Electrohydrodynamics I the Equilibrium of a Charged Gas in a Container
Electrohydrodynamics I the Equilibrium of a Charged Gas in a Container
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Since (21) contradicts (l5) we conclude that (l^ is false. But the alterna- tive is (21), which must therefore apply since we have shown that the right side of (^ is zero for some value of a and that it increases monotonically to 00 as a does* In summary, we have shown that equations (1) - (^ have one and only one solution for any M > 0» In this solution p > 0, p > p(0) if M > and both p and p are constant on S and attain their maximxan values on S# Furthermore, both p and p are increasing fxin...ctions of M at each point of D. By slightly modifying the foregoing proofs, the corresponding results can be obtained for two-dimensional containers. For a one-dimensional container (i*e. , the region between two parallel planes) the boundary is not connected. Never- theless the same results are obtained if it is assumed that p has the same value on the two planes. - 9 - ii. Boxmds on the solution We have just seen that p and p increase as M increaseso We will now show that for a certain class of equations of state, both p and p are bounded above at every inner point of D, independently of M« Th\is although p and p increase with M, they both tend to finite limits as M becomes infinite, at all inner points of D, This result is a consequence of the follovdng theorem, which will not be proved here (see [I4]) : Theorem I ; If v is a solution of (8) in D and if f(v) is positive and increasing and 1-1/2 (22) / / f(z)d dx
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To quickly assess the difficulty of the text, read a short excerpt:
Since (21) contradicts (l5) we conclude that (l^ is false. But the alterna- tive is (21), which must therefore apply since we have shown that the right side of (^ is zero for some value of a and that it increases monotonically to 00 as a does* In summary, we have shown that equations (1) - (^ have one and only one solution for any M > 0» In this solution p > 0, p > p(0) if M > and both p and p are constant on S and attain their maximxan values on S# Furthermore, both p and p are increasing fxin...ctions of M at each point of D. By slightly modifying the foregoing proofs, the corresponding results can be obtained for two-dimensional containers. For a one-dimensional container (i*e. , the region between two parallel planes) the boundary is not connected. Never- theless the same results are obtained if it is assumed that p has the same value on the two planes. - 9 - ii. Boxmds on the solution We have just seen that p and p increase as M increaseso We will now show that for a certain class of equations of state, both p and p are bounded above at every inner point of D, independently of M« Th\is although p and p increase with M, they both tend to finite limits as M becomes infinite, at all inner points of D, This result is a consequence of the follovdng theorem, which will not be proved here (see [I4]) : Theorem I ; If v is a solution of (8) in D and if f(v) is positive and increasing and 1-1/2 (22) / / f(z)d dx
What to read after Electrohydrodynamics I the Equilibrium of a Charged Gas in a Container?
You can find similar books in the "Read Also" column, or choose other free books by Joseph Bishop Keller to read onlineMoreLess
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