Fluid

Continuum mechanics

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A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. It is a subset of the phases of matter and includes liquids, gases, plasmas and, to some extent, plastic solids.

Fluids are also divided into liquids and gases. Liquids form a free surface (that is, a surface not created by their container) while gases do not. The distinction between solids and fluids is not so obvious. The distinction is made by evaluating the viscosity of the matter: for example silly putty can be considered either a solid or a fluid, depending on the time period over which it is observed.

Fluids share the properties of not resisting deformation and the ability to flow (also described as their ability to take on the shape of their containers). These properties are typically a function of their inability to support a shear stress in static equilibrium. While in a solid, stress is a function of strain, in a fluid, stress is a function of rate of strain. A consequence of this behavior is Pascal's law which entails the important role of pressure in characterizing a fluid's state.

Based on how the stress depends on the rate of strain and its derivatives, fluids can be characterized as:

• Newtonian fluids : where stress is directly proportional to rate of strain, and
• Non-Newtonian fluids : where stress is proportional to rate of strain, its higher powers and derivatives (basically everything other than Newtonian fluid).

The behavior of fluids can be described by a set of partial differential equations, which are based on the conservation of mass, linear and angular momentum (Navier-Stokes equations) and energy.

The study of fluids is fluid mechanics, which is subdivided into fluid dynamics and fluid statics depending on whether the fluid is in motion or not.

Newtonian fluids

A Newtonian fluid (named for Isaac Newton) is a fluid that flows like water—its shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. The constant of proportionality is known as the viscosity.

A simple equation to describe Newtonian fluid behavior is

$\tau =\mu {\frac {dv}{dx}}$

where

$\tau$ is the shear stress exerted by the fluid ("drag") [Pa]

$\mu$ is the fluid viscosity - a constant of proportionality [Pa·s]

${\frac {dv}{dx}}$ is the velocity gradient perpendicular to the direction of shear [s⁻¹]

In common terms, this means the fluid continues to flow, regardless of the forces acting on it. For example, water is Newtonian, because it continues to exemplify fluid properties no matter how fast it is stirred or mixed. Contrast this with a non-Newtonian fluid, in which stirring can leave a "hole" behind (that gradually fills up over time - this behaviour is seen in materials such as pudding, oobleck, or, to a less rigorous extent, sand), or cause the fluid to become thinner, the drop in viscosity causing it to flow more (this is seen in non-drip paints).

For a Newtonian fluid, the viscosity, by definition, depends only on temperature and pressure (and also the chemical composition of the fluid if the fluid is not a pure substance), not on the forces acting upon it.

If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress, in the Cartesian coordinate system, is

$\tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)$

where

$\tau _{ij}$ is the shear stress on the $i^{th}$ face of a fluid element in the $j^{th}$ direction

$v_{i}$ is the velocity in the $i^{th}$ direction

$x_{j}$ is the $j^{th}$ direction coordinate

If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types.

Non-Newtonian fluids

A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. As a result, non-Newtonian fluids may not have a well-defined viscosity.

Although the concept of viscosity is commonly used to characterize a material, it can be inadequate to describe the mechanical behavior of a substance, particularly non-Newtonian fluids. They are best studied through several other rheological properties which relate the relations between the stress and strain tensors under many different flow conditions, such as oscillatory shear, or extensional flow which are measured using different devices or rheometers. The rheological properties are better studied using tensor-valued constitutive equations, which are common in the field of continuum mechanics.

Common examples

An inexpensive, non-toxic sample of a non-Newtonian fluid can be made very easily by adding corn starch to a cup of water. Add the starch in small portions and stir it in slowly. When the suspension nears the critical concentration - becoming like single cream in consistency - the so called "shear thickening" property of this non-Newtonian fluid becomes apparent. The application of force - for example by stabbing the surface with a finger, or rapidly inverting the container holding it - leads to the fluid behaving like a solid rather than a liquid. More gentle treatment, such as slowly inserting a spoon, will leave it in its liquid state. Trying to jerk the spoon back out again, however, will trigger the return of the temporary solid state. A person moving quickly and/or applying sufficient force with his feet can literally walk across such a liquid (see video link below).

Shear thickening fluids of this sort are being researched for bullet resistant body armor, useful for their ability to absorb the energy of a high velocity projectile impact but remain soft and flexible while worn.

A familiar example of the opposite, a shear-thinning fluid, is paint: one wants the paint to flow readily off the brush when it is being applied to the surface being painted, but not to drip excessively.

Types of Non-Newtonian fluids

Principal types of non-Newtonian fluids are given in the table below.

Type of fluid	Behaviour	Characteristics	Examples
Plastic solids	Perfectly plastic	Strain does not result in opposing stress	Ductile metals past the yield point
	Bingham plastic	Linear relationship between shear stress and rate of strain once threshold shear stress exceeded	Mud, some colloids
	Yield pseudo-plastic	Pseudo-plastic above some threshold shear stress
	Yield dilatent	Dilatent above some threshold shear stress
Power-law fluids	Pseudoplastic or "shear thinning"	Apparent viscosity reducing with rate of shear	Some colloids, clay, milk, gelatin, blood and liquid cement
	Dilatant or "shear thickening"	Apparent viscosity increasing with rate of shear	Concentrated solution of sugar in water, suspensions of rice starch or corn starch
Viscoelastic - having both viscous and elastic properties	Maxwell material	"Series" linear combination of elastic and viscous effects	metals, composite materials
	Oldroyd-B fluid	Linear combination of Maxwell and Newtonian behaviour	Bitumen, dough, nylon, and Silly Putty
	Kelvin material	"Parallel" linear combination of elastic and viscous effects
	Anelastic	Material returns to a well-defined "rest shape"
Time-dependent viscosity	Rheopectic	Apparent viscosity increases with duration of stress	Some lubricants
	Thixotropic	Apparent viscosity decreases with duration of stress	Non-drip paints and tomato ketchup and most honey varieties.
Generalized Newtonian fluids		Stress depends on normal and shear strain rates and also the presuure applied on it	Blood, Custard

See also

• Rheology
• Thermodynamics
• Superfluid
• Perfect fluid
• Viscosity

External links

• Viscosity Chart

• A pool filled with non-newtonian fluid

• Narrated scientific experiments on the effects of vibrations on a non-newtonian fluid

• Amateur experiments on the effects of vibrations on a non-newtonian fluid

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