Extrema of advected scalars

That no temperature field, being a solution of the steady-state energy equation (2.55), can possess a strong relative maximum or minimum at an interior point of its domain of existence follows from the essential elliptic nature of the equation and the absence of sources or sinks.

Theorem 1 (Nonexistence of extrema of advected scalars)   If a scalar field, $s$, is a regular solution of the steady-state advection-diffusion equation

$$\mbox{\boldmath$\nabla$}\cdot [-\mathsf{A}\cdot\mbox{\boldmath$\nabla$}s+\sum_{i=1}^N h_{(i)}(s)\mathbf{v}_{(i)}]=0$$ (2.76)

in some domain for some scalar functions $h_{(i)}$, solendoidal vector fields, $\mathbf{v}_{(i)}$, and symmetric, positive definite tensor field, $\mathsf{A}$; all continuously differentiable; then there are no interior strong relative extrema of $s$.

Proof: The idea for this proof, suggested by Prof. Bob Street (1999, pers. comm., 4 Feb.), is to recast the equation in quasilinear elliptic form, for which the result is known.

Carrying out the divergence,

$$\textsf{A}\raisebox{0.4ex}{\textbf{:}}\mbox{\boldmath$\nabla... ... \mathbf{v}_{(i)}\cdot h'_{(i)}(s)\mbox{\boldmath$\nabla$}s =0$$ (2.77)

or

$$\mathsf{A}\raisebox{0.4ex}{\textbf{:}}\mbox{\boldmath$\nabla... ...boldmath$\nabla$}s)+\mathbf{v}\cdot\mbox{\boldmath$\nabla$}s=0$$ (2.78)

where

$$\mathbf{v}\equiv\mbox{\boldmath$\nabla$}\cdot \mathsf{A}-\sum_{i=1}^N h'_{(i)}(s)\mathbf{v}_{(i)}.$$ (2.79)

In Cartesian tensor notation with the summation convention in force, this is

$$A_{jk}s_{,jk} + v_{j}s_{,j}=0,$$ (2.80)

which is of the form for which Hopf's Maximum Principle is shown to hold in treatises on partial differential equations (Courant & Hilbert 1962, pp. 320-8; Garabedian 1964, pp. 227-38).$\Box$

Alternative Proof: I offer here an original and quite different proof which I hope, by avoiding the artifice of a comparison function and using vectorial concepts rather than a general calculus of several variables, is more conducive to physical intuition. The use of vectors also ensures that the result is not tied to any particular coordinate system. The reader uncomfortable with a tensorial diffusivity may replace $\textsf{A}\cdot\mbox{\boldmath$\nabla$}s$ by $A\mbox{\boldmath$\nabla$}s$ wherever it appears; this is the special case of isotropy.

The proof is by contradiction: assume that there does exist an interior relative extremum. For definiteness, and without loss of generality, take this to be a minimum.

Construct a family of rays originating at the minimum and terminating when they encounter either:

(i)

a boundary point of the domain; or

(ii)

a stationary point, with respect to the ray, of $s$; i.e. $\mbox{\boldmath$\hat{r}$}\cdot\mbox{\boldmath$\nabla$}s=0$, where $\hat{r}$ is the unit radial vector from the minimum.

Except at the origin, and possibly the rays' termini, $s$ is strictly increasing along the rays:

$$\mbox{\boldmath$\hat{r}$}\cdot\mbox{\boldmath$\nabla$}s> 0,$$ (2.81)

by the definitions of a minimum and the rays (ii). Choose a value $s_1$ of $s$ between that at the minimum and the least of those at the rays' termini. Let $S$ be the set of points with $s=s_1$ passed through by the rays.

Each ray intersects $S$ exactly once, and, since $s$ possesses at least two continuous spatial derivatives (Gresho 1988), $S$ is closed and smooth enough to have a well-defined unit outward normal, $\hat{n}$. No ray is tangent to $S$, since then the ray should have terminated, by (ii); thus,

$$\mbox{\boldmath$\hat{r}$}\cdot\mbox{\boldmath$\hat{n}$}> 0.$$ (2.82)

Now, by definition of the vector triple product,

$$\mbox{\boldmath$\hat{r}$}\times(\mbox{\boldmath$\hat{n}$}\ti... ...{r}$}\cdot\mbox{\boldmath$\hat{n}$})\mbox{\boldmath$\nabla$}s,$$ (2.83)

but $\mbox{\boldmath$\hat{n}$}\times\mbox{\boldmath$\nabla$}s=0$, since the normal of a level surface is parallel to the gradient; therefore,

$$\mbox{\boldmath$\hat{n}$}\cdot(\textsf{A}\cdot\mbox{\boldmat... ...h$\nabla$}s\cdot(\textsf{A}\cdot\mbox{\boldmath$\nabla$}s) > 0$$ (2.84)

by (2.81), (2.82) and since $\textsf{A}$ is positive definite.

Integrate the steady-state advection-diffusion equation (2.76) over the volume $V$ enclosed by $S$:

$$\int\!\!\!\int\!\!\!\int _{V} \mbox{\boldmath$\nabla$}\cdot ... ...$}s +\sum_{i=1}^N h_{(i)}(s) \mathbf{v}_{(i)}] \,\mathrm{d}V.$$ (2.85)

Applying the divergence theorem gives:

$$\sum_{i=1}^N\int\!\!\!\int _S \mbox{\boldmath$\hat{n}$}\cdot... ...}$}\cdot\mathsf{A}\cdot\mbox{\boldmath$\nabla$}s\,\mathrm{d}S,$$ (2.86)

of which the right hand side is positive by (2.84). The left hand side, however, vanishes;

$$\int\!\!\!\int _S \mbox{\boldmath$\hat{n}$}\cdot h_{(i)}(s)\mathbf{v}_{(i)} \,\mathrm{d}S = h_{(i)}(s_1)\int\!\!\!\int _S \mbox{\boldmath$\hat{n}$}\cdot\mathbf{v}_{(i)} \,\mathrm{d}S$$ (2.87)

$$= h_{(i)}(s_1)\int\!\!\!\int\!\!\!\int _V \mbox{\boldmath$\nabla$}\cdot\mathbf{v}_{(i)} \,\mathrm{d}V=0;$$ (2.88)

by virtue of the hypotheses on the $\mathbf{v}_{(i)}$.

This is a contradiction, so that the theorem is proved.$\Box$

Notes:

• There would always be a net diffusion through a closed level surface surrounding a strong relative extremum, but the net advection would vanish.
• The application to the multicomponent energy equation is clear (cf. Bird et al. 1960, pp. 561-6). The variables $s$, $\mathsf{A}$, $h_{(i)}$ and $\mathbf{v}_{(i)}$ are the temperature, (tensor) conductivity, partial specific enthalpies and absolute species fluxes, respectively. The required assumption is that the partial specific enthalpies are independent of pressure and composition.
• Often, as in this project, the diffusivity is isotropic; i.e. a product of a (positive) scalar field and the Kronecker delta; and so is symmetric and positive definite, as required.
• The diffusivity and velocities can depend on $s$, so that the equation is only quasilinear. In the proof, $s$ is assumed given, so that $\mathsf{A}$ and the $\mathbf{v}_{(i)}$ can be re-expressed as functions of position.
• Completely analogous theorems hold in one and two dimensions.
• In the special case $\mathbf{v}_{(i)}=\mathbf{0}$ and $A_{ij}=A\delta_{ij}$, where $A$ is a constant, the steady-state advection-diffusion equation (2.76) reduces to Laplace's equation, for which the corresponding result is classical (Lamb 1932, p. 39).
• The species (2.53) and energy (2.55) equations both fall under the hypotheses of the theorem.
  • For (2.53), take $s=m$, $A_{ij}=\delta_{ij}$, $N=1$, $h_{(1)}(m)=\mbox{\textit{Sc}}\;m$ and $\mathbf{v}_{(1)}=\mbox{\textit{Gr}}(1+N)\;\mathbf{u}$. The hypothesis on $\mathbf{v}_{(1)}$ follows from (2.52).
  • For (2.55), take $s=T$, $A_{ij}=\delta_{ij}$, $N=2$ and
    

    $$h_{(1)}(T) = \mbox{\textit{Pr}}_r T,$$ (2.89)

    $$\mathbf{v}_{(1)} = \mbox{\textit{Gr}}(1+N)\;\mathbf{u},$$ (2.90)

    $$h_{(2)}(T) = \frac{\mbox{\textit{Pr}}_r}{\mbox{\textit{Sc}}}\left(1-\mathrm{e}^{-\varPhi }\right)T, \quad\mbox{and}$$ (2.91)

    $$\mathbf{v}_{(2)} = \mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\;m\;\mathbf{u}-\mbox{\boldmath$\nabla$}m.$$ (2.92)

    
    
    The hypotheses on the $\mathbf{v}_{(i)}$ follow from the equations of continuity of the mixture (2.52) and the vapour (2.53).
• Thus, no extrema of vapour mass fraction or temperature are expected in the solutions of our system of equations (2.52)-(2.55), and one can conclude that the temperature minimum discovered by Weaver & Viskanta (1991a; referred to in §2.1.3) was erroneous.
• Extrema might occur if there were source or sink terms in the equations, such as if the vapour condensed in the domain; the Dufour effect were appreciable; or there were viscous heating.
• Extrema are of course possible in transient advection-diffusion, as for example they may be specified as part of the initial conditions. I leave unanswered the question of whether strong local extrema can arise in the evolution of a scalar field; this was predicted in the two-dimensional numerical solutions of Bergman and Hyun (1996) for the mass fraction of tin in a nonisothermal amalgam with lead.