On the Theory of Flood Waves in Rivers
On the Theory of Flood Waves in Rivers
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10.
2 Hence the ("^. , X ) transformation defined by (13) implies that in (2) the time deri^'ati^''e terms are of lower order of magnitude (or higher order in \) than the space deri^'^atl^e terms which are in turn of lower order of magnitude than the non-derivative terms.
Historically, such arfruments on orders of magnitude of various terms in the Na^ier-otokes equations of motion based on physical considerations led to Prandtl's concept of the boundary layer. For \ = in Uk) = (15a) [h(v - U)] ...(l5b) ■^f - 63 = which is satisfied by the steady flow solution (h, v) = (h, v ) constant, where h and v are related oo o o by (6). Therefore we make perturbation expansions with respect to the parameter X of (h, v) on the constant solution (h, v ) by expressing (16a) h(d, t) = h^ + nS^\c3, r) + \V^Uu), r) + ...
(16b) v(cy, t) = v^ + \v(l^(tv, r) + X^v(2^(a), r) + ...
Making use of the given expansion of U/v with respect o to \ we put (9a) and (16) into (lU) and equate coefficients of like-powers of A in the usual way.
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To quickly assess the difficulty of the text, read a short excerpt:
10.
2 Hence the ("^. , X ) transformation defined by (13) implies that in (2) the time deri^'ati^''e terms are of lower order of magnitude (or higher order in \) than the space deri^'^atl^e terms which are in turn of lower order of magnitude than the non-derivative terms.
Historically, such arfruments on orders of magnitude of various terms in the Na^ier-otokes equations of motion based on physical considerations led to Prandtl's concept of the boundary layer. For \ = in Uk) = (15a) [h(v - U)] ...(l5b) ■^f - 63 = which is satisfied by the steady flow solution (h, v) = (h, v ) constant, where h and v are related oo o o by (6). Therefore we make perturbation expansions with respect to the parameter X of (h, v) on the constant solution (h, v ) by expressing (16a) h(d, t) = h^ + nS^\c3, r) + \V^Uu), r) + ...
(16b) v(cy, t) = v^ + \v(l^(tv, r) + X^v(2^(a), r) + ...
Making use of the given expansion of U/v with respect o to \ we put (9a) and (16) into (lU) and equate coefficients of like-powers of A in the usual way.
What to read after On the Theory of Flood Waves in Rivers?
You can find similar books in the "Read Also" column, or choose other free books by G K Morikawa to read onlineMoreLess
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