1.4: Dynamical Processes and Characteristic Times
- Page ID
- 28120
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Rate processes familiar from electrical circuits are the discharge of a capacitor (\(C\)) or an inductor (\(L\)) through a resistor (\(R\)), or the oscillation of energy between a capacitor and an inductor. One way to characterize the dynamics is in terms of the times \(RC\), \(L/R\) and \(\sqrt{LC}\), respectively.
Characteristic times describing rate processes on a continuum basis are a recurring theme. The electromagnetic times summarized in Table 1.4.1 are the field analogues of those familiar from circuit theory. Rather than defining the variables, reference is made to the section where the characteristic times are introduced. Some of the mechanical and thermal ones also have lumped parameter counterparts. For example, the viscous diffusion time, which represents the mechanical damping of ponderable material, is the continuum version of the damping rate for a dash-pot connected to a mass.
The electromechanical characteristic times represent the competition between electric or magnetic forces and viscous or inertial forces. In specialized areas, they may appear in a different guise. For example, with the electric field intensity \(\overrightarrow{E}\) that due to the bunching of electrons in a plasma, the electro-inertial time is the reciprocal plasma frequency. In a highly conducting fluid stressed by a magnetic field intensity \(H\), the magneto-inertial time is the transit time for an Alfvyn wave.
Especially in fluid mechanics, these characteristic times are often brought into play as dimensionless ratios of times. Table 1.4.2 gives some of these ratios, again with references to the sections where they are introduced.
| Time | Nomenclature | Section reference |
|---|---|---|
| Electromagnetic | ||
| \(\tau_{em} = l/c\) | Electromagnetic wave transit time | 2.3 |
| \(\tau_{e} = \varepsilon/\sigma\) | Charge relaxation time | 2.3, 5.10 |
| \(\tau_{m} = \mu\sigma l^{2}\) | Magnetic diffusion time | 2.3, 6.2 |
| \(\tau_{mig} = l/bE\) | Particle migration time | 5.9 |
| Mechanical and thermal | ||
| \(\tau_{a} = l/a\) | Acoustic wave transit time | 7.11 |
| \(\tau_{v} = \rho l^2/\eta\) | Viscous diffusion time | 7.18, 7.24 |
| \(\tau_{c} = \eta/\rho a^2\) | Viscous relaxation time | 7.24 |
| \(\tau_{D} = l^2/\kappa \) | Molecular diffusion time | 10.2 |
| \(\tau_{T} = l^2 \rho c_{v}/k_T\) | Thermal diffusion time | 10.2 |
| Electromechanical | ||
| \(\tau_{EV} = \eta/\varepsilon E^2\) | Electro-viscous time | 8.7 |
| \(\tau_{MV} = \eta/\mu H^2\) | Magneto-viscous time | 8.6 |
| \(\tau_{EI} = l \sqrt{\rho/\varepsilon E^2}\) | Electro-inertial time | 8.7 |
| \(\tau_{MI} = l \sqrt{\rho/\mu H^2}\) | Magneto-inertial time | 8.6 |
| Time | Symbol | Nomenclature | Sec. ref. |
|---|---|---|---|
| Electromagnetic | |||
| \(\tau_{e}/\tau = \varepsilon U/l \sigma\) | \(R_e\) | Electric Reynolds number | 5.11 |
| \(\tau_{m}/\tau = \mu \sigma l U\) | \(R_m\) | Magnetic Reynolds number | 6.2 |
| Mechanical and thermal | |||
| \(\tau_{a}/\tau = U/a\) | \(M\) | Mach number | 9.19 |
| \(\tau_{v}/\tau = \rho l U/\eta\) | \(R_y\) | Reynolds number | 7.18 |
| \(\tau_{D}/\tau = l U/\kappa\) | \(R_D\) | Molecular Peclet number | 10.2 |
| \(\tau_{T}/\tau =\rho c_p l U/ k_T\) | \(R_T\) | Thermal Peclet number | 10.2 |
| \(\tau_{D}/\tau_{v} =\eta / \rho k_D\) | \(P_D\) | Molecular-viscous Prandtl number | 10.2 |
| \(\tau_{T}/\tau_{v} =c_p \eta/ k_T\) | \(P_T\) | Thermal-viscous Prandtl number | 10.2 |
| Electromechanical | |||
| \(\sqrt{\tau_{m}/\tau_{MV}} =\mu H l \sqrt{\sigma/\eta}\) | \(H_m\) | Magnetic Hartmann number | 8.6 |
| \(\sqrt{\tau_{mig}/\tau_{EV}} ; \tau_{e}/\tau_{EV}\) | \(H_e\) | Electric Hartmann number | 9.12 |
| \(\tau_{m}/\tau_{v} = \eta \mu \sigma/ \rho\) | \(P_m\) | Magnetic-viscous Prandtl number | 8.6 |