Abstract
The current understanding of solidification phenomena is based largely on the differential formulation of the mechanics of the continuum and the associated conservation of mass, energy, species, and momentum equations. Thus, a short introduction to the fundamentals of transport phenomena as applied to solidification is deemed necessary to facilitate engaging in the intricacies of the subject.
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 del operator (or the gradient) of a function u is: \(\nabla u\equiv \frac{\partial u}{\partial x}\mathbf{i}+\frac{\partial u}{\partial y}\mathbf{j}+\frac{\partial u}{\partial z}\mathbf{k}\). Note that ∇u is ∇ operating on u, while ∇ · A is the vector dot product of del with A (or the divergence): \(\nabla \cdot \mathbf{A}=\partial {{\mathbf{A}}_{x}}/\partial x+\partial {{\mathbf{A}}_{y}}/\partial y+\partial {{\mathbf{A}}_{z}}/\partial z\). Furthermore, ∇2 u is the product of the del operator with itself, or \(\nabla \cdot \nabla =\frac{\partial }{\partial x}\left(\frac{\partial }{\partial x} \right)+\frac{\partial }{\partial y}\left(\frac{\partial }{\partial y} \right)+\frac{\partial }{\partial z}\left(\frac{\partial }{\partial z} \right)\) operating on u.
- 2.
The substantial derivative of a function u is given by: \(\frac{Du}{Dt}=\frac{\partial u}{\partial t}+{{u}_{x}}\frac{\partial {{u}_{x}}}{\partial x}+{{u}_{y}}\frac{\partial u}{\partial y}+{{u}_{z}}\frac{\partial u}{\partial z}\). It can be applied to any property of a fluid, the magnitude of which varies with time and position.
References
Beckermann C, Viskanta R (1988) PhysicoChem. Hydrodyn. 10:195
Bennon WD, Incropera FP (1987) Int. J. Heat Mass Transfer 30:2161, 2171
Chang S, Stefanescu DM (1996) Metall. Mater. Trans. 27A:2708
Ganesan S, Poirier DR (1990) Metall. Trans. 21B:173
Ni J, Beckermann C (1991) Metall. Trans. 22B:349
Wang CY, Beckermann C (1996) Metall. Mater. Trans. 27A:2754
Wu MG, Ludwig A (2006) Metall. Mater. Trans. 37A:1613–1631
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Stefanescu, D. (2015). Fundamentals of Transport Phenomena as Applied to Solidification Processing. In: Science and Engineering of Casting Solidification. Springer, Cham. https://doi.org/10.1007/978-3-319-15693-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-15693-4_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15692-7
Online ISBN: 978-3-319-15693-4
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)