{"id":3411,"date":"2024-09-04T19:57:10","date_gmt":"2024-09-04T17:57:10","guid":{"rendered":"https:\/\/test.kint.cz\/?p=3411"},"modified":"2025-06-04T19:14:40","modified_gmt":"2025-06-04T17:14:40","slug":"rheology","status":"publish","type":"post","link":"https:\/\/test.kint.cz\/en\/rheology\/","title":{"rendered":"Rheology"},"content":{"rendered":"\t\t
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This article builds on the articles on thermodynamics<\/a> (article 2<\/a>) and briefly introduces the issue of rheology, which is simply the study of material flow. In the following paragraphs, you will learn, for example, how water vortices are created or how the viscosity of substances is measured, and so on.<\/p>

The following paragraphs provide a summary and introduction to the topic based on research from the following sources:<\/p>

PRAGOLAB. Rheology Seminar. Prague: Pragolab 2015. Available at: http:\/\/www.pragolab.cz\/files\/download\/Seminar_reologie_2015.pdf<\/a><\/p>

PRU\u0160KA, J. Rheology, MH 8. Lecture.<\/em> Prague: FS \u010cVUT 2008. Available at: http:\/\/departments.fsv.cvut.cz\/k135\/data\/wp-upload\/2008\/05\/reologie.pdf<\/a><\/p>

HOLUBOV\u00c1, Renata. Basics of Rheology and Rheometry of Liquids<\/em>. Olomouc: Palack\u00fd University Olomouc, 2014. ISBN 978-80-244-4178-8<\/p>

Rheology is the study of the deformation and flow of materials; the movement of viscous (Newtonian) liquids and the transformation of masses. These masses are not perfectly elastic (Hooke’s material), nor fully ductile (St. Venant’s fluid), or flexible, but exhibit various combinations of these properties. Rheology is divided into macro-rheology and micro-rheology.<\/p>

Macro-rheology<\/em><\/strong> \u2013 examines the deformable properties of matter from a general perspective.<\/p>

Micro-rheology<\/em><\/strong> \u2013 studies the deformable properties of individual parts of matter.<\/p>

Rheology deals with:<\/p>

\u2013 The relationships between different types of deformation of materials and investigates the causes and effects of deformations,<\/p>

\u2013 Relationships between shear stress and shear rate,<\/p>

\u2013 The boundaries between liquids and solids.<\/p>

Properties of Liquids<\/h2>

To describe the properties of ideal liquids<\/em><\/strong>, we define an ideal liquid as a liquid with no internal friction (non-viscous), with zero volume expansion and compressibility, zero gas solubility, and no evaporation.<\/p>

Liquids are further divided according to (Holubov\u00e1 R.) into:<\/p>

\u2013 Newtonian (e.g. water), where viscosity at a given temperature and pressure is a physical constant,<\/em><\/p>

\u2013 Non-Newtonian (e.g. emulsions, mixtures of solid particles with liquids, paints), whose viscosity is not a physical constant.<\/em><\/p>

Flow of real liquids<\/strong> is:<\/em><\/p>

Laminated<\/strong> \u2013 particles move in layers that are mutually parallel, with no movement of particles perpendicular to the direction of motion;<\/em><\/p>

Turbulent<\/strong> \u2013 particles have, in addition to the progressive speed, a turbulent (fluctuating) velocity, which is used to move along the cross-section.<\/em><\/p>

Further in the area of liquid properties, the following relationships for stress are defined.<\/p>

Normal stress<\/strong> (pressure) p \u2013 is given as the ratio of normal elemental force to the size of the given area:<\/em><\/p>

p = dFn<\/sub>\/ dS,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1)<\/p>

where dS is the elemental area inside the liquid, and dFn<\/sub> is the normal component (acting perpendicular to the considered area) of the elemental force dF.<\/p>

In a liquid, tensile stress cannot be induced, so pressure is measured as positive. When measuring pressure from zero value, it is referred to as the so-called absolute pressure. Sometimes it is advantageous to measure pressure from a certain reference pressure (usually atmospheric pressure). Pressure differences above or below this pressure are called overpressure and vacuum, respectively.<\/p>

Shear stress<\/em><\/strong> \u03c4 \u2013 is given as the ratio of shear elemental force to the size of the given area:<\/p>

\u03c4 = dFt<\/sub>\/ dS,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2)<\/p>

where dFt<\/sub> is the shear component (causing particle displacement in the liquid) of the elemental force dF acting on the elemental area dS.<\/p>

For an elementary prism with height dy, where the lower wall moves with speed v and the upper wall moves with speed v + dv according to [11], the following applies:<\/p>

\u03c4 = \u03b7 dv\/dy,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3)<\/p>

Dynamic viscosity<\/em><\/strong> of liquids generally depends on temperature (decreases with increasing temperature) and pressure (the dependence is negligible). It manifests as resistance to particle movement in the liquid. Based on the dependence of dynamic viscosity on shear stress, liquids are divided into Newtonian (independent \u2013 equation (2)) and non-Newtonian (dependent \u2013 equation 3).<\/p>

The dependence of viscosity on temperature can be expressed by the relationship:<\/p>

\u03b7(T) = k * eb\/(T+\u0398)<\/sup>,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (4)<\/p>

where constant k has the dimension of viscosity (Pa \u00b7 s), b and \u0398 are constants characteristic for a given fluid, their unit is Kelvin.<\/em><\/p>

The dependence of viscosity on pressure is expressed by the relationship:<\/p>

\u03b7(p) = \u03b70<\/sub> * e\u03b1p<\/sup>,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 (5)<\/p>

\u03b1 is a coefficient dependent on temperature.<\/p>

In addition to dynamic viscosity, we also introduce the quantity kinematic viscosity<\/strong><\/em>, which is defined by the relationship:<\/p>

\u03bd = \u03b7 \/ \u03c1,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6)<\/p>

where \u03c1 is the density of the liquid.<\/p>

Flow of liquid in a capillary \u2013<\/em><\/strong> characteristic quantities for describing flow are: pressure difference \u0394p, capillary length l, and its radius r. In the stationary case, the pressure force Fp<\/sub> must equal the frictional force FR<\/sub>. For flow through a capillary, the velocity distribution is parabolic, meaning that the velocity v(r) forms a rotational paraboloid. The relationships for velocity profile and the amount of liquid flowing per unit time t are provided in the work.<\/p>

Navier-Stokes Equation \u2013 <\/em><\/strong>represents the motion equation of a real flowing liquid. A real liquid is subject to gravitational, pressure, and frictional forces. The total force can be expressed as the sum of all acting forces. The goal is to find the distribution of velocities and pressures. Therefore, it is necessary to know the external acceleration, the density of the liquid, and the boundary conditions. The individual terms of the Navier-Stokes equation represent:<\/p>

\u2013 external acceleration due to the effect of gravitational force,<\/em><\/p>

\u2013 acceleration from surface (pressure) force,<\/em><\/p>

\u2013 acceleration required to overcome viscous friction in liquids,<\/em><\/p>

\u2013 acceleration due to viscosity in compressible liquids,<\/em><\/p>

\u2013 convective acceleration from inertia force,<\/em><\/p>

\u2013 local acceleration from inertia force.<\/em><\/p>

For incompressible liquids, the fourth term of the equation (viscous acceleration) drops out due to the continuity equation. In a non-viscous liquid, the equation transitions to the Euler equation of hydrodynamics. Using the Navier-Stokes equation, classical hydrodynamics can be described. Unlike the Euler equation, it contains a term that expresses internal friction in the fluid.<\/p>

The Navier-Stokes equation is a nonlinear differential equation. Its solution provides the distribution of velocities in the fluid depending on position and time. Mathematics does not have an analytical solution procedure for this equation; only special cases can be solved.<\/p>

Laminated flow around a ball \u2013<\/em><\/strong> viscous liquids exert force on any object moving within the liquid. It can be examined, for example, how to describe the fall of an object in a viscous liquid. Under the effect of gravitational force Fg<\/sub>, the object is accelerated, and its initial speed v0<\/sub> = 0 continuously increases. The phenomenon lasts until the object reaches a constant velocity of fall v, i.e., the acceleration is zero. Then there is a balance between the downward gravitational force and the upward forces \u2013 buoyancy and friction: Fg<\/sub> = Fvz<\/sub> + FT<\/sub>.<\/p>

Based on experiments with balls of different sizes and using various liquids, it was found that the frictional force depends on the coefficient \u03b7, final speed v, and the radius of the ball r, i.e. Stokes’ Law<\/strong>:<\/p>

FT<\/sub> = -6\u03c0\u03b7rv.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (7)<\/p>

Vortices \u2013<\/em><\/strong> a characteristic feature of turbulence is the formation of vortices. Imagine a cylinder being surrounded by fluid. From a certain flow velocity vG<\/sub>, stationary vortices with opposite rotation direction begin to form behind the cylinder in the so-called dead water zone. The vortex can be divided into two regions \u2013 the core and the circulation area. The explanation of vortex formation is carried out using a cylinder placed in the fluid, which moves from left to right and “collides” with the object \u2013 see the diagram.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t

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Physics \u2013 Rheology: Mechanismus pr\u00e1zn\u00ed v\u00edr\u016f viz (HOLUBOV\u00c1, R.)<\/figcaption>\n\t\t\t\t\t\t\t\t\t\t<\/figure>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t
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\n\t\t\t\t\t\t\t\t\tAt point S1<\/sub> (impact point), the velocity is minimal, which means that according to Bernoulli’s equation, the static pressure at this point is maximal. A pressure gradient forms between points S1<\/sub> and K at the top of the object. At the top of the object, in point K, the static pressure is minimal, and the velocity is maximal, only reduced by the frictional effect (v < vmax<\/sub>). Now, the fluid must overcome the existing pressure increase at the rear of the cylinder, i.e., between point K and point S2<\/sub>. Since v < vmax<\/sub>, the kinetic energy is insufficient to reach point S2<\/sub>. The flow has zero velocity at the turnaround point W; however, since a pressure force acts from point S2<\/sub> towards point K, the decelerated fluid particles are pushed against the flow direction of the outer layer. This causes the fluid element around point W to swirl, creating a vortex. A vortex also develops on the bottom side of the cylinder, but with the opposite rotational direction. Both vortices detach from the cylinder and are replaced by new ones. This creates what is called the K\u00e1rm\u00e1n vortex street<\/strong>.<\/em>\n\nPressure and velocity changes before and after the body are estimated using Bernoulli\u2019s equation.\n\nBased on these considerations, the resistive pressure force acting on the body can be determined.\n\nInside the vortex, there is an area around the core where the fluid rotates like a rigid body, i.e., with a constant angular velocity \u03c9. The tangential velocity of the rotation v = \u03c9xr increases linearly with the distance r from the center. Additionally, all particles have their own rotation. When a particle rotates once around the core, it simultaneously rotates once around its own axis. For areas outside the core (for r > rk<\/sub>), the rotational velocity of the particles decreases with increasing distance. The rotation is only around the core and not around its own axis\u2026 circulation area<\/em> [11].<\/em>\n\nLaminary<\/strong> flow transitions into turbulent<\/strong> flow when the so-called critical Reynolds number is exceeded. This depends on the viscosity, flow velocity of the fluid, and flow geometry. The transition from laminar to turbulent flow also depends on the geometric shape of the flow sections, the rounding of edges at the beginning of the pipe, pipe wall roughness, the turbulence of the incoming flow, etc. In real fluid flow, there is a transition area where both laminar and turbulent flows may occur under specific conditions.<\/em> (e.g., in a pipe)\n\nAccording to Hagen-Poiseuille’s law<\/em><\/strong>, the mean velocity of fluid flow is determined from the volumetric flow, see (HOLUBOV\u00c1, R.).\n\n

Newtonian Fluids<\/em><\/h3> The viscosity of a Newtonian fluid depends only on temperature and follows a direct proportionality between shear stress and velocity gradient (Newton’s law of viscosity) (e.g., water, milk, sugar solution, mineral oils). In the case of an ideally viscous material, the classical Newtonian law holds for shear stress, see equation (3).<\/em>

Non-Newtonian Fluids<\/em><\/h3> Time-dependent:<\/em><\/strong>\n\u2013 thixotropic (becoming thinner over time, viscosity decreases with time) \u2013 used in chemistry, food industry (yogurt),<\/em>\n\n\u2013 rheopectic (becoming thicker over time, viscosity increases with time) \u2013 not common, examples include plaster,<\/em>\n\nTime-independent:<\/em><\/strong> depends on temperature:\n\n\u2013 pseudoplastic (shear-thinning) \u2013 viscosity decreases with increasing shear stress (shampoo, concentrated juices, ketchup),\n\n\u2013 dilatant (shear-thickening) \u2013 viscosity increases with increasing shear stress (wet sand, concentrated starch suspensions),\n\n\u2013 plastic \u2013 have a yield point (curd, toothpaste).\n\n

Rheological Measurement Methods<\/h2> a) Absolute measurement \u2013 from Poiseuille’s law<\/a>, we measure all other quantities,\nb) Relative measurement \u2013 comparison with a fluid whose dynamic viscosity is known \u2013 Ostwald viscometer, H\u00f6ppler viscometer;<\/em>\n\nFlow, falling, and rotational viscometers are commonly used to measure viscosity, with only the last type and special capillary viscometers providing sufficient characterization of the flow curve for non-Newtonian fluids. Proper measurement conditions always require laminar flow across the entire measurement range and a well-defined flow geometry (ability to determine D and \u03c4) for non-Newtonian fluids.\n\n

Measuring Devices Used in Rheology:<\/h3> \u2013 Basic devices,<\/em>\n\u2013 Capillary viscometers,\n\n\u2013 Falling ball viscometers,\n\n\u2013 Rotational viscometers,\n\n\u2013 Rotational rheometers,\n\n\u2013 Sensors \u2013 geometries,\n\n\u2013 Extensional rheometers,\n\n\u2013 Extrusion rheometers.\n\n
Principle<\/em><\/strong><\/td> Device<\/em><\/strong><\/td> Measured quantity<\/em><\/strong><\/td> <\/tr>
<\/td> <\/td> <\/td> <\/tr>
Volumetric flow<\/td> Ford funnel; capillary viscometer<\/td> Time; time (pressure, displacement)<\/td> <\/tr>
Falling ball<\/td> H\u00f6ppler viscometer<\/td> Time<\/td> <\/tr>
Compression<\/td> Compression viscometer<\/td> Force, displacement<\/td> <\/tr>
Rotation<\/td> Rotational viscometer, rheometer<\/td> Force, displacement<\/td> <\/tr> <\/tbody> <\/table> <\/figure>

Applications of Rheology<\/em><\/h2> The field of rheology deals with the study of the internal response of substances (both solid and liquid) to external forces, or their deformability and flow properties. The relationship between microstructure and rheological properties is examined by microrheology. For the needs (not only) of chemical engineering, phenomenological rheology (macrorheology) of fluids is especially important, which treats them as a continuum and formulates the laws of viscous flow.\nThe mathematical expression of the flow properties of fluids is given by rheological state equations, which typically express the relationship between the shear (tangential, viscous) stress \u03c4 and the deformation of the fluid. Their graphical representation is the flow curve (HOLUBOV\u00c1, R).<\/em>\n\nRheological behavior of liquid materials plays an important role in many technological operations. Knowledge of basic rheological quantities, viscosity, yield point, and elasticity moduli is necessary not only for characterizing raw materials or products but also for solving many technical problems and engineering calculations when designing, improving, and controlling various manufacturing and transport equipment (HOLUBOV\u00c1, R).<\/em>\n\nRheology plays a significant role in technical tasks and engineering calculations, especially in safety-related fields, where there is substantial room for errors and neglect that can lead to major failures and accidents in the addressed area.\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"

This article builds on the articles on thermodynamics (article 2) and briefly introduces the issue of rheology, which is simply […]<\/p>\n","protected":false},"author":1,"featured_media":1211,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[243],"tags":[],"class_list":["post-3411","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physics-and-safety"],"_links":{"self":[{"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/posts\/3411","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/comments?post=3411"}],"version-history":[{"count":8,"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/posts\/3411\/revisions"}],"predecessor-version":[{"id":3782,"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/posts\/3411\/revisions\/3782"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/media\/1211"}],"wp:attachment":[{"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/media?parent=3411"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/categories?post=3411"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/test.kint.cz\/en\/wp-json\/wp\/v2\/tags?post=3411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}