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## Rheology

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**Rheology**LMM**Introduction to Rheology**• Rheology describes the deformation of a body under the influence of stresses. • “Bodies” in this context can be either solids, liquids, or gases. • Ideal solids deform elastically. • The energy required for the deformation is fully recovered when the stresses are removed.**Introduction to Rheology**• Ideal fluids such as liquids and gases deform irreversibly -- they flow. • The energy required for the deformation is dissipated within the fluid in the form of heat and cannot be recovered simply by removing the stresses.**Introduction to Rheology**• The real bodies we encounter are neither ideal solids nor ideal fluids. • Real solids can also deform irreversibly under the influence of forces of sufficient magnitude • They creep, they flow. • Example: Steel -- a typical solid -- can be forced to flow as in the case of sheet steel when it is pressed into a form, for example for automobile body parts.**Introduction to Rheology**• Only a few liquids of practical importance come close to ideal liquids in their behavior. • The vast majority of liquids show a rheological behavior that classifies them to a region somewhere between the liquids and the solids. • They are in varying extents both elastic and viscous and may therefore be named “visco-elastic”. • Solids can be subjected to both tensile and shear stresses while liquids such as water can only be sheared.**Introduction to Rheology**• τ = G ⋅ dL/dy = G ⋅ tan γ ≈ G ⋅ γ • τ = shear stress = force/area, N/m 2 = Pa • G = Young’s modulus which relates to the stiffness of the solid, N/m 2 =Pa • γ = dL/y = strain (dimensionless) • y = height of the solid body [m] • ΔL = deformation of the body as a result of shear stress [m].**Introduction to Rheology**• The Young’s modulus G in this equation is a correlating factor indicating stiffness linked mainly to the chemical-physical nature of the solid involved. • It defines the resistance of the solid against deformation.**Introduction to Rheology**• The resistance of a fluid against any irreversible positional change of its’ volume elements is called viscosity. • To maintain flow in a fluid, energy must be added continuously.**Introduction to Rheology**• While solids and fluids react very differently when deformed by stresses, there is no basic difference rheologically between liquids and gases. • Gases are fluids with a much lower viscosity than liquids. • For example hydrogen gas at 20°C has a viscosity a hundredth of the viscosity of water.**Introduction to Rheology**• Instruments which measure the visco-elastic properties of solids, semi-solids and fluids are named “rheometers”. • Instruments which are limited in their use for the measurement of the viscous flow behavior of fluids are described as “viscometers”.**Shear induced flow in liquids can occur in 4 laminar flow**model cases:**Flow between two parallel flat plates**• When one plate moves and the other is stationary. • This creates a laminar flow of layers which resembles the displacement of individual cards in a deck of cards.**Flow in the annular gap between two concentric cylinders.**• One of the two cylinders is assumed to be stationary while the other can rotate. • This flow can be understood as the displacement of concentric layers situated inside of each other. • A flow of this type is realized for example in rotational rheometers with coaxial cylinder sensor systems.**Flow through pipes, tubes, or capillaries.**• A pressure difference between the inlet and the outlet of a capillary forces a Newtonian liquid to flow with a parabolic speed distribution across the diameter. • This resembles a telescopic displacement of nesting, tube-like liquid layers sliding over each other.**Flow through pipes, tubes, or capillaries.**• A variation of capillary flow is the flow in channels with a rectangular cross-section such as slit capillaries. • If those are used for capillary rheometry the channel width should be wide in comparison to the channel depth to minimize the side wall effects.**Flow between two parallel-plates or between a cone-and-plate**sensor • Where one of the two is stationary and the other rotates. • This model resembles twisting a roll of coins causing coins to be displaced by a small angle with respect to adjacent coins. • This type of flow is caused in rotational rheometers with the samples placed within the gap of parallel-plate or cone-and-plate sensor systems.**The basic law**• The measurement of the viscosity of liquids first requires the definition of the parameters which are involved in flow. • Then one has to find suitable test conditions which allow the measurement of flow properties objectively and reproducibly.**The basic law**• Isaac Newton was the first to express the basic law of viscometry describing the flow behavior of an ideal liquid: • shear stress = viscosity ⋅ shear rate**The parallel-plate model helps to define both shear stress**and shear rate**Shear stress**• A force F applied tangentially to an area A being the interface between the upper plate and the liquid underneath, leads to a flow in the liquid layer. • The velocity of flow that can be maintained for a given force is controlled by the internal resistance of the liquid, i.e. by it’s viscosity.**Shear stress**• τ =F (force)/A (area) • N (Newton)/m 2 = Pa [Pascal]**Shear rate**• The shear stress τ causes the liquid to flow in a special pattern. • A maximum flow speed Vmax is found at the upper boundary. • The speed drops across the gap size y down to Vmin = 0 at the lower boundary contacting the stationary plate.**Shear rate**• Laminar flow means that infinitesimally thin liquid layers slide on top of each other, similar to the cards in a deck of cards. • One laminar layer is then displaced with respect to the adjacent ones by a fraction of the total displacement encountered in the liquid between both plates. • The speed drop across the gap size is named “shear rate” and in it’s general form it is mathematically defined by a differential.**Shear rate**In case of the two parallel plates with a linear speed drop across the gap the differential in the equation reduces to:**Shear rate**• In the scientific literature shear rate is denoted as • The dot above the γ indicates that shear rate is the time-derivative of the strain caused by the shear stress acting on the liquid lamina.**Solids vs Liquids**• Comparing equations [1] and [7] indicates another basic difference between solids and liquids: • Shear stress causes strain in solids but in liquids it causes the rate of strain. • This simply means that solids are elastically deformed while liquids flow. • The parameters G and η serve the same purpose of introducing a resistance factor linked mainly to the nature of the body stressed.**Dynamic viscosity**• Solving equation [2] for the dynamic viscosity η gives:**Dynamic viscosity**• The unit of dynamic viscosity η is the “Pascal ⋅ second” [Pa⋅s]. • The unit “milli-Pascal ⋅ second” [mPa⋅s] is also often used. • 1 Pa ⋅ s = 1000 mPa ⋅ s • It is worthwhile noting that the previously used units of “centiPoise” [cP] for the dynamic viscosity η are interchangeable with [mPa⋅s]. • 1 mPa⋅s = 1 cP**Kinematic viscosity**• When Newtonian liquids are tested by means of some capillary viscometers, viscosity is determined in units of kinematic viscosity υ. • The force of gravity acts as the force driving the liquid sample through the capillary. • The density of the sample is one other additional parameter.**Kinematic viscosity**• Kinematic viscosity υ and dynamic viscosity η are linked.**Flow and viscosity curves**• The correlation between shear stress and shear rate defining the flow behavior of a liquid is graphically displayed in a diagram of τ on the ordinate and on the abscissa. • This diagram is called the “Flow Curve”. • The most simple type of a flow curve is shown In Figure 4. • The viscosity in equation(2) is assumed to be constant and independent of .**Viscosity Curve**• Another diagram is very common: η is plotted versus • This diagram is called the “Viscosity Curve”. • The viscosity curve shown in Fig. 5 corresponds to the flow curve of Fig. 4. • Viscosity measurements always first result in the flow curve. • It’s results can then be rearranged mathematically to allow the plotting of the corresponding viscosity curve. • The different types of flow curves have their counterparts in types of viscosity curves.**Viscosity parameters**• Viscosity, which describes the physical property of a liquid to resist shear-induced flow, may depend on 6 independent parameters:**Viscosity Parameters**• “S” - This parameter denotes the physical-chemical nature of a substance being the primary influence on viscosity, i.e. whether the liquid is water, oil, honey, or a polymer melt etc. • “T” - This parameter is linked to the temperature of the substance. Experience shows that viscosity is heavily influenced by changes of temperature. As an example: The viscosity of some mineral oils drops by 10% for a temperature increase of only 1°C.**Viscosity Parameters**• “p” - This parameter “pressure” is not experienced as often as the previous ones. • Pressure compresses fluids and thus increases intermolecular resistance. • Liquids are compressible under the influence of very high pressure-- similar to gases but to a lesser extent. • Increases of pressure tend to increase the viscosity. • As an example: Raising the pressure for drilling mud from ambient to 1000 bar increases it’s viscosity by some 30%.**Viscosity Parameters**• -Parameter “shear rate” is a decisive factor influencing the viscosity of very many liquids. • Increasing shear rates may decrease or increase the viscosity. • “t” Parameter “time” denotes the phenomenon that the viscosity of some substances, usually dispersions, depends on the previous shear history, i.e. on the length of time the substance was subjected to continuous shear or was allowed to rest before being tested.**Viscosity Parameters**• “E” - Parameter “electrical field” is related to a family of suspensions characterized by the phenomenon that their flow behavior is strongly influenced by the magnitude of electrical fields acting upon them. • These suspensions are called either “electro-viscous fluids” (EVF) or “electro-rheological fluids” (ERF). • They contain finely dispersed dielectric particles such as aluminum silicates in electro-conductive liquids such as water which may be polarized in an electrical field. • They may have their viscosity changed instantaneously and reversibly from a low to a high viscosity level, to a dough-like material or even to a solid state as a function of electrical field changes, caused by voltage changes.**Newtonian Liquids**• Newton assumed that the graphical equivalent of his equation [2] for an ideal liquid would be a straight line starting at the origin of the flow curve and would climb with a slope of an angle α. • Any point on this line defines pairs of values for τ and . • Dividing one by the other gives a value of η ([8]). • This value can also be defined as the tangent of the slope angle α of the flow curve: η = tan α .**Newtonian Liquids**• Because the flow curve for an ideal liquid is straight, the ratio of all pairs of τ and -values belonging to this line are constant. • This means that η is not affected by changes in shear rate. • All liquids for which this statement is true are called “Newtonian liquids” (both curves 1 in Fig. 6). • Examples: water, mineral oils, bitumen, molasses.**Non-Newtonian Liquids**• All other liquids not exhibiting this “ideal” flow behavior are called “Non-Newtonian Liquids”. • They outnumber the ideal liquids by far.**Pseudo-plastic Liquids**• Liquids which show pseudo-plastic flow behavior under certain conditions of stress and shear rates are often just called “pseudo-plastic liquids” (both curves 2 in Fig. 6) • These liquids show drastic viscosity decreases when the shear rate is increased from low to high levels.