Question

ar,a7, that V2u V.Vu 6.4. Verify directly from the gradient operator that V uK +uyy-see Definition 6.5 Definition 6.5 (Two-Dimensional Heat or Diffusion Equation). Consider the open do- main (x, y) W. Using the continuity equation (1.4) the flux rule (6.13) yields DV u+R (6.14) where V2u V.Vu u +uyy is the linear Laplacian operator The boundary conditions come in the three types: conditions on u, conditions on flux, and mixed as we are familiar with from Chapter 4. The region W is some open set, and the boundary is denoted aW. The conditions on u are termed Dirichlet boundary conditions: u(x, y, t)(y)eaw = f(x, y) u-condition: (6.15) where f is some specified function Conditions on flux are termed Neumann boundary conditions. Let nw be the outward normal vector to the boundary. That is, naw is perpendicular to the contour defining aW A flux condition through aW is then expressed in terms of the flux o-DVu -DVu nawlxy)eaw g(x,y), Flux-condition: (6.16) where g is a given flux condition-note that -D can be divided on both sides in the above for a more simplified expression: -g(x),/Dg(x). Schematic depictions of Dirichlet and Neumann boundary conditions are Mixed conditions, termed Robin boundary conditions are cooling-type relation between flux and the difference between u and a specified value f at the boundary: shown in Figure 6.4 given by a Netwon's law of Mixed-condition: KVu nwly)eaw + Ky(u-f)ly)eaw 0 (6.17) where f is specified, and the conductivity Ke Dcp and film coefficient K parameters are used here in lieu of diffusivity D

Answer 1

au do urify T that V.u AH.S u Us1 + unty + cs Scanned with CamScanner