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Newtonian

dv    dy 2  r 2 (P  P )  (2 rL) (r ) Pr 2 1   2  = P P x  r   (2 rL)  2 rL 2L

2 4 P R2   r   PR v(r)  1     4 L   R   =   8LV0 Definition of a

F  du    yx      yx A  dy 

For Newtonian behaviour (1)  is proportional to  and a plot passes through the origin; and (2) by definition the constant of proportionality, Newtonian

dv    dy 2  r 2 (P  P )  (2 rL) (r ) Pr 2 1   2  = P P x  r   (2 rL)  2 rL 2L

2 4 P R2   r   PR v(r)  1     4 L   R   =   8LV0 Newtonian

2 From (r ) Pr d u  = P    2rL 2L d y 4 and PR 8LQ  = 0  4 8LV P = R

  = 8v/D 5 6 Non-Newtonian Flow Characteristic of Non-Newtonian Fluid

• Fluids in which shear is not directly proportional to rate are non-Newtonian flow: and Lucite

(Casson Plastic) () changes with . Apparent viscosity (a or ) is always defined by the relationship between and shear rate. Model Fitting - Shear Stress vs. Shear Rate

Summary of Viscosity Models      Newtonian

 n   ( n  1) Pseudoplastic  K

 n   K  ( n  1)

 n       Bingham y

1 1 1  1 Casson  2   2   2  2 0 c

 Herschel-Bulkley      n y K

 or  = shear stress, º = shear rate, a or  = apparent viscosity m or K or K'= consistency index, n or n'= flow behavior index Herschel-Bulkley model (Herschel and Bulkley , 1926) n du    m    0 dy  Values of coefficients in Herschel-Bulkley fluid model Fluid m n Typical examples 0 Herschel-Bulkley >0 00 Minced fish paste, raisin paste

Newtonian >0 1 0 ,fruit juice, , , vegetable Shear-thinning >0 00 1

(1) Fluids for which the rate of shear at any point is determined only by the value of the shear stress at that point at that instant; these fluids are variously known as “ independent”, “purely viscous”, “inelastic”, or “Generalised Newtonian Fluids” (GNF). (2) More complex fluids for which the relation between shear stress and shear rate depends, in addition, on the duration of shearing and their kinematic history; they are called “time- dependent fluids”. (3) Substances exhibiting characteristics of both ideal fluids and elastic and showing partial elastic recovery, after deformation; these are characterised as “visco-elastic” fluids. Time-Independent Fluid Behaviour 1. or pseudoplastic fluids Viscosity decrease with shear stress. Over a limited range of shear- rate (or stress) log (t) vs. log (g) is approximately a straight line of negative slope. Hence

n yx = m(yx) (*) where m = fluid consistency coefficient n = flow behaviour index Re-arrange Eq. (*) to obtain an expression for apparent viscosity

app (=yx/yx) Pseudoplastics

Flow of pseudoplastics is consistent with the random coil model of and melts. At low stress, flow occurs by random coils moving past each other w/o coil deformation. At moderate stress, the coils are deformed and slip past each other more easily. At high stress, the coils are distorted as much as possible and offer low resistance to flow. Entanglements between chains and the model also are consistent with the observed viscosity changes. Why Shear Thinning occurs

Unsheared Sheared

Aggregates Anisotropic Particles align break up with the Flow Streamlines

Random coil elongate and Courtesy: TA Instruments break Shear Thinning Behavior

 Shear thinning behavior is often a result of:  Orientation of non-spherical particles in the direction of flow. An example of this phenomenon is the pumping of fiber slurries.  Orientation of polymer chains in the direction of flow and breaking of polymer chains during flow. An example is polymer melt extrusion  Deformation of spherical droplets to elliptical droplets in an . An industrial application where this phenomenon can occur is in the production of low fat margarine.  Breaking of particle aggregates in suspensions. An example would be stirring paint. Courtesy: TA Instruments 2. Viscoplastic Fluid Behaviour

Viscoplastic fluids behave as if they have a yield stress (0). Until  is exceeded they do not appear to flow. A Bingham plastic fluid has a constant plastic viscosity B B  yx 0  B yx for  yx  0    B  yx  0 for yx 0

B Often the two model parameters 0 and  are treated as curve fitting constants, even when there is no true yield stress.

3. Shear-thickening or Dilatant Fluid Behaviour Eq. (*) is applicable with n>1. Viscosity increases with shear stress. Dilatant: shear thickening fluids that contain suspended solids. Solids can become close packed under shear. Source: Faith A. Morrison, Michigan Tech U. Source: Faith A. Morrison, Michigan Tech U. Source: Faith A. Morrison, Michigan Tech U. Time-dependent Fluid Behaviour The response time of the material may be longer than response time of the measurement system, so the viscosity will change with time. Apparent viscosity depends not only on the rate of shear but on the “time for which fluid has been subject to shearing”.

Thixotropic : Material structure breaks down as shearing continues : e.g. , cream, shortening, salad dressing.

Rheopectic : Structure build up as shearing continues (not common in food : e.g. highly concentrated solution over long periods of time Thixotropic Shear stress

Rheopectic

Shear rate Time independent Time dependent

+ _ + _

A B C D E F G

Non - newtonian

Rheological curves of Time - Independent and Time – Dependent Visco-elastic Fluid Behaviour A visco-elastic fluid displays both elastic and viscous properties. A true visco-elastic fluid gives time dependent behaviour. Newtonian Pseoudoplastic Dilatant s

s s s

s s e

e e r

r r t

t t s

s s

r

r r a

a a e

e e h

h h S

S S

Shear rate Shear rate Shear rate y y y t t t i i i s s s o o o c c c s s s i i i V V V

Shear rate Shear rate Shear rate Common flow behaviours Examples Newtonian flow occurs for simple fluids, such as water, petrol, and . The Non-Newtonian flow behaviour of many microstructured products can offer real advantages. For example, paint should be easy to spread, so it should have a low apparent viscosity at the high shear caused by the paintbrush. At the same time, the paint should stick to the wall after its brushed on, so it should have a high apparent viscosity after it is applied. Many cleaning fluids and furniture should have similar properties. Examples

The causes of Non-Newtonian flow depend on the chemistry of the particular product. In the case of water-based latex paint, the shear-thinning is the result of the breakage of bonds between the surfactants used to stabilise the latex. For many cleaners, the shear thinning behaviour results from disruptions of crystals formed within the products. It is the produced by these chemistries that are responsible for the unusual and attractive properties of these microstructured products. Newtonian Foods

Shear stress

Shear rate

Examples: • Water • Milk • Vegetable • Fruit juices • Sugar and salt solutions Pseudoplastic (Shear thinning) Foods

Shear stress

Shear rate Examples: • Applesauce • Banana puree • Orange juice concentrate • Oyster sauce • CMC solution Dilatant (Shear thickening) Foods

Shear stress

Shear rate

Examples: • Liquid Chocolate • 40% Corn starch solution Bingham Plastic Foods

Shear stress

Shear rate

Examples: • Tooth paste • Tomato paste Non-Newtonian Fluids Newtonian Fluid du    z rz dr

Non-Newtonian Fluid du    z rz dr η is the apparent viscosity and is not constant for non-Newtonian fluids. η - Apparent Viscosity The shear rate dependence of η categorizes non-Newtonian fluids into several types. Law Fluids:  Pseudoplastic – η (viscosity) decreases as shear rate increases (shear rate thinning)  Dilatant – η (viscosity) increases as shear rate increases (shear rate thickening)

Bingham Plastics:

 η depends on a critical shear stress (0) and then becomes constant Modeling Fluids Oswald - de Waele n  n1   duz   duz   duz   rz  K   K      dr    dr    dr 

where: K = flow consistency index n = flow behavior index eff

Note: Most non-Newtonian fluids are pseudoplastic n<1. Modeling Bingham Plastics

Yield stress du    z  rz  dr 0

 rz  0 Frictional Losses Non-Newtonian Fluids

Recall:

L V 2 h  2 f f D g

Applies to any type of fluid under any flow conditions Power Law Fluid n  duz   rz  K   dr 

1 n du  1 p  z    r1 n dr  2 KL 

Boundary Condition

r  R uz  0 Profile of Power Law Fluid Circular Conduit

Upon Integration and Applying Boundary Condition We can derive the expression for u(r)

1 n  n1 n1   1 p   n  n n uz     R  r   2 KL   n 1  Power Law Results ()

n n 2  3n 1 n 2    LKV n p     Dn1 ↑ Hagen-Poiseuille (laminar Flow) for Power Law Fluid ↑

Recall 1  D  2  p f      4  L V 2   Laminar Factor Power Law Fluid n 16 n 1 3n 1 2    K f  n RePL f    V 2n Dn 

Define a Power Law or Generalized Reynolds number (GRe) n 2n n 3n  n  V D  RePL  2    3n 1 K Turbulent Flow

flow behavior index Power Law Fluid Example

A coal slurry is to be transported by horizontal pipeline. It has been determined that the slurry may be described by the power law model with a flow index of 0.4, an apparent viscosity of 50 cP at a shear rate of 100 /s, and a of 90 lb/ft3. What horsepower would be required to pump the slurry at a rate of 900 GPM through an 8 in. Schedule 40 pipe that is 50 miles long ?

P = 1atm P = 1atm

L = 50 miles n'  V   V  K'   app    r   r 

10.4 100  kg K' 50cP   0.792  s  m s1.6

~  900gal   1ft3  1min   1   m  m V        1.759  min   7.48gal   60s   0.3474 ft2   3.281ft  s

1.6  0.4  kg  m   0.4 0.202 m 1442 1.759    0.4  m3 s RE  230.4        7273 N    kg   3(0.4) 1 0.792  1.6   m s  P V 2 gZ Wp     hf  2gc gc

 L V 2 Wp  hf  4 f    D  2

f  0.0048 Fig 5.11

2  m  1.760  2  80460m   s  m Wp  hf  40.0048  11,845  0.202m  2 s2

 m  kg  kg m 1.759 0.0323 m2 1442   81.9 s  m3  s

kg  m2  81.9 11,845  s  s2  Power     970.1 kW 1300 Hp 1000 Bingham Plastics

Bingham plastics exhibit Newtonian behavior after the shear stress exceeds

o. For flow in circular conduits Bingham plastics behave in an interesting fashion. Sheared Annular Region

Unsheared Core Bingham Plastics Unsheared Core

 0 2 r  rc uz  uc  R  rc  2rc

Sheared Annular Region

R  r  rz  r   r  rc uz   1   0    2  R   Laminar Bingham Plastic Flow 16  He He4  f  1  (Non-linear)  3 7  ReBP  6ReBP 3 f ReBP  

D2  0 Hedstrom Number He  2  DV ReBP   Turbulent Bingham Plastic Flow a 0.193 f 10 ReBP 5 a  1.3781 0.146e2.9x10 He  Bingham Plastic Example Drilling mud has to be pumped down into an oil well that is 8000 ft deep. The mud is to be pumped at a rate of 50 GPM to the bottom of the well and back to the surface through a pipe having an effective diameter of 4 in. The at the bottom of the well is 4500 psi. What pump head is required to do this ? The drilling mud has the properties of a Bingham plastic with a yield stress of 100 dyn/cm2, a limiting (plastic) viscosity of 35 cP, and a density of 1.2 g/cm3.

P = 14.7 psi

L = 8000 ft

P = 4500 psi 4 D  ft  0.3333 ft  0.0873 ft 2 12

 min   ft 3   1  ft V  50     1.276      2  min  60s   7.48gal   0.0873 ft  s

lb lb  1.262.4 m  74.88 m ft 3 ft 3

 lb   6.719710  4 m  ft s lb   35 cP    0.0235 m  cP  ft s    

 ft   lbm  0.3333 ft 1.276 74.88   s  ft3 N    1355 RE lb 0.0235 m ft s

dyn g  100 100 o cm2 s 2cm 2   2.54 cm   g  100g  4in   1.2 3  2    in   cm   s cm  5 N HE  2 1.0110  g  0.35   cms 

f  0.14

P V 2 gZ Wp     h f  2gc gc

lb 144 in2   2  4500 14.7 f     ft   2  2   1.276   in  ft  ft lb f 40.148000 ft  s  W  8000    p lb m lbm 0.3333 ft   32.2 ft lbm   74.88 3 2  ft   lb s2     f  

ft lb Wp  8626 8000  339  965 f lbm

In order to get meaningful (universal) values for the viscosity, we need to use geometries that give the viscosity as a scalar invariant of the shear stress or shear rate. Generalized Newtonian models are good for these steady flows: tubular, axial annular, tangential annular, helical annular, parallel plates, rotating disks and cone- and-plate flows. Capillary, Couette and cone-and-plate viscometers are common.

Non-newtonian fluid

• from  = 2r2L dω  = r dr n Ω rdω  -μ  2Lr2  dr 

Integrate from r = Ro Ri and  = 0i Non-newtonian fluid  or a º

• obtain n 4N  n n 1   -K    -K.n 4N  4N   n  n n1   -K.n 4N  ln   ln K -n ln n (n -1) ln 4N  Linear : y = y-intercept + slope (x) K and n

4 y = -0.7466x + 3.053 3.5 R2 = 0.9985 3 2.5 a U

2 n = 1.7466 n l 1.5 1 K = 56.09 0.5 0 -1 -0.5 0 0.5 1 1.5 2 (shear thinning) ln 4TTN