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Lecture 7: Non-Newtonian Fluids

• Classification of Non-Newtonian Fluids• Laminar Flow of a Non-Newtonian fluid in

Circular Pipes• Recommended text-book: W.F. Hughes, J.A.

Brighton, Schaum's outline of theory and problems of fluid dynamics, New York: McGraw Hill, 1999

Examples• Water and simple liquids; air and simple gases… are Newtonian

fluids.

• Fluids in food industry, gels, polymers, slurries, drilling muds, blood… are Non-Newtonian fluids.

• The Non-Newtonian behaviour is frequently associated with complex internal structure: fluid has large complex molecules (like a polymer) or fluid is a heterogeneous solution (like a suspension)...

1: Coal slurries having consistency of over 80% by volume of powdered or crushed coal in water can be pumped long distances with much less power requirements for pumping than pure water.

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Examples

2: In the fracturing treatment of oil wells, materials have been developed which when added to water make a fluid so thick that it suspends sand, glass or metal pellets. Yet the same fluid can be pumped down a well at enormous rates with less than half the friction loss of water.

Such materials are used for fracturing of an oil reservoir. Fracturing is used to increase the production of the well. First, a crack is initiated in the producing zone. Then, fluid pumped down the well under high pressure greatly extends this crack. The sand, or pellets act as propping agents to hold the fracture open after the treatment. Fluid must be delivered at a rapid rate to overcome the loss by diffusion of fluid within the pores of the fractured rock.

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Definition: Newtonian/Non-Newtonian fluids

The viscous stress tensor for incompressible Newtonian fluid is

ikik -- rate of shear strain

i

k

k

i

ik xv

xv

Newtonian fluids: linear proportionality between the shearing tensor and the shearing rate.Non-Newtonian fluids: any different relation between the shearing stress and the shearing rate.

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Classification• Time Independent Fluids (the relation between shearing stress and rate is

unique but non-linear)• Bingham plastics• Pseudoplastic fluids• Dilatant plastics

• Time Dependent Fluids (the shear rate depends on the shearing time or on the previous shear rate history)

• Thixotropic fluids• Reopectic fluids

• Viscoelastic fluids (the shear stress is determined by the shear strain and the rate of shear strain)

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Time Independent Fluids

ikikf

τ

slope η

0

2

3

10 – Newtonian fluid1 – Bingham plastics2 – pseudoplastic fluids3 – dilatant fluids

1. Bingham plastics

y

Examples: slurries, plastics, emulsions such as paints, and suspensions of finely solids in a liquid (e.g. drilling muds, which consist primarily of clays suspended in water).

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A progressively decreasing slope of shear stress vs. shear rate. The slope can be defined as apparent viscosity:At very high rates of shear in real fluids the apparent viscosity becomes constant.

Examples: paper pulp in water, latex paint, blood, syrup, molasses, ketchup, whipped cream, nail polishThe simplest empirical model is the power law due to Ostwald:

The power law model is popular due to its simplicity. For the cases where power law model does not give an adequate representation, it might be practical to use the actual measured properties of the fluid.

a

11 nk n

a,

2. Pseudoplastic (shear thinning) fluids

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3. Dilatant (shear thickening) fluids

The apparent viscosity increases with increasing shear rate. Can be represented by the power law model with n>1.Less common (suspensions of corn starch or sand in water). Applications:Traction control: some all-wheel drive systems use a viscous coupling unit full of dilatant fluid to provide power transfer between front and rear wheels. On high traction surfacing, the relative motion between primary and secondary drive wheels is the same, so the shear is low and little power is transferred. When the primary drive wheels start to slip, the shear increases, causing the fluid to thicken. As the fluid thickens, the torque transferred to the secondary drive wheels increases, until the maximum amount of power possible in the fully thickened state is transferred. Body armour: application of shear thickening fluids for use as body armour, allowing the wearer flexibility for a normal range of movement, yet providing rigidity to resist piercing by bullets, stabbing knife blows, and similar attacks.

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Time Dependent FluidsThixotropic fluids (the shear stress decreases with time as the fluid is sheared)As the fluid is sheared from the state of rest, it breaks down (on molecular scale), but then the structural reformation will increase with time. An equilibrium situation is eventually reached where the breakdown rate is equal to build-up rate. If allowed to rest, the fluid builds up slowly and eventually regains its original consistency.Examples: many gels or colloidshttp://www.youtube.com/watch?v=oJaLV_r_kNAhttp://www.youtube.com/watch?v=zn2uonRGKYk

Rheopectic fluids (the shear stress increases with time as the fluid is sheared).Molecular structure is formed by shear and behaviour is opposite to that of thixotropy. Example: beating and thickening of egg whites, inks.http://www.youtube.com/watch?v=cuzn8wh8Fys

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Viscoelastic FluidsA viscoelastic material exhibits both elastic and viscous properties. The simplest type is one which is Newtonian in viscosity and obeys Hooke’s law for the elastic part:

λ is a rigidity modulus.

Simplest and popular model -- Maxwell liquids:

Under steady flow, . If the motion is stopped the stress relaxes as

Movies: •http://www.youtube.com/watch?v=B1C4qNyrUjU•http://www.youtube.com/watch?v=zwz3R0IG9Xc•http://www.youtube.com/watch?v=KcNWLIpv8gc•http://www.youtube.com/watch?v=wmUx-1o3Lzs•http://www.youtube.com/watch?v=nX6GxoiCneY•http://www.youtube.com/watch?v=HQ8FP0sa_hk

texp

Examples: polymers, metals at temperature close to their melting point

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Laminar flow of Bingham plastics

R

zrp

We consider steady plane parallel flow. The governing equations are reduced to

rzrrrz

pz

1

0:projection (*)

By denoting and integrating (*) we obtainAzp

rcAr

rz1

2

c1 = 0 as the stress tensor must be bounded at r = 0, i.e.

2Ar

rz

11

τrz rRrp

τy

y

z

yrz

y

z

ττrv

rA

ττrv

for

for

21

0

(**)

,:

,:

yrzy

y

0For Bingham plastics, the rate of stress tensor is related to the shearing stress as

or

Ar

rv

rr y

p

z

p

20

:at

12

Integration of (**) gives

RrrrRrRA

drrAvp

yr

Ryz

,

22

4211

Setting givesA

rr y

p

2

01

22

rr

rR

Av

p

p

p,

The volumetric flow rate

4

4 2

312

34

18 RARA

ARQ yy

At τy we will have the formulae earlier obtained for the Poiseuille flow.

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Eugene Cook Bingham, born 8 December 1878, died 6 November 1945. Bingham made many contributions to rheology.

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Lecture 8: Flow fields with negligible inertia forces

• Flow in slowly-varying channel• Lubrication theory

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Flow in slowly-varying channels

For Poiseuille flow: 0tv

0 vv 0v

Steady flow Plane-parallel flow

For steady flow in a slowly varying channel

0 vv

but small

We can always make the ratio

by choosing a sufficiently slow rate of variation of the cross-section.

1

v

vv

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Consider a steady flow along a circular tube with R(x), with

xdxdR and xA

dxdp

In such a flow, . Andzr vv ~ ,~ 2RV

u

RV

vv2

~

1~

RV

vvv

Hence, , i.e. inertia force is negligible.

This approximation is useful in many different circumstances e.g. the flow of a fluid squeezed out radially by pressing close together two plane disks.

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4, rR

Arxvz

8

4ARQ

Thus, the flow profile and the volumetric flow flux are

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Lubrication theory

It is a matter of common experience that two solid bodies can slide over one another easily when there is a thin layer of fluid between them and that under certain conditions a high positive pressure is set up in the fluid layer. This is used as a means of substituting fluid-solid friction for the much larger friction between two solid bodies in contact. In some case the fluid layer is used to support a useful load, and is then called a lubrication bearing.

α

h2hh1

Reynolds’ theory, 1886

U

1

RU

which is usually satisfied under practical conditions of lubrication

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h

yhUyhy

Au

2

Flow profile:

Poiseuille flow Couette flow

Volumetric flow flux:

UhAh

yuQh

21

12d

3

0

From here, the pressure gradient is

32

26

hQ

hU

Axp

Q must be independent of x. In addition, xhh 1

(1)

Integration of (1) gives

2

12

10

11116hh

Qhh

Upp

20

Suppose the sliding block is completely immersed in the fluid, so p=p0 when h=h2, which enables us to determine Q,

21

21

hhhh

UQ

and

212

210

6hhh

hhhhUpp

120 if0 hhpp A lubrication layer will be able to support a load normal to the layer only when the layer is so arranged that the relative motion of the two surfaces tends to drag fluid from the wider to the narrow end.

21

max hLU

p pmax can be high if h1 is small

(L is a layer length)

p-p0

x

21

The total normal force exerted on either of the two boundaries by the fluid layer is

21

21

2

12

0

0 2ln6

hhhh

hhU

dxppL

The total tangential force exerted by the fluid on the lower plate

2

1

21

21

0 0

ln232

hh

hhhhU

dxyuL

y

The tangential force on the upper boundary is

2

1

21

21

0

ln32

hh

hhhhU

dxyuL

hy

22

L

hhh

f 1

2

1 ~block on the force normal

block on the force tangential

This ratio is independent of the viscosity, and can be made indefinitely small by reduction of h1 with h1/h2 held constant.

α is regarded as a given quantity, although in any case in which the sliding block is free to move, α may be a variable, but consideration of this is beyond of our scope.