Two theorems on fully developed flow

Theorem 2 (Unidirectionality of fully developed flow)   Let $\mathbf{u}$ be a regular stationary solution of the incompressible Navier-Stokes equations with a purely vertical body force. If $\mathbf{u}$ vanishes on the boundary of a vertically prismatic domain and has zero vertical gradient then $\mathbf{u}$ is purely vertical.

Proof: Since $\mbox{\boldmath$\hat{\jmath}$}\cdot\mbox{\boldmath$\nabla$}\mathbf{u}=0$, the equation of continuity reduces to

$$\mbox{\boldmath$\nabla$}_{\perp}\cdot\mathbf{u}_{\perp}=0$$ (7.86)

and the horizontal components of the equation of motion are

$$\mathit{Re}\;\mathbf{u}_{\perp}\cdot\mbox{\boldmath$\nabla$}... ...bla$}p + \mbox{\boldmath$\nabla$}_{\perp}^2\mathbf{u}_{\perp},$$ (7.87)

where $\mathit{Re}$ is the Reynolds number.

The remnant of the equation of continuity (7.86) implies the existence of a scalar function, $\psi$ (see Lamb 1932, pp. 62-3 for a construction) such that

$$u = -\frac{1}{\mbox{$\mathcal S$}}\frac{\partial \psi}{\partial Z}$$ (7.88)

$$w = \frac{\partial \psi}{\partial X},$$ (7.89)

so that the horizontal momentum problem becomes

$$\frac{\mathit{Re}}{\mbox{$\mathcal S$}} \left(\frac{\partial... ...partial Z}\right) \nabla_{\perp}^2\psi = -\nabla_{\perp}^4\psi$$ (7.90)

subject to

$$\psi = \frac{\partial \psi}{\partial n}=0 \qquad\mbox{on}\quad \partial\Omega,$$ (7.91)

where $\partial/\partial n$ is the outward normal derivative. On multiplying (7.90) through by $\varphi$, an arbitrary function also satisfying the boundary conditions (7.91), and integrating over the section, the variational form is obtained (Girault & Raviart 1979, p. 120):

$${\frac{\mbox{\textit{Gr}}(1+N)}{\mbox{$\mathcal S$}} \int\!\!\!\i... ...al Z} \right]\, \,\mathrm{d}X\,\mbox{$\mathcal S$}\,\mathrm{d}Z =}\hspace{50mm}$$

$$\int\!\!\!\int\limits_{\Omega} (\nabla_{\perp}^{2}\psi)(\nabla_{\perp}^{2}\varphi)\, \,\mathrm{d}X\,\mbox{$\mathcal S$}\,\mathrm{d}Z.$$

If $\varphi$ is then set equal to $\psi$, the integrand on the left hand side vanishes and the equation reduces to

$$\int\!\!\!\int\limits_{\Omega} \left(\nabla_{\perp}^{2}\psi\right)^{2}\, \,\mathrm{d}X\,\mbox{$\mathcal S$}\,\mathrm{d}Z=0$$ (7.93)

which is only possible if $\nabla_{\perp}^{2}\psi=0$. With the boundary conditions (7.91), and the well-known uniqueness of harmonic functions in bounded domains (Lamb 1932, pp. 41, 64), we have $\psi=0$ throughout $\Omega$ and so $u=w=0$, uniquely. $\Box$

Note: The horizontal components of velocity satisfy a two-dimensional Navier-Stokes problem with no body force and homogeneous boundary conditions. The volume integral (7.93) is proportional to the net rate of viscous dissipation of energy by this horizontal flow per unit length in the vertical direction (Lamb 1932, p. 580). Since, under the hypothesis $\mbox{\boldmath$\hat{\jmath}$}\cdot\mbox{\boldmath$\nabla$}\mathbf{u}=0$, the only available source for this power is the kinetic energy of the fluid, any existing horizontal motion would have to continually diminish in magnitude; thus, the entire proof is similar to that of the uniqueness of creeping flows with prescribed boundary velocities (Lamb 1932, pp. 617-8; Rayleigh 1913), even though the inertial terms of the equation of motion were not neglected here.

Theorem 3 (Vortex-lines in fully developed flow)   If a flow has zero gradient in some direction, the component of velocity in this direction is constant along vortex-lines.

Proof: The hypothesis may be written

$$\mbox{\boldmath$\hat{e}$}\cdot\mbox{\boldmath$\nabla$}\mathbf{u} = 0,$$

where $\mathbf{e}$ is a unit vector. Adding three terms (the first and third of which are identically zero due to the uniformity of $\hat{e}$) to each side of the hypothesis and forming the scalar product with the vorticity gives

$$(\mbox{\boldmath$\nabla$}\times\mathbf{u})\cdot[ \mbox{\bol... ...th$\hat{e}$}\times(\mbox{\boldmath$\nabla$}\times\mathbf{u})].$$

The triple scalar product on the right hand side vanishes, while the term in brackets on the left hand side is simply $\mbox{\boldmath$\nabla$}(\mbox{\boldmath$\hat{e}$}\cdot\mathbf{u})$; thus,

$$(\mbox{\boldmath$\nabla$}\times\mathbf{u})\cdot\mbox{\boldmath$\nabla$}(\mbox{\boldmath$\hat{e}$}\cdot\mathbf{u}) = 0;$$

i.e. the component of velocity in the direction of zero gradient is constant along vortex-lines. $\Box$

Note: The hypothesis of the theorem includes all two-dimensional flows and all unidirectional solenoidal flows. The theorem is entirely kinematical; the only restriction placed on the velocity field is differentiability.