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Journal articles on the topic 'Orr-Sommerfeld equation'

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Published: 4 June 2021

Last updated: 5 February 2022

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1

Bridges, Thomas J. "The Orr–Sommerfeld equation on a manifold." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455, no. 1988 (1999): 3019–40. http://dx.doi.org/10.1098/rspa.1999.0437.

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2

Herron, Isom H. "The Orr–Sommerfeld Equation on Infinite Intervals." SIAM Review 29, no. 4 (1987): 597–620. http://dx.doi.org/10.1137/1029113.

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3

Puri, Pratap. "Stability and eigenvalue bounds of the flow of a dipolar fluid between two parallel plates." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2057 (2005): 1401–21. http://dx.doi.org/10.1098/rspa.2004.1434.

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In this article, we derive the Orr–Sommerfeld equation for the stability of parallel flows of a dipolar fluid. The classical results found by Squire, for viscous Newtonian fluids, are generalized to the case of dipolar fluids. A sufficient condition for stability is obtained for dipolar fluids and eigenvalue bounds for the Orr–Sommerfeld equation are found.

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4

Alvarez, L. M., and V. V. Ditkin. "On the numerical solution of the Orr-Sommerfeld equation." USSR Computational Mathematics and Mathematical Physics 30, no. 2 (1990): 183–86. http://dx.doi.org/10.1016/0041-5553(90)90094-9.

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5

Pierucci, Mauro, and Pedro G. Morales. "Effect of Finite Thickness Flexible Boundary Upon the Stability of a Poiseuille Flow." Journal of Applied Mechanics 57, no. 4 (1990): 1056–60. http://dx.doi.org/10.1115/1.2897625.

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The stability behavior, the stress, and velocity distributions for a plane Poiseuille flow bounded by a finite thickness elastic layer is studied. The analysis is performed by utilizing the coupled relationships between the Orr-Sommerfeld stability equation for the fluid and the Navier equations for the solid. The numerical instabilities experienced in the solution of the Orr-Sommerfeld equation have been overcome with the use of Davey’s orthonormalization technique. This study focuses only on the Tollimen-Schlichting instabilities. This mode is the most unstable of the three different types of instabilities. The results show that certain combinations of parameters can lead to improved stability conditions. Under these conditions the normal and shear stress distributions may behave completely different in certain regions of the fluid.

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6

Trinh, Ngoc Anh, and Dong Vuong Lap Tran. "Calculation of the Orr-Sommerfeld stability equation for the plane Poiseuille flow." Science and Technology Development Journal - Natural Sciences 2, no. 5 (2019): 122–29. http://dx.doi.org/10.32508/stdjns.v2i5.787.

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The stability of plane Poiseuille flow depends on eigenvalues and solutions which are generated by solving Orr-Sommerfeld equation with input parameters including real wavenumber and Reynolds number . In the reseach of this paper, the Orr-Sommerfeld equation for the plane Poiseuille flow was solved numerically by improving the Chebyshev collocation method so that the solution of the Orr-Sommerfeld equation could be approximated even and odd polynomial by relying on results of proposition 3.1 that is proved in detail in section 2. The results obtained by this method were more economical than the modified Chebyshev collocation if the comparison could be done in the same accuracy, the same collocation points to find the most unstable eigenvalue. Specifically, the present method needs 49 nodes and only takes 0.0011s to create eigenvalue while the modified Chebyshev collocation also uses 49 nodes but takes 0.0045s to generate eigenvalue with the same accuracy to eight digits after the decimal point in the comparison with , see [4], exact to eleven digits after the decimal point.

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7

Lie, K. H., and D. N. Riahi. "Numerical solution of the Orr-Sommerfeld equation for mixing layers." International Journal of Engineering Science 26, no. 2 (1988): 163–74. http://dx.doi.org/10.1016/0020-7225(88)90102-4.

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8

Zakharchuk, V. T., N. P. Savenkova, and S. L. Chernyshov. "An approach to the solution of the Orr?Sommerfeld equation." Computational Mathematics and Modeling 4, no. 2 (1993): 135–39. http://dx.doi.org/10.1007/bf01131207.

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9

Banerjee, Mihir B., R. G. Shandil, and Balraj Singh Bandral. "Eigenvalue bounds for Orr-Sommerfeld equation ‘No backward wave’ theorem." Proceedings Mathematical Sciences 106, no. 3 (1996): 281–87. http://dx.doi.org/10.1007/bf02867436.

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10

Sengupta, Tapan K. "Solution of the Orr-Sommerfeld equation for high wave numbers." Computers & Fluids 21, no. 2 (1992): 301–3. http://dx.doi.org/10.1016/0045-7930(92)90027-s.

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11

Tan, Ying, and Weidong Su. "A trigonometric series expansion method for the Orr-Sommerfeld equation." Applied Mathematics and Mechanics 40, no. 6 (2019): 877–88. http://dx.doi.org/10.1007/s10483-019-2484-9.

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12

Lahmann, J., and M. Plum. "On the Spectrum of the Orr-Sommerfeld Equation on the Semiaxis." Mathematische Nachrichten 216, no. 1 (2000): 145–53. http://dx.doi.org/10.1002/1522-2616(200008)216:1<145::aid-mana145>3.0.co;2-0.

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13

Jacobs, Robert G., and Paul A. Durbin. "Shear sheltering and the continuous spectrum of the Orr–Sommerfeld equation." Physics of Fluids 10, no. 8 (1998): 2006–11. http://dx.doi.org/10.1063/1.869716.

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14

Ortiz de Zárate, J. M., and J. V. Sengers. "Hydrodynamic Fluctuations in Laminar Fluid Flow. I. Fluctuating Orr-Sommerfeld Equation." Journal of Statistical Physics 144, no. 4 (2011): 774–92. http://dx.doi.org/10.1007/s10955-011-0256-1.

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15

Malcolm Brown, B., Matthias Langer, Marco Marletta, Christiane Tretter, and Markus Wagenhofer. "Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics." LMS Journal of Computation and Mathematics 13 (March 24, 2010): 65–81. http://dx.doi.org/10.1112/s1461157008000466.

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AbstractIn this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr–Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds numberR=5772.221818; the Orr–Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixedRand wave numberα; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire’s problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators.

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16

Monschke, Jason A., Matthew S. Kuester, and Edward B. White. "Acoustic Receptivity Measurements Using Modal Decomposition of a Modified Orr–Sommerfeld Equation." AIAA Journal 54, no. 3 (2016): 805–15. http://dx.doi.org/10.2514/1.j054043.

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17

Damaren, Christopher J. "Laminar–Turbulent Transition Control Using Passivity Analysis of the Orr–Sommerfeld Equation." Journal of Guidance, Control, and Dynamics 39, no. 7 (2016): 1602–13. http://dx.doi.org/10.2514/1.g001763.

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18

Long, Lyle N. "Uniformly-valid asymptotic solutions to the Orr-Sommerfeld equation using multiple scales." Journal of Engineering Mathematics 21, no. 3 (1987): 167–78. http://dx.doi.org/10.1007/bf00127462.

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19

Kirchner, N. P. "Computational aspects of the spectral Galerkin FEM for the Orr-Sommerfeld equation." International Journal for Numerical Methods in Fluids 32, no. 1 (2000): 105–21. http://dx.doi.org/10.1002/(sici)1097-0363(20000115)32:1<105::aid-fld938>3.0.co;2-x.

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20

Walker, Ira J., David S. Torain II, and Morris H. Morgan III. "An eigenvalue search method using the Orr–Sommerfeld equation for shear flow." Journal of Computational and Applied Mathematics 236, no. 11 (2012): 2795–802. http://dx.doi.org/10.1016/j.cam.2012.01.012.

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21

Ishkin, Khabir Kabirovich, and Rustem Il'darovich Marvanov. "On localization conditions for spectrum of model operator for Orr - Sommerfeld equation." Ufimskii Matematicheskii Zhurnal 12, no. 4 (2020): 64–77. http://dx.doi.org/10.13108/2020-12-4-64.

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22

Tsugé, Shunichi, and Hiroshi Sakai. "Uniformly valid solution of the Orr-Sommerfeld equation by a modified Heisenberg method." Journal of Fluid Mechanics 153, no. -1 (1985): 167. http://dx.doi.org/10.1017/s0022112085001197.

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23

Lahmann, J. R., and M. Plum. "A computer-assisted instability proof for the Orr-Sommerfeld equation with Blasius profile." ZAMM 84, no. 3 (2004): 188–204. http://dx.doi.org/10.1002/zamm.200310093.

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24

Danabasoglu, Gökhan, and Sedat Biringen. "A Chebyshev matrix method for the spatial modes of the Orr-Sommerfeld equation." International Journal for Numerical Methods in Fluids 11, no. 7 (1990): 1033–37. http://dx.doi.org/10.1002/fld.1650110709.

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25

Jacobi, I., and B. J. McKeon. "Dynamic roughness perturbation of a turbulent boundary layer." Journal of Fluid Mechanics 688 (October 27, 2011): 258–96. http://dx.doi.org/10.1017/jfm.2011.375.

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AbstractThe zero-pressure-gradient turbulent boundary layer over a flat plate was perturbed by a temporally oscillating, spatial impulse of roughness, and the downstream response of the flow field was interrogated by hot-wire anemometry and particle-image velocimetry. The key features common to impulsively perturbed boundary layers, as identified in Jacobi & McKeon (J. Fluid Mech., 2011), were investigated, and the unique contributions of the dynamic perturbation were isolated by contrast with an appropriately matched static impulse of roughness. In addition, the dynamic perturbation was decomposed into separable large-scale and small-scale structural effects, which in turn were associated with the organized wave and roughness impulse aspects of the perturbation. A phase-locked velocity decomposition of the entire downstream flow field revealed strongly coherent modes of fluctuating velocity, with distinct mode shapes for the streamwise and wall-normal velocity components. Following the analysis of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), the roughness perturbation was treated as a forcing of the Navier–Stokes equation and a linearized analysis employing a modified Orr–Sommerfeld operator was performed. The experimentally ascertained wavespeed of the input disturbance was used to solve for the most amplified singular mode of the Orr–Sommerfeld resolvent. These calculated modes were then compared with the streamwise and wall-normal velocity fluctuations. The discrepancies between the calculated Orr–Sommerfeld resolvent modes and those experimentally observed by phase-locked averaging of the velocity field were postulated to result from the violation of the parallel flow assumption of Orr–Sommerfeld analysis, as well as certain non-equilibrium effects of the roughness. Additionally, some difficulties previously observed using a quasi-laminar eigenmode analysis were also observed under the resolvent approach; however, the resolvent analysis was shown to provide reasonably accurate predictions of velocity fluctuations for the forced Orr–Sommerfeld problem over a portion of the boundary layer, with potential applications to designing efficient flow control strategies. The combined experimental and analytical effort provides a new opportunity to examine the non-equilibrium and forcing effects in a dynamically perturbed flow.

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26

Benyza, J., M. Lamine, and A. Hifdi. "Transient energy growth of channel flow with cross-flow." MATEC Web of Conferences 286 (2019): 07008. http://dx.doi.org/10.1051/matecconf/201928607008.

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The effect of a uniform cross flow (injection/ suction) on the transient energy growth of a plane Poiseuille flow is investigated. Non-modal linear stability analysis is carried out to determine the two-dimensional optimal perturbations for maximum growth. The linearized Navier-Stockes equations are reduced to a modified Orr Sommerfeld equation that is solved numerically using a Chebychev collocation spectral method. Our study is focused on the response to external excitations and initial conditions by examining the energy growth function G(t) and the pseudo-spectrum. Results show that, the transient energy of the optimal perturbation grows rapidly at short times and decline slowly at long times when the cross-flow rate is low or strong. In addition, the maximum energy growth is very pronounced in low injection rate than that of the strong one. For the intermediate cross-flow rate, the transient energy growth of the perturbation, is only possible at the long times with a very high-energy gain. Analysis of the pseudo-spectrum show that the non-normal character of the modified Orr-Sommerfeld operator tends to a high sensitivity of pseudo-spectra structures.

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27

Nizamova, A. D. "The influence of temperature dependence of viscosity on stability liquid flows in a plane channel." Proceedings of the Mavlyutov Institute of Mechanics 10 (2014): 90–94. http://dx.doi.org/10.21662/uim2014.1.017.

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The influence problem of temperature dependence of viscosity on stability of liquid flow in a plane channel under non-uniform temperature field was considered. The system of two ordinary differential equations for the perturbation amplitudes of velocity and temperature was received. In the case of isothermal flow, derived system can be reduced to the Orr-Sommerfeld equation. The spectra of eigenvalues for Poiseuille flow with different temperature dependence of viscosity were numerical studied. It is shown that temperature dependence of viscosity has an influence on the stability of the liquid flow.

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28

Georgievskii, D. V., W. H. Müller, and B. E. Abali. "Eigenvalue problems for the generalized Orr-Sommerfeld equation in the theory of hydrodynamic stability." Doklady Physics 56, no. 9 (2011): 494–97. http://dx.doi.org/10.1134/s1028335811090023.

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29

Ziadé, Paul, and Pierre E. Sullivan. "Sensitivity of the Orr–Sommerfeld equation to base flow perturbations with application to airfoils." International Journal of Heat and Fluid Flow 67 (October 2017): 122–30. http://dx.doi.org/10.1016/j.ijheatfluidflow.2017.06.005.

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30

Craik, A. D. D. "The continuous spectrum of the Orr-Sommerfeld equation: note on a paper of Grosch & Salwen." Journal of Fluid Mechanics 226 (May 1991): 565–71. http://dx.doi.org/10.1017/s0022112091002513.

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Grosch & Salwen (1978) discuss the continuous-spectrum contribution in both temporal and spatial stability problems that are governed by the linear Orr–Sommerfeld equation. Their computed temporal continuum eigenfunctions for the Blasius boundary layer and for a laminar jet profile show surprising differences. This note provides an improved physical understanding of the results through a simple model, and shows that these differences are more apparent than real.

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31

Ray, R. N., A. Samad, and T. K. Chaudhury. "Low Reynolds number stability of MHD plane Poiseuille flow of an Oldroyd fluid." International Journal of Mathematics and Mathematical Sciences 23, no. 9 (2000): 617–25. http://dx.doi.org/10.1155/s0161171200002040.

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The linear stability of plane Poiseuille flow at low Reynolds number of a conducting Oldroyd fluid in the presence of a transverse magnetic field has been investigated numerically. Spectral tau method with expansions in Chebyshev polynomials is used to solve the Orr-Sommerfeld equation. It is found that viscoelastic parameters have destabilizing effect and magnetic field has a stabilizing effect in the field of flow. But no instabilities are found.

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32

ABDULLAH, A. A., and K. A. LINDSAY. "SOME REMARKS ON THE COMPUTATION OF THE EIGENVALUES OF LINEAR SYSTEMS." Mathematical Models and Methods in Applied Sciences 01, no. 02 (1991): 153–65. http://dx.doi.org/10.1142/s0218202591000095.

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This paper has been prompted by some recent computations of eigenvalues of the Orr-Sommerfeld equation for very high Reynolds numbers. We have used a spectral analysis to emulate these calculations and our results have motivated some general remarks on the suitability of tracking and spectral methods as numerical eigenvalue schemes in the context of stability theory. Our remarks are further supported by illustrating the development of eigenvalues in the Magnetic Benard.

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33

Ache, Gerardo A., and Debora Cores. "Note on the Two-Point Boundary Value Numerical Solution of the Orr-Sommerfeld Stability Equation." Journal of Computational Physics 116, no. 1 (1995): 180–83. http://dx.doi.org/10.1006/jcph.1995.1016.

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34

Bower, W. W., J. T. Kegelman, A. Pal, and G. H. Meyer. "A numerical study of two-dimensional instability-wave control based on the Orr–Sommerfeld equation." Physics of Fluids 30, no. 4 (1987): 998. http://dx.doi.org/10.1063/1.866287.

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35

Chu, W. Kwang-Hua. "Spectra of the Orr-Sommerfeld equation: the plane Poiseuille flow for the normal fluid revisited." Journal of Physics A: Mathematical and General 34, no. 16 (2001): 3389–92. http://dx.doi.org/10.1088/0305-4470/34/16/306.

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36

Motylev, L. YU. "Asymptotic solutions of the Orr-Sommerfeld equation with a point of rotation of high order." USSR Computational Mathematics and Mathematical Physics 26, no. 6 (1986): 15–20. http://dx.doi.org/10.1016/0041-5553(86)90141-2.

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37

Luchini, Paolo, and François Charru. "On the large difference between Benjamin’s and Hanratty’s formulations of perturbed flow over uneven terrain." Journal of Fluid Mechanics 871 (May 24, 2019): 534–61. http://dx.doi.org/10.1017/jfm.2019.312.

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Flow over an uneven terrain is a complex phenomenon that requires a chain of approximations in order to be studied. In addition to modelling the intricacies of turbulence if present, the problem is classically first linearized about a flat bottom and a locally parallel flow, and then asymptotically approximated into an interactive representation that couples a boundary layer and an irrotational region through an intermediate inviscid but rotational layer. The first of these steps produces a stationary Orr–Sommerfeld equation; since this is a one-dimensional problem comparatively easy for any computer, it would seem appropriate today to forgo the second sweep of approximation and solve the Orr–Sommerfeld problem numerically. However, the results are inconsistent! It appears that the asymptotic approximation tacitly restores some of the original problem’s non-parallelism. In order to provide consistent results, Benjamin’s version of the Orr–Sommerfeld equation needs to be modified into Hanratty’s. The large difference between Benjamin’s and Hanratty’s formulations, which arises in some wavenumber ranges but not in others, is here explained through an asymptotic analysis based on the concept of admittance and on the symmetry transformations of the boundary layer. A compact and accurate analytical formula is provided for the wavenumber range of maximum laminar shear-stress response. We highlight that the maximum turbulent shear-stress response occurs in the quasi-laminar regime at a Reynolds-independent wavenumber, contrary to the maximum laminar shear-stress response whose wavenumber scales with a power of the boundary-layer thickness. A numerical computation involving an eddy-viscosity model provides a warning against the inaccuracy of such a model. We emphasize that the range $k\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}<10^{-3}$ of the spectrum remains essentially unexplored, and that the question is still open whether a fully developed turbulent regime, similar to the one predicted by an eddy-viscosity model, ever exists for open flow even in the limit of infinite wavelength.

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38

Jovanovic, Milos. "Vorticity evolution in perturbed Poiseuille flow." Theoretical and Applied Mechanics 40, no. 1 (2013): 71–86. http://dx.doi.org/10.2298/tam1301071j.

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We consider numerical simulation of temporal hydrodynamic instability with finite amplitude perturbations in plane incompressible Poiseuille flow. Two dimensional Navier Stokes equations have been used and reduced to vorticity-stream function form. Trigonometric polynomials have been used in homogeneous direction and Chebyshev polynomials in inhomogeneous direction. The problem of boundary conditions for vorticity has been solved by using the method of influence matrices. The Orr-Sommerfeld equation has been solved by Chebyshev polynomials, and linear combination of the obtained eigenfunctions has been optimized with regard to the corresponding eigenvalue. We present here the results of simulation for the perturbations optimized in regard to the least stable eigenvalue for the Reynolds number Re=1000.

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39

Nijimbere, Victor. "Asymptotic Approximation of the Eigenvalues and the Eigenfunctions for the Orr-Sommerfeld Equation on Infinite Intervals." Advances in Pure Mathematics 09, no. 12 (2019): 967–89. http://dx.doi.org/10.4236/apm.2019.912049.

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40

Mamou, M., and M. Khalid. "Finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements: plane Poiseuille flow." International Journal for Numerical Methods in Fluids 44, no. 7 (2004): 721–35. http://dx.doi.org/10.1002/fld.661.

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41

McBain, G. D., T. H. Chubb, and S. W. Armfield. "Numerical solution of the Orr–Sommerfeld equation using the viscous Green function and split-Gaussian quadrature." Journal of Computational and Applied Mathematics 224, no. 1 (2009): 397–404. http://dx.doi.org/10.1016/j.cam.2008.05.040.

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42

GASTER, M. "On the growth of waves in boundary layers: a non-parallel correction." Journal of Fluid Mechanics 424 (November 16, 2000): 367–77. http://dx.doi.org/10.1017/s002211200000197x.

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The estimation of the growth of propagating instability waves in laminar boundary layers is considered when the Reynolds number is sufficiently large for the mean flow to deviate only slightly from a truly parallel flow. An approximate solution for the linear perturbation is sought in the form of a scaled solution of the related locally parallel flow problem. The amplitude scaling is chosen so as to satisfy the full linearized perturbation equations as closely as possible by making the mean-square deviation of the remainder a minimum. By re-arranging the terms in the equations so that some of the small correction terms arising from the non-parallel mean flow are contained in the ordinary differential equation (ODE) defining the quasi-parallel flow solution, a useful simplification is obtained for the scaling function. Then a modified Orr–Sommerfeld equation defines the base solution and the differential expression for the scaling that can be integrated forms a simple conservation relation.

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43

Auteri, F., and L. Quartapelle. "Galerkin-Laguerre Spectral Solution of Self-Similar Boundary Layer Problems." Communications in Computational Physics 12, no. 5 (2012): 1329–58. http://dx.doi.org/10.4208/cicp.130411.230911a.

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AbstractIn this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory. An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown. A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows. The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation. The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.

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44

Lamine, M., and A. Hifdi. "Dominant modes in hydromagnetic stability of channel flow with porous walls." MATEC Web of Conferences 286 (2019): 07009. http://dx.doi.org/10.1051/matecconf/201928607009.

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A linear stability analysis of a plane channel flow with porous walls under a uniform cross-flow and an external transverse magnetic field is explored. The physical problem is governed by a system of combined equations of the hydrodynamic and those of Maxwell. The perturbed problem of base state leads to a modified classical Orr-Sommerfeld equation which is solved numerically using the Chebyshev spectral collocation method. The combined eﬀects of the cross-flow Reynolds number and the Hartmann number on the dangerous mode of hydromagnetic stability are investigated.The study shows that, the magnetic field tends to suppress the instability occurred by cross-flow. This stabilizing eﬀect becomes perceptible when the magnetic field produces a mode transition from walls mode to that of the center.

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45

Nizamova, A. D., V. N. Kireev, and S. F. Urmancheev. "Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation." Multiphase Systems 14, no. 1 (2019): 52–58. http://dx.doi.org/10.21662/mfs2019.1.007.

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The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

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46

Banerjee, Mihir B., R. G. Shandil, M. G. Gourla, and S. S. Chauhan. "Eigenvalue Bounds for the Orr-Sommerfeld Equation and Their Relevance to the Existence of Backward Wave Motion." Studies in Applied Mathematics 103, no. 1 (1999): 43–50. http://dx.doi.org/10.1111/1467-9590.00119.

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47

Ishkin, Kh K. "Conditions for localization of the limit spectrum of a model operator associated with the Orr-Sommerfeld equation." Doklady Mathematics 86, no. 1 (2012): 549–52. http://dx.doi.org/10.1134/s1064562412040357.

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48

Theofilis, V. "The discrete temporal eigenvalue spectrum of the generalised Hiemenz flow as solution of the Orr-Sommerfeld equation." Journal of Engineering Mathematics 28, no. 3 (1994): 241–59. http://dx.doi.org/10.1007/bf00058439.

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49

Lam, T. T., and Y. Bayazitoglu. "Solution to the orr-sommerfeld equation for liquid film flowing down an inclined plane: An optimal approach." International Journal for Numerical Methods in Fluids 6, no. 12 (1986): 883–94. http://dx.doi.org/10.1002/fld.1650061203.

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50

Fan, Xiao Chao, Rui Jing Shi, and Bo Wei. "Stability in the Special Saturated Liquid Film along an Inclined Heated Plate." Advanced Materials Research 516-517 (May 2012): 202–7. http://dx.doi.org/10.4028/www.scientific.net/amr.516-517.202.

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Abstract:

Stable analysis of flow and heat transfer in the saturated liquid film of liquid low boiling point gases falling down an inclined heated plate is investigated. Firstly, the boundary value problem of linear stability differential equation (Orr–Sommerfeld equation) on small perturbation is derived representing surface tension by nonlinear relationship on temperature. Then, the expression of the wave velocity is got by solving the boundary value problem of O–S equation using the perturbation method. The effects of the inclined angle and some other factors, such as Reynolds number, wave number, temperature of the plate and the parameter for the physical property, on stability in the saturated liquid film of liquid low boiling point gas N2 are numerically analyzed by MATLAB software. Finally, it is shown and analyzed a new critical Reynolds number which is actually the extension of Yih’s.

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