Sediment Transport in Alluvial Systems

Abstract

Sediment transport arises in alluvial lake-river systems in two different forms: (i) as bed load, comprising the moving detritus of the river bed and of the shallow, often only near-shore regions, and (ii) the suspended sediment load of the finer fractions. In river hydraulics the latter are often neglected; so, the bed load transport is treated without back-coupling with the wash-load. This is justified on decadal time scales. In the deeper parts of lakes wind-induced shearing in the benthic boundary layer hardly mobilizes the bed material, which stays immobile for most time and may be set in motion only interruptedly. However, the particle laden fluid transports the suspended material, which is advected and may on longer time scales settle in deposition-prone regions. In general, the deposition to and erosion from the basal surface occur concurrently. This environmental interplay is studied in this article. The slurry—a mixture of the bearer fluid and particles of various sizes—is treated as a mixture of class I, in which mass, momentum and energy balances for the mixture as a whole are formulated to describe the geophysical fluid mechanical setting, whilst the suspended solid particles move through the bearer medium by diffusion. The governing equations of this problem are formulated, at first for a compressible, better non-density preserving, mixture. They thus embrace barotropic and baroclinic processes. These equations, generally known as Navier–Stokes–Fourier–Fick (NSFF) fluids, are subjected to turbulent filter operations and complemented by zeroth and first order closure schemes . Moreover, simplified versions, e.g. the (generalized) Boussinesq, shallow water and hydrostatic pressure assumptions are systematically derived and the corresponding equations presented in both conservative and non-conservative forms. Beyond the usual constitutive postulates of NSFF-fluids and turbulent closure schemes the non-buoyant suspended particles give rise to settling velocities; these depend on the particle size, expressed by a nominal particle diameter. A review of the recent hydraulic literature of terminal settling velocities is given. It shows that the settling velocity depends on the particle diameter and on the particle Reynolds number. A separate section is devoted to the kinematic and dynamic boundary conditions on material and non-material singular surfaces as preparation for the mathematical-physical description of the sediment transport model, which follows from an analysis of jump transition conditions at the bed. The simplest description of bed load transport does not use the concept of the motion of a thin layer of sediments. It treats it as a singular surface, which is equipped with surface grains of various grain size diameters. Such a simplified theoretical level is also used in this chapter; it implies that solid mass exchange, as erosion and deposition of different particle size fractions, is the only physical quantity relevant in the description of the sediment transport. It entails formulation of surface mass balances of an infinitely thin detritus layer for the sediment and surface momentum balance of the mixture. The deposition rate of the various grain fractions, expressed as grain classes, follows from a parameterization of the free fall velocity of isolated particles in still water, but is in general coupled with the local flow and then follows from the solution of the hydrodynamic equations and the processes at the basal surface. The erosion rate is governed by two statements, (a) a fracture criterion determining the threshold value of a stress tensor invariant at the basal surface, which separates existence and absence regimes of erosion, and (b) determination of the amount of erosion beyond the threshold value of the mentioned stress invariant.

This chapter is a slightly extended version of an article with the title ‘A global view of sediment transport in alluvial systems’ and co-authored by K.H. and Prof. Dr. Ioana Luca, Department of Mathematics II, University Politehnica of Bucharest, and published within a report on Sediment Transport of the Institut für Wasserbau at the Technische Universität, München in 2013 [18]. We thank Professor Dr. sc. tech. Peter Rutschmann for the permission of copyright in this extended form.

Appendices

Appendix A: Implications from the Second Law of Thermodynamics

This appendix gives a justification for the approximation (32.44). The results which are presented can be taken from any book on thermodynamics, e.g. Hutter (2003) [16]. The basis of the considerations is the so-called Gibbs relation of a heat conducting fluid,

$$\begin{aligned} {\text {d}}\eta = \frac{1}{T}\left( {\text {d}}\epsilon - \frac{p}{\rho ^{2}}\,{\text {d}}\rho \right) , \end{aligned}$$

(32.210)

in which $\eta$ is the entropy, $T$ the Kelvin temperature, $\epsilon$ the internal energy, $p$ the pressure and $\rho$ the fluid density; (32.210) is a consequence of the second law of thermodynamics. Solving (32.210) for ${\text {d}}\epsilon$,

$$\begin{aligned} {\text {d}}\epsilon = T{\text {d}}\eta + \frac{p}{\rho ^{2}}\,{\text {d}}\rho , \end{aligned}$$

(32.211)

identifies $\epsilon$ as a function of $\eta$ and $\rho$, so that, alternatively and with $\epsilon = \hat{\epsilon }(\eta , \rho )$,

$$\begin{aligned} {\text {d}}\epsilon = \frac{\partial \hat{\epsilon }}{\partial \eta }\,{\text {d}}\eta + \frac{\partial \hat{\epsilon }}{\partial \rho }\,{\text {d}}\rho . \end{aligned}$$

(32.212)

Comparison of (32.211) and (32.212) implies

$$\begin{aligned} T = \frac{\partial \hat{\epsilon }}{\partial \eta }, \quad p = \rho ^{2}\,\frac{\partial \hat{\epsilon }}{\partial \rho }. \end{aligned}$$

(32.213)

The internal energy, interpreted as a function of entropy $\eta$ and density $\rho$, is a thermodynamic potential for the absolute temperature and the pressure.

With the functions

$$\begin{aligned} \begin{array}{ll} \displaystyle \psi = \epsilon - T\eta &{} {\textsc {Helmholtz}}\,\text {free energy}, \\ &{} \\ \displaystyle h = \epsilon + \frac{p}{\rho } &{} \text {enthalpy}, \\ &{} \\ \displaystyle g = h - T\eta &{} {\textsc {Gibbs}}\,\text {free energy}, \end{array} \end{aligned}$$

(32.214)

(these are Legendre transformations) the Gibbs relation (32.211) takes the alternative forms

$$\begin{aligned} \begin{array}{ccc} \displaystyle {\text {d}}\psi = - \eta {\text {d}}T + \frac{p}{\rho ^{2}}\,{\text {d}}\rho &{} \longrightarrow &{} \displaystyle \psi = \hat{\psi }(T, \rho ), \\ \displaystyle {\text {d}}h = - T{\text {d}}\eta + \frac{1}{\rho }\,{\text {d}}p &{} \longrightarrow &{} \displaystyle h = \hat{h}(\eta , p), \\ \displaystyle {\text {d}}g = - \eta {\text {d}}T + \frac{1}{\rho }\,{\text {d}}p &{} \longrightarrow &{} \displaystyle g = \hat{g}(T, p). \end{array} \end{aligned}$$

(32.215)

With the indicated different dependencies and the obvious potential properties, analogous to (32.213), we have

$$\begin{aligned} \begin{array}{ll} \displaystyle \eta = - \frac{\partial \hat{\psi }}{\partial T}, &{} \displaystyle p = \rho ^{2}\frac{\partial \hat{\psi }}{\partial \rho } , \\ \displaystyle T = - \frac{\partial \hat{h}}{\partial \eta } , &{} \displaystyle \frac{1}{\rho } = \frac{\partial \hat{h}}{\partial p}, \\ \displaystyle \eta = - \frac{\partial \,\hat{g}}{\partial T}, &{}\displaystyle \frac{1}{\rho } = \frac{\partial \hat{g}}{\partial p}, \end{array} \end{aligned}$$

(32.216)

and the integrability conditions

$$\begin{aligned} \begin{array}{ll} \displaystyle -\frac{\partial \eta }{\partial \rho } \equiv \frac{\partial }{\partial T}\left( \frac{p}{\rho ^{2}}\right) &{} \displaystyle \text {for}\quad \hat{\psi }(T, \rho ), \\ \displaystyle -\frac{\partial T}{\partial p} \equiv \frac{\partial }{\partial T}\left( \frac{1}{\rho }\right) &{} \displaystyle \displaystyle \text {for}\quad \hat{h}(T, p),\\ \displaystyle -\frac{\partial \eta }{\partial p} \equiv \frac{\partial }{\partial T}\left( \frac{1}{\rho }\right) &{} \displaystyle \text {for}\quad \hat{g}(T, p). \end{array} \end{aligned}$$

(32.217)

Internal energy formulation. If we regard $T$ and $\rho$ as the independent thermodynamic variables, then according to (32.214)$_1$ we have

$$\begin{aligned} \epsilon = \psi - T\frac{\partial \psi }{\partial T} = - T^{2}\frac{\partial }{\partial T}\left( \frac{\psi }{T}\right) , \end{aligned}$$

(32.218)

and therefore,

$$\begin{aligned} \begin{array}{c} \displaystyle \rho \frac{{\text {d}}\epsilon }{{{\text {d}}}{t}} = \rho \,c_{v}\frac{{\text {d}}T}{{{\text {d}}}{t}} + \rho \,c_{T\rho }\frac{{\text {d}}\rho }{{{\text {d}}}{t}},\\ \displaystyle c_{v}: = - \frac{\partial }{\partial T}\left( T^{2}\frac{\partial }{\partial T} \left( \frac{\hat{\psi }}{T}\right) \right) = \frac{\partial \hat{\epsilon }}{\partial T},\\ \displaystyle c_{T\rho } := - T^{2}\frac{\partial }{\partial T}\left( \frac{\partial \hat{\psi }/\partial \rho }{T}\right) = \frac{\partial \hat{\epsilon }}{\partial \rho }. \end{array} \end{aligned}$$

(32.219)

With the separation assumption

$$\begin{aligned} \psi = \hat{\psi }_{T}(T) + \hat{\psi }_{\rho }(\rho ), \end{aligned}$$

(32.220)

$c_{v} = \hat{c}_{v}(T)$ and $c_{T\rho } = \hat{c}_{T\rho }(\rho ) = {\text {d}}\hat{\psi }_{\rho }/{{\text {d}}}\rho$. Therefore, (32.219)$_1$ can be written as

$$\begin{aligned} \rho \frac{{\text {d}}\epsilon }{{{\text {d}}}{t}} = \rho \,\hat{c_{v}}(T)\frac{{{\text {d}}}T}{{{\text {d}}}{t}} + \underbrace{\rho \frac{{\text {d}}\hat{\psi }_{\rho }}{{{\text {d}}}\rho } \frac{{{\text {d}}}\rho }{{{\text {d}}}{t}}}_{\text {nearly}\; 0} \approx \rho \hat{c}_{v}(T)\frac{{{\text {d}}}T}{{{\text {d}}}{t}}. \end{aligned}$$

(32.221)

The second term on the right-hand side of (32.221) can be ignored since density variations in a nearly incompressible fluid are minute.

Enthalpy formulation. If we regard $T$ and $p$ as the independent thermodynamic variables, the Gibbs free energy is the thermodynamic potential and the enthalpy the adequate internal energy function. In view of (32.215) we now have

$$\begin{aligned} h = g - T\frac{\partial g}{\partial T }= - T^{2}\frac{\partial }{\partial T}\left( \frac{g}{T}\right) , \end{aligned}$$

(32.222)

and therefore,

$$\begin{aligned} \begin{array}{c} \displaystyle \rho \frac{{\text {d}}h}{{{\text {d}}}{t}} =\rho \,c_{p}\frac{{\text {d}}t}{{{\text {d}}}{t}} + \rho \,c_{Tp}\frac{{\text {d}}p}{{{\text {d}}}{t}}, \\ \displaystyle c_{p}: = - \frac{\partial }{\partial T}\left( T^{2}\frac{\partial }{\partial T} \left( \frac{\hat{g}}{T}\right) \right) = \frac{\partial \hat{h}}{\partial T},\\ \displaystyle c_{Tp} := - T^{2}\frac{\partial }{\partial T}\left( \frac{1}{T}\frac{\partial g}{\partial p}\right) = \frac{\partial \hat{h}}{\partial p}. \end{array} \end{aligned}$$

(32.223)

With the separation assumption

$$\begin{aligned} h = \hat{g}_{T}(T) + \hat{g}_{p}(p), \end{aligned}$$

(32.224)

$c_{p} = \hat{c}_{p}(T)$ and $c_{Tp} = \hat{c}_{Tp}(p) = {\text {d}}\hat{g}_{p}/{{\text {d}}}p$. Therefore, (32.210)$_1$ can be written as

$$\begin{aligned} \rho \frac{{\text {d}}h}{{{\text {d}}}{t}} = \rho \,\hat{c_{p}}(T)\frac{{{\text {d}}}T}{{{\text {d}}}{t}} + \underbrace{\rho \frac{{\text {d}}\hat{g}_{p}}{{{\text {d}}}p}\frac{{{\text {d}}}p}{{{\text {d}}}{t}}}_{\text {nearly}\; 0} \, \approx \, \rho \hat{c}_{p}(T)\frac{{\text {d}}t}{{{\text {d}}}{t}}. \end{aligned}$$

(32.225)

Here the second term on the right-hand side can be ignored, since ${\text {d}}\hat{g}_{p}/{\text {d}}p$ must be very small, the growth of the enthalpy due to a pressure rise cannot be large as its working is due to dilatational deformations, which are small.

Parameterizations. Because the temperature range of lake or ocean water is small, $0\,^\circ$C$\, \leqslant T \leqslant 50\,^\circ$C, the coefficients $c_v$ and $c_p$ exhibit a constrained variability and may well be assumed to be constant or linear functions of $T$. This then suggests to use

• for constant specific heats,

  $$\begin{aligned} \begin{array}{c} \displaystyle \epsilon = \int _{T_0}^{T}c_{v}(\bar{T})\,{\text {d}}\bar{T} = c_{v}^{0}(T - T_0) + \epsilon _0,\quad \\ h = \displaystyle \int _{T_0}^{T}c_{p}(\bar{T})\,{\text {d}}\bar{T} = c_{p}^{0}(T - T_0) + h_0, \end{array} \end{aligned}$$

  (32.226)

• for specific heats as linear functions of T:

  $$\begin{aligned} \displaystyle \epsilon&= \int _{T_0}^{T}[c_{v}^0 + c_{v}^{\prime }(\bar{T} - T_0)]\,{\text {d}}\bar{T} = c_{v}^{0}(T - T_0) + \frac{1}{2}c_{v}^{\prime }(T - T_0)^{2} + \epsilon _0,\nonumber \\ \displaystyle h&= \int _{T_0}^{T}[c_{p}^0 + c_{p}^{\prime }(\bar{T} - T_0)]\,{\text {d}}\bar{T} = c_{p}^{0}(T - T_0) + \frac{1}{2}c_{p}^{\prime }(T - T_0)^{2} + h_0. \end{aligned}$$

  (32.227)

The expressions (32.221), (32.225), (32.226), (32.227) provide a thermodynamic justification of relations (32.44).

Appendix B: Turbulent Closure by Large Eddy Simulation

Large Eddy Simulation (LES) is another popular approach for simulating turbulent flows. In this technique the large, geometry-dependent eddies are explicitly accounted for by using a subgrid-scale (SGS) model. Equations (32.76)–(32.80) are now interpreted as resolved field equations obtained by applying a non-statistical filter to the Navier–Stokes equations. ³³

The effect of the small eddies on the resolved filtered field is included in the SGS-parameterization of the stress ${\varvec{R}}$, given by (32.52) but by

$$\begin{aligned} {\varvec{R}}= 2\rho \nu _{SGS}{\varvec{D}}, \quad \mathrm {tr}\,{\varvec{D}}= 0, \end{aligned}$$

(32.228)

where $\nu _{SGS}$ is the SGS-turbulent viscosity,

$$\begin{aligned} \nu _{SGS} \equiv (C_{s}\varDelta )^{2}\left( \mathrm {tr}\,(2{\varvec{D}}^{2})\right) ^{1/2}. \end{aligned}$$

(32.229)

This parameterization is due to Smagorinsky (1963) [42]. $C_{s}$ is a dimensionless coefficient, called Smagorinsky constant, and $\varDelta$ is a length scale, equal to the local grid spacing. Thus, (32.228) with (32.229) is the classical viscous power law relating stress and stretching. According to Kraft et al. [24], the above ‘model is found to give acceptable results in LES of homogeneous and isotropic turbulence. With $C_{s} \approx 0.17$ according to Lilly (1967) [26], it is too dissipative [$\ldots$] in the near wall region because of the excessive eddy-viscosity arising from the mean shear (Moin and Kim (1982) [30]). The eddy viscosity predicted by Smagorinsky is nonzero in laminar flow regions; the model introduces spurious dissipation which damps the growth of small perturbations and thus restrains the transition to turbulence (Piomelli and Zang (1991) [34]).

The limitations of the Smagorinsky model have led to the formulation of more general SGS models. The best known of these newer models may be the dynamic SGS (DSGS) model of Germano et al. (1991) [12]. In this model $C_{s}$ is not a fixed constant but is calculated as a function of position and time, $C_{s}(\mathbf{x}, t)$, which vanishes near the boundary with the correct behaviour (Piomelli (1993) [24, 33]).

The parameterisations for the energy flux, ${\varvec{Q}}_{\epsilon }$ and constituent mass fluxes, ${\varvec{J}}_{\alpha }$, are the same as stated in (32.52)$_{2,3}$, however, with $\nu _{SGS}$ evaluated as given in (32.229). It is also evident from this presentation that the $(k-\varepsilon )$-equations are not needed.

Appendix C: Justification for (32.157)

In this appendix we provide a derivation of formula (32.157) for erosion inception on the basis of dimensional analysis. We consider sediment transport at a lake basal surface. It is rather intuitive that the erosion inception will likely depend on a stress (the shear stress) on the lake side of the basal surface, $\tau _c$, the true densities, $\rho _s$, $\rho _f$, of the sediment grains and the fluid, the solid concentration, $c_s$, gravity acceleration, $g$, mixture kinematic viscosity, $\nu$, and the nominal diameter, ${\mathfrak d}$, of the sediment corn, all evaluated at the base. So, inception of sediment transport can likely be described by an equation of the form

$$\begin{aligned} f(\tau _c, \rho _s, \rho _f, g, {\mathfrak d}, \nu , c_s) = 0. \end{aligned}$$

(32.230)

The dimensional matrix of the above 7 variables has rank 3; so, there are 4 dimensionless $\pi$-products, which we choose as follows:

$$\begin{aligned} \pi _1 = \frac{\tau _c}{\varDelta \rho \,g\,{\mathfrak d}}, \quad \pi _2 = \frac{\rho _s}{\rho _f}, \quad \pi _3 = c_s, \quad \pi _4 = \left( \frac{g}{\varDelta \nu ^{2}}\right) ^{1/3}{\mathfrak d}, \end{aligned}$$

(32.231)

where $\rho$ is the mixture density and $\varDelta \equiv (\rho _s/\rho - 1)$. Here, $\tau _c$ has been scaled with the ‘submerged’ density $(\rho _s - \rho )$. Furthermore, it is not difficult to see that for small $c_s$ the mixture density in (32.231) may approximately be replaced by $\rho _f$. We may thus write

$$\begin{aligned} f(\pi _1, \pi _2, \pi _3, \pi _4) = 0 \quad \text {or} \quad \frac{\tau _c}{\varDelta \,\rho \,g{\mathfrak d}} = \tilde{f}(\pi _2, \pi _3, \pi _4). \end{aligned}$$

(32.232)

The number of variables is now reduced from 7 to 4, a dramatic reduction. However, even further reduction is possible. For sediment transport in the geophysical environment $\pi _2$ is very nearly a constant on the entire Globe, and $\pi _3$ is very small ($\leqslant \!\!10^{-2}$); so, the $\pi _3$-dependence may be dropped (i.e. expressed in a Taylor series expansion of $\pi _3$ and restricted to the term $\tilde{f}(\pi _2, 0, \pi _4))$. Thus, we may assume

$$\begin{aligned} \theta _c \equiv \frac{\tau _c}{\varDelta \,\rho \,g{\mathfrak d}} = \tilde{f}(Re_c^{*}) = \tilde{f}({\mathfrak d}^{*}),\quad \pi _4 = Re_{c}^{*} = {\mathfrak d}^*\equiv \left( \frac{g}{\varDelta \,\nu ^{2}}\right) ^{1/3}{\mathfrak d} . \end{aligned}$$

(32.233)

This derivation assumes that only a single sediment fraction is present. It is important to note that the viscosity $\nu$ of the mixture is present in the variables describing the erosion inception. If it is dropped, then $\tilde{f}$ in (32.233) reduces to a constant and

$$\begin{aligned} \tau _c = \text {const.}\times \varDelta \,\rho \,g{\mathfrak d}^*, \end{aligned}$$

which is not supported by experiments. Omitting $g$ as a governing parameter is disastrous, because $\pi _1$ and $\pi _4$ are then missing as $\pi$-products. In this case $\tilde{f}(\pi _2, \pi _3) = 0$ is simply meaningless.

Appendix D: Justification for (32.177), (32.178) and (32.182), (32.183)

Justification for (32.177), (32.178). For the constituent masses, noting that

$$\begin{aligned} {\rho _{\alpha }({\varvec{v}}_{\alpha } - {\varvec{w}})}=\underbrace{\rho _{\alpha }({\varvec{v}}_{\alpha } - {\varvec{v}})}_{\equiv \,{\displaystyle {{\mathfrak {J}}}}_{\alpha },\;\, \text {see eq. (8)}}{ \displaystyle +\rho _{\alpha }({\varvec{v}} - {\varvec{w}})}, \end{aligned}$$

the non-averaged balance (32.140), in which $f_{\mathcal {S}}=\mu _\alpha$, ${{\varvec{\phi }}}^{f_{\mathcal {S}}}=\varvec{ {0}}$, $f=\rho _\alpha$, ${\varvec{v}}={\varvec{v}}_\alpha$, ${{\varvec{\phi }}}^{f}=\varvec{ {0}}$, can be written as

$$\begin{aligned} \begin{array}{c} \displaystyle \frac{\partial \mu _\alpha }{\partial t}+\left( \mu _{\alpha } {\varvec{v}}_{{\mathcal {S}}\alpha }\right) ^{\mathfrak a}_{\;;\mathfrak a} -\frac{\partial \mu _\alpha }{\partial \xi ^{\mathfrak a}}\,w^{\mathfrak a}- 2\mu _{\alpha }\,{\mathcal U}_{b} K= - [\![{\varvec{{\mathfrak {J}}}}_{\alpha } + \rho _{\alpha }({\varvec{v}} - {\varvec{w}}) ]\!]\cdot {\varvec{n}}_b. \end{array} \end{aligned}$$

(32.234)

Analogously, for the fluid we deduce

$$\begin{aligned} \displaystyle \frac{\partial \mu _f}{\partial t}+\left( \mu _{f} {\varvec{v}}_{{{\mathcal {S}}}f}\right) ^{\mathfrak a}_{\;;\mathfrak a} -\frac{\partial \mu _f}{\partial \xi ^{\mathfrak a}}\,w^{\mathfrak a}- 2\mu _{f}\,{\mathcal U}_{b} K =- [\![{\varvec{{\mathfrak {J}}}}_{f} + \tilde{\rho }_{f}({\varvec{v}} - {\varvec{w}}) ]\!]\cdot {\varvec{n}}_b, \end{aligned}$$

(32.235)

where ${\varvec{{\mathfrak {J}}}}_{f}\equiv \tilde{\rho }_f({\varvec{v}}_{f} - {\varvec{v}})$, with $\tilde{\rho }_f$ and ${\varvec{v}}_f$ the mass density and velocity of the fluid ($\tilde{\rho }_f=\rho -\sum _\alpha \rho _\alpha$). Now we sum Eqs. (32.234) and (32.235) over all constituents. Using relation

$$\begin{aligned} \sum _{\alpha }{\varvec{{\mathfrak {J}}}}_{\alpha } + {\varvec{{\mathfrak {J}}}}_{f} = \varvec{ {0}}, \end{aligned}$$

(32.236)

and definitions

$$\begin{aligned} \mu \equiv \sum _{\alpha }\mu _{\alpha } + \mu _{f},\quad \mu {\varvec{v}}_{{\mathcal {S}}} \equiv \sum _{\alpha }\mu _{\alpha }{\varvec{v}}_{{{\mathcal {S}}}\alpha } + \mu _{f}{\varvec{v}}_{{{\mathcal {S}}}f} \end{aligned}$$

(32.237)

for the mixture surface density $\mu$ and mixture velocity ${\varvec{v}}_{{\mathcal {S}}}$, we obtain the mass balance for the mixture by summation of (32.234) and (32.235):

$$\begin{aligned} \displaystyle \frac{\partial \mu }{\partial t}+\left( \mu {\varvec{v}}_{{\mathcal {S}}}\right) ^{\mathfrak a}_{\;;\mathfrak a} -\frac{\partial \mu }{\partial \xi ^{\mathfrak a}}\,w^{\mathfrak a}- 2\mu \,{\mathcal U}_{b} K = - [\![\rho ({\varvec{v}} - {\varvec{w}}) ]\!]\cdot {\varvec{n}}_b . \end{aligned}$$

(32.238)

We now average Eqs. (32.234) and (32.235). In so doing we assume that the interface does not perform any fluctuations, whence necessarily $\langle \,{\varvec{n}}_b\,\rangle = {\varvec{n}}_b$, $\langle \,K\,\rangle = K$, $\langle \,{\varvec{w}}\,\rangle = {\varvec{w}}$ and $\langle {\mathcal U}_{b}\rangle = {\mathcal U}_{b}$. Thus, for the averaged equations we get

$$\begin{aligned} \begin{array}{r} \displaystyle \frac{\partial \langle \mu _\alpha \rangle }{\partial t}+\left( \langle \mu _{\alpha }\rangle \, \langle \,{\varvec{v}}_{{\mathcal {S}}\alpha }\,\rangle \right) ^{\mathfrak a}_{\;;\mathfrak a} +\left( \langle \,\mu _{\alpha }^{\prime } \,\left( {\varvec{v}}_{{{\mathcal {S}}}\alpha }\right) ^{\prime }\,\rangle \right) ^{\mathfrak a}_{\;;\mathfrak a}-\frac{\partial \langle \mu _\alpha \rangle }{\partial \xi ^{\mathfrak a}}\,w^{\mathfrak a}- 2\langle \mu _{\alpha }\rangle \,{\mathcal U}_{b} K\\ = - [\![\langle \,{\varvec{{\mathfrak {J}}}}_{\alpha }\rangle +\langle \,\rho _{\alpha }^{\prime }{\varvec{v}} ^{\prime }\,\rangle + \langle \,\rho _{\alpha }\,\rangle ({\langle \,\varvec{v}}\,\rangle - {\varvec{w}}) ]\!]\cdot {\varvec{n}}_b, \end{array} \end{aligned}$$

(32.239)

$$\begin{aligned} \begin{array}{r} \displaystyle \frac{\partial \langle \mu _f\rangle }{\partial t}+\left( \langle \mu _{f}\rangle \, \langle \,{\varvec{v}}_{{{\mathcal {S}}}f}\,\rangle \right) ^{\mathfrak a}_{\;;\mathfrak a} +\left( \langle \,\mu _{f}^{\prime } \,\left( {\varvec{v}}_{{{\mathcal {S}}}f}\right) ^{\prime }\,\rangle \right) ^{\mathfrak a}_{\;;\mathfrak a} -\frac{\partial \langle \mu _f\rangle }{\partial \xi ^{\mathfrak a}}\,w^{\mathfrak a}- 2\langle \mu _{f}\rangle \,{\mathcal U}_{b} K \;\,\\ = - [\![\langle \,{\varvec{{\mathfrak {J}}}}_{f}\rangle + \langle \,\tilde{\rho }_{f}^{\prime }{\varvec{v}} ^{\prime }\,\rangle + \langle \,\tilde{\rho }_{f}\,\rangle ({\langle \,\varvec{v}}\,\rangle - {\varvec{w}}) ]\!]\cdot {\varvec{n}}_b. \end{array} \end{aligned}$$

(32.240)

If we sum (32.239) and (32.240), because of (32.236), (32.237) we obtain

$$\begin{aligned} \displaystyle \frac{\partial \langle \mu \rangle }{\partial t}&+\left( \langle \mu \rangle \, \langle \,{\varvec{v}}_{{\mathcal {S}}}\,\rangle \right) ^{\mathfrak a}_{\;;\mathfrak a} +\left( \langle \,\mu ^{\prime } \,\left( {\varvec{v}}_{{\mathcal {S}}}\right) ^{\prime }\,\rangle \right) ^{\mathfrak a}_{\;;\mathfrak a} -\frac{\partial \langle \mu \rangle }{\partial \xi ^{\mathfrak a}}\,w^{\mathfrak a}- 2\langle \mu \rangle \,{\mathcal U}_{b} K \nonumber \\&\qquad \quad = - [\![\qquad \qquad \underbrace{\langle \,\rho ^{\prime }{\varvec{v}}^{\prime }\,\rangle }\qquad +\, \langle \,\rho \,\rangle (\langle {\varvec{v}}\,\rangle - {\varvec{w}})]\!]\cdot {\varvec{n}}_b.\\&\qquad \qquad \quad \quad \equiv {{\varvec{\phi }}}^{\rho } z {\text {in Table 32.6}}\nonumber \end{aligned}$$

(32.241)

Of course, (32.241) is the average of (32.238), and only two of (32.239)–(32.241) are independent. For computations of initial boundary value problems we recommend to use (32.239) and (32.241) and to infer $\langle \mu _{f}\rangle$ a posteriori from $\langle \mu _{f}\rangle =\langle \mu \rangle - \sum \nolimits _{\alpha }\langle \mu _{\alpha }\rangle$.

It follows: with Reynolds averaging we have a non-vanishing mass flux in the mass balance (32.241). A Favre-type averaging would have to be performed. However, if $\rho ^{\prime }$ is small on both sides of the basal surface we can drop $\langle \,\rho ^{\prime }{\varvec{v}}^{\prime }\,\rangle$ in (32.241). Moreover, with $\rho ^{\prime }\approx 0$, $\rho _\alpha =\rho c_\alpha$, decomposition (32.9) and definition of ${\varvec{J}}_\alpha$ [see (32.43)], for the constituent class $\alpha$ the mass flux $\langle \,{\varvec{{\mathfrak {J}}}}_{\alpha }\rangle +\langle \,\rho _{\alpha }^{\prime }{\varvec{v}} ^{\prime }\,\rangle$ takes the form

$$\begin{aligned} \langle \,{\varvec{{\mathfrak {J}}}}_{\alpha }\rangle +\langle \,\rho _{\alpha }^{\prime }{\varvec{v}} ^{\prime }\,\rangle ={\varvec{J}}_{\alpha }-\rho \,\langle \,c_{\alpha }\rangle \langle {\varvec{w}}^{s}_\alpha \rangle , \end{aligned}$$

which explains Table 32.6 for Model 2. The main text, formulae (32.177), (32.178) [as deduced from (32.239), (32.241)] and Table 32.6 show the averaged fields without the averaging operator $\langle \cdot \rangle$ and with negligible correlations

$$\begin{aligned} \langle \,\mu _{\alpha }^{\prime } \,\left( {\varvec{v}}_{{{\mathcal {S}}}\alpha }\right) ^{\prime }\,\rangle ,\quad \langle \,\mu ^{\prime } \,\left( {\varvec{v}}_{{\mathcal {S}}}\right) ^{\prime }\,\rangle . \end{aligned}$$

Justification for (32.182) and (32.183). Now we consider (32.139), in which $f_{\mathcal {S}}=\mu _\alpha {\varvec{v}}_{{\mathcal {S}}\alpha }$, ${{\varvec{\phi }}}^{f_{\mathcal {S}}}=-{{\varvec{\sigma }}}_{{\mathcal {S}}\alpha }$, $\pi ^{f_{\mathcal {S}}}=0$, $s^{f_{\mathcal {S}}}=\mu _\alpha {{\varvec{g}}}$, $f=\rho _\alpha {\varvec{v}}_\alpha$, ${\varvec{v}}={\varvec{v}}_\alpha$, ${{\varvec{\phi }}}^{f}=-{{\varvec{\sigma }}}_\alpha$, for each $\alpha =1,\ldots ,N$:

$$\begin{aligned}&\displaystyle \quad \frac{\partial }{\partial t}\left( \mu _\alpha {\varvec{v}}_{{\mathcal {S}}\alpha }\right) +\text {Div}\,\left( \mu _\alpha {\varvec{v}}_{{\mathcal {S}}\alpha }\otimes {\varvec{v}}_{{\mathcal {S}}\alpha }-{{\varvec{\sigma }}}_{{\mathcal {S}}\alpha }\right) -\frac{\partial }{\partial \xi ^{\mathfrak a}}(\mu _\alpha {\varvec{v}}_{{\mathcal {S}}\alpha })\,w^{\mathfrak a} \nonumber \\ \displaystyle&= -[\![\rho _\alpha {\varvec{v}}_\alpha \otimes ({\varvec{v}}_\alpha -{\varvec{w}})-{\varvec{\sigma }}_\alpha ]\!]\,{\varvec{n}}_b+\mu _\alpha {{\varvec{g}}}. \end{aligned}$$

(32.242)

A similar equation holds for the interstitial fluid:

$$\begin{aligned}&\displaystyle \frac{\partial }{\partial t}\left( \mu _f{\varvec{v}}_{{\mathcal {S}}f}\right) +\text {Div}\left( \mu _f{\varvec{v}}_{{\mathcal {S}}f}\otimes {\varvec{v}}_{{\mathcal {S}}f }-{{\varvec{\sigma }}}_{{\mathcal {S}}f}\right) - \frac{\partial }{\partial \xi ^{\mathfrak a}}(\mu _f{\varvec{v}}_{{\mathcal {S}}f})\,w^{\mathfrak a} \nonumber \\&=\displaystyle -[\![\tilde{\rho }_f {\varvec{v}}_f\otimes ({\varvec{v}}_f-{\varvec{w}})-{\varvec{\sigma }}_f]\!]\,{\varvec{n}}_b+\mu _f{{\varvec{g}}}. \end{aligned}$$

(32.243)

Summing (32.242), (32.243) and using definitions (32.237) we obtain

$$\begin{aligned}&\displaystyle \frac{\partial }{\partial t}\left( \mu {\varvec{v}}_{{\mathcal {S}}}\right) +\text {Div}\left( \mu {\varvec{v}}_{{\mathcal {S}}}\otimes {\varvec{v}}_{{\mathcal {S}}}-{{\varvec{\sigma }}}_{{\mathcal {S}}}\right) - \frac{\partial }{\partial \xi ^{\mathfrak a}}(\mu {\varvec{v}}_{{\mathcal {S}}})\,w^{\mathfrak a} \nonumber \\&\quad = \displaystyle -[\![\rho {\varvec{v}}\otimes ({\varvec{v}}-{\varvec{w}})-{\varvec{\sigma }}]\!]\,{\varvec{n}}_b+\mu {{\varvec{g}}}, \end{aligned}$$

(32.244)

where the bulk, ${\varvec{\sigma }}$, and surface, ${\varvec{\sigma }}_{\mathcal {S}}$, mixture stress tensors are defined by

$$\begin{aligned} \rho {\varvec{v}}\otimes {\varvec{v}}-{{\varvec{\sigma }}} \equiv \sum _\alpha \left( \rho _\alpha {\varvec{v}}_{\alpha }\otimes {\varvec{v}}_{\alpha }-{{\varvec{\sigma }}}_{\alpha }\right) +\tilde{\rho }_f {\varvec{v}}_{ f}\otimes {\varvec{v}}_{{\mathcal {S}}f}-{{\varvec{\sigma }}}_{ f}, \end{aligned}$$

(32.245)

$$\begin{aligned} \mu {\varvec{v}}_{{\mathcal {S}}}\otimes {\varvec{v}}_{{\mathcal {S}}} -{{\varvec{\sigma }}}_{\mathcal {S}}\equiv \sum _\alpha \left( \mu _\alpha {\varvec{v}}_{{\mathcal {S}}\alpha }\otimes {\varvec{v}}_{{\mathcal {S}}\alpha }-{{\varvec{\sigma }}}_{{\mathcal {S}}\alpha }\right) +\mu _f {\varvec{v}}_{{\mathcal {S}}f}\otimes {\varvec{v}}_{{\mathcal {S}}f}-{{\varvec{\sigma }}}_{{\mathcal {S}}f}. \end{aligned}$$

(32.246)

Averaging (32.244) under the assumptions $\mu ^\prime \approx 0$, $\rho ^{\prime }\approx 0$, recalling definition (32.43)$_1$ of the Reynolds stress tensor ${\varvec{R}}$ and introducing the laminar and turbulent surface mixture stress tensor ${{\varvec{R}}}_{{\mathcal {S}}}$ according to

$$\begin{aligned} {{\varvec{R}}}_{{\mathcal {S}}}\equiv \langle {{\varvec{\sigma }}}_{{\mathcal {S}}}\rangle - \mu \,\langle {\varvec{v}}_{{\mathcal {S}}}^\prime \otimes {\varvec{v}}_{{\mathcal {S}}}^\prime \rangle , \end{aligned}$$

(32.247)

we deduce (we omit the angular brackets)

$$\begin{aligned} \begin{array}{r} \displaystyle \frac{\partial }{\partial t}\left( \mu {\varvec{v}}_{{\mathcal {S}}}\right) +\text {Div}\left( \mu {\varvec{v}}_{{\mathcal {S}}}\otimes {\varvec{v}}_{{\mathcal {S}}}-{{\varvec{R}}}_{{\mathcal {S}}}\right) - \frac{\partial }{\partial \xi ^{\mathfrak a}}\left( \mu {\varvec{v}}_{{\mathcal {S}}}\right) w^{\mathfrak a}=\quad \quad \quad \\ \displaystyle -[\![\rho {\varvec{v}}\otimes ({\varvec{v}}-{\varvec{w}})+ p\varvec{I}-{\varvec{R}}]\!]\,{\varvec{n}}_b+\mu {{\varvec{g}}}, \end{array} \end{aligned}$$

(32.248)

which explains the last line in Table 32.6.

Next we want to write (32.248) using the components of vectors and tensors with respect to the local basis $\{{{\varvec{\tau }}}_1,{{\varvec{\tau }}}_2,{\varvec{n}}_b\}$, which will give (32.182) and (32.183). To this end we use the formulae (for simplicity in this derivation we omit the lower index $b$ in ${\mathcal U}_b$ and ${\varvec{n}}_b$ referring to the basal surface)

$$\begin{aligned} \begin{array}{c} \displaystyle \frac{\partial {{\varvec{\tau }}}_{\mathfrak a}}{\partial \xi ^{\mathfrak b}}=\varGamma ^{\mathfrak c}_{\mathfrak a\mathfrak b}{{\varvec{\tau }}}_{\mathfrak c} + b_{\mathfrak a\mathfrak b}{\varvec{n}},\quad \frac{\partial {{\varvec{n}}}}{\partial \xi ^{\mathfrak a}}=- b_{\mathfrak a\mathfrak b}{{\varvec{\tau }}}^{\mathfrak b} ,\quad \frac{\partial {\varvec{n}}}{\partial \,t}=-g^{\mathfrak a\mathfrak b} \left\{ \frac{\partial {\mathcal U}}{\partial \xi ^{\mathfrak a}}+b_{\mathfrak c\mathfrak b}w^{\mathfrak c}\right\} {{\varvec{\tau }}_{\mathfrak b}},\\ \displaystyle \frac{\partial {{\varvec{\tau }}}_{\mathfrak a}}{\partial \,t}=\frac{\partial {\varvec{w}}}{\partial \xi ^{\mathfrak a}}=\left\{ \frac{\partial w^{\mathfrak b}}{\partial \xi ^{\mathfrak a}}+w^{\mathfrak c}\varGamma ^{\mathfrak b}_{\mathfrak c\mathfrak a}-{\mathcal U}b_{\mathfrak a\mathfrak c}g^{\mathfrak c\mathfrak b}\right\} {{\varvec{\tau }}_{\mathfrak b}}+ \left\{ \frac{\partial \mathcal U}{\partial \xi ^{\mathfrak a}}+ w^{\mathfrak b}b_{\mathfrak b\mathfrak a} \right\} {\varvec{n}}.\\ \end{array} \end{aligned}$$

(32.249)

The first two relations are derived in (32.111) and (32.112). The third and fourth are proved in an analogous way. Indeed, starting with the representations

$$\begin{aligned}&\frac{\partial n^{k}}{\partial t} = \dot{a}n^{k} + \dot{b}_{{\mathfrak a}}\tau ^{{\mathfrak a}k}\; \left( = \dot{a}n^{k} + \dot{b}^{{\mathfrak b}}\tau _{{\mathfrak b}}^{k}\right) ,\nonumber \\ \\&\frac{\partial \tau _{{\mathfrak a}}^{k}}{\partial t} = \dot{d}_{{\mathfrak a}}n^{k} + \dot{e}_{{\mathfrak a}}^{{\mathfrak b}}\tau _{{\mathfrak b}}^{k},\nonumber \end{aligned}$$

(32.250)

with yet unknown $\dot{a}$, $\dot{b}^{{\mathfrak b}}$, $\dot{e}_{\mathfrak a}^{\mathfrak b}$, we readily deduce

$$\begin{aligned}&\dot{a} =\frac{\partial n^{k}}{\partial t}n_{k} = \frac{1}{2}\frac{\partial (n^{k}\cdot n_{k})}{\partial t} = \frac{1}{2}\frac{\partial (1)}{\partial t} = 0 \nonumber \\&\dot{b}^{{\mathfrak b}} = \frac{\partial n^{k}}{\partial t}\tau _{k}^{{\mathfrak b}} = \dot{b}^{{\mathfrak a}} \underbrace{\tau _{{\mathfrak a}}^{k}\tau ^{{\mathfrak b}}_{k}}_{\delta _{{\mathfrak a}}^{{\mathfrak b}}} = \dot{b}^{{\mathfrak a}}\delta _{{\mathfrak a}}^{{\mathfrak b}}, \\&\frac{\partial \tau _{{\mathfrak a}}^{k}}{\partial t}n^{k} = \dot{d}_{{\mathfrak a}}, \nonumber \\&\frac{\partial \tau _{{\mathfrak a}}^{k}}{\partial t}\tau _{k}^{{\mathfrak c}} = \dot{e}_{{\mathfrak a}}^{{\mathfrak b}} \tau _{{\mathfrak b}}^{k} \tau _{k}^{{\mathfrak c}} = \dot{e}_{\mathfrak a} ^{{\mathfrak b}}g_{\mathfrak b\mathfrak c} = \dot{e}_{{\mathfrak a}}^{\mathfrak c}.\nonumber \end{aligned}$$

(32.251)

Explicit evaluation of $\dot{b}_{\mathfrak b}$ follows by recognizing that

$$\begin{aligned} \frac{\partial }{\partial t}\left( n^{k}\tau _{{\mathfrak b}k}\right) = 0 \quad \longrightarrow \quad \underbrace{\frac{\partial n^{k}}{\partial t} \tau _{{\mathfrak b}k}}_{\dot{b}_{{\mathfrak b}}} + n^{k}\frac{\partial \tau _{{\mathfrak b}}}{\partial t} = 0. \end{aligned}$$

(32.252)

So,

$$\begin{aligned} \dot{b}_{{\mathfrak b}}&= - n^{k}\frac{\partial \tau _{{\mathfrak b}k}}{\partial t} = -n^{k}\left\{ \frac{\partial }{\partial t} \left( \frac{\partial r_{k}}{\partial \xi ^{{\mathfrak b}}}\right) \right\} = - n^{k}\left\{ \frac{\partial }{\partial \xi ^{{\mathfrak b}}} \underbrace{\left( \frac{\partial r_{k}}{\partial t}\right) }_{=w^{{\mathfrak c}}\tau _{k{\mathfrak c}}+{\mathcal U}n_{k}}\right\} \nonumber \\&= - n^{k}\left\{ \frac{\partial w^{{\mathfrak c}}}{\partial \xi ^{{\mathfrak b}}}\tau _{k{\mathfrak b}} + w^{{\mathfrak c}}\underbrace{\frac{\partial \tau _{k{\mathfrak c}}}{\partial \xi ^{{\mathfrak b}}}}_{\varGamma _{{\mathfrak b\mathfrak c}}^{{\mathfrak d}}\tau _{{\mathfrak d}k}+b_{{\mathfrak b\mathfrak c}}n_{k}} + \frac{\partial {\mathcal U}}{\partial \xi ^{{\mathfrak b}}}n_{k} + {\mathcal U}\frac{\partial n_{k}}{\partial \xi ^{{\mathfrak b}}}\right\} \nonumber \\&= -\left\{ \frac{\partial {\mathcal U}}{\partial \xi ^{{\mathfrak b}}} + b_{\mathfrak b\mathfrak c}w^{{\mathfrak c}}\right\} , \end{aligned}$$

(32.253)

proving (32.249)$_3$. Similarly,

$$\begin{aligned} \dot{d}_{{\mathfrak a}}&= \frac{\partial \tau _{{\mathfrak a}}^{k}}{\partial t}n_{k} = \frac{\partial }{\partial t}\left\{ \frac{\partial r^{k}}{\partial \xi ^{{\mathfrak a}}}\right\} n_{k} = \frac{\partial }{\partial \xi ^{{\mathfrak a}}}\left\{ \frac{\partial r_{k}}{\partial t}\right\} n_{k} = \frac{\partial }{\partial \xi ^{{\mathfrak a}}}\left( w^{{\mathfrak b}}\tau _{{\mathfrak b}}^{k} + {\mathcal U}n^{k} \right) \nonumber \\&= \left\{ \frac{\partial w^{{\mathfrak b}}}{\partial \xi ^{{\mathfrak a}}}\tau _{{\mathfrak b}}^{k} + w^{{\mathfrak b}} \underbrace{\frac{\partial \tau _{\mathfrak b}^{k}}{\partial \xi ^{{\mathfrak a}}}}_{(\varGamma _{{\mathfrak a\mathfrak b}}^{\mathfrak c}\tau _{\mathfrak c}^{k}+b_{{\mathfrak a\mathfrak b}}n^{k}), \; \text {see} \; (32.111)_1} + \frac{\partial {\mathcal U}}{\partial \xi ^{{\mathfrak a}}}n^{k} + {\mathcal U}\frac{\partial n^{k}}{\partial \xi ^{{\mathfrak a}}}\right\} n^{k} \nonumber \\&= \left( w^{{\mathfrak b}}b_{{\mathfrak a\mathfrak b}} + \frac{\partial {\mathcal U}}{\partial \xi ^{{\mathfrak a}}}\right) , \end{aligned}$$

(32.254)

as well as

$$\begin{aligned} \dot{e}_{{\mathfrak a}}^{{\mathfrak c}}&= \frac{\partial \tau _{{\mathfrak a}}^{k}}{\partial t}\tau _{k}^{{\mathfrak c}} = \frac{\partial }{\partial t} \left\{ \frac{\partial r^{k}}{\partial \xi ^{{\mathfrak a}}}\right\} \tau _{k}^{{\mathfrak c}} = \frac{\partial }{\partial \xi ^{{\mathfrak a}}}\left( w^{{\mathfrak b}}\tau _{\mathfrak b}^{k} + {\mathcal U}n^{k}\right) \tau _{k}^{{\mathfrak c}} \nonumber \\&= \left\{ \frac{\partial w^{{\mathfrak b}}}{\partial \xi ^{{\mathfrak a}}}\tau _{{\mathfrak b}}^{k} + w^{{\mathfrak b}}\underbrace{ \frac{\partial \tau _{{\mathfrak b}}^{k}}{\partial \xi ^{{\mathfrak a}}}}_{\varGamma _{{\mathfrak a\mathfrak b}}^{{\mathfrak d}}\tau _{\mathfrak d}^{k}+b_{{\mathfrak a\mathfrak b}}n^{k}} + \frac{\partial {\mathcal U}}{\partial \xi ^{{\mathfrak a}}}n^{k} + {\mathcal U}\underbrace{\frac{\partial n^{k}}{\partial \xi ^{\mathfrak a}}}_{-b_{\mathfrak a\mathfrak d}\tau ^{{\mathfrak d}k}}\right\} \tau _{k}^{\mathfrak c} \nonumber \\&= \frac{\partial w^{{\mathfrak b}}}{\partial \xi ^{{\mathfrak a}}}\tau _{\mathfrak b}^{k}\tau _{k}^{{\mathfrak c}} + \left\{ w^{{\mathfrak b}}\left( \varGamma _{{\mathfrak a\mathfrak b}}^{\mathfrak d}\tau _{\mathfrak d}^{k} + b_{{\mathfrak a\mathfrak b}}n^{k}\right) + \frac{\partial {\mathcal U}}{\partial \xi ^{\mathcal a}}n^{k} + {\mathcal U}(-b_{{\mathcal ad}}\tau ^{{\mathfrak d}k})\right\} \tau _{k}^{\mathfrak c} \nonumber \\&= \frac{\partial w^{{\mathfrak b}}}{\partial \xi ^{{\mathfrak a}}}\delta _{{\mathfrak b}}^{{\mathfrak d}} + w^{{\mathfrak b}}\varGamma _{{\mathfrak a\mathfrak b}}^{{\mathfrak d}}\underbrace{\tau _{{\mathfrak d}}^{k}\tau _{k}^{{\mathfrak c}}}_{\delta _{{\mathfrak d}}^{{\mathfrak c}}} - b_{{\mathfrak a\mathfrak d}}{\mathcal U} \underbrace{\tau ^{{\mathfrak d}k}\tau _{k}^{{\mathfrak c}}}_{g^{{\mathfrak dc}}} \nonumber \\&= \frac{\partial w^{{\mathfrak c}}}{\partial \xi ^{{\mathfrak a}}} + w^{{\mathfrak b}}\varGamma _{{\mathfrak a\mathfrak b}}^{\mathfrak c} - b_{{\mathfrak a\mathfrak d}}g^{\mathfrak dc}. \end{aligned}$$

(32.255)

Relations (32.254) and (32.255) prove (32.249)$_4$.

We also need for a scalar valued $f$, for vector fields ${\varvec{u}}, {\varvec{v}}$ and for a second order tensor field ${\varvec{T}}$, defined on the surface ${\mathcal S}$ the following rules of differentiation

$$\begin{aligned} \begin{array}{c} \displaystyle \mathrm {Div}\,(f{\varvec{v}})=f \mathrm {Div}\,{\varvec{v}}+{\mathrm {Grad}}\,f\cdot {\varvec{v}}, \quad \mathrm {Div}\,(f{\varvec{T}})=f \mathrm {Div}\, {{\varvec{T}}} +{{\varvec{T}}}{\mathrm {Grad}}\,f, \\ \displaystyle \mathrm {Div}\,({{\varvec{u}}}\otimes {{\varvec{v}}})=v^{\mathfrak a}\frac{\partial {\varvec{u}}}{\partial \xi ^{\mathfrak a}}+ (\mathrm {Div}\,{{\varvec{v}}})\,{{\varvec{u}}}, \quad \mathrm {Div}\,{\varvec{n}}=-2K,\quad \mathrm {Div}\, ({\varvec{n}}\otimes {\varvec{n}})=-2K{\varvec{n}},\\ \displaystyle \mathrm {Div}\,({\varvec{n}}\otimes {{\varvec{\tau }}}_{\mathfrak a})=-b_{\mathfrak a\mathfrak b}{{\varvec{\tau }}}^{\mathfrak b}+\varGamma ^{\mathfrak b}_{\mathfrak a\mathfrak b}{\varvec{n}},\quad \mathrm {Div}\,( {{\varvec{\tau }}}_{\mathfrak a}\otimes {\varvec{n}})=-2K {{\varvec{\tau }}}_{\mathfrak a}, \end{array} \end{aligned}$$

(32.256)

where

$$\begin{aligned} {\mathrm {Grad}}\,f\equiv \frac{\partial \,f}{\partial \xi ^{\mathfrak a}}\,{{\varvec{\tau }}}^{\mathfrak a}, \quad \mathrm {Div}\, {\varvec{v}}\equiv \frac{\partial {\varvec{v}}}{\partial \xi ^{\mathfrak a}}\cdot {{\varvec{\tau }}}^{\mathfrak a},\quad \mathrm {Div}\,{{\varvec{T}}}\equiv \frac{\partial {{\varvec{T}}}}{\partial \xi ^{\mathfrak a}}\,{{\varvec{\tau }}}^{\mathfrak a}. \end{aligned}$$

Thus, using the decomposition

$$\begin{aligned} {\varvec{v}}_{\mathcal {S}}={\varvec{v}}_{{\mathcal {S}}\parallel }+{\mathcal U}{\varvec{n}}=v^{\mathfrak a}{{\varvec{\tau }}}_{\mathfrak a}+{\mathcal U}{\varvec{n}}, \end{aligned}$$

we obtain

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial }{\partial \,t}(\mu {\varvec{v}}_{\mathcal {S}})\!=\!\frac{\partial \mu v^{\mathfrak a}}{\partial \,t}\,{{\varvec{\tau }}}_{\mathfrak a}+ \mu v^{\mathfrak b}\left\{ \frac{\partial w^{\mathfrak a}}{\partial \xi ^{\mathfrak b}}+w^{\mathfrak c}\varGamma ^{\mathfrak a}_{\mathfrak c\mathfrak b} - {\mathcal U}b_{\mathfrak b\mathfrak c} g^{\mathfrak c\mathfrak a}\right\} {{\varvec{\tau }}}_{\mathfrak a} -\\ \displaystyle \quad \quad \quad \quad \;\; \mu {\mathcal U} g^{\mathfrak a\mathfrak b}\left\{ \frac{\partial \mathcal U}{\partial \xi ^{\mathfrak b}} +b_{\mathfrak b\mathfrak c}w^{\mathfrak c} \right\} {{\varvec{\tau }}}_{\mathfrak a}+ \left\{ \frac{\partial \mu \,\mathcal U}{\partial t}+\mu v^{\mathfrak a}\frac{\partial \mathcal U}{\partial \xi ^{\mathfrak a}} +\mu b_{\mathfrak b\mathfrak a}v^{\mathfrak a} w^{\mathfrak b}\right\} {\varvec{n}}. \end{array} \end{aligned}$$

(32.257)

Then,

$$\begin{aligned}&\quad \mathrm {Div}\,(\mu {\varvec{v}}_{{\mathcal {S}}}\otimes {\varvec{v}}_{{\mathcal {S}}})\nonumber \\&= \mathrm {Div}\,(\mu {\varvec{v}}_{{\mathcal {S}}\parallel }\otimes {\varvec{v}}_{{\mathcal {S}}\parallel })+\mathrm {Div}\,(\mu {\mathcal U}{\varvec{v}}_{{\mathcal {S}}\parallel }\otimes {\varvec{n}})+\mathrm {Div}\,(\mu {\mathcal U}{\varvec{n}}\otimes {\varvec{v}}_{{\mathcal {S}}\parallel })+\mathrm {Div}\,(\mu {\mathcal U}^2{\varvec{n}}\otimes {\varvec{n}})\nonumber \\&=\mathrm {Div}\,(\mu {\varvec{v}}_{{\mathcal {S}}\parallel }\otimes {\varvec{v}}_{{\mathcal {S}}\parallel })-\mu {\mathcal U}\,v^{\mathfrak b} \,b_{\mathfrak b\mathfrak c}\,g^{\mathfrak c\mathfrak a}\,{{\varvec{\tau }}}_{\mathfrak a}+\mathrm {Div}\,(\mu {\mathcal U}{\varvec{v}}_{{\mathcal {S}}\parallel })\,{\varvec{n}}-2\mu K{\mathcal U}{\varvec{v}}_{\mathcal {S}}, \end{aligned}$$

(32.258)

and with the notations (32.181) for the components of ${{\varvec{R}}}_{\mathcal {S}}$,

$$\begin{aligned} \displaystyle \mathrm {Div}{{\varvec{R}}}_{\mathcal {S}}&= \mathrm {Div}\,(S^{\mathfrak a\mathfrak b}{{\varvec{\tau }}}_{\mathfrak a}\otimes {{\varvec{\tau }}}_{\mathfrak b}) - \left\{ S^{\mathfrak c} b_{\mathfrak c\mathfrak b}g^{\mathfrak b\mathfrak a} + 2KS^{\mathfrak a}\right\} {{\varvec{\tau }}}_{\mathfrak a}\nonumber \\&+\left\{ \mathrm {Div}\,(S^{\mathfrak a}{{\varvec{\tau }}}_{\mathfrak a})-2SK\right\} {\varvec{n}}. \end{aligned}$$

(32.259)

Finally, we have

$$\begin{aligned} \displaystyle \frac{\partial }{\partial \xi ^{\mathfrak b}}(\mu {\varvec{v}}_{\mathcal {S}})\,w^{\mathfrak b}&= w^{\mathfrak b}\left\{ \frac{\partial \mu v^{\mathfrak a}}{\partial \xi ^{\mathfrak b}}+ \mu v^{\mathfrak c}\,\varGamma ^{\mathfrak a}_{\mathfrak c\mathfrak b} -\mu {\mathcal U}\,b_{\mathfrak b\mathfrak c} g^{\mathfrak c\mathfrak a} \right\} {{\varvec{\tau }}}_{\mathfrak a}\nonumber \\&+ \displaystyle w^{\mathfrak b}\left\{ \mu v^{\mathfrak c}\,b_{\mathfrak c\mathfrak b}+\frac{\partial \mu \mathcal U}{\partial \xi ^{\mathfrak b}}\right\} {\varvec{n}}. \end{aligned}$$

(32.260)

Now, substituting (32.257)–(32.260) into (32.248) and separating the tangential and normal parts of the emerging relation yields (32.182) and (32.183).