# Rheology

Continuum mechanics Key topics
Conservation of mass
Conservation of momentum
Navier-Stokes equations
Classical mechanics
Stress · Strain · Tensor
Solid mechanics
Solids · Elasticity
Fluid mechanics
Fluids · Fluid statics
Fluid dynamics · Viscosity · Newtonian fluids
Non-Newtonian fluids
Surface tension
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Rheology is the study of the deformation and flow of matter under the influence of an applied stress, which might be shear stress or extensional stress. Rheology dealing with shear stress is called shear rheology. Rheology dealing with extensional stress is called extensional rheology. Shear rheology is much more widely used than extensional rheology. That is why it is often referred to as simply rheology without specfying type of the stress.

The term rheology was coined by Eugene Bingham, a professor at Lehigh University, in 1920, from a suggestion by a colleague, Markus Reiner. The term was inspired by Heraclitus's famous expression panta rei, "everything flows".

## Scope

In practice, rheology is principally concerned with extending the "classical" disciplines of elasticity and (Newtonian) fluid mechanics to materials whose mechanical behaviour cannot be described with the classical theories. It is also concerned with establishing predictions for mechanical behaviour (on the continuum mechanical scale) based on the micro- or nanostructure of the material, e.g. the molecular size and architecture of polymers in solution or the particle size distribution in a solid suspension.

Rheology unites the seemingly unrelated fields of plasticity and non-Newtonian fluids by recognizing that both these types of materials are unable to support a shear stress in static equilibrium. In this sense, a plastic solid is a fluid. Granular rheology refers to the continuum mechanical description of granular materials.

One of the tasks of rheology is to empirically establish the relationships between deformations and stresses, respectively their derivatives by adequate measurements. These experimental techniques are known as rheometry. Such relationships are then amenable to mathematical treatment by the established methods of continuum mechanics.

## Applications

Rheology has important applications in engineering, geophysics, pharmaceutics and physiology. In particular, hemorheology, the study of blood flow, has an enormous medical significance. In geology, solid Earth materials that exhibit viscous flow over long time scales are known as rheids. In engineering, rheology has had its predominant application in the development and use of polymeric materials (plasticity theory has been similarly important for the design of metal forming processes, but in the engineering community is often not considered a part of rheology). Rheology modifiers are also a key element in the development of paints in achieving paints that will level but not sag on vertical surfaces.

## Elasticity, viscosity, solid- and liquid-like behaviour, and plasticity

One is used to associating liquid with viscous (a thick oil is a viscous liquid) as well as solid with elastic (an elastic string is an elastic solid). In fact, when one tries to deform a piece of material, some of the above properties appear at short times (relative to the duration of the experiment), others at long times.

Liquid and solid characters are long-time properties

Let us attempt to deform the material by applying a continuous, weak, constant stress:

• if the material, after some deformation , eventually resists further deformation, it is a solid ;
• if, by contrast, the material eventually flows, it is a liquid.
By contrast, elastic and viscous characters (or intermediate, viscoelastic behaviours) appear at short times

Again, let us attempt to deform the material by applying a weak stress varying in time:

• if the material deformation follows the applied force or stress, then the material is elastic;
• if the time-derivative of the deformation (deformation rate) follows the force or stress, then the material is viscous.
Plasticity appears at high stresses

Liquid, solid, viscous and elastic characters can be detected under weak applied stresses. If a high stress is applied, a material that behaves as a solid under low applied stresses may start to flow. It then reveals a plastic character: it is a plastic solid. Plasticity is thus characterised by a threshold stress (called plasticity threshold or yield stress) beyond which the material flows.

The term plastic solid is used when the plasticity threshold is rather high, while yield stress fluid is used when the threshold stress is rather low. There is no fundamental difference, however, between both concepts.

## Dimensionless numbers in rheology

Deborah number

When the rheological behaviour of a material includes a transition from elastic to viscous as the time scale increase (or, more generally, a transition from a more resistant to a less resistant behaviour), one may define the relevant time scale as a relaxation time of the material. Correspondingly, the ratio of the relaxation time of a material to the timescale of a deformation is called Deborah number. Small Deborah numbers correspond to situations where the material has time to relax (and behaves in a viscous manner), while high Deborah numbers correspond to situations where the material behaves rather elastically.

Note that the Deborah number is relevant for materials that flow on long time scales (like a Maxwell fluid) but not for the reverse kind of materials (like the Voigt or Kelvin model) that are viscous on short time scales but solid on the long term.

Reynolds number

In fluid mechanics, the Reynolds number is the ratio of inertial forces (vsρ) to viscous forces (μ/L) and consequently it quantifies the relative importance of these two types of forces for given flow conditions. Thus, it is used to identify different flow regimes, such as laminar or turbulent flow.

It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining dynamic similitude. When two geometrically similar flow patterns, in perhaps different fluids with possibly different flowrates, have the same values for the relevant dimensionless numbers, they are said to be dynamically similar.

Typically it is given as follows:

${\mathit {Re}}={\rho v_{s}^{2}/L \over \mu v_{s}/L^{2}}={\rho v_{s}L \over \mu }={v_{s}L \over \nu }={\frac {\mbox{Inertial forces}}{\mbox{Viscous forces}}}$ where:

• vs - mean fluid velocity, [m s-1]
• L - characteristic length, [m]
• μ - (absolute) dynamic fluid viscosity, [N s m-2] or [Pa s]
• ν - kinematic fluid viscosity: ν = μ / ρ, [m² s-1]
• ρ - fluid density, [kg m-3]. 