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For real base-flow velocity and positive Reynolds number, multiply the homogeneous Squire equation by the conjugate mode and integrate with zero endpoint values. Taking real parts gives the displayed negative temporal growth rate. This proves decay of homogeneous Squire modes; the coupled shear-flow system can still be unstable or show transient growth because wall-normal velocity can force the Squire variable.

An Orr-Sommerfeld mode has nonzero wall-normal velocity satisfying the Orr-Sommerfeld equation. Its wall-normal vorticity satisfies the accompanying forced Squire equation. The triangular coupling separates the velocity eigenproblem from the vorticity forcing; the vorticity need not vanish in three dimensions.

Multiply the homogeneous Squire equation by $\bar{\eta }$ and integrate over the channel. The zero boundary values remove the integration by parts boundary term, givingFor real $U$ and $k$, the middle term is purely imaginary. Taking real parts yields the dissipation identity for a homogeneous Squire mode:The strict inequality holds for a nonzero mode with zero endpoint values and positive Reynolds number. Hence all homogeneous Squire modes are temporally stable. In the complete shear-flow perturbation system, the Squire variable can also be forced by the wall-normal velocity; this homogeneous-mode result does not rule out instability of that coupled system or its transient growth.

Use incompressible Euler equations with pressure divided by the constant reference density. Writing $L={\partial }_t+U(y){\partial }_x$, their linearization about the parallel shear flow isTaking the divergence requires the commutator $[{\partial }_y,L]=U^{\prime }{\partial }_x$. Both this commutator and the divergence of $U^{\prime }v$ contribute, so ${\nabla }^{2}p=-2U^{\prime }{v}_x$. Applying the Laplacian to $Lv=-p_y$ givesHence the wall-normal velocity equation isTaking ${\partial }_z$ of the streamwise momentum equation minus ${\partial }_x$ of the spanwise equation cancels pressure and yields the inviscid Squire equation,This is the vorticity component in the original PDF; the converted TeX corrupts its definition. Impermeable inviscid walls impose $v(\pm 1,t)=0$. They do not impose a viscous no-slip condition on the tangential perturbation velocity.

For a Fourier mode, set $k^2={\alpha }^2+{\beta }^2$, $D=d/dy$ and $q=(D^2-k^2)\tilde{v}$. The equations becomeFor Couette flow $U=\lambda y$, $U^{\prime \prime }=0$, so $q(y,t)=e^{-i\alpha \lambda yt}{q}_0(y)$ with $q_0=(D^2-k^2){\tilde{v}}_0$. Invert this using the Dirichlet Green function of $D^2-k^2$ on $[-1,1]$. If $y_{<}=\min (y,s)$ and $y_{>}=\max (y,s)$, it isIt vanishes at both walls and its first derivative has jump one at $y=s$. ThusAt $t=0$, uniqueness of the Dirichlet inversion recovers the compatible initial velocity. This inviscid Couette initial-value Green function directly solves the initial-value problem without a Laplace transform.

The Squire equation is now a first-order forced equation in time. Its integrating factor givesFor nonzero $\alpha$, the velocity formula contains shearing phases and the inviscid Couette continuous spectrum. Velocity cancellation can occur even though the transported $q$ has constant magnitude; it must not be confused with exponential modal instability.

For the separate case $\alpha =0$, $\beta \ne 0$, the Dirichlet operator $D^2-{\beta }^2$ is invertible and $q_t=0$. Hence $\tilde{v}={\tilde{v}}_0$ andSince $\tilde{\eta }=i\beta \tilde{u}$ in this case, $\tilde{u}={\tilde{u}}_0-\lambda t{\tilde{v}}_0$. This is the lift-up effect: a transverse disturbance transports base-flow momentum and creates a streamwise perturbation growing linearly in time, with perturbation kinetic energy generically growing as $t^2$. The growth is algebraic and is possible without an exponentially unstable normal mode. For nonzero initial normal velocity and nonzero shear, nonlinear effects eventually limit the validity of the linearized calculation. The same streamwise-independent result holds with $\lambda$ replaced by $U^{\prime }(y)$ for a general parallel profile.

Choose a length $\ell$, velocity $U_s$, time $\ell /U_s$, and pressure $\rho U_s^2$, giving Reynolds number $Re=U_s\ell /\nu$. Write perturbation velocity as $(u,v,w)$ and let $\mathcal{L}={\partial }_t+U{\partial }_x-R{e}^{-1}{\nabla }^2$. Linearizing the incompressible Navier-Stokes equations givesTaking the divergence gives ${\nabla }^{2}p=-2U^{\prime }{v}_x$. Applying ${\nabla }^2$ to the wall-normal momentum equation and usingthen cancels the pressure derivatives and yields $\mathcal{L}{\nabla }^{2}v-U^{\prime \prime }{v}_x=0$. Define wall-normal vorticity $\eta =u_z-w_x$. Applying ${\partial }_z$ to the streamwise equation minus ${\partial }_x$ to the spanwise equation gives $\mathcal{L}\eta =-U^{\prime }{v}_z$.

For the assumed normal modes, set $D=d/dy$ and ${\kappa }^2={\alpha }^2+{\beta }^2$. The two equations become the Orr-Sommerfeld equation and Squire equation:At rigid no-slip walls the conditions are $\hat{v}=D\hat{v}=\hat{\eta }=0$ for nonzero horizontal wavenumber.

An Orr-Sommerfeld mode has $\hat{v}\ne 0$ satisfying the Orr-Sommerfeld equation, with accompanying vorticity satisfying the forced Squire equation. A Squire mode has $\hat{v}=0$ and $\hat{\eta }\ne 0$, so its Squire equation is homogeneous.

For a Squire mode multiply that homogeneous equation by $\overline{\eta }$ and integrate over the channel. Under the no-slip condition $\eta =0$ at the walls, integration by parts givesSince $U,\alpha$ are real, taking the real part provesThe numerator is strictly positive for a nontrivial finite-channel mode, and $Re>0$. Thus Squire modes are exponentially damped in the convention $e^{-i\omega t}$. This does not rule out transient growth from non-normal modes in the coupled system.

A Squire mode has zero wall-normal velocity and nonzero wall-normal vorticity, satisfying the homogeneous Squire equation. For real base velocity, positive finite Reynolds number, and zero boundary terms,for a nontrivial mode in a finite channel with no-slip walls. This follows by multiplying the homogeneous equation by $\overline{\eta }$, integrating by parts and taking the real part. Such modes are damped even when the coupled velocity-vorticity system can exhibit transient growth from non-normal modes.