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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You can find similar books in the "Read Also" column, or choose other free books by Joseph Bishop Keller to read onlineMoreLess
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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