Viscosity

Viscosity

What is it?

Viscosity is a liquid's resistance to flowing smoothly due to forces holding the molecules together. Viscosity increases as temperature decreases.

Give me an example!

Have you ever heard someone say that something is "slower than molasses in January"? That refers to viscosity. In other words, when it's cold (like in January), molasses doesn't flow as readily as it does during warmer months (like July).

Shear Strain,Shear Stress, Shear Rate,Viscosity

Shear Strain:-

• To define the term STRAIN we will consider a cube of material with its base fixed to a surface as shown below in figure-1.

• If we now apply a constant 'pushing' force, F, to the upper part of the cube, assuming the material behaves as an ideal solid, it will obey Hooke's law of elastic deformation and will deform to a new position as shown in figure-1.
• This type of deformation (lower fixed upper moving) is defined as a SHEAR DEFORMATION.
• The deformation δ u and h are used to define the SHEAR STRAIN as :

                             Shear Strain =  δ u/h

• The shear strain is simply a ratio of two lengths (displacement / gap) and so has no units. It is important since it enables us to quote pre-defined deformations without having to specify sizes of sample etc

Shear Stress:-

• The SHEAR STRESS is defined as F/A (A is the area of the upper surface of the cube l x w) Since the units of force are Newtons and the units of area are m^2 it follows that the units of Shear Stress are N/M2 This is referred to as the PASCAL (i.e. 1 N/m^2 = 1 Pascal) and is denoted by the symbol σ (in older textbooks you may see it denoted as τ)

Shear Rate:-

• Consider the case of a cube of material that behaves as an ideal fluid. When we apply a shear stress (force) the material will continually deform at a constant rate as illustrated in figure-2.

• The rate of change of strain is referred to as the SHEAR STRAIN RATE often abbreviated to SHEAR RATE and is found by the rate of change of strain as a function of time i.e. the differential d.

       SHEAR STRAIN / d.TIME

Viscosity

• The Shear Rate obtained from an applied Shear Stress will be dependant upon the materials resistance to flow i.e. its VISCOSITY
• Since the flow resistance ≡ force / displacement it follows that ;

               VISCOSITY = SHEAR STRESS / SHEAR RATE

• The units of viscosity are Nm^2s   Which are better known as Pascal Seconds (Pa.s)
• If a material has a viscosity which is independent of shear stress then it is referred to as an ideal or NEWTONIAN fluid. The mechanical analogue of a Newtonian fluid is a viscous dashpot which moves at a constant rate when a load is applied as seen in figure-3.

Viscosity

The resistance of a fluid to flow

1 Ns/m^2 = 1 Pa.s = 10 Poise
Viscosity in Poise / Density = Kinematic viscosity in Stokes

Shear Stress

1 Pa = 1 N/m^2 = 10 dyn cm^-2

Kinematic Viscosity

The dynamic viscosity divided by density

Shear Rate

The rate of change of shear stress.  The velocity gradient perpendicular to the direction of shear flow (dv/dx).  Units 1/s or s-1

Shear Stress

The shear force per unit area

Shear Strain

A unit-less quantity, the relative displacement of the faces of a sheared body (for example a layer of fluid) divided by the distance between them.

Zero-shear Viscosity

The viscosity at the limit of low shear rate. The viscosity a product will ultimately attain when at rest and undisturbed.

Viscosity

• Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. 
• In everyday terms (and for fluids only), viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity. Put simply, the less viscous the fluid is, the greater its ease of movement .

Shear stress in fluids

• Any real fluids (liquids and gases included) moving along solid boundary will incur a shear stress on that boundary. 
• The speed of the fluid at the boundary (relative to the boundary) is zero, but at some height from the boundary the flow speed must equal that of the fluid. 
• The region between these two points is aptly named the boundary layer. For all Newtonian fluids in laminar flow the shear stress is proportional to the strain rate in the fluid where the viscosity is the constant of proportionality. 
• However for Non Newtonian fluids, this is no longer the case as for these fluids the viscosity is not constant. 
• The shear stress is imparted onto the boundary as a result of this loss of velocity. The shear stress, for a Newtonian fluid, at a surface element parallel to a flat plate, at the point y, is given by:

where

μ is the dynamic viscosity of the fluid;

u is the velocity of the fluid along the boundary;

y is the height above the boundary.

• A shear stress,  is applied to the top of the square while the bottom is held in place. This stress results in a strain, or deformation, changing the square into a parallelogram.

Specifically, the wall shear stress is defined as:

In case of wind, the shear stress at the boundary is called wind stress.

Viscosity:

• Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries causes the fluid to shear. The force required for this action is a measure of the fluid's viscosity. This type of flow is known as a Couette flow. 
• In general, in any flow, layers move at different velocities and the fluid's viscosity arises from the shear stress between the layers that ultimately opposes any applied force.
• The relationship between the shear stress and the velocity gradient can be obtained by considering two plates closely spaced at a distance y, and separated by a homogeneous substance. 
• Assuming that the plates are very large, with a large area A, such that edge effects may be ignored, and that the lower plate is fixed, let a force F be applied to the upper plate. If this force causes the substance between the plates to undergo shear flow with a velocity gradient u(as opposed to just shearing elastically until the shear stress in the substance balances the applied force), the substance is called a fluid.
• The applied force is proportional to the area and velocity gradient in the fluid and inversely proportional to the distance between the plates. Combining these three relations results in the equation:

,

where μ is the proportionality factor called viscosity.

• This equation can be expressed in terms of shear stress

                        . 

• Thus as expressed in differential form by Isaac Newton for straight, parallel and uniform flow, the shear stress between layers is proportional to the velocity gradient in the direction perpendicular to the layers:

• Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.
• Note that the rate of shear deformation is  which can be also written as a shear velocity, .
• James Clerk Maxwell called viscosity fugitive elasticity because of the analogy that elastic deformation opposes shear stress in solids, while in viscous fluids, shear stress is opposed by rate of deformation.