# Difference between revisions of "Stokes equations"

From Interactive System for Ice sheet Simulation

(New page: ===Stokes Equations=== In Cartesian coordinates, the Stokes equations are given by, <math>\begin{align} & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+...) |
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===Stokes Equations=== | ===Stokes Equations=== | ||

− | In Cartesian coordinates, | + | The conservation equations that describe the balance of momentum for an ice sheet (assuming that accelerations are negligible) are the non-linear Stokes equations. In Cartesian coordinates, these are given by, |

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− | where ''P'' is the pressure and <big>τ</big> is the deviatoric stress tensor. The latter is given by | + | where ''P'' is the pressure and ''<big>τ</big>'' is the deviatoric stress tensor. The latter is given by |

− | + | <math>\quad\tau _{ij}=\sigma _{ij}+P\delta _{ij}</math>, | |

− | where <big>σ</big> is the full stress tensor and <big><math>\delta _{ij}</math></big> is the Kronecker delta (or identity matrix). | + | where ''<big>σ</big>'' is the full stress tensor and ''<big><math>\delta _{ij}</math></big>'' is the Kronecker delta (or identity matrix). |

+ | |||

+ | The "nonlinearity" is not obvious here, but comes in when we invoke a constitutive law that links stresses and strain rates. A standard Newtonian (constant viscosity) constitutive law for ice looks like | ||

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+ | <math>\begin{align}\quad \tau _{ij}=2\eta \dot{\varepsilon }_{ij} \\ | ||

+ | \end{align}</math> | ||

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+ | where ''η'' is the viscosity. We can write the constitutive law the same way, but define the viscosity instead as an "effective" viscosity such that | ||

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+ | <math>\begin{align}\quad \eta \equiv \frac{1}{2}B\dot{\varepsilon }_{e}^{\frac{1-n}{n}} \\\end{align}</math> |

## Revision as of 12:30, 3 August 2009

### Stokes Equations

The conservation equations that describe the balance of momentum for an ice sheet (assuming that accelerations are negligible) are the non-linear Stokes equations. In Cartesian coordinates, these are given by,

,

where *P* is the pressure and *τ* is the deviatoric stress tensor. The latter is given by

,

where *σ* is the full stress tensor and is the Kronecker delta (or identity matrix).

The "nonlinearity" is not obvious here, but comes in when we invoke a constitutive law that links stresses and strain rates. A standard Newtonian (constant viscosity) constitutive law for ice looks like

where *η* is the viscosity. We can write the constitutive law the same way, but define the viscosity instead as an "effective" viscosity such that