tailieunhanh - Advanced Engineering Mathematics 2011 Part 13

Tham khảo tài liệu 'advanced engineering mathematics 2011 part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Laplace s Equation 469 not unique. This follows from the fact that if u x y is a solution so is u x y c where c is any constant. Finally we note that we must specify the boundary conditions along each side of the boundary. These sides may be at infinity as in problems with semi-infinite domains. We must specify values along the entire boundary because we could not have an equilibrium solution if any portion of the domain was undetermined. SEPARATION OF VARIABLES As in the case of the heat and wave equations separation of variables is the most popular technique for solving Laplace s equation. Although the same general procedure carries over from the previous two chapters the following examples fill out the details. Example Groundwater flow in a valley Over a century ago a French hydraulic engineer named Henri-Philibert-Gaspard Darcy 1803-1858 published the results of a laboratory experiment on the flow of water through sand. He showed that the apparent fluid velocity q relative to the sand grains is directly proportional to the gradient of the hydraulic potential kVp where the hydraulic potential ip equals the sum of the elevation of the point of measurement plus the pressure potential p pg . In the case of steady flow the combination of Darcy s law with conservation of mass Vq 0 yields Laplace s equation v2 p 0 if the aquifer is isotropic same in all directions and homogeneous. To illustrate how separation of variables may be used to solve Laplace s equation we shall determine the hydraulic potential within a small drainage basin that lies in a shallow valley. See Figure . Following Tóth 2 the governing equation is the two-dimensional Laplace equation ữ x L ũ y z0 ox1 dy2 along with the boundary conditions u x z0 - gz0 gcx Uj o y UX L y 0 and Uj a o o where u x y is the hydraulic potential g is the acceleration due to gravity and c gives the slope of the topography. The conditions UX L y 0 2 Tóth J. J. Geophys. Res. 67 4375-4387 .