By Rita Meyer-Spasche (auth.)
It turns out uncertain even if we will be able to anticipate to appreciate absolutely the instability of fluid circulation with no acquiring a mathematical representa tion of the movement of a fluid in a few specific case during which instability can really be ob served, in order that an in depth comparability will be made among the result of research and people of scan. - G.l. Taylor (1923) notwithstanding the equations of fluid dynamics are rather complex, there are configurations which enable uncomplicated circulation styles as desk bound options (e.g. flows among parallel plates or among rotating cylinders). those move styles will be received in simple terms in sure parameter regimes. For parameter values now not in those regimes they can not be acquired, as a rule for 2 various purposes: • The mathematical life of the suggestions is parameter established; or • the recommendations exist mathematically, yet they aren't solid. for locating good regular states, steps are required: the regular states need to be discovered and their balance should be determined.
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Additional resources for Pattern Formation in Viscous Flows: The Taylor-Couette Problem and Rayleigh-Bénard Convection
Example text
43) and is thus available at each step. 45) was used to terminate the iterations, but IIG(u ll ,{3l1)11 was watched. 45) was. 46) a factor of 10- 4 was still missing. 46) as an additional condition for the termination of the iterations. This turned out to be very valuable when we discovered that the increase of the necessary number of iterations was caused by a programming error. 34). 2. Details of a Numerical Method the residuals computed were correct. 46) thus saved us the work of redoing all the computations that had been performed with the wrong Jacobian.
Also, the resulting expressions were simpler. Thus only this version is discussed here. t. T gives l/ = 1, ... 18) gives p~ + (l/k)2u v + l/kw~ = Re h(u, u', v, w; k; l/, T). ·. ,T1 +"2) ="2 (11( ... ,T1) + h( ... ,T1 + h)). 23) with h/2 for the first derivatives in h. We thus used the implicit midpoint rule. In our computations for wide ranges of parameters we never had any problems with numerical stability. 42 Chapter 2. 30) 1 I + r2 1 Va ) [r] = Re h(u, v, v,I w; k; 0, r), -va/I - ~va 1 I -Vv/I - ~vv ( 0, + (( vk )2 + r21)) Vv [r] ~W~ + (vk)2w v - I = Re h(u, V, v, w; k;v,r), Vk Pv ) [r] = Re h(u, V, W, Wi; k; v, r), r = 1 + I1h, 11 = 1, ...
1 J= h(u, v, Vi, w; k; 1/, r) - + ~Vj) r jkvjwj), = UlIV~ - ~UlIVO - ~ (~( UjV~_j + ~UjVlI-j + (1/ - j)kvlI-jWj) r 2 r j=l j=1I+1 - (j - v)kvj-vwj - jkvjwj-v) ), h(u, W, Wi; k; 1/, r) = 1 - - 2 + (1I-1 L(UjW~-j + jkwlI-jwj) t j=l (wjUj-lI - Ujwj_lI - I/kWj_lIWj )) . 24). If r is not a grid point (first and last eqs. 28) and the implicit midpoint rule, as explained in the previous section. 52). 6). 30). The first elements of x are the PlI at r = rl + h/2, followed by PlI' UlI , VlI , WlI at r = rl + h, .