hillslopes l05 continuummechanics p4 conservation laws
TRANSCRIPT
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Continuum Mechanics for Hillslopes:Part IV
Focus on conservation laws
Homework: Translate and improve one of the lectures
that has already been given based on the
reading by Major, 2013. Add good,physical examples of how these conceptsare applied. The best of these will be usedin future course offerings.
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Conservation of Mass
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For a stationary elemental volume (one of fixed size having a position fixed in space),The rate of mass accumulating within the volume
is equal to(1) the mass flux into the volume
minus(2) the mass flux out of the volume.
We shall assume that no mass is generated (or consumed) within the volume
(i.e., we ignore relativistic effects);Hence, the only way to have mass accumulate in the volume is to have more flow in than out.
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Conservation of Mass
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For a stationary elemental volume (one of fixed size having a position
fixed in space),
The rate of mass accumulating within the volume
is equal to
the mass flux into the volume minus
the mass flux out of the volume.
We shall assume that no mass is generated (or consumed)
within the volume(i.e., we ignore relativistic effects);
Hence, the only way to have massaccumulate in the volume is tohave it flow across the volume boundary.
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Conservation of Mass
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Qm = Qm =
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Conservation of Mass
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Because the volume is stationary , we can recast this as:
The change in the volume is equal tothe total mass flux through all the faces.
Changein mass
[Inputs Outputs]
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Conservation of Mass
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Dividing through by x y z and taking the limit as thesedimensions go to zero (creating the derivative) yields theContinuity Equation.
Or: the rate of change of density within an elemental volumefixed in space is equal to the net rate of mass flux across its boundariesdivided by its volume
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The negative sign in front of the right-hand side of the equationindicates that if the net gradient of mass flux decreasesalong the coordinate directions within the volume,then the change of density with time is positive(because mass accumulates).
If, however,the net gradient of mass flux increases along the coordinate directions within the volume, the change of density with time is negative(because mass is lost).
This assumes that the volume is stationary ( Eulerian perspective)
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Conservation of Mass
If we allow the reference volume to float along with the mediumthat is transporting mass across the volume boundary,this is the Lagrangian perspective. This is know as theconvective rate of change.
The term on the left-hand side describes rate changes of density detected from the perspective of an observer floating along with themotion of the medium that is transporting the mass.
The first term describes the rate of change observed from general variations with time , and the following three terms describe anadditional change related to any spatial gradients of density.
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This combination of the temporal and spatial gradients is called thesubstantial or material derivative and is designated by a capital D.
Allowing:
To be rewritten as:
Conservation of Mass
Divergence in velocity field
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Conservation of Mass
If incompressible: THUS,
Our earlier expressions can then be written using theDivergence operator which takes the derivative in all directions.
(Eulerian perspective)
(Lagrangian perspective)
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Divergence of Sediment Transport Rate
z
Downslope
dq s
dx = 0
z
Downslope
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z
Downslope
Must erode
dq s
dx > 0
z
Downslope
Divergence of Sediment Transport Rate
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z
Downslope
dq s
dx < 0
z
Downslope
Must deposit
Divergence of Sediment Transport Rate
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How does this differ for a bedrock landscape?
Soil mantledlandscape
Bedrock landscape
Bill Dietrich
Bill Dietrich
= P -dh
dt q s
= -dhdt
q s
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The time rate of change of momentumin an elemental control volume equals
the flow of momentum into the control volumeminus
the flow of momentum leaving the volumeplus
the sum of the forces acting on the volume.
Based on:
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Conservation of linear momentum
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Conservation of linear momentum
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Conservation of linear momentum
the net convective flux of the x-direction component of momentum
across all six faces of the volume boundary is :
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the rate of change of momentum is also affected by the forces actingon the volume , of which there are two types:
body forces that affect all parts of the volume equally andsurface forces related to the stresses acting on the volume.
the sum of thex-direction components of these forces can be written as:
(the negative terms result from positively defined forces acting in the
negative x-direction, compression)
Conservation of linear momentum
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the time rate of change of momentum in the x-direction can be written as
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Conservation of linear momentum
Fluxes
Surface
Body
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If we assume the spatial dimensions of the elemental volume are constant , then we can divide the previous largeequation by x y z, take the limit as each of those dimensionsgo to zero (the derivative), and write the x-directioncomponent of the conservation of momentum as:
Conservation of linear momentum
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The same in the y and z directions:
Conservation of linear momentum
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These three components of momentum conservation in a morecompact manner using vector and tensor notation.
in whichu represents the velocity vector field,
the stress tensor, andg the vector field for acceleration of gravity.
Conservation of linear momentum
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Constitutive Relations
Linearly Viscous FluidLinearly Elastic Material Relationships between stress and normal
strain Relationships between shear stress and
shear strain
Relationship between pressure anddilatationPlasticity the Coulomb Failure Rule
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