Abstract
The reader would have realized from the earlier chapters that the spectral method is a powerful tool to solve a large variety of problems. Among these problems are eigenvalue problems, problems on diffusion and problems with closed flows. However the method is not restricted to merely these types of problems and so to give the reader an appreciation of the power of the spectral method we show how it can be applied to non-linear problems that have the following characteristics: a solution exists for a large range of a given parameter which is termed the control parameter. In some cases this solution may become unstable to even the smallest disturbance when the parameter exceeds a critical value and in other cases it may become unstable to large disturbances once the parameter exceeds a critical value. Needless to say there are several examples where even more exotic situations can occur. These problems, in general, are termed stability problems in transport phenomena. Our interest in this chapter is to expose the reader to a few examples involving physical instabilities which ought to be distinguished from the numerical instabilities that he might have come across earlier.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The Case (II) \(r\left(u_{max}\right)\) curve has a simple analytical expression, viz., \(r=2\left(\frac{\mathrm{{arc}}\cosh (\zeta ) }{\zeta }\right)^2\) with \(\zeta =\mathrm{{e}}^{u_{max}/2}\) (see Nahme 1940).
- 2.
The reader should realize that the ratio \(\tau ^{(k+1)}\) is very close to the exponential growth rate \(\sigma ^{\prime }\) defined by (4.13).
- 3.
Notice that the same notation, \(\delta T\), is adopted for the polynomial field \(\delta T^{(L,N)}(x,z)\)) and for the corresponding column vector \(\delta T\). There ought to be no confusion given the context in which the notation is used.
References
Chénier E, Delcarte C, Labrosse G (1999) Stability of the axisymmetric buoyant-capillary flows in a laterally heated liquid bridge. Phys Fluids 11:527–541
Johns L, Narayanan R (2002) Interfacial instability. Springer-Verlag, New York
Lorenz E (1963) Deterministic non-periodic flow. J Atmos Sci 20:130–141
Nahme R (1940) Beitre zur Hydrodynamischen Theorie der Lagerreibung. Ing-Arch 11:191–209
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Guo, W., Labrosse, G., Narayanan, R. (2012). Applications to Transport Instabilities. In: The Application of the Chebyshev-Spectral Method in Transport Phenomena. Lecture Notes in Applied and Computational Mechanics, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34088-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-34088-8_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34087-1
Online ISBN: 978-3-642-34088-8
eBook Packages: EngineeringEngineering (R0)