TY - JOUR

T1 - The instability of sheared liquid layers

AU - Smith, Marc K.

AU - Davis, Stephen H.

N1 - Funding Information:
This research has been supported in part by grants from 'Fundacao de Amparo a Pesquisa do Estado de Sao Paulo-FAPESP', 'Conselho Nacional de Desenvolvimento Cientifico e Tecnolcgico-e-CblPq", 'Coordenacao de Aperfeicoamento do Pessoal de Ensino SuperiorCAPES', Brazil.

PY - 1982

Y1 - 1982

N2 - A prescribed shear stress applied to the free surface of a thin liquid layer sets up a steady shear flow. When the shear flow has a linear velocity profile, Miles, using asymptotic analysis, finds critical values Rc of the Reynolds number above which unstable travelling waves exist. However, Miles omits a term in the normal-stress boundary condition. We correct this omission and solve the appropriate Orr-Sommer-feld system numerically to obtain the critical conditions. For the case of a zero-surface-tension interface, we find that Rc = 34–2, as compared with Miles’ value of Rc = 203. As surface tension increases, Re asymptotes to the inviscid limit developed by Miles. The critical Reynolds number, critical wavenumber and critical phase speed are presented as functions of a non-dimensional surface tension. We investigate the mechanism of the instability through an examination of the disturbance-energy equation. When the shear flow has a parabolic velocity profile, we find a long-wave instability at small values of the Reynolds number. Numerical methods are used to extend these results to larger values of the wavenumber. Examination is made of the relation between this long-wave instability and profile curvature.

AB - A prescribed shear stress applied to the free surface of a thin liquid layer sets up a steady shear flow. When the shear flow has a linear velocity profile, Miles, using asymptotic analysis, finds critical values Rc of the Reynolds number above which unstable travelling waves exist. However, Miles omits a term in the normal-stress boundary condition. We correct this omission and solve the appropriate Orr-Sommer-feld system numerically to obtain the critical conditions. For the case of a zero-surface-tension interface, we find that Rc = 34–2, as compared with Miles’ value of Rc = 203. As surface tension increases, Re asymptotes to the inviscid limit developed by Miles. The critical Reynolds number, critical wavenumber and critical phase speed are presented as functions of a non-dimensional surface tension. We investigate the mechanism of the instability through an examination of the disturbance-energy equation. When the shear flow has a parabolic velocity profile, we find a long-wave instability at small values of the Reynolds number. Numerical methods are used to extend these results to larger values of the wavenumber. Examination is made of the relation between this long-wave instability and profile curvature.

UR - http://www.scopus.com/inward/record.url?scp=84972295962&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972295962&partnerID=8YFLogxK

U2 - 10.1017/S0022112082001852

DO - 10.1017/S0022112082001852

M3 - Article

AN - SCOPUS:84972295962

SN - 0022-1120

VL - 121

SP - 187

EP - 206

JO - Journal of fluid Mechanics

JF - Journal of fluid Mechanics

ER -