Difference between revisions of "Stokes equations"

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(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 Stokes equations are given by,
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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>&tau;</big> is the deviatoric stress tensor. The latter is given by
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where ''P'' is the pressure and ''<big>&tau;</big>'' is the deviatoric stress tensor. The latter is given by
  
  
<big><math>\tau _{ij}=\sigma _{ij}+P\delta _{ij}</math></big>,
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<math>\quad\tau _{ij}=\sigma _{ij}+P\delta _{ij}</math>,
  
  
where <big>&sigma;</big> is the full stress tensor and <big><math>\delta _{ij}</math></big> is the Kronecker delta (or identity matrix).
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where ''<big>&sigma;</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} \\
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\end{align}</math>
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 +
 
 +
where ''&eta;'' 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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 +
 
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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 13:30, 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,


\begin{align}
  & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=0 \\ 
 & y:\quad \frac{\partial \tau _{yy}}{\partial y}-\frac{\partial P}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=0 \\ 
 & z:\quad \frac{\partial \tau _{zz}}{\partial z}-\frac{\partial P}{\partial z}+\frac{\partial \tau _{zy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial x}=\rho g \\ 
\end{align},


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


\quad\tau _{ij}=\sigma _{ij}+P\delta _{ij},


where σ is the full stress tensor and \delta _{ij} 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


\begin{align}\quad \tau _{ij}=2\eta \dot{\varepsilon }_{ij} \\ 
\end{align}


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


\begin{align}\quad \eta \equiv \frac{1}{2}B\dot{\varepsilon }_{e}^{\frac{1-n}{n}} \\\end{align}