onepager - MIT OpenCourseWare http://ocw.mit.edu 18.085...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Gradient and Divergence / Parallel Table Gradient v = grad u = �u Potential u(x, y ): v1 = Test on v : �u �u , v2 = �x �y Divergence div w = � · w = 0 Stream function s(x, y ): w1 = � w1 � w2 + =0 �x �y Solenoidal: zero source Zero flux through loops: � � Test on w: w · n ds = �s �s , w2 = − �y �x � v2 � v1 − =0 �x �y Irrotational: zero vorticity Zero circulation around loops: � � v · t ds = v1 dx + v2 dy = 0 w1 dy − w2 dx = 0 Kirchhoff’s Voltage Law Equip otentials u(x, y ) = constant v is p erpendicular to equip otentials Kirchhoff’s Current Law Streamlines s(x, y ) = constant w is tangent to streamlines Green-Gauss Formula �� w · grad u dx dy = �� u(− div w) dx dy + � u w · n ds (grad)T = − div from integration by parts: (Au)T w = uT (AT w ) Connections when (v1 , v2 ) = (w1 , w2 ) 1. Equip otentials are p erpendicular to streamlines �� �� � �u � �u 2. Laplace’s equation div(grad u) = + = � · �u = 0 �x �x �y �y 3. Cauchy-Riemann equations 4. Laplace’s equation for s �u �s �u �s = and =− connecting u to s �x �y �y �x �2s �2s �2u �2u + 2 =− + =0 � x2 � y � x�y � y �x 5. Zero vorticity and zero source: Ideal p otential flow 6. In two dimensions: u(x, y ) + is(x, y ) is a function f (x + iy ) ...
View Full Document

Ask a homework question - tutors are online