By P. G. Drazin, N. Riley
ISBN-10: 0521681626
ISBN-13: 9780521681629
The Navier-Stokes equations have been firmly confirmed within the nineteenth Century because the method of nonlinear partial differential equations which describe the movement of most typically taking place fluids in air and water, and because that point detailed strategies were sought by means of scientists. jointly those suggestions let a transparent perception into the habit of fluids, offering a motor vehicle for novel mathematical tools and an invaluable cost for computations in fluid dynamics, a box during which theoretical study is now ruled through computational tools. This booklet attracts jointly certain recommendations from commonly differing resources and provides them in a coherent demeanour, partly by means of classifying strategies through their temporal and geometric constraints. it is going to end up to be a necessary source to all who've an curiosity within the topic of fluid mechanics, and particularly to those that are studying or educating the topic on the senior undergraduate and graduate degrees.
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Extra resources for The Navier-Stokes Equations: A Classification of Flows and Exact Solutions
Example text
The profiles may or may not be symmetrical with respect to the central line of the channel. If π > α > π/2 pure outflow is impossible, and there is a range of values of small Reynolds number in which pure inflow is impossible. The effect of increasing R in outflow is to exclude, progressively, more and more of the simpler types of flow. No such exclusion is introduced when |R| is increased in inflow. With increasing Reynolds number in pure inflow, and with small values of α, the velocity profile exhibits all the well-known characteristics of boundary layers near the walls, and an approximately constant velocity across the rest of the channel.
7 we show, for various values of B − A, profiles g(η). In their analyses Stuart and Tamada take B = A, and Dorrepaal takes B = 0. 44) close to the boundary y = 0, it can be shown that the dividing streamline ψ = 0 meets the boundary at ν 1/2 D , x = xs = −ζ0 3 k C where its slope is given by − 3kC 2 , ζ0 {(B − A)C + D} which is independent of the kinematic viscosity ν. Furthermore, the ratio of this slope to that of the dividing streamline far from the boundary, where viscous effects are unimportant, is C2 3 2 {(B − A)C + D} which is, in addition, independent of k and ζ0 , depending only upon the constant pressure gradient parallel to the boundary through both B and D.
One may note that a = r vr |max /ν may be interpreted as a Reynolds number of the flow, and is used as such by Millsaps and Pohlhausen (1953). 69) where E is the second complete elliptic integral. 11. 42/R. 67), as −b increases the flow tends to become uniform, except for boundary layers close to the walls. 11 The critical curve αc = αc (R). Below this pure inflow or outflow is possible, above it regions of reversed flow must be present. √ where β = tanh−1 2/3, from which we see that F ≈ b except in boundary layers of thickness O{(−b)−1/2 }.
The Navier-Stokes Equations: A Classification of Flows and Exact Solutions by P. G. Drazin, N. Riley
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