering, one is often interested in understanding the forces involved in the deformation of a material. In other words, one wishes to know what stresses arise in a material as the result of some given deformation. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is

title: Viscosity

text: water is about 0.89 mPa·s at room temperature (25 °C). As a function of temperature, the viscosity can be estimated using the semi-empirical relation: where = , = 247.8 K, and = 140 K. Experimentally determined values of the viscosity at various temperatures are given below. Under standard atmospheric conditions (25 °C and pressure of 1 bar), the viscosity of air is 18.5 μPa·s, roughly 50 times smaller than the viscosity of water at the same temperature. Except at very high pressure, the viscosity of air depends mostly on the temperature. Viscosity The viscosity of a fluid is the measure

title: Viscosity

text: the sensor is submerged in the fluid and is made to resonate at a specific frequency. As the surface of the sensor shears through the liquid, energy is lost due to its viscosity. This dissipated energy is then measured and converted into a viscosity reading. A higher viscosity causes a greater loss of energy. "Extensional viscosity" can be measured with various rheometers that apply extensional stress. Volume viscosity can be measured with an acoustic rheometer. Apparent viscosity is a calculation derived from tests performed on drilling fluid used in oil or gas well development. These calculations and tests help engineers

title: Viscosity

text: used as the singular form. The reciprocal of viscosity is "fluidity", usually symbolized by formula_51 or formula_52, depending on the convention used, measured in "reciprocal poise" (P, or cm·s·g), sometimes called the "rhe". Fluidity is seldom used in engineering practice. Nonstandard units include the reyn, a British unit of dynamic viscosity. In the automotive industry the viscosity index is used to describe the change of viscosity with temperature. At one time the petroleum industry relied on measuring kinematic viscosity by means of the Saybolt viscometer, and expressing kinematic viscosity in units of Saybolt universal seconds (SUS). Other abbreviations such as

title: Intrinsic viscosity

text: Intrinsic viscosity Intrinsic viscosity formula_1 is a measure of a solute's contribution to the viscosity formula_2 of a solution. It should not be confused with inherent viscosity, which is the ratio of the natural logarithm of the relative viscosity to the mass concentration of the polymer. Intrinsic viscosity is defined as formula_3 where formula_4 is the viscosity in the absence of the solute, formula_2 is (dynamic or kinematic) viscosity of the solution and formula_6 is the volume fraction of the solute in the solution. As defined here, the intrinsic viscosity formula_1 is a dimensionless number. When the solute particles are

title: Viscosity printing

text: felt blankets. It is run through the press where great pressure (approximately 8 tons to the square inch) pushes the dampened paper down into the engraved or etched grooves to pick up ink. In other words, in intaglio we see printed what is below the surface of the plate and the ink is now embossed on the paper. Among the greatest masters of engraving and etching are Albrecht Dürer, Hans Holbein the Younger, Rembrandt, Goya, and Picasso. Three to four colors of ink are mixed, each of a different viscosity. This viscosity is adjusted by the addition of uncooked linseed

title: Viscosity

text: the fluid is essential to acquire accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C. For some fluids, the viscosity is constant over a wide range of shear rates (Newtonian fluids). The fluids without a constant viscosity (non-Newtonian fluids) cannot be described by a single number. Non-Newtonian fluids exhibit a variety of different correlations between shear stress and shear rate. One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer. In coating industries, viscosity may be measured with a cup in which the efflux time is

title: Volume viscosity

text: Volume viscosity Volume viscosity (also called second coefficient of viscosity or dilatational viscosity or bulk viscosity) becomes important only for such effects where fluid compressibility is essential. Volume viscosity is mainly related to the vibrational energy of the molecules. It is zero for monatomic gases at low density, but can be large for fluids with larger molecules. The volume viscosity is important in describing sound attenuation (as in e.g. Stokes's law), and the absorption of sound energy into the fluid depends on the sound frequency i.e. the rate of fluid compression and expansion. Volume viscosity is also important in describing

title: Apparent viscosity

text: Apparent viscosity Apparent viscosity (sometimes denoted η) is the shear stress applied to a fluid divided by the shear rate (formula_1). For a Newtonian fluid, the apparent viscosity is constant, and equal to the Newtonian viscosity of the fluid, but for non-Newtonian fluids, the apparent viscosity depends on the shear rate. Apparent viscosity has the SI derived unit Pa·s (Pascal-second, but the centipoise is frequently used in practice: (1 mPa·s = 1 cP). A single viscosity measurement at a constant speed in a typical viscometer is a measurement of the apparent viscosity of a fluid. In the case of non-Newtonian

title: Viscosity

text: the viscosity formula_10 is frequently called the "dynamic viscosity". Transport theory provides an alternate interpretation of viscosity in terms of momentum transport: viscosity is the material property which characterizes momentum transport within a fluid, just as thermal conductivity characterizes heat transport, and (mass) diffusivity characterizes mass transport. To see this, note that in Newton's law of viscosity, formula_41, the shear stress formula_42 has units equivalent to a momentum flux, i.e. momentum per unit time per unit area. Thus, formula_42 can be interpreted as specifying the flow of momentum in the formula_8 direction from one fluid layer to the next. Per

title: Viscosity index

text: "L" and "H" are values based on the oil's kinematic viscosity at . "L" and "H" are the values of viscosity at 40°C for oils of VI 0 and 100 respectively, having the same viscosity at 100°C as the oil whose VI we are trying to determine. These "L" and "H" values can be found in ASTM D2270. Viscosity index The viscosity index (VI) is an arbitrary, unitless measure of the change of viscosity with temperature, mostly used to characterize the viscosity-temperature behavior of lubricating oils. The lower the VI, the more the viscosity is affected by changes in temperature.

title: Volume viscosity

text: used as working fluids in power systems having non-fossil fuel heat sources, wind tunnel testing, and pharmaceutical processing. Volume viscosity Volume viscosity (also called second coefficient of viscosity or dilatational viscosity or bulk viscosity) becomes important only for such effects where fluid compressibility is essential. Volume viscosity is mainly related to the vibrational energy of the molecules. It is zero for monatomic gases at low density, but can be large for fluids with larger molecules. The volume viscosity is important in describing sound attenuation (as in e.g. Stokes's law), and the absorption of sound energy into the fluid depends on

title: Apparent viscosity

text: formula_2, can be modeled by where These fluids are called power-law fluids. To ensure that formula_2 has the same sign as "du/dy", this is often written as where the term gives the apparent viscosity. Apparent viscosity Apparent viscosity (sometimes denoted η) is the shear stress applied to a fluid divided by the shear rate (formula_1). For a Newtonian fluid, the apparent viscosity is constant, and equal to the Newtonian viscosity of the fluid, but for non-Newtonian fluids, the apparent viscosity depends on the shear rate. Apparent viscosity has the SI derived unit Pa·s (Pascal-second, but the centipoise is frequently used

title: Intrinsic viscosity

text: formula_28. The intrinsic viscosity formula may also be generalized to include a frequency dependence. The intrinsic viscosity is very sensitive to the axial ratio of spheroids, especially of prolate spheroids. For example, the intrinsic viscosity can provide rough estimates of the number of subunits in a protein fiber composed of a helical array of proteins such as tubulin. More generally, intrinsic viscosity can be used to assay quaternary structure. In polymer chemistry intrinsic viscosity is related to molar mass through the Mark–Houwink equation. A practical method for the determination of intrinsic viscosity is with a Ubbelohde viscometer. Intrinsic viscosity Intrinsic

title: Viscosity

text: "fugitive elasticity" for fluid viscosity.) However, many liquids (including water) will briefly react like elastic solids when subjected to sudden stress. Conversely, many "solids" (even granite) will flow like liquids, albeit very slowly, even under arbitrarily small stress. Such materials are therefore best described as possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation); that is, being viscoelastic. Indeed, some authors have claimed that amorphous solids, such as glass and many polymers, are actually liquids with a very high viscosity (greater than 10 Pa·s). Viscoelastic solids may exhibit both shear viscosity and bulk viscosity. The extensional

title: Viscosity

text: involve rapid expansions and compressions. It is worth emphasizing that the above expressions are not fundamental laws of nature, but rather "definitions" of viscosity. As such, their utility for any given material, as well as means for measuring or calculating the viscosity, must be established using separate and independent means. In fluid dynamics, it is common to work in terms of the "kinematic viscosity" (also called "momentum diffusivity"), defined as the ratio of the viscosity to the density of the fluid . It is usually denoted by the Greek letter nu () and has units formula_38: Consistent with this nomenclature,

title: Viscosity index

text: Viscosity index The viscosity index (VI) is an arbitrary, unitless measure of the change of viscosity with temperature, mostly used to characterize the viscosity-temperature behavior of lubricating oils. The lower the VI, the more the viscosity is affected by changes in temperature. The VI was originally measured on a scale from 0 to 100; however, advancements in lubrication science have led to the development of oils with much higher VIs. The viscosity of a lubricant is closely related to its ability to reduce friction in solid body contacts. Generally, the least viscous lubricant which still forces the two moving surfaces

title: Viscosity

text: formula_23 is a viscosity tensor that maps the strain rate tensor formula_24 onto the viscous stress tensor formula_25. Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" formula_23 in total. However, due to spatial symmetries these coefficients are not all independent. For instance, for isotropic Newtonian fluids, the 81 coefficients can be reduced to 2 independent parameters. The most usual decomposition yields the standard (scalar) viscosity formula_10 and the bulk viscosity formula_28: where formula_30 is the unit tensor, and the dagger formula_31 denotes the transpose. This equation can be thought of as

title: Viscosity solution

text: obtained most often through the Perron method. The vanishing viscosity method is not practical for second order equations in general since the addition of artificial viscosity does not guarantee the existence of a classical solution. Moreover, the definition of "viscosity solutions" does not involve any viscosity of any kind. The theory of viscosity solutions is completely unrelated to viscous fluids. Thus, it has been suggested that the name "viscosity solution" does not represent the concept appropriately. Yet, the name persists because of the history of the subject. Other names that were suggested were "Crandall–Lions solutions", in honor to their pioneers,

title: Viscosity

text: calculate a large-scale motion with an eddy viscosity that characterizes the transport and dissipation of energy in the smaller-scale flow (see large eddy simulation). Values of eddy viscosity used in modeling ocean circulation may be from to depending upon the resolution of the numerical grid. Observed values of viscosity vary over several orders of magnitude, even for common substances. For instance, a 70% sucrose (sugar) solution has a viscosity over 400 times that of water, and 26000 times that of air. More dramatically, pitch has been estimated to have a viscosity 230 billion times that of water. The viscosity of

title: Viscosity

text: measured. There are several sorts of cup – such as the Zahn cup and the Ford viscosity cup – with the usage of each type varying mainly according to the industry. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through the conversion equations. Also used in coatings, a Stormer viscometer uses load-based rotation in order to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers. Vibrating viscometers can also be used to measure viscosity. Resonant, or vibrational viscometers work by creating shear waves within the liquid. In this method,

title: Viscosity printing

text: Viscosity printing Viscosity printing is a multi-color printmaking technique that incorporates principles of relief printing and intaglio printing. It was pioneered by Stanley William Hayter. The process uses the principle of viscosity to print multiple colors of ink from a single plate, rather than relying upon multiple plates for color separation. It is a fine art printmaking technique, making original prints in limited editions, as it is slow and allows too much variation between proofs to make large editions feasible. Color viscosity printing is among the latest developments in "intaglio" printmaking. Color viscosity printing was developed by a group working

title: Volume viscosity

text: volume viscosity are formula_2 and formula_3. Volume viscosity appears in the classic Navier–Stokes equation if it is written for compressible fluid, as described in the most books on general hydrodynamics and acoustics. where formula_5 is the shear viscosity coefficient and formula_2 is the volume viscosity coefficient. The parameters formula_5 and formula_2 was originally called the first and second viscosity coefficients, respectively. The operator formula_9 is the material derivative. By introducing the tensors (matrices) formula_10, formula_11 and formula_12, which describes crude shear flow, pure shear flow and compression flow, respectively, the classic Navier–Stokes equation gets a lucid form. Note that the

title: Fuel viscosity control

text: the command is sent to the heater to adjust the temperature of the fuel. Fuel viscosity control Fuel viscosity control is a technique to control viscosity and temperature of fuel oil (FO) for efficient combustion in diesel engines of motor vessels and generators of oil-fired power plants. Fuel oil's viscosity strongly depends on the temperature, the higher is the temperature the lower is the viscosity. For optimal combustion the viscosity of the fuel should be in the range of 10–20cSt. To maintain this value a combination of viscometer, PID controller and heater is used. Viscometer measures the actual viscosity of

title: Fuel viscosity control

text: Fuel viscosity control Fuel viscosity control is a technique to control viscosity and temperature of fuel oil (FO) for efficient combustion in diesel engines of motor vessels and generators of oil-fired power plants. Fuel oil's viscosity strongly depends on the temperature, the higher is the temperature the lower is the viscosity. For optimal combustion the viscosity of the fuel should be in the range of 10–20cSt. To maintain this value a combination of viscometer, PID controller and heater is used. Viscometer measures the actual viscosity of the fuel, this value is compared with the set point in the controller and

title: Viscosity

text: SSU ("Saybolt seconds universal") or SUV ("Saybolt universal viscosity") are sometimes used. Kinematic viscosity in centistokes can be converted from SUS according to the arithmetic and the reference table provided in ASTM D 2161. In general, the viscosity of a system depends in detail on how the molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. The simplest exact expressions are the Green–Kubo relations for the linear shear viscosity or the "transient time correlation function" expressions derived by Evans and Morriss in 1985. Although these expressions are each exact, calculating the

title: Saybolt universal viscosity

text: Saybolt universal viscosity Saybolt universal viscosity (SUV), and the related Saybolt FUROL viscosity (SFV), are specific standardised tests producing measures of kinematic viscosity. "FUROL" is an acronym for "fuel and road oil". Saybolt universal viscosity is specified by the ASTMD2161. Both tests are considered obsolete to other measures of kinematic viscosity, but their results are quoted widely in technical literature. In both tests, the time taken for 60ml of the liquid, held at a specific temperature, to flow through a calibrated tube, is measured, using a Saybolt viscometer. The Saybolt universal viscosity test occurs at , or more recently, .

title: Viscosity

text: develop and maintain the properties of the drilling fluid to the specifications required. The SI unit of dynamic viscosity is Pa·s or kg·m·s. The cgs unit is called the poise (P), named after Jean Léonard Marie Poiseuille. It is commonly expressed, particularly in ASTM standards, as "centipoise" (cP) since the latter is equal to the SI multiple millipascal seconds (mPa·s). The SI unit of kinematic viscosity is m/s, whereas the cgs unit for kinematic viscosity is the "stokes" (St), named after Sir George Gabriel Stokes. It is sometimes expressed in terms of "centistokes" (cSt). In U.S. usage, "stoke" is sometimes

title: Viscosity

text: viscosity is a linear combination of the shear and bulk viscosities that describes the reaction of a solid elastic material to elongation. It is widely used for characterizing polymers. In geology, earth materials that exhibit viscous deformation at least three orders of magnitude greater than their elastic deformation are sometimes called rheids. Viscosity is measured with various types of viscometers and rheometers. A rheometer is used for those fluids that cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case for a viscometer. Close temperature control of

title: Viscosity

text: near its walls; therefore some stress (such as a pressure difference between the two ends of the tube) is needed to overcome the friction between particle layers to keep the fluid moving. For a given velocity pattern, the stress required is proportional to the fluid's viscosity. A fluid that has no resistance to shear stress is known as an "ideal" or "inviscid" fluid. Zero viscosity is observed only at very low temperatures in superfluids. Otherwise, the second law of thermodynamics requires all fluids to have positive viscosity; such fluids are technically said to be viscous or viscid. A fluid with

title: Rotational viscosity

text: a pseudovector) and formula_15 is the rotational viscosity coefficient. Rotational viscosity Viscosity is usually described as the property of a fluid which determines the rate at which local momentum differences are equilibrated. Rotational viscosity is a property of a fluid which determines the rate at which local angular momentum differences are equilibrated. It is only appreciable if there are rotational degrees of freedom for the fluid particles. In the classical case, by the equipartition theorem, at equilibrium, if particle collisions can transfer angular momentum as well as linear momentum, then these degrees of freedom will have the same average energy.

title: Apparent viscosity

text: fluids, measurement of apparent viscosity without knowledge of the shear rate is of limited value: the measurement cannot be compared to other measurements if the speed and geometry of the two instruments is not identical. An apparent viscosity that is reported without the shear rate or information about the instrument and settings (e.g. speed and spindle type for a rotational viscometer) is meaningless. Multiple measurements of apparent viscosity at different, well-defined shear rates, can give useful information about the non-Newtonian behaviour of a fluid, and allow it to be modeled. In many non-Newtonian fluids, the shear stress due to viscosity,

title: Viscosity

text: viscosity of a dense fluid using these relations currently requires the use of molecular dynamics computer simulations. On the other hand, much more progress can be made for a dilute gas. Even elementary assumptions about how gas molecules move and interact lead to a basic understanding of the molecular origins of viscosity. More sophisticated treatments can be constructed by systematically coarse-graining the equations of motion of the gas molecules. An example of such a treatment is Chapman–Enskog theory, which derives expressions for the viscosity of a dilute gas from the Boltzmann equation. Momentum transport in gases is generally mediated by

title: Saybolt universal viscosity

text: The Saybolt FUROL viscosity test occurs at , or more recently, , and uses a larger calibrated tube. This provides for the testing of more viscous fluids, with the result being approximately of the universal viscosity. The test results are specified in seconds (s), more often than not referencing the test: Saybolt universal seconds (SUS); seconds, Saybolt universal (SSU); seconds, Saybolt universal viscosity (SSUV); and Saybolt FUROL seconds (SFS); seconds, Saybolt FUROL (SSF). The precise temperature at which the test is performed is often specified as well. Saybolt universal viscosity Saybolt universal viscosity (SUV), and the related Saybolt FUROL viscosity

title: Intrinsic viscosity

text: rigid spheres at infinite dilution, the intrinsic viscosity equals formula_8, as shown first by Albert Einstein. In practical settings, formula_6 is usually solute mass concentration (c, g/dL), and the units of intrinsic viscosity formula_1 are deciliters per gram (dL/g), otherwise known as inverse concentration. Generalizing from spheres to spheroids with an axial semiaxis formula_11 (i.e., the semiaxis of revolution) and equatorial semiaxes formula_12, the intrinsic viscosity can be written where the constants are defined The formula_19 coefficients are the Jeffery functions It is possible to generalize the intrinsic viscosity formula from spheroids to arbitrary ellipsoids with semiaxes formula_11, formula_12 and

title: Viscosity solution

text: goes to zero, while formula_74 is the limit solution of the vanishing viscosity problem formula_79 . One can readily confirm that formula_80 solves the PDE formula_76 for each epsilon. Further, the family of solutions formula_82 converge toward the solution formula_83 as formula_77 vanishes (see Figure). The three basic properties of viscosity solutions are "existence", "uniqueness" and "stability". The term "viscosity solutions" first appear in the work of Michael G. Crandall and Pierre-Louis Lions in 1983 regarding the Hamilton–Jacobi equation. The name is justified by the fact that the existence of solutions was obtained by the vanishing viscosity method. The definition

title: Viscosity models for mixtures

text: simply selects a known correlation for the dilute gas viscosity formula_53, and substracts this contribution from the total viscosity which is measured in the laboratory. This gives a residual viscosity term, often denoted formula_165, which represents the contribution of the dense fluid, formula_166. The dense fluid viscosity is thus defined as the viscosity in excess of the dilute gas viscosity. This technique is often used in developing mathematical models for both purely empirical correlations and models with a theoretical support. The dilute gas viscosity contribution becomes important when the zero density limit (i.e. zero pressure limit) is approached. It is

title: Viscosity models for mixtures

text: skin: where the symmetric gradient matrix with non-zero trace is How much the volume viscosity contributes to the flow characteristics in e.g. a choked flow such as convergent-divergent nozzle or valve flow is not well known, but the shear viscosity is by far the most utilized viscosity coefficient. The volume viscosity will now be abandoned, and the rest of the article will focus on the shear viscosity. Another application of shear viscosity models is Darcy's law for multiphase flow. and formula_22 and formula_23 are absolute and relative permeability, respectively. These 3 (vector) equations models flow of water, oil and natural

title: Viscosity models for mixtures

text: [molefractions]: Liquid phase of n mole in two-phase region where the liquid composition is formula_35 [molefractions]: where Since nearly all input to this viscosity model is provided by the EOS and the equilibrium calculations, this SS model (or TS model) for viscosity should be very simple to use for fluid mixtures. The viscosity model also have some empirical parameters that can be used as tuning parameters to compensate for imperfect EOS models and secure high accuracy also for fluid mixtures. Viscosity models for mixtures The shear viscosity (short viscosity) of a fluid is a material property that describes the friction

title: Viscosity solution

text: at formula_46 with formula_49 and formula_68 near formula_46. This implies that formula_4 is a viscosity solution. The previous boundary value problem is an Eikonal equation in a single spatial dimension with formula_71, where the solution is known to be the signed distance function to the boundary of the domain. Note also in the previous example, the importance of the sign of formula_72. In particular, the viscosity solution to the PDE formula_73 with the same boundary conditions is formula_74. This can be explained by observing that the solution formula_43 is the limiting solution of the vanishing viscosity problem formula_76 as formula_77

title: Viscosity printing

text: press, which prints all of the colors simultaneously. This is of a certain advantage, as in some other multi-color printing processes, correct registration of the blocks presents a difficulty. Viscosity printing Viscosity printing is a multi-color printmaking technique that incorporates principles of relief printing and intaglio printing. It was pioneered by Stanley William Hayter. The process uses the principle of viscosity to print multiple colors of ink from a single plate, rather than relying upon multiple plates for color separation. It is a fine art printmaking technique, making original prints in limited editions, as it is slow and allows too

title: Rotational viscosity

text: Rotational viscosity Viscosity is usually described as the property of a fluid which determines the rate at which local momentum differences are equilibrated. Rotational viscosity is a property of a fluid which determines the rate at which local angular momentum differences are equilibrated. It is only appreciable if there are rotational degrees of freedom for the fluid particles. In the classical case, by the equipartition theorem, at equilibrium, if particle collisions can transfer angular momentum as well as linear momentum, then these degrees of freedom will have the same average energy. If there is a lack of equilibrium between these

title: Ford viscosity cup

text: Ford viscosity cup The Ford viscosity cup is a simple gravity device that permits the timed flow of a known volume of liquid passing through an orifice located at the bottom. Under ideal conditions, this rate of flow would be proportional to the kinematic viscosity (expressed in stokes and centistokes) that is dependent upon the specific gravity of the draining liquid. However, the conditions in a simple flow cup are seldom ideal for making true measurements of viscosity. It is important when using a Ford Cup and when retesting liquids that the temperature of the cup and the liquid is

title: Viscosity printing

text: oil. Metal plates, usually copper or zinc, are used, as in the intaglio processes. The artist produces images on the plate by etching lines or textures. The plate is then inked in several stages. The first ink would be fairly dense—of a relatively high viscosity. The application of the high-viscosity ink is carried out as in any intaglio process: by forcing it into the recesses of the plate and then wiping off the plate's surface with a tarlatan. Ink of a second color, and the thinnest viscosity, is then applied to the surface of the plate with a hard rubber

title: Volume viscosity

text: term in the momentum equation that contains the volume viscosity disappears for an "incompressible fluid" because the divergence of the flow equals 0. There are cases where formula_17, which are explained below. And also it should be noted that formula_2 is not just a property of the fluid in the classic thermodynamic sense, but also depends on the process, for example the compression/expansion rate. The same goes for shear viscosity. For a Newtonian fluid the shear viscosity is a pure fluid property, but for a non-Newtonian fluid it is not a pure fluid property due to its dependence on the

title: Viscosity models for mixtures

text: Viscosity models for mixtures The shear viscosity (short viscosity) of a fluid is a material property that describes the friction between internal neighboring fluid surfaces (or sheets) flowing with different fluid velocities. This friction is the effect of (linear) momentum exchange caused by molecules with sufficient energy to move (or "to jump") between these fluid sheets due to fluctuations in their motion. The viscosity is not a material constant, but a material property that depends on temperature, pressure, fluid mixture composition, local velocity variations. This functional relationship is described by a mathematical viscosity model called a constitutive equation which is

title: Viscosity solution

text: a second derivative almost everywhere (in modern versions of the proof this is achieved with sup-convolutions and Alexandrov theorem). In subsequent years the concept of viscosity solution has become increasingly prevalent in analysis of degenerate elliptic PDE. Based on their stability properties, Barles and Souganidis obtained a very simple and general proof of convergence of finite difference schemes. Further regularity properties of viscosity solutions were obtained, especially in the uniformly elliptic case with the work of Luis Caffarelli. Viscosity solutions have become a central concept in the study of elliptic PDE. In the modern approach, the existence of solutions is

title: Volume viscosity

text: gives From Jacobian matrix, tensor derivative, tensor index notation, divergence and vector calculus identities we get the relations The classic Navier–Stokes equation can then be written in two familiar ways: The equivalent alternative variants of Navier–Stokes equations are: These last four equations are valid for compressible fluids that is subject to so small variations in pressure, temperature and velocity gradients that both shear and volume viscosity can be treated as constants. The volume viscosity of many fluids is inaccurately known, despite its fundamental role for fluid dynamics at high frequencies. The only values for the volume viscosity of simple Newtonian

title: Viscosity models for mixtures

text: a function and becomes The viscosity model proposed by Allal et al.(2001b) is thus A digression is that the self-diffusion coefficient of Boned et al. (2004) becomes Local nomenclature list: The mixture viscosity is The dilute gas viscosity formula_445 is taken from Chung et al.(1988) which is displayed in the section on SS theory. The dense fluid contribution to viscosity in FV theory is where formula_447 are three characteristic parameters of the fluid w.r.t. viscosity calculations. For fluid mixtures are these three parameters calculated using mixing rules. If the self-diffusion coefficient is included in the governing equations, probably via the

title: Temperature dependence of liquid viscosity

text: and applies over a large temperature and viscosity range: where "T" is absolute temperature in kelvins, formula_26 is the kinematic viscosity in centistokes, formula_27 is the zero order modified Bessel function of the second kind, and "A" and "B" are liquid specific values. This form should not be applied to ammonia or water viscosity over a large temperature range. For liquid metal viscosity as a function of temperature, Seeton proposed: Temperature dependence of liquid viscosity The temperature dependence of liquid viscosity is the phenomenon by which liquid viscosity tends to decrease (or, alternatively, its "fluidity" tends to increase) as its

title: Viscosity

text: (based on existing viscosity measurements) appear to be the only consistently reliable means of calculating viscosity in liquids. The same molecular-kinetic picture of a single component gas can also be applied to a gaseous mixture. For instance, in the Chapman-Enskog approach the viscosity formula_91 of a binary mixture of gases can be written in terms of the individual component viscosities formula_92, their respective volume fractions, and the intermolecular interactions. As for the single-component gas, the dependence of formula_91 on the parameters of the intermolecular interactions enters through various collisional integrals which may not be expressible in terms of elementary functions.

title: Viscosity models for mixtures

text: new analogy models for viscosity based on PR EOS (Peng and Robinson 1976) and PRPT EOS (Patel and Teja 1982) respectively. The following year T.-M. Guo modified the PR based viscosity model slightly, and it is this version that will be presented below as a representative of EOS analogy models for viscosity. PR EOS is displayed on the next line. The viscosity equation of Guo (1998) is displayed on the next line. To prepare for the mixing rules, the viscosity equation is re-written for a single fluid component i. Details of how the composite elements of the equation are related

title: Viscosity models for mixtures

text: viscosity value if the fluid is a non-Newtonian fluid such as paint, but it will give the same viscosity value for a Newtonian fluid such as water, petroleum oil or gas. If another parameter like temperature, formula_33, is changed, and the experiment is repeated with the same force, a new value for viscosity will be the calculated, for both non-Newtonian and Newtonian fluids. The great majority of material properties varies as a function of temperature, and this goes for viscosity also. The viscosity is also a function of pressure and, of course, the material itself. For a fluid mixture, this

title: Viscosity models for mixtures

text: the scaling parameter formula_107 alone is not a true or complete scaling factor unless all fluid components have a fairly similar (and preferably spherical) shape. The most important result of this kinetic derivation is perhaps not the viscosity formula, but the semi-empirical parameter formula_107 that is used extensively throughout the industry and applied science communities as a scaling factor for (shear) viscosity. The literature often reports the reciprocal parameter and denotes it as formula_112. It should be noted that the dilute gas viscosity contribution to the total viscosity of a fluid will only be important when predicting the viscosity of

title: Viscosity models for mixtures

text: of these two classes of molecules The gas-like viscosity contribution is taken from the viscosity model of Chung et al.(1984, 1988), which is based on the Chapman-Enskog(1964) kinetic theory of viscosity for dilute gases and the empirical expression of Neufeld et al.(1972) for the reduced collision integral, but expanded empirical to handle polyatomic, polar and hydrogen bonding fluids over a wide temperature range. The viscosity model of Chung et al.(1988) is where Local nomenclature list: In the 2000s, the development of the solid-like viscosity contribution started with Macías-Salinas et al.(2003) who used the Eyring equation in TS theory as an

title: Viscosity models for mixtures

text: to n-octadecane (CH to CH). The empirical parameters in the reduced friction functions above are treated as universal constants, and they are listed in the following table. For convenience are critical viscosities included in the tables for models with 5- and 7-parameters that was presented further up. The mixture viscosity is given by The mixture viscosity of well defined components is given by The mixture viscosity function of uncertain components is given by The mixture viscosity can be tuned to measured viscosity data by optimizing (regressing) the parameter formula_379. where the mixture friction coefficients are obtained by eq(I.7.45) through eq(I.7.47)

title: Viscosity models for mixtures

text: pressure dominated scaling parameter formula_137 : Zéberg-Mikkelsen (2001) proposed an empirical correlation for critical viscosity η parameter for n-alkanes, which is The unit equations for the two constitutive equations above by Zéberg-Mikkelsen (2001) are Inserting the critical temperature in the three viscosity equations from elementary kinetic theory gives three parameter equations. The three viscosity equations now coalesce to a single viscosity equation because a nondimensional scaling is used for the entire viscosity equation. The standard nondimensionality reasoning goes like this: Creating nondimensional variables (with subscript D) by scaling gives Claiming nondimensionality gives The collision cross section and the critical molar

title: Viscosity models for mixtures

text: vapors at low pressures or the viscosity of dense fluids at high temperatures. The viscosity model for dilute gas, that is shown above, is widely used throughout the industry and applied science communities. Therefore, many researchers do not specify a dilute gas viscosity model when they propose a total viscosity model, but leave it to the user to select and include the dilute gas contribution. Some researchers do not include a separate dilute gas model term, but propose an overall gas viscosity model that cover the entire pressure and temperature range they investigated. In this section our central macroscopic variables

title: Viscosity

text: a generalized form of Newton's law of viscosity. The bulk viscosity expresses a type of internal friction that resists the shearless compression or expansion of a fluid. Knowledge of formula_28 is frequently not necessary in fluid dynamics problems. For example, incompressible liquids satisfy formula_33 and so the term containing formula_28 is absent. Moreover, formula_28 is often assumed to be negligible for gases since it is formula_2 in a monoatomic ideal gas. One situation in which formula_28 can be important is the calculation of energy loss in sound and shock waves, described by Stokes' law of sound attenuation, since these phenomena

title: Viscosity index

text: chosen arbitrarily for reference are . The scale was originally interpolated between 0 for a naphthenic Texas Gulf crude and 100 for a paraffinnic Pennsylvania crude. Since the inception of the scale better oils have also been produced, leading to VIs greater than 100 (see below). VI improving additives and higher quality base oils are widely used nowadays which increase the VIs attainable beyond the value of 100. The Viscosity Index of synthetic oils ranges from 80 to over 400. The viscosity index can be calculated using the following formula: where "U" is the oil's kinematic viscosity at , and

title: Viscosity models for mixtures

text: Zéberg-Mikkelsen (2001) proposed a FF-model for light gas viscosity as follows The friction functions for light gases are simple The FF-model for light gas is valid for low, normal, critical and super critical conditions for these gases. Although the FF-model for viscosity of dilute gas is recommended, any accurate viscosity model for dilute gas can also be used with good results. The unit equations for viscosity and temperature are This article started with viscosity for mixtures by displaying equations for dilute gas based on elementary kinetic theory, hard core (kinetic) theory and proceeded to selected theories (and models) that aimed

title: Time-dependent viscosity

text: uses of rheopectic fluids can be gained through further research into the physics of hysteresis. Time-dependent viscosity Some non-Newtonian pseudoplastic fluids show a time-dependent change in viscosity and a non-linear stress-strain behavior in which the longer the fluid undergoes shear stress, the lower its viscosity becomes. A thixotropic fluid is one that takes time to attain viscosity equilibrium when introduced to a step change in shear rate. When shearing in a thixotropic fluid exceeds a certain threshold, it results in a breakdown of the fluid's microstructure and the exhibition of a shear thinning property. Certain gels or fluids that are

title: Time-dependent viscosity

text: Time-dependent viscosity Some non-Newtonian pseudoplastic fluids show a time-dependent change in viscosity and a non-linear stress-strain behavior in which the longer the fluid undergoes shear stress, the lower its viscosity becomes. A thixotropic fluid is one that takes time to attain viscosity equilibrium when introduced to a step change in shear rate. When shearing in a thixotropic fluid exceeds a certain threshold, it results in a breakdown of the fluid's microstructure and the exhibition of a shear thinning property. Certain gels or fluids that are thick (viscous) under static conditions will begin to thin and flow as they are shaken,

title: Viscosity

text: mixture: where formula_58 is an empirical parameter, and formula_97 and formula_92 are the respective mole fractions and viscosities of the component liquids. Since blending is an important process in the lubricating and oil industries, a variety of empirical and propriety equations exist for predicting the viscosity of a blend, besides those stemming directly from molecular theory. In a suspension of solid particles (e.g. micron-size spheres suspended in oil), an effective viscosity formula_99 can be defined in terms of stress and strain components which are "averaged" over a volume large compared with the distance between the suspended particles, but small with

title: Viscosity

text: no simple yet accurate picture for the molecular origins of viscosity in liquids. At the simplest level of description, the relative motion of adjacent layers in a liquid is opposed primarily by attractive molecular forces acting across the layer boundary. In this picture, one (correctly) expects viscosity to decrease with increasing temperature. This is because increasing temperature increases the random thermal motion of the molecules, which makes it easier for them to overcome their attractive interactions. Building on this visualization, a simple theory can be constructed in analogy with the discrete structure of a solid: groups of molecules in a

title: Viscosity

text: approximation that holds in some materials and fails in others. A fluid that behaves according to Newton's law, with a viscosity that is independent of the stress, is said to be Newtonian. Gases, water, and many common liquids can be considered Newtonian in ordinary conditions and contexts. There are many non-Newtonian fluids that significantly deviate from that law in some way or other. For example: Shear-thinning liquids are very commonly, but misleadingly, described as thixotropic. Even for a Newtonian fluid, the viscosity usually depends on its composition and temperature. For gases and other compressible fluids, it depends on temperature and

title: Viscosity

text: Newton's law of viscosity, this momentum flow occurs across a velocity gradient, and the "magnitude" of the corresponding momentum flux is determined by the viscosity. The analogy with heat and mass transfer can be made explicit. Just as heat flows from high temperature to low temperature and mass flows from high density to low density, "momentum "flows from "high velocity" to "low velocity". These behaviors are all described by compact expressions, called constitutive relations, whose one-dimensional forms are given here: where formula_46 is the density, formula_47 and formula_48 are the mass and heat fluxes, and formula_49 and formula_50 are the

title: Viscosity models for mixtures

text: also very common to scale the dense fluid viscosity by the critical viscosity, or by an estimate of the critical viscosity, which is a characteristic point far into the dense fluid region. The simplest model of the dense fluid viscosity is a (trunkated) power series of reduced mole density or pressure. Jossi et al. (1962) presented such a model based on reduced mole density, but its most widespread form is the version proposed by Lohrenz et al. (1964) which is displayed below. The LBC-function is then expanded in a (truncated) power series with empirical coefficients as displayed below. The final

title: Viscosity models for mixtures

text: the following sections. This means that some known contributions of research and development directions are not included. For example, is the group contribution method applied to a shear viscosity model not displayed. Even though it is an important method, it is thought to be a method for parameterization of a selected viscosity model, rather than a viscosity model in itself. The microscopic or molecular origin of fluids means that transport coefficients like viscosity can be calculated by time correlations which are valid for both gases and liquids, but it is computer intensive calculations. Another approach is the Boltzmann equation which

title: Viscosity models for mixtures

text: and repulsive pressure functions by reduced pressure functions. This will of course, affect the friction functions also. New friction functions are therefore introduced. They are called reduced friction functions, and they are of a more universal nature. The reduced frictional viscosity is Returning to the unreduced frictional viscosity and rephrasinge the formula, gives Critical viscosity is seldom measured and attempts to predict it by formulas are few. For a pure fluid, or component i in a fluid mixture, a formula from kinetic theory is often used to estimate critical viscosity. where formula_352 is a constant, and critical molar volume V

title: Viscosity models for mixtures

text: says that two fluids (subscript a and z) of a group (e.g. fluids of non-polar molecules) have approximately the same reduced molar volume (or reduced compressibility factor) when compared at the same reduced temperature and reduced pressure. In mathematical terms this is When the common CS principle above is applied to viscosity, it reads Note that the CS principle was originally formulated for equilibrium states, but it is now applied on a transport property - viscosity, and this tells us that another CS formula may be needed for viscosity. In order to increase the calculation speed for viscosity calculations based

title: Viscosity models for mixtures

text: volume which are both difficult to access experimentally, are avoided or circumvented. On the other hand, the critical viscosity has appeared as a new parameter, and critical viscosity is just as difficult to access experimentally as the other two parameters. Fortunately, the best viscosity equations have become so accurate that they justify calculation in the critical point, especially if the equation is matched to surrounding experimental data points. Wilke (1950) derived a mixing rule based on kinetic gas theory The Wilke mixing rule is capable of describing the correct viscosity behavior of gas mixtures showing a nonlinear and non-monotonical behavior,

title: Viscosity models for mixtures

text: viscosity is generally not an equation of state, so the term constitutional equation will in general be used for viscosity equations (or functions). The free variables in the two equations above, also indicates that specific constitutive equations for shear viscosity will be quite different from the simple defining equation for shear viscosity that is shown further up. The rest of this article will show that this is certaintly true. Non-Newtonian fluids will therefore be abandoned, and the rest of this article will focus on Newtonian fluids. In textbooks on elementary kinetic theory one can find results for dilute gas modeling

title: Viscosity models for mixtures

text: usually fare more complex than the defining equation of shear viscosity. One such complicating feature is the relation between the viscosity model for a pure fluid and the model for a fluid mixture which is called mixing rules. When scientists and engineers use new arguments or theories to develop a new viscosity model, instead of improving the reigning model, it may lead to the first model in a new class of models. This article will display one or two representative models for different classes of viscosity models, and these classes are: Selected contributions from these development directions is displayed in

title: Viscosity models for mixtures

text: with the statistical approach with respect to gas viscosity models for simple fluids (simple molecules). The gas viscosity model of Chung et alios (1988) is combination of the Chapman-Enskog(1964) kinetic theory of viscosity for dilute gases and the empirical expression of Neufeld et alios (1972) for the reduced collision integral, but expanded empirical to handle polyatomic, polar and hydrogen bonding fluids over a wide temperature range. This viscosity model illustrates a successful combination of kinetic theory and empiricism, and it is displayed in the section of Significant structure theory and its model for the gas-like contribution to the total fluid

title: Viscosity models for mixtures

text: classic linear mole weighted mixing rules which are displayed below for a mixture of N fluid components. The three characteristic viscosity parameters formula_452 are usually established by optimizing the viscosity formula against measured viscosity data for pure fluids (i.e. single component fluids). The three characteristic viscosity parameters formula_453 are usually established by optimizing the viscosity formula against measured viscosity data for pure fluids (i.e. single component fluids). Data for these parameters can then be stored in databases together with data for other chemical and physical material properties and information. This happens more often if use of the equation becomes widespread.

title: Viscosity solution

text: of solution had actually been given earlier by Lawrence C. Evans in 1980. Subsequently the definition and properties of viscosity solutions for the Hamilton–Jacobi equation were refined in a joint work by Crandall, Evans and Lions in 1984. For a few years the work on viscosity solutions concentrated on first order equations because it was not known whether second order elliptic equations would have a unique viscosity solution except in very particular cases. The breakthrough result came with the method introduced by Robert Jensen in 1988 to prove the comparison principle using a regularized approximation of the solution which has

title: Viscosity index

text: much in viscosity over the temperature range. This allows for consistent engine performance within the normal working conditions. Historically, there were two different oil types, recommended for usage in different weather conditions. For winter use, the engines required motor oil with the viscosity rating 5, while for warm and hot season it had to be changed to oil with 30-40 viscosity rating. Later on, there appeared multi-grade motor oils, which contained polymers that made the oil film thinner at higher temperatures without thickening the oil. The VI scale was set up by the Society of Automotive Engineers (SAE). The temperatures

title: Constant viscosity elastic fluid

text: aluminum giant Alcoa developed a way to convert the waste into a thick matter that could still flow down pipes. Using this method, the dangers of tubes erupting was cut out, leading to a more sustainable practice. Constant viscosity elastic fluid Constant viscosity elastic liquids, also known as Boger fluids are elastic fluids with constant viscosity. This creates an effect in the fluid where it flows like a liquid, yet behaves like an elastic solid when stretched out. Most elastic fluids exhibit shear thinning (viscosity decreases as shear strain is applied), because they are solutions containing polymers. But Boger fluids

title: Constant viscosity elastic fluid

text: Constant viscosity elastic fluid Constant viscosity elastic liquids, also known as Boger fluids are elastic fluids with constant viscosity. This creates an effect in the fluid where it flows like a liquid, yet behaves like an elastic solid when stretched out. Most elastic fluids exhibit shear thinning (viscosity decreases as shear strain is applied), because they are solutions containing polymers. But Boger fluids are exceptions since they are highly dilute solutions, so dilute that shear thinning caused by the polymers can be ignored. Boger fluids are made primarily by adding a small amount of polymer to a Newtonian fluid with

title: High viscosity mixer

text: High viscosity mixer High viscosity mixers are mixers designed for mixing materials with laminar mixing processes because the ingredients have such high viscosities that a turbulent mixing phase cannot be obtained at all or cannot be obtained without a high amount of heat. The process can be used for high viscosity liquid to liquid mixing or for paste mixing combining liquid and solid ingredients. Some products that may require laminar mixing in a high viscosity mixer include putties, chewing gum, and soaps. The end product usually starts at several hundred thousand centipoise and can reach as high as several million

title: Ford viscosity cup

text: maintained, as ambient temperature makes a significant difference to viscosity and thus flow rate. The original Ford Cup was based on Imperial (US) measurement of the aperture. Many other types of flow cups are being used, depending on the industry or region. Din Cup 4 mm., standard DIN 53211 (cancelled) ISO Cup 2, 3, 4, 5, 6, 8 mm. standard ISO 2431 AFNOR Cup 2,5, 4, 6, 8 mm. standard NF T30-014 ASTM Cup 1,2,3,4,5 standard ASTM D1200 Ford viscosity cup The Ford viscosity cup is a simple gravity device that permits the timed flow of a known volume of

text: Temperature dependence of liquid viscosity The temperature dependence of liquid viscosity is the phenomenon by which liquid viscosity tends to decrease (or, alternatively, its "fluidity" tends to increase) as its temperature increases. This can be observed, for example, by watching how cooking oil appears to move more fluidly upon a frying pan after being heated by a stove. Kinematic viscosity can be estimated as a typical (thermal) velocity times the mean free path. A molecular view of liquids can be used for a qualitative picture of decrease in the shear (or bulk) viscosity of a simple fluid with temperature. As

title: Volume viscosity

text: the change in volume almost immediately unless, of course, the rate of change of volume is very large." He later adds: "It may happen, nevertheless, that the relaxation times of the processes of restoration of equilibrium are long, i.e. they take place comparatively slowly." After an example, he concludes: "Hence, if the relaxation time of these processes is long, a considerable dissipation of energy occurs when the fluid is compressed or expanded, and, since this dissipation must be determined by the second viscosity, we reach the conclusion that formula_2 is large." The denotation "bulk viscosity" is sometimes used for the

title: Volume viscosity

text: the fluid dynamics of liquids containing gas bubbles. For an incompressible liquid the volume viscosity is superfluous, and does not appear in the equation of motion. The negative-one-third of the trace of the Cauchy stress tensor at the equilibrium is often identified with the thermodynamic pressure, which only depends upon the equilibrium state potentials like temperature and density (equation of state). In general, the trace of the stress tensor is the sum of thermodynamic pressure contribution and another contribution which is proportional to the divergence of the velocity field. This coefficient of proportionality is called volume viscosity. Common symbols for

title: Volume viscosity

text: parameter formula_20, especially in older articles and textbooks, but this is not recommended. There exists a momentum density equation that is a variant of Navier–Stokes equation where formula_5 is the shear viscosity coefficient and formula_23 is a second viscosity coefficient. Notice that formula_10 is not traceless, meaning that it will also model outflow/inflow to the control volume (or grid cell in a 3D model) due to pure expansion/compression. It is quite common in theoretical work to add zero to an equation. If we add and subtract formula_25 the equation becomes By comparing the variant of the Navier–Stokes equation displayed just

title: Viscosity models for mixtures

text: motion with the defining equation of shear viscosity gives the well known equation for shear viscosity for dilute gases: where where formula_48 is the Boltzmann constant, formula_49 is the Avogadro constant, formula_50 is the gas constant, formula_51 is the molar mass and formula_52 is the molecular mass. The equation above presupposes that the gas density is low (i.e. the pressure is low), hence the subscript zero in the variable formula_53. This implies that the kinetic translational energy dominates over rotational and vibrational molecule energies. The viscosity equation displayed above further presupposes that there is only one type of gas molecules,

title: Viscosity models for mixtures

text: or showing a characteristic bump shape, when the viscosity is plotted versus mass density at critical temperature, for mixtures containnig molecules of very different sizes. Due to its complexity, it has not gained widespread use. Instead, the slightly simpler mixing rule proposed by Herning and Zipperer (1936), is found to be suitable for gases of hydrocarbon mixtures. The classic Grunberg-Nissan (1949) mixing rule for liquid mixture is where formula_151 is the viscosity of the liquid mixture, formula_152 is the viscosity (equation) for fluid component i when flowing as a pure fluid, and formula_36 is the molfraction of component i in

title: Viscosity models for mixtures

text: (for fluid component i in a mixture) is then written as The formulas from friction theory is then related to well defined and uncertain fluid components. The result is However, in order to obtain the characteristic critical viscosity of the heavy pseudocomponents, the following modification of the Uyehara and Watson (1944) expression for the critical viscosity can be used. The frictional (or residual) viscosity is then written as The unit equations are formula_368 and formula_369 and formula_370. The unit equation of formula_374 is formula_375. The 1-parameter model have been developed based on single component fluids in the series from methane

title: Viscosity solution

text: for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations. There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book or the definition using semi-jets in the Users Guide. Consider the boundary value problem formula_39, or formula_40, on formula_41 with boundary conditions formula_42. The function formula_43 is the unique viscosity solution. To see this, note that the boundary conditions are satisfied, and formula_39 is well-defined on the interior except at formula_45. Thus, it remains to show

title: Volume viscosity

text: above, and the classic Navier–Stokes equation displayed near the top of this article, it follows that The tensors formula_28 and formula_12 describes pure shear flow and pure compression, while the tensor formula_10 describes a crude (or unpure) shear flow. The viscosity parameter formula_31 therefore compensates for the excessive/insufficient compression flow caused by using the tensor formula_10 instead of formula_11. When the denotation "bulk viscosity" is used, it is often given the symbol formula_34 and defined (as a seemingly composite parameter) by the equation Next we return to the classic Navier–Stokes equation and assume that formula_5 and formula_2 are constants. This

title: Viscosity models for mixtures

text: and parameters and their units are temperature formula_113 [K], pressure formula_114 [bar], molar mass formula_51 [g/mol], low density (low pressure or dilute) gas viscosity formula_69 [μP]. It is, however, common in the industry to use another unit for liquid and high density gas viscosity formula_1 [cP]. From Boltzmann's equation Chapman and Enskog derived a viscosity model for a dilute gas. where formula_119 is (the absolute value of) the energy-depth of the potential well (see e.g. Lennard-Jones interaction potential). The term formula_120 is called the collision integral, and it is occurs as a general function of temperature that the user must

title: Viscosity models for mixtures

text: means that the shear viscosity will also vary according to the fluid composition. To map the viscosity as a function of all these variables require a large sequence of experiments that generates an even larger set of numbers called measured data, observed data or observations. Prior to, or at the same time as, the experiments is a material property model (or short material model) proposed to describe or explain the observations. This matemathical model is called the constitutive equation for shear viscosity. It is usually an explicit function that contains some empirical parameters that is adjusted in order to match

title: Viscosity models for mixtures

text: the above equation and the definition of the compressibility factor it follows that Zéberg-Mikkelsen (2001) also proposed an empirical correlation for η, with parameters for n-alkanes, which is The unit equations for the two constitutive equations above by Zéberg-Mikkelsen (2001) are The next step is to split the formulas into formulas for well defined components (designated by subscript d) with respect critical viscosity and formulas for uncertain components (designated by subscript u) where critical viscosity is estimated using formula_361 and the universal constant formula_354 which will be treated as a tuning parameter for the current mixture. The dense fluid viscosity

title: Viscosity models for mixtures

text: acting between two moving layers can be separated into a term formula_281 caused by dilute gas collisions, and a term formula_282 caused by friction in the dense fluid. The dilute gas viscosity (i.e. the limiting viscosity behavior as the pressure, normal stress, goes to zero) and the dense fluid viscsoity (the residual viscosity) can be calculated by where du/dy formula_285 is the local velocity gradient orthogonal to the direction of flow. Thus The basic idea of QZS (2000) is that internal surfaces in a Couette flow acts like (or is analogue to) mechanical slabs with friction forces acting on each

title: Viscosity models for mixtures

text: the parameter formula_79 above where the gas constant formula_50 is absorbed into the empirical constant, will often be encountered in practice. In this case the viscosity equation becomes where the empirical parameter formula_82 is not dimensionless, and a proposed viscosity model for dense fluid will not be dimensionless if formula_79 is the common scaling factor. Notice that Inserting the critical temperature in the equation for dilute viscosity gives The default values of the parameters formula_86 and formula_87 should be fairly universal values, although formula_87 depends on the unit system. However, the critical molar volume in the scaling parameters formula_89 and