This article follows on from the articles on thermodynamics (article 2) and briefly introduces the issue of rheology, which is simply the science of material flow. In the following paragraphs, you will learn, for example, how water vortices are formed or how the viscosity of substances is measured, etc.

The following paragraphs provide a summary and introduction to the issue based on research from the following sources:

PRAGOLAB. Seminář reologie. Praha: Pragolab 2015. Dostupné z: http://www.pragolab.cz/files/download/Seminar_reologie_2015.pdf

PRUŠKA, J. *Reologie, MH 8. Přednáška.* Praha: FS ČVUT 2008. Dostupné z: http://departments.fsv.cvut.cz/k135/data/wp-upload/2008/05/reologie.pdf

HOLUBOVÁ, Renata. *Základy reologie a reometrie kapalin*. Olomouc: Univerzita Palackého v Olomouci,2014. ISBN 978-80-244-4178-8

Rheology is the science of material deformation and flow; on the movement of viscous (Newtonian) liquids and the transformation of masses. These substances are not perfectly elastic (Hooke’s substance), nor completely malleable (St. Venant’s liquid) or ductile, but in which there are various combinations of these properties. Rheology is divided into macrorheology and microrheology.

*Macrorheology – examines the deformable properties of matter from an overall perspective.*

*Microrheology – studies the deformable properties of individual parts of matter.*

Rheology deals with:

– connections between different types of deformation of matter and investigates the causes and manifestations of deformations,

– relations between shear stress and shear rate,

– boundaries between liquid and solid.

## Properties of liquids

To describe the properties of ideal liquids, we define an ideal liquid as a liquid without internal friction (non-viscous), has zero volume expansion and compressibility, zero gas solubility and does not evaporate.

According to (Holubová R.), liquids are further divided into:

*– Newtonian (e.g. water), for which viscosity at a given temperature and pressure is a physical constant,*

*– and non-Newtonian (e.g. emulsions, mixtures of solids with liquids, paints) whose viscosity is not a physical constant.*

**The real fluid flow is:**

**Laminar – particles move in layers that are parallel to each other, while there is no displacement of particles perpendicular to the direction of movement;**

**Turbulent – in addition to the gradual speed, the particles also have a turbulent (fluctuating) speed, with which they move along the cross-section.**

Furthermore, the following relations for stress are defined in the field of fluid properties.

**The normal stress (pressure) p – is given as the ratio of the normal elementary force and the size of the given area:**

p = dF_{n }/ dS, (1)

where dS is the elemental surface inside the liquid, dFn the normal component (acting on the considered surface) of the elemental force dF.

A tensile stress cannot be induced in a liquid, so we measure the pressure as positive. If we measure pressure from a zero value, we speak of the so-called absolute pressure. Sometimes it is convenient to measure the pressure from a certain reference pressure (mostly atmospheric pressure). Pressure difference above or under this pressure we call overpressure or negative pressure.

*Tangential (shear) stress τ – is given as the ratio of the tangential elementary force and the size of the given area:*

τ = dF_{t }/ dS, (2)

where dFt is the tangential component (causes particle displacement in the liquid) of the elementary force dF acting on the elementary surface dS.

For an elementary prism of height dy, the lower wall of which moves with velocity v and the upper wall with velocity v + dv holds:

τ = η dv/dy, (3)

The dynamic viscosity of liquids is generally dependent on temperature (it decreases with increasing temperature) and pressure (the dependence is negligible). It manifests itself as resistance to the movement of liquid particles. According to the dependence of dynamic viscosity on tangential stress, we distinguish between Newtonian (independent – equation (2)) and non-Newtonian (dependent – equation 3) liquids.

The dependence of viscosity on temperature can be expressed using the relation:

η(T) = k * e^{b/(T+Θ)}, (4)

*where the constant k has the dimension of viscosity (Pa · s), b and Θ are constants characteristic of the given fluid, their unit is kelvin.*

The dependence of viscosity on pressure expresses the relation:

η(p) = η_{0} * e^{αp}, (5)

α is a temperature-dependent coefficient.

In addition to dynamic viscosity, we also introduce the quantity kinematic viscosity, which is defined by the relation:

ν = η / ρ, (6)

where ρ is the density of the liquid.

*Liquid flow through a capillary – the characteristic quantities for describing the flow are: pressure difference Δp, capillary distance l and its radius r. In the stationary case, the pressure force Fp must be equal to the friction force FR. The flow through the capillary has a parabolic velocity distribution, i.e. that the velocity v(r) spatially forms a paraboloid of rotation. The relations for the velocity profile as well as the amount of liquid that flows per unit time t are given in the thesis.*

*Navier-Stokes equation – represents the equation of motion of a real flowing liquid. Gravitational force, pressure force and frictional force act on a real liquid. The total force can be expressed as the sum of all acting forces. The goal of the solution is to find the distribution of velocities and pressures. It is therefore necessary to know the external acceleration, the density of the liquid and the external conditions. The individual terms of the Navier-Stokes equation are represented by:*

*– v*external acceleration, which is a consequence of the action of gravity,*,*

*– acceleration from surface (pressure) force,*

*– acceleration required to overcome the viscous friction of fluids,*

*– *acceleration due to viscosity in compressible liquids,

*– *convective acceleration from inertial force,

*– *local acceleration from inertial force.

In an incompressible liquid, the fourth term of the equation (acceleration due to viscosity) is omitted due to the continuity equation, in a non-viscous liquid, the equation turns into the Euler equation of hydrodynamics. Classical hydrodynamics can be described using the Navier-Stokes equation. Unlike the Euler equation, it contains a term that expresses the internal friction in the fluid.

The Navier-Stokes equation represents a nonlinear differential equation. By solving it, we get the distribution of velocities in the fluid depending on place and time. Mathematics does not know the procedure for analytically solving this equation, only special cases can be solved.

*Laminar flow around a ball – viscous liquids exert a force on any object that moves in the liquid. One can investigate, for example, how to describe the fall of an object in a viscous liquid. Due to the gravitational force Fg, the object is accelerated and its initial speed v0 = 0 increases continuously. The phenomenon lasts as long as the object goes into uniform motion with a constant rate of decrease in , i.e. acceleration is zero. Then there is a balance between the downward force of gravity and the upward forces – buoyancy and friction: Fg=Fvz + FT.*

Based on experiments with balls of different sizes and using different liquids, we find that the friction force depends on the coefficient η, the final speed v and the radius of the ball r, i.e. Stokes’ law:

F_{T} = -6πηrv. (7)

*Vortices – a characteristic feature of turbulence is the formation of vortices. Let’s imagine a cylinder surrounded by a liquid. From a certain flow speed in G, stationary eddies with the opposite direction of rotation begin to form behind the cylinder in the area of so-called dead water. The vortex can be divided into two regions – the core and the circulation region. We will explain the formation of vortices using a cylinder that is inserted into a liquid that moves from left to right and “hits” the body – see picture.*

At point S1 (place of impact) the speed is minimal, i.e. that according to Bernoulli’s equation, the static pressure is maximum at this point. A pressure drop is created between points S 1 and K at the top of the body. At the top of the body at point K, the static pressure is minimum and the velocity is maximum, reduced only by the effect of friction (v < vmax ). Now the fluid has to overcome the existing pressure rise at the back of the cylinder, i.e. between point K and point S 2 . Since v < v max , there is not enough kinetic energy to reach point S 2 . The flow has zero velocity at the turning point W , however, since the pressure force acts from point S 2 towards point K, the braked fluid particles are pushed against the flow direction of the outer layer. There is a twisting of the fluid element in the vicinity of point W, which creates a vortex. A vortex also develops on the lower side of the cylinder, but with the opposite direction of rotation. Both faiths disengage from war and are replaced by new ones. The so-called Kármán vortex path is created.

The pressure and velocity changes in front of and behind the flowed body can be estimated using Bernoulli’s equation.

Based on these considerations, it is possible to determine the resistive pressure force that acts on the wrapped body.

*Inside the vortex, there is a region around the core where the fluid rotates like a solid body, i.e. with a constant angular velocity ω. The circumferential speed of rotation v = ωxr increases linearly with the distance r from the center. In addition, all particles have their own spin. If the particle rotates once around the nucleus, it also rotates once around its own axis. For the region outside the core (for r > rk ), the speed of particle rotation decreases with increasing distance. The rotation is only around the nucleus and not around its own axis…circulation region.*

**Laminar flow becomes turbulent when the so-called critical value of the Reynolds number is exceeded. This depends on the viscosity, fluid flow rate and flow geometry. The transition from laminar to turbulent flow also depends (e.g. in a pipe) on the geometric shape of the flow parts, the rounding of the edges at the initial section of the pipe, the roughness of the pipe walls, the degree of turbulence of the inflowing current, etc. In the flow of a real liquid, there is a transition area where, according to specific conditions, conditions, both laminar and turbulent flow can occur.**

*Hagen’s-Poiseuille’s law – determination of the mean velocity of liquid flow from volume flow, see (HOLUBOVÁ, R.).*

*Newtonian fluids*

The viscosity of a Newtonian liquid depends only on temperature, the direct proportionality between the shear stress and the velocity gradient (Newton’s law of viscosity) is fulfilled (e.g. water, milk, sugar solution, mineral oils). In the case of an ideally viscous material, the classical Newton’s law applies to the tangential stress, see relation (3).

*Non-Newtonian fluids*

*Time dependent:*

*– thixotropic (thinner with time, viscosity decreases with time) – used in chemistry, food industry (yogurt),*

*– *rheopetic (they thicken with time, viscosity increases with time) – they do not occur so often, an example is gypsum,

**Time-independent, depends on temperature:**

- pseudoplastic (thinning) – viscosity decreases with increasing shear stress (shampoo, juice concentrates, ketchup),

- dilatant (thickening) – viscosity increases with increasing tangential stress (wet sand, concentrated starch suspensions),

- plastic – have a yield point (cottage cheese, toothpaste).

## Measurement methods in rheology

a) absolute measurement – from Poiseuille’s law, we measure all other quantities,

b) relative measurement – comparison with a liquid whose dynamic viscosity is known

– *Ostwald viscometer, Höppler viscometers;*

Flow, drop and rotary viscometers are commonly used to measure viscosity, but only the last type and special capillary viscometers allow to adequately characterize the flow curve of non-Newtonian liquids. A condition for correct measurement is always laminar flow throughout the measurement range and a well-defined flow geometry (possibility of determining D and τ ) in the case of non-Newtonian liquids.

### Measuring devices used in rheology:

– basic devices,

– capillary viscometers,

– falling ball viscometers,

– rotary viscometers,

– rotational rheometers,

– sensors

– geometry,

– extensional rheometers,

– extrusion rheometers.

lePrincip | Equipment | Measured unit |

Volume flow rate | ford funnel; capillary viscometer | time; time (pressure, dislocation) |

Falling ball | höpplerův viscometer | time |

Compression | compression viscometer | force, dislocation |

Rotation | rotational viscometer, rheometer | force, dislocation |

*Use of rheology*

The scientific field called rheology deals with the study of the internal reaction of substances (solid and liquid) to the action of external forces or their deformability and flow properties. The connection between microstructure and rheological properties is investigated by microrheology. For the needs of (not only) chemical engineering, the phenomenological rheology (macroreology) of liquids is especially important, which looks at them as a continuum and formulates the laws of viscous flow.

*The mathematical expression of the flow properties of liquids are rheological state equations, which usually express the relationship between the deformation shear (tangential, viscous) stress τ and the deformation of the liquid. Their graphic form is flow curves (HOLUBOVÁ, R).*

*The rheological behavior of liquid materials plays an important role in a number of technological operations. Knowledge of basic rheological parameters, viscosity, flow limit and modulus of elasticity is needed not only to characterize raw materials, or products, but also to solve many technical tasks and engineering calculations in the design, improvement and control of various production and transport equipment (HOLUBOVÁ, R).*

Above all, in the areas of technical tasks and engineering calculations in design and control, rheology plays a significant role from the point of view of safety fields, because there is a lot of room for errors and neglect, which can lead to major failures and accidents in the addressed area.