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Abstract

In the first part of the thesis we consider steady state flows in slender tubes. These tubes are characterised by having walls whose equations are r = H(εx), where (r,θ,x) are cylindrical polar co-ordinates is a small parameter proportional to 1/R, where R is the Reynolds number. We obtain an approximation to the steady flow by expanding the stream function in powers of λ= ε R. This approximation is compared with the solutions of Daniels & Eagles (1979) for exponential slender tubes (H = exp(aε x)) and good agreement is found.

We next consider stability of the Daniels-Eagles profiles, by a qua si-parallel theory. Both spatially growing and temporally growing modes are considered ,and, contrary to expectation we have found no evidence of instability. In the course of this work detailed agreement was found with the eigenvalues of Davey & Drazin (1969), all of which remain stable eigenvalues as the parameter γ = eaR varies in the range -6 < γ < +6.

In the second main part of the thesis we consider the stability of the flow in certain channels, taking into account non-parallel effects by means of a new method. Here the equations of the wall are y = ±H(εx), where ε is independent of R. The flow is obtained to order as in Blasius(1910) and the corresponding stability analysis is performed to same order. By comparison with earlier results of Eagles & Weissman (1975) we show that this non-parallel theory appears stability curves that this method earlier methods. to give good results for the neutral for values of ε2 upto about 3, and we claim is more satisfactory mathematically than

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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