Geodynamics www.helsinki.fi/yliopisto
Geodynamics Lecture 9
Basics of fluid mechanicsLecturer: David [email protected]!
!30.9.2014
2
www.helsinki.fi/yliopistoGeodynamics
Paper discussion on Thursday
• Paper discussion on Thursday after lecture!
• Similar to last discussion!
• Read paper in advance!
• Be prepared to present a figure to the class (in groups)
3
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© 2001 Macmillan Magazines Ltd
Goals of this lecture
• Introduce the basic concepts of fluid mechanics!
!
• Investigate examples of fluid flow in one dimension
4
The fluids and the Earth
!
• Fluid: Any material that flows in response to an applied stress!
!
• Differences between solids and fluids!
!
!
!
• Rheological (or constitutive) law: An equation relating stress to strain rates in a fluid
5
Solids Fluids
Strain from being stressed Continuous deformation under applied forces
Stresses related to strains Stresses related to rates of strain
Strain result of displacement gradients Strain result of velocity gradients
Newtonian (or linear) fluid
• A Newtonian fluid is a fluid in which there is a linear relationship between the rate of strain and the applied stress!
• What would this relationship look like as an equation?
6
Material Approximate Viscosity [Pa s]
Air 1x10
Water 1x10
Ice 1x10
Rock Salt 1x10
Granite 1x10
Newtonian (or linear) fluid
• A Newtonian fluid is a fluid in which there is a linear relationship between the rate of strain and the applied stress!
• What would this relationship look like as an equation?
!
• The proportionality constant ! is known as the dynamic (or shear) viscosity!
• Dynamic viscosity has units of Pa s
7
� / "̇ � = ⌘"̇or
• The concepts of fluid mechanics are based on conservation of!
• mass !
• momentum, and!
• energy !
• Conservation of mass, momentum and energy are combined with rheological laws to describe fluid movement under an applied force
Fluid mechanics
8
Hydrothermal fluid flow
9
Geysir in Yellowstone National Park, USA
Magma migration and flow
10
Folding of rock
11
Mantle convection
12
Fig. 1.61, Turcotte and Schubert, 2014
1D channel flows
!
!
!
!
!
• The most simple fluid flow we can consider is flow of a fluid in one direction within a channel of fixed width!
• This kind of flow might occur, for example, as a result of plates moving over the asthenosphere
13
Fig. 6.1, Turcotte and Schubert, 2014
1D channel flows
!
!
!
!
!
• Fluid is flowing with velocity " in the # direction, and the flow velocity " is a function of distance across the channel $!
• Flow results from!
• a pressure gradient (%0 - %1)/&, and/or!
• motion of the side wall of the channel "014
Fig. 6.1, Turcotte and Schubert, 2014
1D channel flows
!
!
!
!
!
• Shear, or a gradient in the velocity, in the channel results in a shear stress ' that is exerted on horizontal planes in the fluid!
• For a Newtonian fluid with a constant dynamic viscosity ! we can state
15
Fig. 6.1, Turcotte and Schubert, 2014
⌧ = ⌘du
dy
1D channel flows
!
!
!
!
!
• Shear, or a gradient in the velocity, in the channel results in a shear stress ' that is exerted on horizontal planes in the fluid!
• For a Newtonian fluid with a constant dynamic viscosity ! we can state
16
Fig. 6.1, Turcotte and Schubert, 2014
⌧ = ⌘du
dystress strain rate
1D channel flows
!
!
!
!
!
• We can now determine the flow in the channel using the equation of motion, based on the force balance on a layer of fluid of thickness ($ and length &!
• The net pressure force on the element in the # direction is
17
Fig. 6.1, Turcotte and Schubert, 2014
(p1 � p0)�y
1D channel flows
!
!
!
!
• Because the shear stress ' and velocity " are both only a function of distance $, the shear force on the upper boundary of the element is!
• The equivalent shear force on the lower boundary is
18
Fig. 6.1, Turcotte and Schubert, 2014
�⌧(y)l
⌧(y + �y)l =
✓⌧(y) +
d⌧
dy�y
◆l
1D channel flows
!
!
!
!
• The net force (or sum of the forces) must be equal to zero, or
• As ($ → 0, the relationship above becomes
19
Fig. 6.1, Turcotte and Schubert, 2014
(p1 � p0)�y +
⌧(y) +
d⌧
dy�y
�l � ⌧(y)l = 0
d⌧
dy= � (p1 � p0)
l
1D channel flows
!
!
!
!
• The right side of the previous equation is the horizontal pressure gradient in the channel
• From which the equation of motion can be written
20
Fig. 6.1, Turcotte and Schubert, 2014
dp
dx
= � (p1 � p0)
l
d⌧
dy
=dp
dx
1D channel flows
!
!
!
!
• Velocity in the channel is found by substituting the rheological law for a Newtonian fluid into the equation of motion
• Integrating the equation above twice yields
21
Fig. 6.1, Turcotte and Schubert, 2014
d⌧
dy
=d
dy
⌘
du
dy
= ⌘
d
2u
dy
2=
dp
dx
u =1
2⌘
dp
dx
y
2 + c1y + c2
1D channel flows
!
!
!
!
• The constants )1 and )2 can be found by applying the boundary conditions that " = 0 at $ = �, and " = "0 at $ = 0 (no slip)
22
Fig. 6.1, Turcotte and Schubert, 2014
u =1
2⌘
dp
dx
(y2 � hy)� u0y
h
+ u0
OK, now what does this equation tell us?
!
!
!
• We’ll now look at two simple end-member fluid flow behaviors!
• Couette flow: Zero pressure gradient (*%/*# = 0)!
• Poiseuille flow: Zero boundary velocity ("0 = 0)!
!
• What are your predictions for what these flows should look like?
23
u =1
2⌘
dp
dx
(y2 � hy)� u0y
h
+ u0
Couette flow
• A Couette flow has no pressure gradient, or *%/*# = 0, reducing the 1D equation for velocity in the channel down to
• Clearly, this predicts a linear increase in velocity from $ = � to $ = 0
24
u = u0
⇣1� y
h
⌘
Poiseuille flow
• Poiseuille flow is driven only by a pressure gradient in the channel with zero boundary velocities ("0 = 0), thus
• In a coordinate system with $ʹ at the middle of the channel we can say $ʹ = $ - �/2, which results in the relationship
25
u =1
2⌘
dp
dx
(y2 � hy)
u =1
2⌘
dp
dx
✓y
02 � h
2
4
◆
Application: Salt tectonics
!
!
!
• Rock salt is a common rock type in sedimentary basins that is nearly incompressible and can be modeled as a Newtonian fluid with a low viscosity (! = ~1018 Pa s; it flows readily)!
• This flow and tendency of salt to migrate when loaded by sedimentary overburden is the focus of studies of salt tectonics!
• Let’s now consider a simple set of experiments for a 2D continental passive margin overlying a layer of rock salt
26
and initial velocity analytically and in the section‘Compar-ison of analytical and numerical results’ we compare theseresults with those from the numerical models.
Thin sheet approximation of the stabilityanalysis
Lehner (2000) uses local balance of stresses to predict in-itial deformation styles of systems with a viscous substrateoverlain by frictional-plastic sediments of laterally varyingthickness. In this section, we re-derive the Lehner (2000)stability criterion using balance of the horizontal bulkforces that act on the transition zonewhere the overburdenis thinning.We consider vertical plane-strain initial geo-metries, like those of Fig.1, inwhich the base is horizontaland the linear viscous layer has a uniform thickness be-neath a variable thickness frictional-plastic overburden.No consideration is given to theway inwhich the geometrywas created or to the ¢nite deformation. Consider the hor-izontal force balance of the overburden transition zoneoutlined by the thick line (Fig. 2). The upper surface isstress free and forces F1 and F2 result from the verticallyintegrated horizontal stresses in the frictional overburden.The di¡erential overburden load also induces a Poiseuille£ow in the viscous layer, which produces shear traction onthe base of the overburden resulting in the horizontalforce Fp.The overburden is stable against outward £ow inthe downdip direction when
F1 þ F2 þ Fp < 0 ð1Þusing the sign convention that forces directed to the rightare positive. In this case the forces, F1 and F2, are below
their respective extensional and contractional yield values.By introducing the yieldvalues ofF1andF2, Eqn. (1) can berewritten as the stability condition for outward £ow of theoverburden
F1e þ F2c þ Fp < 0 system is stable ð2aÞ
F1e þ F2c þ Fp < 0 system is unstable ð2bÞ
where F1e and F2c are the forces that result from limitingextensional and contractional horizontal stresses in theplastic overburden, above locations x1 and x2. The Mohr^Coulomb criterion is used to represent the cohesionlessfrictional-plastic behaviour of the overburden
txz ¼ szz % sxx ¼ &ðsxx þ szzÞ sinf ð3Þ
txz is the shear stress andf is the internal angle of friction.To estimate these limiting horizontal forces, we assumethat the principal stresses in the overburden are horizontal(sxx) andvertical (szz) and that szz is equal to the lithostaticpressure (small angle approach, Dahlen,1990), which yields
sxx ¼ %rgðhc þ hðxÞ % zÞ ð1& sinfÞð1& sinfÞ
ð4Þ
where r is the density, g is the gravitational accelerationand (hc1h(x)% z) is the depth.The resulting forces are
F1e ¼ %Z
h1
minðsxxÞ dz ¼12rg h21
ð1% sinfÞð1þ sinfÞ ð5aÞ
F2c ¼ %Z
h2
maxðsxxÞ dz ¼12rg h22
ð1þ sinfÞð1% sinfÞ
ð5bÞ
(note the di¡erent signs in the force expressions due to theopposite orientations of the outward normal vectors to thetwo surfaces onwhich the forces are acting).
To estimate the basal traction force, Fp, we assume thatthe topography changes slowly with position and thereforethat slopes are small.The thin sheet approximation (Lob-kovsky & Kerchman, 1991) then gives the distribution ofhorizontal velocities, vp, in the viscous substratum subjectto variations of lithostatic pressure as
vp ¼ % rg2Z
@hðxÞ@x
zðhc % zÞ ð6Þ
Fig.1. Deformation styles in systems where a frictional-plasticoverburden (white) of varying thickness overlies a viscoussubstratum (grey). (a) Stable overburden. A pressure-drivenPoiseuille £ow in the viscous channel dominates deformation.(b) Unstable overburden.The Couette £ow-induced overburdenvelocities are smaller than the Poiseuille £ow velocities in theviscous channel. (c) Unstable overburden. Couette £owdominates the deformation pattern.
Fig. 2. Horizontal forces acting on the overburden transitionzone (outlined by a thick solid line) when the model is stable.F1 and F2 are forces related to the horizontal stresses within theoverburden. Fp is the traction force caused by the Poiseuille £owin the viscous channel. h1 and h2 are the updip and downdipoverburden thicknesses and hc is the thickness of the salt.r is thedensity,f is the internal angle of friction and Z is the viscosity.
r 2004 Blackwell Publishing Ltd,Basin Research, 16, 199^218 201
Salt tectonics driven by differential sediment loading
Gemmer et al., 2004
Application: Salt tectonics
!
!
!
• In this scenario, we might expect different flow behavior in the rock salt depending on the deformation of the overlying sediment!
• We predict Poiseuille flow when the sedimentary overburden is stable and does not move horizontally
27
and initial velocity analytically and in the section‘Compar-ison of analytical and numerical results’ we compare theseresults with those from the numerical models.
Thin sheet approximation of the stabilityanalysis
Lehner (2000) uses local balance of stresses to predict in-itial deformation styles of systems with a viscous substrateoverlain by frictional-plastic sediments of laterally varyingthickness. In this section, we re-derive the Lehner (2000)stability criterion using balance of the horizontal bulkforces that act on the transition zonewhere the overburdenis thinning.We consider vertical plane-strain initial geo-metries, like those of Fig.1, inwhich the base is horizontaland the linear viscous layer has a uniform thickness be-neath a variable thickness frictional-plastic overburden.No consideration is given to theway inwhich the geometrywas created or to the ¢nite deformation. Consider the hor-izontal force balance of the overburden transition zoneoutlined by the thick line (Fig. 2). The upper surface isstress free and forces F1 and F2 result from the verticallyintegrated horizontal stresses in the frictional overburden.The di¡erential overburden load also induces a Poiseuille£ow in the viscous layer, which produces shear traction onthe base of the overburden resulting in the horizontalforce Fp.The overburden is stable against outward £ow inthe downdip direction when
F1 þ F2 þ Fp < 0 ð1Þusing the sign convention that forces directed to the rightare positive. In this case the forces, F1 and F2, are below
their respective extensional and contractional yield values.By introducing the yieldvalues ofF1andF2, Eqn. (1) can berewritten as the stability condition for outward £ow of theoverburden
F1e þ F2c þ Fp < 0 system is stable ð2aÞ
F1e þ F2c þ Fp < 0 system is unstable ð2bÞ
where F1e and F2c are the forces that result from limitingextensional and contractional horizontal stresses in theplastic overburden, above locations x1 and x2. The Mohr^Coulomb criterion is used to represent the cohesionlessfrictional-plastic behaviour of the overburden
txz ¼ szz % sxx ¼ &ðsxx þ szzÞ sinf ð3Þ
txz is the shear stress andf is the internal angle of friction.To estimate these limiting horizontal forces, we assumethat the principal stresses in the overburden are horizontal(sxx) andvertical (szz) and that szz is equal to the lithostaticpressure (small angle approach, Dahlen,1990), which yields
sxx ¼ %rgðhc þ hðxÞ % zÞ ð1& sinfÞð1& sinfÞ
ð4Þ
where r is the density, g is the gravitational accelerationand (hc1h(x)% z) is the depth.The resulting forces are
F1e ¼ %Z
h1
minðsxxÞ dz ¼12rg h21
ð1% sinfÞð1þ sinfÞ ð5aÞ
F2c ¼ %Z
h2
maxðsxxÞ dz ¼12rg h22
ð1þ sinfÞð1% sinfÞ
ð5bÞ
(note the di¡erent signs in the force expressions due to theopposite orientations of the outward normal vectors to thetwo surfaces onwhich the forces are acting).
To estimate the basal traction force, Fp, we assume thatthe topography changes slowly with position and thereforethat slopes are small.The thin sheet approximation (Lob-kovsky & Kerchman, 1991) then gives the distribution ofhorizontal velocities, vp, in the viscous substratum subjectto variations of lithostatic pressure as
vp ¼ % rg2Z
@hðxÞ@x
zðhc % zÞ ð6Þ
Fig.1. Deformation styles in systems where a frictional-plasticoverburden (white) of varying thickness overlies a viscoussubstratum (grey). (a) Stable overburden. A pressure-drivenPoiseuille £ow in the viscous channel dominates deformation.(b) Unstable overburden.The Couette £ow-induced overburdenvelocities are smaller than the Poiseuille £ow velocities in theviscous channel. (c) Unstable overburden. Couette £owdominates the deformation pattern.
Fig. 2. Horizontal forces acting on the overburden transitionzone (outlined by a thick solid line) when the model is stable.F1 and F2 are forces related to the horizontal stresses within theoverburden. Fp is the traction force caused by the Poiseuille £owin the viscous channel. h1 and h2 are the updip and downdipoverburden thicknesses and hc is the thickness of the salt.r is thedensity,f is the internal angle of friction and Z is the viscosity.
r 2004 Blackwell Publishing Ltd,Basin Research, 16, 199^218 201
Salt tectonics driven by differential sediment loading
Gemmer et al., 2004
Application: Salt tectonics
!
!
!
• When the sedimentary overburden cannot support the lateral stress due to variations in its thickness, failure will occur in the sediments, leading to horizontal translation of the sediment!
• This produces a dominantly Couette-type of flow in the salt
28
and initial velocity analytically and in the section‘Compar-ison of analytical and numerical results’ we compare theseresults with those from the numerical models.
Thin sheet approximation of the stabilityanalysis
Lehner (2000) uses local balance of stresses to predict in-itial deformation styles of systems with a viscous substrateoverlain by frictional-plastic sediments of laterally varyingthickness. In this section, we re-derive the Lehner (2000)stability criterion using balance of the horizontal bulkforces that act on the transition zonewhere the overburdenis thinning.We consider vertical plane-strain initial geo-metries, like those of Fig.1, inwhich the base is horizontaland the linear viscous layer has a uniform thickness be-neath a variable thickness frictional-plastic overburden.No consideration is given to theway inwhich the geometrywas created or to the ¢nite deformation. Consider the hor-izontal force balance of the overburden transition zoneoutlined by the thick line (Fig. 2). The upper surface isstress free and forces F1 and F2 result from the verticallyintegrated horizontal stresses in the frictional overburden.The di¡erential overburden load also induces a Poiseuille£ow in the viscous layer, which produces shear traction onthe base of the overburden resulting in the horizontalforce Fp.The overburden is stable against outward £ow inthe downdip direction when
F1 þ F2 þ Fp < 0 ð1Þusing the sign convention that forces directed to the rightare positive. In this case the forces, F1 and F2, are below
their respective extensional and contractional yield values.By introducing the yieldvalues ofF1andF2, Eqn. (1) can berewritten as the stability condition for outward £ow of theoverburden
F1e þ F2c þ Fp < 0 system is stable ð2aÞ
F1e þ F2c þ Fp < 0 system is unstable ð2bÞ
where F1e and F2c are the forces that result from limitingextensional and contractional horizontal stresses in theplastic overburden, above locations x1 and x2. The Mohr^Coulomb criterion is used to represent the cohesionlessfrictional-plastic behaviour of the overburden
txz ¼ szz % sxx ¼ &ðsxx þ szzÞ sinf ð3Þ
txz is the shear stress andf is the internal angle of friction.To estimate these limiting horizontal forces, we assumethat the principal stresses in the overburden are horizontal(sxx) andvertical (szz) and that szz is equal to the lithostaticpressure (small angle approach, Dahlen,1990), which yields
sxx ¼ %rgðhc þ hðxÞ % zÞ ð1& sinfÞð1& sinfÞ
ð4Þ
where r is the density, g is the gravitational accelerationand (hc1h(x)% z) is the depth.The resulting forces are
F1e ¼ %Z
h1
minðsxxÞ dz ¼12rg h21
ð1% sinfÞð1þ sinfÞ ð5aÞ
F2c ¼ %Z
h2
maxðsxxÞ dz ¼12rg h22
ð1þ sinfÞð1% sinfÞ
ð5bÞ
(note the di¡erent signs in the force expressions due to theopposite orientations of the outward normal vectors to thetwo surfaces onwhich the forces are acting).
To estimate the basal traction force, Fp, we assume thatthe topography changes slowly with position and thereforethat slopes are small.The thin sheet approximation (Lob-kovsky & Kerchman, 1991) then gives the distribution ofhorizontal velocities, vp, in the viscous substratum subjectto variations of lithostatic pressure as
vp ¼ % rg2Z
@hðxÞ@x
zðhc % zÞ ð6Þ
Fig.1. Deformation styles in systems where a frictional-plasticoverburden (white) of varying thickness overlies a viscoussubstratum (grey). (a) Stable overburden. A pressure-drivenPoiseuille £ow in the viscous channel dominates deformation.(b) Unstable overburden.The Couette £ow-induced overburdenvelocities are smaller than the Poiseuille £ow velocities in theviscous channel. (c) Unstable overburden. Couette £owdominates the deformation pattern.
Fig. 2. Horizontal forces acting on the overburden transitionzone (outlined by a thick solid line) when the model is stable.F1 and F2 are forces related to the horizontal stresses within theoverburden. Fp is the traction force caused by the Poiseuille £owin the viscous channel. h1 and h2 are the updip and downdipoverburden thicknesses and hc is the thickness of the salt.r is thedensity,f is the internal angle of friction and Z is the viscosity.
r 2004 Blackwell Publishing Ltd,Basin Research, 16, 199^218 201
Salt tectonics driven by differential sediment loading
Gemmer et al., 2004
Application: Salt tectonics
!
!
!
• In nature, it is likely that salt flows in this type of environment involve both Couette and Poiseuille components, resulting in a velocity field that is a combination of both
29and initial velocity analytically and in the section‘Compar-ison of analytical and numerical results’ we compare theseresults with those from the numerical models.
Thin sheet approximation of the stabilityanalysis
Lehner (2000) uses local balance of stresses to predict in-itial deformation styles of systems with a viscous substrateoverlain by frictional-plastic sediments of laterally varyingthickness. In this section, we re-derive the Lehner (2000)stability criterion using balance of the horizontal bulkforces that act on the transition zonewhere the overburdenis thinning.We consider vertical plane-strain initial geo-metries, like those of Fig.1, inwhich the base is horizontaland the linear viscous layer has a uniform thickness be-neath a variable thickness frictional-plastic overburden.No consideration is given to theway inwhich the geometrywas created or to the ¢nite deformation. Consider the hor-izontal force balance of the overburden transition zoneoutlined by the thick line (Fig. 2). The upper surface isstress free and forces F1 and F2 result from the verticallyintegrated horizontal stresses in the frictional overburden.The di¡erential overburden load also induces a Poiseuille£ow in the viscous layer, which produces shear traction onthe base of the overburden resulting in the horizontalforce Fp.The overburden is stable against outward £ow inthe downdip direction when
F1 þ F2 þ Fp < 0 ð1Þusing the sign convention that forces directed to the rightare positive. In this case the forces, F1 and F2, are below
their respective extensional and contractional yield values.By introducing the yieldvalues ofF1andF2, Eqn. (1) can berewritten as the stability condition for outward £ow of theoverburden
F1e þ F2c þ Fp < 0 system is stable ð2aÞ
F1e þ F2c þ Fp < 0 system is unstable ð2bÞ
where F1e and F2c are the forces that result from limitingextensional and contractional horizontal stresses in theplastic overburden, above locations x1 and x2. The Mohr^Coulomb criterion is used to represent the cohesionlessfrictional-plastic behaviour of the overburden
txz ¼ szz % sxx ¼ &ðsxx þ szzÞ sinf ð3Þ
txz is the shear stress andf is the internal angle of friction.To estimate these limiting horizontal forces, we assumethat the principal stresses in the overburden are horizontal(sxx) andvertical (szz) and that szz is equal to the lithostaticpressure (small angle approach, Dahlen,1990), which yields
sxx ¼ %rgðhc þ hðxÞ % zÞ ð1& sinfÞð1& sinfÞ
ð4Þ
where r is the density, g is the gravitational accelerationand (hc1h(x)% z) is the depth.The resulting forces are
F1e ¼ %Z
h1
minðsxxÞ dz ¼12rg h21
ð1% sinfÞð1þ sinfÞ ð5aÞ
F2c ¼ %Z
h2
maxðsxxÞ dz ¼12rg h22
ð1þ sinfÞð1% sinfÞ
ð5bÞ
(note the di¡erent signs in the force expressions due to theopposite orientations of the outward normal vectors to thetwo surfaces onwhich the forces are acting).
To estimate the basal traction force, Fp, we assume thatthe topography changes slowly with position and thereforethat slopes are small.The thin sheet approximation (Lob-kovsky & Kerchman, 1991) then gives the distribution ofhorizontal velocities, vp, in the viscous substratum subjectto variations of lithostatic pressure as
vp ¼ % rg2Z
@hðxÞ@x
zðhc % zÞ ð6Þ
Fig.1. Deformation styles in systems where a frictional-plasticoverburden (white) of varying thickness overlies a viscoussubstratum (grey). (a) Stable overburden. A pressure-drivenPoiseuille £ow in the viscous channel dominates deformation.(b) Unstable overburden.The Couette £ow-induced overburdenvelocities are smaller than the Poiseuille £ow velocities in theviscous channel. (c) Unstable overburden. Couette £owdominates the deformation pattern.
Fig. 2. Horizontal forces acting on the overburden transitionzone (outlined by a thick solid line) when the model is stable.F1 and F2 are forces related to the horizontal stresses within theoverburden. Fp is the traction force caused by the Poiseuille £owin the viscous channel. h1 and h2 are the updip and downdipoverburden thicknesses and hc is the thickness of the salt.r is thedensity,f is the internal angle of friction and Z is the viscosity.
r 2004 Blackwell Publishing Ltd,Basin Research, 16, 199^218 201
Salt tectonics driven by differential sediment loading
Gemmer et al., 2004
Application: Salt tectonics
30Gemmer et al., 2004
thick. For this model the basal traction caused by the £ow-ing viscous material is su⁄cient to cause overburdenyielding, and the system is characterised by a combinationof Poiseuille and Couette £ow. Figure 5c shows a modelwith a thin (1.5 km) downdip overburden. In this case, theCouette velocity caused by the unstable, moving overbur-den is signi¢cantly greater than the Poiseuille velocity anda linear velocity pro¢le develops in the viscous layer.Thusthe numerical model results conform to the conceptual£ow regimes illustrated in Fig.1.
COMPARISON OFANALYTICAL ANDNUMERICAL RESULTSStability results
The stability criterion de¢ned byEqn. (8) is shown as a so-lid curve (Fig. 6) as a function of the internal angle of fric-tion f and the downdip overburden thickness, h!2, for aconstant value of the updip overburden thickness,h!1 ¼ 4:5. For a given internal angle of friction, f a mini-mum value of h!2 is needed for the overburden to remainstable. For progressively higher internal angles of friction,the overburden strength increases and the h!2needed tokeep the overburden stable decreases.To test the response
of the ¢nite-element model against the stability criterion,we examine sets of models inwhich h!1 is held constant, andh!2 is varied for a given overburden strength f. Numericalmodel sets of this type span 514f4501 (Fig. 6) to yield asuite of model results for comparison with the non-di-mensional analytical stability criterion. The numericalmodel results are converted to the non-dimensional formusing hc and k as de¢ned in Eqn. (8).The results are codedaccording to the initial velocity pattern predicted for the¢rst 10 time steps (before signi¢cant changes in the geo-metry occur) (Fig. 6). The overall results show a goodagreement between the ¢nite-element models and theanalytically predicted stability criterion. This indicatesthat the numerical model is capable of calculating stressesand overburden stability associated with £ows caused bydi¡erential loading. It is not possible to determine the ab-solute accuracy of the numerical results from the compar-ison of the numerical model and the analytical predictionsbecause the analytical theory is itself approximate.
Initial velocities
A suite of similar numerical models was compared withthe analytical Couette velocity prediction of Eqn. (12). Forthese models, h!1 ¼ 4:5 and the internal angle of friction
Fig. 5. Velocities predicted by the numerical model. Arrows represent horizontal velocity vectors, the magnitude ofwhich is relative tothe scale in each frame. Light grey: Frictional-plastic overburden. Dark grey: viscous substratum. (a) Poiseuille-dominated £ow.(b) Combined Poiseuille and Couette £ow. (c) Couette-dominated £ow. Note in all three cases that £ow is restricted to the transitionzone.
r 2004 Blackwell Publishing Ltd,Basin Research, 16, 199^218 205
Salt tectonics driven by differential sediment loading
Numerical model predictions for variable sediment strength
Asthenospheric counterflow
• One model for mantle flow is that the motion of lithospheric plates on the Earth’s surface produces a counterflow in the uppermost asthenosphere (upper ~100-200 km)!
• If we assume the plate is rigid and moving at velocity "0, and that the velocity at some depth $ = � must be zero, it is clear that the counterflow is opposite in direction to the plate motion in order to conserve mass
31
Asthenospheric counterflow
• Mathematically, we can state that aswhere �, is the thickness of the lithosphere and � is the thickness of the asthenosphere!
• If we insert our equation for 1D channel flow in the second term, we get
32
u0hL +
Z h
0u dy = 0
u0hL +h
3
12⌘
dp
dx
+u0h
2= 0
Asthenospheric counterflow
• If we now solve for the pressure gradient, we find
• And this can be inserted into the equation for 1D channel flow to get the predicted velocity profile for a counterflow
33
dp
dx
=12⌘u0
h
2
✓hL
h
+1
2
◆
u = u0
1� y
h+ 6
✓hL
h+
1
2
◆✓y2
h2� y
h
◆�
Asthenospheric counterflow
!
!
• Looking at this equation for a moment, what strikes you as perhaps somewhat surprising?
• Is there anything missing that you might expect to see?
34
u = u0
1� y
h+ 6
✓hL
h+
1
2
◆✓y2
h2� y
h
◆�
Kinematic viscosity and the Prandtl number
• We’ve dealt thus far with the dynamic viscosity !, but a similar quantity, the kinematic viscosity -, can be quite useful
• One reason this is useful is the units. - has units of m2 s-1, meaning it can be thought of as a diffusivity for momentum, much like ., the thermal diffusivity
35
⌫ =⌘
⇢
Kinematic viscosity and the Prandtl number
!
!
!
!
• The Prandtl number is the ratio of the kinematic viscosity to the thermal diffusivity, giving us a relationship between thermal diffusion and diffusion of momentum
• A fluid with a large Prandtl number will diffuse momentum more quickly than heat and the opposite is true for a fluid with a small Prandtl number
36
Pr ⌘ ⌫
414 Fluid Mechanics
Table 6.1 Transport Properties of Some Common Fluids at 15◦C andAtmospheric Pressure
Kinematic ThermalViscosity µ Viscosity ν Diffusivity κ Prandtl(Pa s) (m2 s−1) (m2 s−1) Number Pr
Air 1.78 × 10−5 1.45 × 10−5 2.02 × 10−5 0.72Water 1.14 × 10−3 1.14 × 10−6 1.40 × 10−7 8.1Mercury 1.58 × 10−3 1.16 × 10−7 4.2 × 10−6 0.028Ethyl alcohol 1.34 × 10−3 1.70 × 10−6 9.9 × 10−8 17.2Carbon tetrachloride 1.04 × 10−3 6.5 × 10−7 8.4 × 10−8 7.7Olive oil 0.099 1.08 × 10−4 9.2 × 10−8 1,170Glycerine 2.33 1.85 × 10−3 9.8 × 10−8 18,880
Newtonian fluid is the constant of proportionality between shear stress andstrain rate or velocity gradient. The more viscous the fluid, the larger thestress required to produce a given shear.
The viscosities of some common fluids are listed in Table 6–1. The SI unitof viscosity is the Pascal second (Pa s). The ratio µ/ρ (ρ is the density ofthe fluid) occurs frequently in fluid mechanics. It is known as the kinematicviscosity ν of a fluid
ν =µ
ρ. (6.2)
The quantity µ is the dynamic viscosity. The SI unit of kinematic viscosityis square meter per second (m2 s−1). The kinematic viscosity is a diffusivity,similar to the thermal diffusivity κ. While κ describes how heat diffuses bymolecular collisions, ν describes how momentum diffuses. The ratio of ν toκ is a dimensionless quantity known as the Prandtl number, Pr
Pr ≡ν
κ. (6.3)
A fluid with a small Prandtl number diffuses heat more rapidly than it doesmomentum; the reverse is true for a fluid with a large value of Pr. Table6–1 also lists the kinematic viscosities, thermal diffusivities, and Prandtlnumbers of a variety of fluids.
The flow in the channel in Figure 6–1 is determined by the equation ofmotion. This is a mathematical statement of the force balance on a layerof fluid of thickness δy and horizontal length l (see Figure 6–1). The netpressure force on the element in the x direction is
(p1 − p0) δy.
Hydraulic head
• Pressure drops in channels are often related to a hydraulic head
• The hydraulic head is the thickness (or height) of a fluid required to generate a hydrostatic pressure equal to %1 - %0, the pressure drop along the channel
37
H ⌘ (p1 � p0)
⇢g
Recap
• Fluid mechanics describes fluid motion based on the conservation of mass, momentum and energy!
!
• 1D channel flows can be divided into two fundamental types that relate to the conditions in the channel!
• Couette flow: Linear velocity gradient across channel, pressure gradient very small!
• Poiseuille flow: Parabolic velocity profile across channel, minimal displacement of channel walls
38
References
Gemmer, L., Ings, S. J., Medvedev, S., & Beaumont, C. (2004). Salt tectonics driven by differential sediment loading: stability analysis and finite-element experiments. Basin Research, 16(2), 199–218.
39