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Flow in an Aquifer. Consider the flow of water in a porous medium, such as sand, in an

ISBN: 9781119256007 392

Solution for problem 19 Chapter 10.8

Elementary Differential Equations and Boundary Value Problems | 11th Edition

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Problem 19

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.

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ISBN: 9781119256007

This full solution covers the following key subjects: . This expansive textbook survival guide covers 75 chapters, and 1655 solutions. The answer to “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.” is broken down into a number of easy to follow steps, and 227 words. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Since the solution to 19 from 10.8 chapter was answered, more than 206 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. The full step-by-step solution to problem: 19 from chapter: 10.8 was answered by , our top Math solution expert on 03/13/18, 08:17PM.

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