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MAE 5130: VISCOUS FLOWS

Lecture 4: Viscosity

August 26, 2010

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

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WHAT IS VISCOSITY?• What does viscosity mean?

– Often related to ‘time to flow’ especially for petroleum products– Viscosity is a measure of a fluid's resistance to flow. A fluid with low viscosity

flows easily and is often called "thin." Water is an example of a fluid with a relatively low viscosity. A fluid with high viscosity is often described as "thick." Maple syrup is an example of a fluid with a relatively high viscosity.

• Remember: Time to flow is not viscosity

• How long does it take for 60 ml of oil at specified temperature (100 ºC or 210 ºF, approx. engine operating temp) to flow out of a 1.76 cm hole in bottom of a cup?

– SAE 10 motor oil takes 10 seconds– SAE 30 motor oil takes 30 seconds– W rating indicates that oil has been tested at a colder temperature– 10-W30 motor oil performs like a SAE 10 motor oil at colder temperatures

(engine start-up) but still has the SAE 30 viscosity at higher temperatures (engine operating conditions)

• What does a viscosity index (VI) number mean? – Measure of relative change in viscosity of oil over a temperature range– HIGHER VI → SMALLER viscosity change over temperature– VI not related to actual viscosity or SAE viscosity, but is measure of rate of

viscosity change– Generally, multigrade oils (0W-40, 10W-30, etc.) will have high viscosity

indexes. Monograde oils (SAE 30, 40, etc.) will have lower viscosity indexes.

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COEFFICIENT OF VISCOSITY, • More fundamental approach to viscosity shows it is

property of fluid which relates applied stress () to resulting strain rate ()

• Consider fluid sheared between two flat plates– Bottom plate is fixed– Top plate moving at constant velocity, V, in

positive x-direction only• u = u(y) only

– Geometry dictates that shear stress, xy, must be constant throughout fluid

• Perform experiment → for all common fluids, applied shear is a unique function of strain rate

– For given V, xy is constant, it follows that du/dy and xy are constant, so that resulting velocity profile is linear across plates

• Newtonian fluids (air, water, oil): linear relationship between applied stress and strain

– Coefficient of viscosity of a Newtonian fluid: – Dimension: Ns/m2 or kg/ms– Thermodynamic property (related to molecular

interactions) that varies with T&P

dy

du

h

V

f

dy

du

y

u

x

v

y

u

xyxy

xyxy

xyxy

xy

2

2

1

2

1

2

1

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VISCOUS BEHAVIOR OF VARIOUS MATERIALS• All true fluids can not resist shear, so must pass

through origin on plot of vs. – Yielding fluids show finite stress at zero

strain rate (part solid and part fluid)– Often called a Bingham plastic– Toothpaste, grease, hand creams

• Pseudoplastic: shear-thinning– Usually solutions of large, polymeric

molecules in a solvent with smaller molecules

– Ketchup: When at rest it is hard to pour, however it has lower viscosity when agitated

– Hair gel: much harder to pour off fingers (a low shear application), but that it produces much less resistance when rubbed between the fingers (a high shear application)

• Dilatant: shear-thickening– Uncooked mix of cornstarch and water: – Under high shear the water is squeezed out

from between the starch molecules, which are able to interact more strongly

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VISCOUS BEHAVIOR OF VARIOUS MATERIALS• Behavior of some non-Newtonian fluids may be

time-dependent

• If strain rate is held constant, shear stress may vary

• Thixotropic: shear stress decreases– The longer the fluid undergoes shear, the

lower its viscosity– Yogurt– Paint– Many clutch-type automatic transmissions use

fluids with thixotropic properties, to engage the different clutch plates inside the transmission housing at specific pressures, which then changes the gearset

– Clay-like ground can practically liquefy under the shaking of a tremor

– Ketchup is frequently thixotropic

• Rheopectic: shear stress increases– The longer the fluid undergoes shear, the

higher its viscosity– Some lubricants, thicken or solidify when

shaken– Gypsum paste

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POWER-LAW APPROXIMATION FOR NONNEWTONIAN FLUIDS

• K and n are material parameters which in general vary with T & P

• K is the flow consistency index

• n is the flow behavior index

– If n < 1: pseudoplastic

– If n = 1: Newtonian (K = m)

– If n > 1: dilatant

• Power law is only a good description of fluid behavior across range of shear rates to which coefficients were fitted

nxy

n

xy Ky

uK 2

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VISCOSITY AS A FUNCTION OF T &P

• Non-dimensionalization performed relative to critical point

– Tr=T/Tc

• General Trends

1. Viscosity of liquids ↓ as T ↑

2. Viscosity of low-pressure gases (or dilute mixtures) ↑ as T ↑

3. Viscosity always ↑ as P ↑

4. Poor accuracy near Pc, Tc

• Usually Pc ~ 10 atm

• Common in many problems to ignore P dependence and consider only T dependence

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CORRELATIONS FOR AND k

ST

ST

T

T

k

k

T

T

k

kn

02

3

00

00

ST

ST

T

T

T

Tn

02

3

00

00

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EQUATIONS OF MOTION: CARTESIAN

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EQUATIONS OF MOTION: CYLINDRICAL (r, , z)

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EQUATIONS OF MOTION: SPHERICAL (r, , )

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PROBLEM: 1-4

• Steady viscous flow enters tube from a reservoir• Wall friction causes a viscous layer, initially

probably laminar, to begin at inlet and grow in thickness downstream, possibly becoming turbulent further inside tube

• Internal flow constrained by solid walls, so viscous layers must coalesce at some distance, xL, at which point tube is completely filled with boundary layer

• Downstream of coalescence, flow profile ceases to change with axial position and is called ‘fully-developed’

• If ReD > 2,000, flow will end up turbulent– See picture above

• At lower ReD flow remains laminar– See pictures to right

22 rRC

u

Poiseuille-Paraboloid Laminar Pipe Flow Formula

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PROBLEM: 1-6

• Vorticity, z, calculated from equations in Appendix B

• Instantaneous velocity and vorticity profiles are shown on the right for C=1, =1.

• At t=0, the flow is a ‘line’ vortex’, irrotational everywhere except at the origin where z=∞

t

r

t

C

t

r

r

Cv

v

z

r

4exp

2

4exp1

0

2

2

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PROBLEM: 1-6

• Vorticity, z, calculated from equations in Appendix B

• Instantaneous velocity and vorticity profiles are shown on the right for C=1, =1.

• At t=0, the flow is a ‘line’ vortex’, irrotational everywhere except at the origin where z=∞

t

r

t

C

t

r

r

Cv

v

z

r

4exp

2

4exp1

0

2

2

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PROBLEM: 1-8

• Inviscid flow past a [non-rotating] cylinder

– 2 stagnation points (r,) = (R,0) and (R,)