Flow in an Aquifer. Consider the flow of water in a porous medium, such as sand, in an aquifer. The flow is driven by the hydraulic head, a measure of the potential energy of the water above the aquifer. Let R : 0 < x < a, 0 < z < b be a vertical section of an aquifer. In a uniform, homogeneous medium, the hydraulic head u( x, z) satisfies Laplaces equation uxx + uzz =0 inR. (39) If water cannot flow through the sides and bottom of R, then the boundary conditions there are ux(0, z) = 0, ux (a, z) = 0, 0 < z < b (40) uz ( x, 0) = 0, 0 < x < a. (41) Finally, suppose that the boundary condition at z = b is u( x, b) = b + x, 0< x < a, (42) where is the slope of the water table. a. Solve the given boundary value problem for u( x, z). G b. Let a = 1000, b = 500, and = 0.1. Draw a contour plot of the solution in R; that is, plot some level curves of u( x, z). G c. Water flows along paths in R that are orthogonal to the level curves of u( x, z) in the direction of decreasing u. Plot some of the flow paths.

HIST 201 WEEK 1 Colliding Worlds Indigenous America The First Americans Series of migrations across Bering Strait land bridge Settlement of Americans: “Indians” settled in the New World between 15,000 and 60,000 years ago Agricultural revolution *Original people were completely isolated from the rest of the world Great Civilizations Maya empire ...