Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 3 / d

Take $k^{\ast }>0$ and positive shear scale $S^{\ast }$, with kinematic viscosity ${\nu }^{\ast }$. ChooseThe velocity scale is $S^{\ast }/k^{\ast }$. A reversed shear can be treated by reversing the corresponding coordinate orientation and mode convention. The dimensional Orr-Sommerfeld equation, with $U^{\ast \prime \prime }=0$, isSubstitution of the scales and cancellation gives

In two dimensions at nondimensional streamwise wavenumber one, incompressibility gives $i\hat{u}+D\hat{v}=0$, so $\hat{u}=iD\hat{v}$. The spanwise vorticity is the $z$ component of the curl:ConsequentlyThe symbol $\hat{\omega }$ here denotes a vorticity amplitude, not the temporal frequency used elsewhere. In dimensional variables, ${\hat{\zeta }}^{\ast }=-i(D_{\ast }^2-(k^{\ast }{)}^2){\hat{v}}^{\ast }/k^{\ast }$, with the same sign convention.

Put $W=(D^2-1)\hat{v}$ and ${\delta }^3=R{e}^{-1}$. The constant-shear Orr-Sommerfeld equation givesLet $a=e^{i\pi /6}$ and $Z=a(y-c-i{\delta }^3)/\delta$. Since $d/dy=(a/\delta )d/dZ$ and $a^3=i$, the equation becomesThis is the Airy ordinary differential equation, yielding the Airy reduction of the Orr-Sommerfeld equation in constant shear. It is distinct from the evolution PDE represented by the existing Airy equation article.

Write $W(s)=a_1\mathrm{Ai}(Z(s))+a_2\mathrm{Bi}(Z(s))$, using the independent Airy functions. To solve $(D^2-1)\hat{v}=W$, the variation of parameters formula, equivalently a direct convolution, gives the velocity reconstruction from the Orr-Sommerfeld vorticity:The reference point $y_0$ is arbitrary. Twice differentiating the integral produces $W(y)$ because $\sinh 0=0$ and its derivative at zero is one, verifying the formula. The four constants are independent: applying $D^2-1$ to an identically zero combination first forces $a_1=a_2=0$, then the complementary solutions force $C_{+}=C_{-}=0$. The boundary conditions would be $\hat{v}(y_j)=D\hat{v}(y_j)=0$, $j=1,2$. Their compatibility selects the eigenvalues; they need not be imposed here.