We prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less than Re-Orr (2 pi/(lambda sin theta))/sin theta when a perturbation is a tilted perturbation in the direction x' which forms an angle theta is an element of(0, pi/2] with the direction i of the basic motion and does not depend on x'. Re-Orr is the critical Orr-Reynolds number for spanwise perturbations which is computed for wave number 2 pi/(lambda sin theta), with lambda being any positive wavelength. By taking the minimum with respect to lambda, we obtain the critical energy Reynolds number for a fixed inclination angle and any wavelength: for plane Couette flow, it is Re-Orr = 44.3/sin theta, and for plane Poiseuille flow, it is Re-Orr = 87.6/sin theta (in particular, for theta = pi/2 we have the classical values Re-Orr = 44.3 for plane Couette flow and Re-Orr = 87.6 for plane Poiseuille flow). Here the nondimensional interval between the planes bounding the channel is [-1, 1]. In particular, these results improve those obtained by Joseph, who found for streamwise perturbations a critical nonlinear value of 20.65 in the plane Couette case, and those obtained by Joseph and Carmi who found the value 49.55 for plane Poiseuille flow for streamwise perturbations. If we fix some wavelengths from the experimental data and the numerical simulations, the critical Reynolds numbers that we obtain are in a very good agreement both with the the experiments and the numerical simulation. These results partially solve the Couette-Sommerfeld paradox

### Nonlinear stability results for plane Couette and Poiseuille flows

#### Abstract

We prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less than Re-Orr (2 pi/(lambda sin theta))/sin theta when a perturbation is a tilted perturbation in the direction x' which forms an angle theta is an element of(0, pi/2] with the direction i of the basic motion and does not depend on x'. Re-Orr is the critical Orr-Reynolds number for spanwise perturbations which is computed for wave number 2 pi/(lambda sin theta), with lambda being any positive wavelength. By taking the minimum with respect to lambda, we obtain the critical energy Reynolds number for a fixed inclination angle and any wavelength: for plane Couette flow, it is Re-Orr = 44.3/sin theta, and for plane Poiseuille flow, it is Re-Orr = 87.6/sin theta (in particular, for theta = pi/2 we have the classical values Re-Orr = 44.3 for plane Couette flow and Re-Orr = 87.6 for plane Poiseuille flow). Here the nondimensional interval between the planes bounding the channel is [-1, 1]. In particular, these results improve those obtained by Joseph, who found for streamwise perturbations a critical nonlinear value of 20.65 in the plane Couette case, and those obtained by Joseph and Carmi who found the value 49.55 for plane Poiseuille flow for streamwise perturbations. If we fix some wavelengths from the experimental data and the numerical simulations, the critical Reynolds numbers that we obtain are in a very good agreement both with the the experiments and the numerical simulation. These results partially solve the Couette-Sommerfeld paradox
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/20.500.11769/365600`
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