hillslopes l05 continuummechanics p4 conservation laws

Upload: benjamin-ledesma

Post on 14-Apr-2018

220 views

Category:

Documents


0 download

TRANSCRIPT

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    1/23

    9/25/2013

    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, 2013. Add good,physical examples of how these conceptsare applied. The best of these will be usedin future course offerings.

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    2/23

    Conservation of Mass

    9/25/2013

    2

    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.

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    3/23

    Conservation of Mass

    9/25/2013

    3

    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.

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    4/23

    Conservation of Mass

    9/25/2013

    4

    Qm = Qm =

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    5/23

    Conservation of Mass

    9/25/2013

    5

    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]

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    6/23

    Conservation of Mass

    9/25/2013

    6

    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

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    7/23

    Conservation of Mass

    9/25/2013

    7

    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)

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    8/23

    9/25/20138

    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.

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    9/23

    9/25/20139

    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

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    10/23

    9/25/201310

    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)

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    11/23

    Divergence of Sediment Transport Rate

    z

    Downslope

    dq s

    dx = 0

    z

    Downslope

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    12/23

    z

    Downslope

    Must erode

    dq s

    dx > 0

    z

    Downslope

    Divergence of Sediment Transport Rate

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    13/23

    z

    Downslope

    dq s

    dx < 0

    z

    Downslope

    Must deposit

    Divergence of Sediment Transport Rate

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    14/23

    How does this differ for a bedrock landscape?

    Soil mantledlandscape

    Bedrock landscape

    Bill Dietrich

    Bill Dietrich

    = P -dh

    dt q s

    = -dhdt

    q s

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    15/23

    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:

    9/25/201315

    Conservation of linear momentum

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    16/23

    9/25/2013

    16

    Conservation of linear momentum

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    17/23

    9/25/2013

    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 :

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    18/23

    9/25/201318

    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

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    19/23

    the time rate of change of momentum in the x-direction can be written as

    9/25/201319

    Conservation of linear momentum

    Fluxes

    Surface

    Body

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    20/23

    9/25/201320

    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

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    21/23

    9/25/201321

    The same in the y and z directions:

    Conservation of linear momentum

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    22/23

    9/25/201322

    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

  • 7/27/2019 Hillslopes L05 ContinuumMechanics p4 Conservation Laws

    23/23

    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

    9/25/2013