• R G Shandil

Articles written in Proceedings – Mathematical Sciences

• A proof of Howard’s conjecture in homogeneous parallel shear flows

A rigorous mathematical proof of Howard's conjecture which states that the growth rate of an arbitrary unstable wave must approach zero, as the wave length decreases to zero, in the linear instability of nonviscous homogeneous parallel shear flows, is presented here for the first time under the restriction of the boundedness of the second derivative of the basic velocity field with respect to the vertical coordinate in the concerned flow domain.

• A proof of Howard's conjecture in homogeneous parallel shear flows—II: Limitations of Fjortoft's necessary instability criterion

The present paper on the linear instability of nonviscous homogeneous parallel shear flows mathematically demonstrates the correctness of Howard's  prediction, for a class of velocity distributions specified by a monotone functionU of the altitudey and a single point of inflexion in the domain of flow, by showing not only the existence of a critical wave numberkc&gt;0 but also deriving an explicit expression for it, beyond which for all wave numbers the manifesting perturbations attain stability. An exciting conclusion to which the above result leads to is that the necessary instability criterion of Fjortoft has the seeds of its own destruction in the entire range of wave numbersk&gt;kc—a result which is not at all evident either from the criterion itself or from its derivation and has thus remained undiscovered ever since Fjortoft enunciated .

• Eigenvalue bounds for Orr-Sommerfeld equation ‘No backward wave’ theorem

Theoretical estimates of the phase velocityCr of an arbitrary unstable, marginally stable or stable wave derived on the basis of the classical Orr-Sommerfeld eigenvalue problem governing the linear instability of plane Poiseuille flow (U(z)=1−z2,−1≤z≤+1), leave open the possibility of these phase velocities lying outside the rangeUmin&lt;Cr&lt;Umax, but not a single experimental or numerical investigation in this regard, which are concerned with unstable or marginally stable waves has supported such a possibility as yet,Umin andUmax being respectively the minimum and the maximum value ofU(z) forz∈[−1, +1]. This gap between the theory on one side and the experiment and computation on the other has remained unexplained ever since Joseph derived these estimates, first, in 1968, and has even led to the speculation of a negative phase velocity (or rather,Cr&lt;Umin=0) and hence the possibility of a ‘backward’ wave as in the case of the Jeffery-Hamel flow in a diverging channel with back flow (). A simple mathematical proof of the non-existence of such a possibility is given herein by showing that the phase velocityCr of an arbitrary unstable or marginally stable wave must satisfy the inequalityUmin&lt;Cr&lt;Umax. It follows as a consequence stated here in this explicit form for the first time to the best of our knowledge, that ‘overstability’ and not the ‘principle of exchange of stabilities’ is valid for the problem of plane Poiseuille flow.