1. 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, Add good, physical examples of how these concepts are applied. The best of these will be used in future course offerings.
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2. Conservation of Mass 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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3. Conservation of Mass 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 mass accumulate in the volume is to have it flow across the volume boundary.
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4. Conservation of Mass
Qm =
Qm =
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5. Conservation of Mass The change in the volume is equal to
the total mass flux through all the faces.
[Inputs – Outputs]
Change
in mass
Because the volume is stationary, we can recast this as:
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6. Conservation of Mass
Dividing through by δxδyδz and taking the limit as these
dimensions go to zero (creating the derivative) yields the
“Continuity Equation.”
Or: the rate of change of density within an elemental volume
fixed in space is equal to the net rate of mass flux across its boundaries divided by its volume
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7. Conservation of Mass
The negative sign in front of the right-hand side of the equation
indicates that if the net gradient of mass flux decreases
along 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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8. Conservation of Mass
If we allow the reference volume to float along with the medium that is transporting mass across the volume boundary,
this is the Lagrangian perspective. This is know as the convective 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 the motion 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 an additional change related to any spatial gradients of density.
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9. Conservation of Mass
This combination of the temporal and spatial gradients is called the ‘substantial’ or ‘material’ derivative and is designated by a capital D.
Allowing:
To be rewritten as:
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Divergence in velocity field

10. Conservation of Mass
Our earlier expressions can then be written using the
‘Divergence operator’ which takes the derivative in all directions.
(Eulerian perspective)
(Lagrangian perspective)
If incompressible: THUS,
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11. Divergence of Sediment Transport Rate
dqs dx
= 0 z
Downslope z
Downslope

12. Divergence of Sediment Transport Rate
dqs dx
> 0 z
Downslope
Must erode z
Downslope

13. Divergence of Sediment Transport Rate
dqs dx
< 0 z
Downslope
Must deposit z
Downslope

14. How does this differ for a bedrock landscape?
Bill Dietrich
Bedrock landscape
= P - dh dt ∙∆ qs
Soil mantled landscape
= - dh dt ∙∆ qs
Bill Dietrich

15. Conservation of linear momentum
The time rate of change of momentum in an elemental control volume
equals
the flow of momentum into the control volume
minus
the flow of momentum leaving the volume
plus
the sum of the forces acting on the volume.
Based on:
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16. Conservation of linear momentum
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17. 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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18. Conservation of linear momentum
the rate of change of momentum is also affected by the forces acting on the volume, of which there are two types:
body forces that affect all parts of the volume equally
and
surface forces related to the stresses acting on the volume.
the sum of the
x-direction components of these forces can be written as:
(the negative terms result from positively defined forces acting in the negative x-direction, compression)
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19. Conservation of linear momentum
the time rate of change of momentum in the x-direction can be written as
Fluxes
Surface
Body
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20. Conservation of linear momentum
If we assume the spatial dimensions of the elemental volume are constant, then we can divide the previous large equation by δxδyδz, take the limit as each of those dimensions go to zero (the derivative), and write the x-direction component of the conservation of momentum as:
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21. Conservation of linear momentum
The same in the y and z directions:
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22. Conservation of linear momentum
These three components of momentum conservation in a more compact manner using vector and tensor notation.
in which
u represents the velocity vector field,
the stress tensor, and
g the vector field for acceleration of gravity.
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23. Constitutive Relations
Linearly Viscous Fluid
Linearly Elastic Material
Relationships between stress and normal strain
Relationships between shear stress and shear strain
Relationship between pressure and dilatation
Plasticity – the Coulomb Failure Rule
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