Willingness to pay

2013-10-09T20:04:44Z

37.100.131.254: /* Formal definition */

{{Multiple issues|unreferenced = November 2006|orphan = November 2009|context = May 2011|technical = May 2011|
{{Underlinked|date=May 2013}}
}}

'''Streamline diffusion''', given an [[advection equation|advection]]-[[diffusion equation]], refers to all diffusion going on along the advection direction.

==Explanation==
If we take an advection equation, for simplicity of writing we have assumed <math>\nabla\cdot{\bold u}=0</math>, and <math>||{\bold u}||=1</math>
:<math>
\frac{\partial\psi}{\partial t}
+{\bold u}\cdot\nabla\psi=0.
</math>

we may add a diffusion term, again for simplicty, we assume the diffusion to be constant over the entire field.

:<math>D\nabla^2\psi</math>,

Giving us an equation of the form:

:<math>
\frac{\partial\psi}{\partial t}
+{\bold u}\cdot\nabla\psi
+D\nabla^2\psi
=0
</math>

We may now rewrite the equation on the following form:

:<math>
\frac{\partial\psi}{\partial t}
+{\bold u}\cdot \nabla\psi
+{\bold u}({\bold u}\cdot D\nabla^2\psi)
+(D\nabla^2\psi-{\bold u}({\bold u}\cdot D\nabla^2\psi))
=0
</math>

The term below is called streamline diffusion.
:<math>{\bold u}({\bold u}\cdot D\nabla^2\psi)</math>

===Crosswind diffusion===
Any diffusion orthogonal to the streamline diffusion is called crosswind diffusion, for us this becomes the term:
:<math>
(D\nabla^2\psi-{\bold u}({\bold u}\cdot D\nabla^2\psi))
</math>

[[Category:Fluid dynamics]]
[[Category:Diffusion]]
[[Category:Partial differential equations]]

{{applied-math-stub}}

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