Viscosity

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Viscosity
Fluid Flow Characteristics
Viscous Fluids
Dependency of Viscosity on Temperature
Non-Newtonian Fluids
Select one subject!!
Laboratory exercise
Examples
Viscometer
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Section 1 Viscous Fluids
Horizontal flow of viscous fluids
Continuity equation for viscous flow
Viscous flow through a porous medium made up of a
bundle of identical tubes
Viscous flow in a cylindrical tube
Exercise capillary tube viscosity measurement
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Horizontal flow of Viscous Fluids
Lower plate is set in motion with constant
velocity
The fluid start to move due to motion of the plate
After a while the fluid enter a steady state
velocity profile
Y a very small distance
To maintain this steady state motion, a constant
force F is required
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The force may be expressed
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Also used is the so called Newtons equation of
viscosity
The shear stress
The fluid velocity in x- direction
The fluid viscosity
The dimensions
And
By definition
Explicit
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Continuity equation for viscous flow
The flow will have the biggest velocity at the
top (the surface )
A liquid is flowing in a open channel
Fluid flows in an open channel
The flow velocity will approximately be zero at
the bottom, due to retardation when liquid
molecules colliding with the non-moving bottom
A graphic presentation of this phenomena is shown
here
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Transferred momentum pr. time pr. area
The change in fluid flow velocity pr. distance
between the two layers
Viscosity is here defined as s proportionality
constant, similar to what was done in the case of
defining absolute permeability
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dy is the width of the box and S S is the
cross-section
Momentum transfer between layers S and S in
Newtonian viscous flow
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Combining with the equation from last page gives
us
Substituting
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Viscous flow in a cylindrical tube
To develop the continuity equation for this
example, dr (a thin layer), of liquid is
considered at a radius r.
Cross-section of viscous flow through a
cylindrical tube
Viscous flow in a tube is characterised by a
radial decreasing flow velocity, because of the
boundary effect of the tube wall
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The momentum flux through the cylindrical volume
The momentum intensity
jp is redefined according to the geometric
conditions in this example
The minus sign show that the flow velocity vx is
decreasing when the radius r is increasing
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The continuity equation for viscous flow in a
cylindrical tube is
This is under stationary conditions.
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The general solution of the previous equation,
found by integrating twice, is
The general constants are found by considering
the boundary conditions. For this example the
flow is directed along the x-axis
The solution is then
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Defines a the laminar viscous flow pattern
This flow velocity profile in the tube is given
by the formula above.
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Viscous flow through a porous medium made up of a
bundle of identical tubes
The incremental flow rate through a fraction of
the cross-section of a capillary tube can be
expressed
The total flow rate can be found by integration
For the sake of convenience we may present the
last equation in the following form
This is also known as the Poiseuilles equation,
where A is the cross-section of the capillary tube
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Considering a porous medium as a bundle of
identical capillary tubes, the total flow qp
through the medium is
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Exercise Capillary tube viscosity measurement
Fluid viscosity may also be estimated by the
measuring the volume of fluid flowing through a
capillary tube pr. time (as the figure below).
Rewriting the Poiseuilles equation an
expression for the dynamic fluid viscosity is
written
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Which means that if the relative accuracy in the
tube-radius is /- 2-3 , then the relative
accuracy in the viscosity is about /- 10 . A
small variation in capillary tube fabrication
induces large uncertainty in the viscosity
measurement.
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Section 2 Fluid Flow Characteristics
For laminar flow in a cylindrical tube we have
derived Poiseuilles equation
For turbulent flow in a cylindrical tube, an
empirical law, called Fannings equation has been
found
This factor is dependent of the tube surface
roughness, but also on the flow regime
established in the tube
F the Fanning friction factor
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2R is the spatial dimension where the flow occur
the diameter of a capillary tube or width of an
open channel.
From experimental studies an upper limit for
laminar flow has been defined at a Reynolds
number Re 2000. Above this number, turbulent
flow will dominate. (This limit is not absolute
and may therefore change somewhat depending on
the experimental conditions.)
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Using the previous table we find Reynolds number
Re 1 (ca.), for laboratory core flow, which is
far below the limit of turbulent flow.
For gas, turbulent flow may occur if the
potentials are steep enough. If the formula for
Reynolds number and Poiseuilles law is compared
The only fluid dependent parameters are the
density and the viscosity
Comparing the Reynolds number for typical values
of gas and oil
Demonstrates the possibilities for turbulent when
gas is flowing in a porous medium
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Section 3 Dependency of Viscosity on Temperature
The Viscosity of liquids decreases with
increasing temperature. For gas its the
opposite viscosity increase with increasing
temperature.
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Temperature depending viscosity of gases
expressed by the Satterlands equation
Where K and C are constants depending on the type
of gas.
Another commonly equation
Where n depends on the type of gas (1 lt n lt 0,75)
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Section 4 Non-Newtonian Fluids
Viscous-Plastic fluids Bingham (1916) and
Shvedov (1889) investigated the rheology of
viscous-plastic fluids. These fluids also feature
elasticity in addition to viscosity. Equation
describing viscous-plastic fluids
Some oils, drilling mud and cements slurries
represent viscous-plastic fluids.
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Pseudo-Plastic fluids Some fluids do not have
breaking shear stress but rather, their apparent
viscosity depends on a shear rate
Which means that their apparent viscosity
decreases when dv/dy grows
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Section 5 Examples
6.2.1 Water viscosity at reservoir conditions
6.2.2 Falling sphere viscosity measurement
6.2.4 Rotating cylinder viscosity measurement
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Water viscosity at reservoir conditions
Correlation for estimation of water viscosity at
res. temp.
Water viscosity is measured in cP and temperature
in Fahrenheit
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Falling sphere viscosity measurement
A metal sphere falling in viscous fluid reaches a
constant velocity vs
Then the viscous retarding force plus the
buoyancy force equals the weight of the sphere
The weight of the sphere will balance the viscous
force plus the buoyancy force at the terminal
sphere velocity when the sum of forces acting on
the sphere is zero
g the constant of gravitation
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C characteristic constant, determined through
calibration with a fluid of known viscosity
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Rotating cylinder viscosity measurement
Here we will also Look at the motion of a fluid
between two coaxial cylinders
Due to the elastic force (in the fluid) a
viscosity shear will exist there
One cylinder is rotating with an angular velocity
The other one is kept constant
h the fluid height level on the two cylinders

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For a certain viscometer, the viscosity as
function of angular momentum and angular velocity
is
C the characteristic constant for the
viscometer
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Section 6 Laboratory exercise
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Section 6
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