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2012 | OriginalPaper | Chapter

4. Fluid Dynamics

Author : Rainer Kimmich

Published in: Principles of Soft-Matter Dynamics

Publisher: Springer Netherlands

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Abstract

Treatments of molecular dynamics in general and especially if they are based on Langevin equations of motions are hardly possible without referring to hydrodynamics. Basic hydrodynamic concepts such as Stokes’ friction law of particles in a viscous medium are ubiquitously employed in molecular dynamics. The present treatise is unique in the sense that it juxtaposes the principles of the analytical formalism in the form of computer simulations with real experiments. If the topology of objects is known or predetermined by suitable sample preparations, hydrodynamics can be simulated and measured under identical conditions. This option opens a promising application field of utmost importance for chemical engineering. The term fluid dynamics is moreover understood in a generalized sense. Apart from pressure-driven flow, a wealth of related transport phenomena will be addressed in this chapter. The scope covers examples as different as the spatially resolved probing of thermal convection, heat conduction, electroosmosis, and ionic currents.

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previous chapter Noninvasive Methods
next chapter Molecular Dynamics in Polymers
Footnotes
1
Note that the transport uniformity refers in principle to uncoupled particles here. This is in contrast to the less stringent version of coherence, namely, collective relaxation modes, that will be discussed in Chap.​ 5 (polymer dynamics) and in Chap.​ 6 (dynamics in liquid crystals). The coherence is then a consequence of chain connectivity and interactions among molecular entities.
 
2
A more detailed and modified picture of the conditions at fluid–surface interfaces will be outlined in Sect. 7.5.7 in the context of the NMR flow-relaxation effect.
 
3
The (total) differential operator \( {{{\rm d}} \left/ {{{\rm d}t}} \right.} \) refers to derivations by implicit as well as by explicit time dependences, whereas the partial derivative \( {{\partial } \left/ {{\partial t}} \right.} \) is restricted to explicit time dependences only.
 
4
As already mentioned in Sect. 2.3.4, fluids with a viscosity \( \eta \) independent of the shear rate are called Newtonian liquids. This is what we anticipate here and in the following. In contrast to this ordinary viscous behavior, one distinguishes shear thinning and shear thickening in cases where the viscosity decreases or increases with the shear rate, respectively. Respective examples are blood and granular suspensions. In some relatively exotic systems, the viscosity may also increase or decrease with the duration of the applied stress. One then speaks of rheopectic and thixotropic fluids, respectively. Usually, these are extremely viscous materials.
 
5
In principle, a velocity potential can also be introduced for the description of irrotational flow of compressible fluids. The formalism is however much more complicated [3].
 
6
Details of the conversion to spherical coordinates can be found in mathematical-physics textbooks such as Ref. [4], for instance.
 
7
These results can be cross-checked by inserting them in Eqs. (4.49) and (4.47). However, the expressions to be handled tend to be extremely lengthy. It may therefore be recommendable to employ a computer algebra software package for this endeavor.
 
8
In gaseous media where the finite compressibility needs to be taken into account, it may be necessary to add a further diagonal term of the form \( \lambda \left( {\nabla \cdot {v}} \right){{\delta}_{{ik}}} \) with \( \lambda \) the second viscosity coefficient analogous to the first Lamé constant [1].
 
9
The thermal conductivity \( \kappa \) (SI unit: Wm−1 K−1) must be distinguished from the thermal diffusivity \( \alpha = \kappa /\left( {\rho {{c}_v}} \right) \) (SI unit: m2s−1), where \( \rho \) is the mass density (SI unit: kg m−3) and \( {{c}_v} \) is the specific heat at constant volume (Si unit: J kg−1 K−1).
 
10
Turbulent fluctuations would also contribute to this process.
 
11
Equation (4.87) coincides with the equation of creeping motion, Eq. (4.43).
 
12
Actually, the pore space assumed here corresponds to an Ising-correlated percolation cluster with a porosity of 0.5672 which is above the percolation threshold. According to Ref. [15], the percolation threshold of this sort of network is somewhat lower than in ordinary random-site percolation clusters.
 
13
Note, however, that the non-slipping boundary condition \( {{v}_{{\rm electrodes}}} = 0 \) must be assumed at the surfaces of the electrodes in closed systems blocking any in- and outflow. This is in contrast to open systems defined by unrestricted flow across the electrode boundaries.
 
14
As before, quasi-two-dimensional means that the pore space is restricted to a thin layer, the thickness of which is negligible relative to the lateral extension.
 
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Metadata
Title
Fluid Dynamics
Author
Rainer Kimmich
Copyright Year
2012
Publisher
Springer Netherlands
Book
Principles of Soft-Matter Dynamics

Print ISBN: 978-94-007-5535-2

Electronic ISBN: 978-94-007-5536-9

Copyright Year: 2012

DOI
https://doi.org/10.1007/978-94-007-5536-9_4