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Shear Stress

Newton’s Law of Viscosity(Laminar Flow)

• Solids –Modulus of elasticity– Resistance of deformation

• Liquids – Similar concept to solids– Newton’s Law of Viscosity

Strain ShearStress Shear

Modulus Shear =

µ== Strain Shear of Rate

Stress Shear Viscosity

Viscosity

• Resistance to RATE of deformation• Large viscosity (µ) – more resistance to

deformation– Deformation related to flow

• µ function of T,P, and composition• Independent of rate of shear strain

– Newtonian Fluid Only!

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Rate of Shear Strain

• Consider an element of fluid “cube”• Rate of shear strain

– Geometrical change of fluid element as a function of time

dtd- Strain Shear of Rate δ

=

Deformation of Fluid Element

∆x

∆y

Element attime t

Element attime t+∆t

(V|y+∆y- V|y )∆t

δt δt+Dt

y

V,x

Quantifying Deformation

t2

- yt

yvy yvarctan - 2

limit -

t

tt t limit - dtd

-

-

0 t y, x,

0 t y, x,

∆⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡∆∆

⎟⎠⎞⎜

⎝⎛

∆+=

=∆∆+==−

ππ

δδδ

→∆∆∆

→∆∆∆

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Quantifying Deformation

• In the limit, the rate of deformation is

dydv

dtd

=−δ

• Newton’s Law of Viscosity

dydv µ=τ

• Above is development for 1-D flow

1D Slot Flow• Law only valid for fluids that have a linear

relation between yield stress and rate of strain

h

Velocity and shear stress profile

( )2x y-hy A V =

( )2y -h A µτ =

y

x

dydV xµ=τ

Non-Newtonian Fluids

• Newton’s law of viscosity does not predict the shear stress in all fluids. Only fluids where the relationship between the shear stress and the rate of shearing strain are linear.

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*Non Slip condition

• Fluid layer adjacent to a boundary wall has the same velocity as the boundary.

• Stationary boundary V = 0• Moving boundary V = Vboundary

• Result due to experimental observation.• Fails when fluid is treated as a particle not a

continuum

Shear Stress in Multidimensional Laminar flows of Newtonian Fluids

• Previous analysis is valid only for 1D parallel flow laminar flows

• General definition still valid

µ StrainShear of Rate

StressShear Viscosity ==

• Examine shear stress for 3D body.• Shear Stress in 3D is a tensor quantity• f (magnitude, direction, and orientation)

Multi-Dimensional Shear Stress

• Shear stress is represented as a tensor: τij• 1st Subscript = direction of axis to which plane of

action of shear is normal• 2nd Subscript = direction of action of shear stress• Example

– Consider the shear stress: τxy• acts on plane normal to the x axis (in the yz plane)• acts in the y direction

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τyx

τxy

τyx

τxy

y

x

τzy

τ yz

τzy

τ yz

z

y

τxz

τzx

τxz

τzx

x

z

Shear Stress Figures

Stokes’ Viscosity Relations

• Shear Stress (Laminar Flow)

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

i

x v

x v µτ ij

• Normal Stress

P - 32

x v2

i

i⎟⎟⎠

⎞⎜⎜⎝

⎛•−= ∂

∂ v∇µσ ii

Navier Stokes

• Utilizes Stokes’ relations of viscosity• Conservation of momentum• Incompressible flow (liquids)

v Pg vv tv 2 ∇∇−ρ∇∂∂ρ µ+=⎟

⎠⎞

⎜⎝⎛ ⋅+