# Rheology

**Rheology** is the science that studies the behavior of **fluids** subject to a flow or a **deformation**. One of its main objectives is to comprehend how the **microstructure** of a fluid effects the response of that fluid to a deformation.

Rheology is thus a discipline relevant to a wide range of fundamental processes in nature as well as many idustrial relevalnt products including pharmaceutics, paints, inks, polymers or cement. But it is also relevant to everyday consumer products such as food, cosmetics, house- and home-care products.

*blood: a precious fluid with amazing rheological properties*

Understanding the rheological properties of a fluid not only helps us to understand nature better, but also facilitates the **optimization** of manufacturing processes and stimulates the discovery of **novel intelligent ****materials **that can often be used for many different **applications.**

The **rheological investigation** of a fluid is typically conducted by applying a deformation to the fluid and analyzing the response to that deformation. Depending on the applied strain, the response of the system can be very different and conclusions can be drawn accordingly. This common approach is also referred to as **mechanical or bulk rheology. **It reveals the **macrorheology** of the fluid.

Alternatively, the rheological properties can be measured by studying the Brownian motion of** **particles imbedded in the system. This approach is called **microrheology **and is complementary to macrorheology [1, 2]. If the motion of the imbedded particles is only caused by Browning motion, it is called **passive **microrheology. If an external force such as a magnetic or electric field is applied to displace the particles, one calls this **active** microrheology.

In case of a macrorheological experiment, deformations can always be related to a combination of the two most simple deformations: **shear** and **elongation**.

## Shear motion

The simplest case of a **shear motion** is represented in Figure 1a, where a fluid is confined between a fixed plate and a moving one.

In this case, the **deformation** \(\gamma\), is given by the ratio between the displacement of the material in the direction of the flow, s, and the distance between the two plates, \(h: \gamma=\frac{s}{h}\).

The rate of the deformation, named** shear rate**, is given by the ratio between the rate of the displacement \(v\), and \(h: \dot{\gamma}=\frac{v}{h}\).

Under shear motion, the **shear stress**, \(\sigma\), and the **first normal stress difference** \(N_1\), arise.

The former acts in the same direction of the motion and it is proportional to the force \(F\), necessaire to move the plate as: \(\sigma=\frac{F}{A}, A \) being the surface of the moving plate.

*N _{1}* is the result of two different normal stresses, one acting perpendicular to the shearing plane \(\sigma_{n,1}\), and the other acting perpendicular to the direction of the motion, \(\sigma_{n,2}\) (Figure 1b). However, it is worth highlighting that normal stresses are only present in

**elastic fluids**.

*Figure 1. (a) Sketch of shear flow between parallel plates; (b) Shear normal stresses.*

## Elongational motion

When a fluid is subjected to an elongational motion, it undergoes **stretching** and **compression**.

This is the case of all the materials going under extrutions processes, such as plastics, ceramics, pasta etc.

Figure 2 shows a monoaxial elongational flow of a material which is held from one side and stretched from the other. This induces an elongation along the flow direction and a compression in the transversal ones.

As for the shear motion, it is possible to define the **deformation**, called **Hencky strain**, \(\varepsilon\), calculated as: \(\varepsilon\)=\(ln( \lambda )\), where \(\lambda\) is the elongational ratio (thus the final length \(L\), on the initial one \(L_0\)), and the stretching rate as the time derivative of \(\varepsilon\).

The normal stress, \(\sigma\), is defined as the ratio between the tractive force \(T\), and the surface,\(S\), to which it is applied as \(\sigma=\frac{T}{S}\).

The normal stress, in the elongational motion, can also be seen as the difference between the stress along the direction of the motion \(\sigma_N\), and that acting on the side surfaces, perpendicularly to the motion, \(\sigma_L\).

*Figure 2. Extensional deformation.*

[1] C.W. Macosko, Rheology. Principles, Measurements and Applications, VCH Publishers, New York 1994.

[2] Barnes H.A.; A Handbook of Elementary Rheology, The University of Wales Institute of Non-Newtonian Fluid Mechanics (2000).