Viscosity
Discussion
definitions
Informally, viscosity is the quantity that describes a fluid's resistance to flow. Fluids resist the relative motion of immersed objects through them as well as to the motion of layers with differing velocities within them.
Formally, viscosity (represented by the symbol η "eta") is the ratio of the shearing stress (F/A) to the velocity gradient (Δv_{x}/Δz or dv_{x}/dz) in a fluid.
η =  F̅/A 
Δv_{x}/Δz 
or
η =  F/A 
dv_{x}/dz 
The more usual form of this relationship, called Newton's equation, states that the resulting shear of a fluid is directly proportional to the force applied and inversely proportional to its viscosity. The similarity to Newton's second law of motion (F = ma) should be apparent.

⇔ 

Or if you prefer calculus symbols (and who doesn't)…

⇔ 

The SI unit of viscosity is the pascal second [Pa s], which has no special name. Despite its selfproclaimed title as an international system, the International System of Units has had little international impact on viscosity. The pascal second is rarely used in scientific and technical writing today. The most common unit of viscosity is the dyne second per square centimeter [dyne s/cm^{2}], which is given the name poise [P] after the French physiologist Jean Poiseuille (1799–1869). Ten poise equal one pascal second [Pa s] making the centipoise [cP] and millipascal second [mPa s] identical.
1 Pa s =  10 P 
1000 mPa s =  10 P 
1 mPa s =  0.01 P 
1 mPa s =  1 cP 
There are actually two quantities that are called viscosity. The quantity defined above is sometimes called dynamic viscosity, absolute viscosity, or simple viscosity to distinguish it from the other quantity, but is usually just called viscosity. The other quantity called kinematic viscosity (represented by the Greek letter ν "nu") is the ratio of the viscosity of a fluid to its density.
ν =  η 
ρ 
Kinematic viscosity is a measure of the resistive flow of a fluid under the influence of gravity. It is frequently measured using a device called a capillary viscometer — basically a graduated can with a narrow tube at the bottom. When two fluids of equal volume are placed in identical capillary viscometers and allowed to flow under the influence of gravity, the more viscous fluid takes longer than the less viscous fluid to flow through the tube. Capillary viscometers will be discussed in more detail later in this section.
The SI unit of kinematic viscosity is the square meter per second [m^{2}/s], which has no special name. This unit is so large that it is rarely used. A more common unit of kinematic viscosity is the square centimeter per second [cm^{2}/s], which is given the name stokes [St] after the Irish mathematician and physicist George Stokes (1819–1903). One square meter per second is equal to ten thousand stokes.
1 cm^{2}/s =  1 St 
1 m^{2}/s =  10,000 cm^{2}/s 
1 m^{2}/s =  10,000 St 
Even this unit is a bit too large, so the most common unit is probably the square millimeter per second [mm^{2}/s] or the centistokes [cSt]. One square meter per second is equal to one million centistokes.
1 mm^{2}/s =  1 cSt 
1 m^{2}/s =  1,000,000 mm^{2}/s 
1 m^{2}/s =  1,000,000 cSt 
The stokes is a rare example of a word in the English language where the singular and plural forms are identical. Fish is the most immediate example of a aword thatbehaves like this. 1 fish, 2 fish, red fish, blue fish; 1 stokes, 2 stokes, some stokes, few stokes.
factors affecting viscosity
Viscosity is first and foremost a function of material. The viscosity of water at 20 ℃ is 1.0020 millipascal seconds (which is conveniently close to one by coincidence alone). Most ordinary liquids have viscosities on the order of 1 to 1,000 mPa s, while gases have viscosities on the order of 1 to 10 μPa s. Pastes, gels, emulsions, and other complex liquids are harder to summarize. Some fats like butter or margarine are so viscous that they seem more like soft solids than like flowing liquids. Molten glass is extremely viscous and approaches infinite viscosity as it solidifies. Since the process is not as well defined as true freezing, some believe (incorrectly) that glass may still flow even after it has completely cooled, but this is not the case. At ordinary temperatures, glasses are as solid as true solids.
From everyday experience, it should be common knowledge that viscosity varies with temperature. Honey and syrups can be made to flow more readily when heated. Engine oil and hydraulic fluids thicken appreciably on cold days and significantly affect the performance of cars and other machinery during the winter months. In general, the viscosity of a simple liquid decreases with increasing temperature. As temperature increases, the average speed of the molecules in a liquid increases and the amount of time they spend "in contact" with their nearest neighbors decreases. Thus, as temperature increases, the average intermolecular forces decrease. The actual manner in which the two quantities vary is nonlinear and changes abruptly when the liquid changes phase.
Viscosity is normally independent of pressure, but liquids under extreme pressure often experience an increase in viscosity. Since liquids are normally incompressible, an increase in pressure doesn't really bring the molecules significantly closer together. Simple models of molecular interactions won't work to explain this behavior and, to my knowledge, there is no generally accepted more complex model that does. The liquid phase is probably the least well understood of all the phases of matter.
While liquids get runnier as they get hotter, gases get thicker. (If one can imagine a "thick" gas.) The viscosity of gases increases as temperature increases and is approximately proportional to the square root of temperature. This is due to the increase in the frequency of intermolecular collisions at higher temperatures. Since most of the time the molecules in a gas are flying freely through the void, anything that increases the number of times one molecule is in contact with another will decrease the ability of the molecules as a whole to engage in the coordinated movement. The more these molecules collide with one another, the more disorganized their motion becomes. Physical models, advanced beyond the scope of this book, have been around for nearly a century that adequately explain the temperature dependence of viscosity in gases. Newer models do a better job than the older models. They also agree with the observation that the viscosity of gases is roughly independent of pressure and density. The gaseous phase is probably the best understood of all the phases of matter.
Since viscosity is so dependent on temperature, it shouldn't never be stated without it.
simple liquids  T (℃)  η (mPa s)  gases  T (℃)  η (μPa s)  

alcohol, ethyl (grain)  20  1.1  air  15  17.9  
alcohol, isopropyl  20  2.4  hydrogen  0  8.42  
alcohol, methyl (wood)  20  0.59  helium (gas)  0  18.6  
blood  37  3–4  nitrogen  0  16.7  
ethylene glycol  25  16.1  oxygen  0  18.1  
ethylene glycol  100  1.98  complex materials  T (℃)  η (Pa s)  
freon 11 (propellant)  −25  0.74  caulk  20  1000  
freon 11 (propellant)  0  0.54  glass  20  10^{18}–10^{21}  
freon 11 (propellant)  +25  0.42  glass, strain pt.  504  10^{15.2}  
freon 12 (refrigerant)  15  ?  glass, annealing pt.  546  10^{12.5}  
freon 12 (refrigerant)  0  ?  glass, softening pt.  724  10^{6.6}  
freon 12 (refrigerant)  +15  0.20  glass, working pt.  10^{3}  
glycerin  20  1420  glass, melting pt.  10^{1}  
glycerin  40  280  honey  20  10  
helium (liquid)  4 K  0.00333  ketchup  20  50  
mercury  15  1.55  lard  20  1000  
milk  25  3  molasses  20  5  
oil, vegetable, canola  25  57  mustard  25  70  
oil, vegetable, canola  40  33  peanut butter  20  150–250  
oil, vegetable, corn  20  65  sour cream  25  100  
oil, vegetable, corn  40  31  syrup, chocolate  20  10–25  
oil, vegetable, olive  20  84  syrup, corn  25  2–3  
oil, vegetable, olive  40  ?  syrup, maple  20  2–3  
oil, vegetable, soybean  20  69  tar  20  30,000  
oil, vegetable, soybean  40  26  vegetable shortening  20  1200  
oil, machine, light  20  102  
oil, machine, heavy  20  233  
oil, motor, SAE 20  20  125  
oil, motor, SAE 30  20  200  
oil, motor, SAE 40  20  319  
propylene glycol  25  40.4  
propylene glycol  100  2.75  
water  0  1.79  
water  20  1.00  
water  40  0.65  
water  100  0.28 
motor oil
Motor oil is like every other fluid in that its viscosity varies with temperature and pressure. Since the conditions under which most automobiles will be operated can be anticipated, the behavior of motor oil can be specified in advance. In the United States, the organization that sets the standards for the performance of motor oils is the Society of Automotive Engineers (SAE). The SAE numbering scheme describes the behavior of motor oils under low and high temperature conditions — conditions that correspond to starting and operating temperatures. The first number, which is always followed by the letter W for winter, describes the low temperature behavior of the oil at start up while the second number describes the high temperature behavior of the oil after the engine has been running for some time. Lower SAE numbers describe oils that are meant to be used under lower temperatures. Oils with low SAE numbers are generally runnier (less viscous) than oils with high SAE numbers, which tend to be thicker (more viscous).
For example, 10W40 oil would have a viscosity no greater than 7,000 mPa s in a cold engine crankcase even if its temperature should drop to −25 ℃ on a cold winter night and a viscosity no less than 2.9 mPa s in the high pressure parts of an engine near the point of overheating (150 ℃).
low temperature specifications  

sae prefix 
dynamic viscosity cranking maximum 
dynamic viscosity pumping maximum 

00W  06,200 mPa s  (−35 ℃)  60,000 mPa s  (−40 ℃) 
05W  06,600 mPa s  (−30 ℃)  60,000 mPa s  (−35 ℃) 
10W  07,000 mPa s  (−25 ℃)  60,000 mPa s  (−30 ℃) 
15W  07,000 mPa s  (−20 ℃)  60,000 mPa s  (−25 ℃) 
20W  09,500 mPa s  (−15 ℃)  60,000 mPa s  (−20 ℃) 
25W  13,000 mPa s  (−10 ℃)  60,000 mPa s  (−15 ℃) 
high temperature specifications  
sae suffix 
kinematic viscosity low shear rate 
dynamic viscosity high shear rate 

08  04.0–6.10 mm^{2}/s  (100 ℃)  >1.7 mPa s  (150 ℃) 
12  05.0–7.10 mm^{2}/s  (100 ℃)  >2.0 mPa s  (150 ℃) 
16  06.1–8.20 mm^{2}/s  (100 ℃)  >2.3 mPa s  (150 ℃) 
20  05.6–9.30 mm^{2}/s  (100 ℃)  >2.6 mPa s  (150 ℃) 
30  09.3–12.5 mm^{2}/s  (100 ℃)  >2.9 mPa s  (150 ℃) 
*40*  12.5–16.3 mm^{2}/s  (100 ℃)  >2.9 mPa s  (150 ℃) 
^{†}40^{†}  12.5–16.3 mm^{2}/s  (100 ℃)  >3.7 mPa s  (150 ℃) 
50  16.3–21.9 mm^{2}/s  (100 ℃)  >3.7 mPa s  (150 ℃) 
60  21.9–26.1 mm^{2}/s  (100 ℃)  >3.7 mPa s  (150 ℃) 
capillary viscometer
The the mathematical expression describing the flow of fluids in circular tubes was determined by the French physician and physiologist Jean Poiseuille (1799–1869). Since it was also discovered independently by the German hydraulic engineer Gotthilf Hagen (1797–1884), it should be properly known as the HagenPoiseuille equation, but it is usually just called Poiseuille's equation. I will not derive it here. (Please don't ask me to.) For nonturbulent, nonpulsatile fluid flow through a uniform straight pipe, the volume flow rate (q_{m}) is…
 directly proportional to the pressure difference (ΔP) between the ends of the tube,
 inversely proportional to the length (ℓ) of the tube,
 inversely proportional to the viscosity (η) of the fluid, and
 proportional to the fourth power of the radius (r^{4}) of the tube.
q_{m} =  πΔPr^{4} 
8ηℓ 
Solve for viscosity if that's what you want to know.
η =  πΔPr^{4} 
8q_{m}ℓ 
capillary viscometer… keep writing…
falling sphere
The mathematical expression describing the viscous drag force on a sphere was determined by the Nineteenth Century British physicist George Stokes. I will not derive it here. (Once again, don't ask.)
R = 6πηrv
The formula for the buoyant force on a sphere is accredited to the Ancient Greek engineer Archimedes of Syracuse, but equations weren't invented back then.
B = ρ_{fluid}gV_{displaced}
The formula for weight had to be invented by someone, but I don't know who.
W = mg = ρ_{object}gV_{object}
Let's combine all these things together for a sphere falling in a fluid. Weight goes down, buoyancy goes up, drag goes up. After a while, the sphere will fall with constant velocity. When it does, all these forces cancel. When a sphere is falling through a fluid it is completely submerged, so there is only one volume to talk about — the volume of a sphere. Let's work through this.
B  +  R  =  W  
ρ_{fluid}gV  +  6πηrv  =  ρ_{object}gV  
6πηrv  =  (ρ_{object} − ρ_{fluid})gV  
6πηrv  =  Δρg ^{4}_{3}πr^{3} 
And here we are.
η =  2Δρgr^{2} 
9v 
Drop a sphere into a liquid. If you know the size and density of the sphere and the density of the liquid, you can determine the viscosity of the liquid. If you don't know the density of the liquid you can still determine the kinematic viscosity. If you don't know the density of the sphere, but you know its mass and radius, well then you do know its density. Why are you talking to me? Go back several chapters and get yourself some education.
Should I write more?
nonnewtonian fluids
Newton's equation relates shear stress and velocity gradient by means of a quantity called viscosity. A newtonian fluid is one in which the viscosity is just a number. A nonnewtonian fluid is one in which the viscosity is a function of some mechanical variable like shear stress or time. (Nonnewtonian fluids that change over time are said to have a memory.)
Some gels and pastes behave like a fluid when worked or agitated and then settle into a nearly solid state when at rest. Such materials are examples of shearthinning fluids. House paint is a shearthinning fluid and it's a good thing, too. Brushing, rolling, or spraying are means of temporarily applying shear stress. This reduces the paint's viscosity to the point where it can now flow out of the applicator and onto the wall or ceiling. Once this shear stress is removed the paint returns to its resting viscosity, which is so large that an appropriately thin layer behaves more like a solid than a liquid and the paint does not run or drip. Think about what it would be like to paint with water or honey for comparison. The former is always too runny and the latter is always too sticky.
Toothpaste is another example of a material whose viscosity decreases under stress. Toothpaste behaves like a solid while it sits at rest inside the tube. It will not flow out spontaneously when the cap is removed, but it will flow out when you put the squeeze on it. Now it ceases to behave like a solid and starts to act like a thick liquid. when it lands on your toothbrush, the stress is released and the toothpaste returns to a nearly solid state. You don't have to worry about it flowing off the brush as you raise it to your mouth.
Shearthinning fluids can be classified into one of three general groups. A material that has a viscosity that decreases under shear stress but stays constant over time is said to be pseudoplastic. A material that has a viscosity that decreases under shear stress and then continues to decrease with time is said to be thixotropic. If the transition from high viscosity (nearly semisolid) to low viscosity (essentially liquid) takes place only after the shear stress exceeds some minimum value, the material is said to be a bingham plastic.
Materials that thicken when worked or agitated are called shearthickening fluids. An example that is often shown in science classrooms is a paste made of cornstarch and water (mixed in the correct proportions). The resulting bizarre goo behaves like a liquid when squeezed slowly and an elastic solid when squeezed rapidly. Ambitious science demonstrators have filled tanks with the stuff and then run across it. As long as they move quickly the surface acts like a block of solid rubber, but the instant they stop moving the paste behaves like a liquid and the demonstrator winds up taking a cornstarch bath. The shearthickening behavior makes it a difficult bath to get out of. The harder you work to get out, the harder the material pulls you back in. The only way to escape it is to move slowly.
Materials that turn nearly solid under stress are more than just a curiosity. They're ideal candidates for body armor and protective sports padding. A bulletproof vest or a kneepad made of of shearthickening material would be supple and yielding to the mild stresses of ordinary body motions, but would turn rock hard in response to the traumatic stress imposed by a weapon or a fall to the ground.
Shearthickening fluids are are also divided into two groups: those with a timedependent viscosity (memory materials) and those with a timeindependent viscosity (nonmemory materials). If the increase in viscosity increases over time, the material is said to be rheopectic. If the increase is roughly directly proportional to the shear stress and does not change over time, the material is said to be dilatant.
shearthinning  shearthickening  

timedependent (memory materials) 
thixotropic ketchup, honey, quicksand, snake venom, polymeric thick film inks 
rheopectic cream being whipped 
timeindependent (nonmemory materials) 
pseudoplastic paint, styling gel, whipped cream, cake batter, applesauce, ballpoint pen ink, ceramicmetal inks 
dilatant starch pastes, silly putty, synovial fluid, chocolate syrup, viscous coupling fluids, liquid armor 
with a yield stress  bingham plastic toothpaste, drilling mud, blood, cocoa butter, mayonnaise, yoghurt, tomato puree, nail polish, sewage sludge 
n/a 
With a bit of adjustment, Newton's equation can be written as a power law (the Ostwaldde Waele equation) that handles the pseudoplastics and the dilantants.
F  = k  ⎛ ⎝ 
dv_{x}  ⎞^{n} ⎠ 
A  dz 
Where η the viscosity is replaced with k the flow consistency index [Pa s^{n}] and the velocity gradient is raised to some power n called the flow behavior index [dimensionless]. The latter number varies with the class of fluid.
n < 1  n = 1  n > 1 
pseudoplastic  newtonian  dilatant 
A different modification to Newton's equation is needed to handle Bingham plastics (the Bingham equation).
F  = σ_{y} + η_{pl}  dv_{x} 
A  dz 
Where σ_{y} is the yield stress [Pa] and η_{pl} is the plastic viscosity [Pa s]. The former number separates Bingham plastics from newtonian fluids.
σ_{y} < 0  σ_{y} = 0  σ_{y} > 0 
impossible  newtonian  bingham plastic 
Combining the Ostwaldde Waele power law with the Bingham yield stress gives us the more general HerschelBulkley equation. Again, σ_{y} is the yield stress [Pa], k is the flow consistency index [Pa s^{n}], and n is the flow behavior index [dimensionless].
F  = σ_{y} + k  ⎛ ⎝ 
dv_{x}  ⎞^{n} ⎠ 
A  dz 
viscoelasticity
When a force (F) is applied to an object, one of four things can happen.
 It could accelerate as a whole, in which case Newton's second law of motion would apply…
F = ma
This term is not interesting to us right now. We've already discussed this kind of behavior in earlier chapters. Mass (m) is resistance to acceleration (a), which is the second derivative of position (x). Let's move on to something new.
 It could flow like a fluid, which could be described by this relationship…
F = −bv
This is the simplified model where drag is directly proportional to speed (v), the first derivative of position (x). We used this in terminal velocity problems just because it gave differential equations that were easy to solve. We also used it in the damped harmonic oscillator, again because it gave differential equations that were easy to solve (relatively easy, anyway). The proportionality constant (b) is often called the damping factor.
 It could deform like a solid according to Hooke's law…
F = −kx
The proportionality constant (k) is the spring constant. Position (x) is not the part of any derivative nor is it raised to any power.
 It could get stuck…
F = −f
That symbol f makes it look like we're discussing static friction. In fluids (nonnewtonian fluids, to be specific) a term like this is associated with yield stress. Position (x) is not involved in any way.
Put everything together and state acceleration and velocity as derivatives of position.
F = m  d^{2}x  − b  dx  − kx − f 
dt^{2}  dt 
This differential equation summarizes the possible behaviors of an object. The interesting thing is that it mixes up the behaviors of fluids and solids. The more interesting thing is that there are occasions when both behaviors will be present in one thing. Materials that both flow like fluids and deform like solids are said to be viscoelastic — an obvious mashup of viscosity and elasticity. The study of materials with fluid and solid properties is called rheology, which comes from the Greek verb ρέω (reo), to flow.
What old book gave me this idea? What should I write next?