VISCOUS CORRECTION FOR POTENTIAL FLOW ANALYSIS OF CAPILLARY AND KELVIN-HELMHOLTZ INSTABILITY

Abstract

In inviscid potential theory, viscosity is neglected when considering irrotational solutions of Navier-Stokes equations. However, in viscous potential flow (VPF) theory, viscosity is not neglected and irrotational flow of a viscous fluid satisfies Navier-Stokes equations. The viscous stresses enter into the viscous potential flow analysis of free surface problems through normal stress balance at the interface and tangential stresses are not considered in this theory. Viscous correction for the viscous potential flow (VCVPF) analysis is based on the irrotational motion of the fluids and it contains the effect of both normal as well as shearing stresses at the interface. The VCVPF is an irrotational flow analysis which differs from VPF only by additional viscous pressure. Viscous correction for the viscous potential flow analysis of stability problems with heat and mass transfer has not been considered in earlier studies. The VCVPF theory is also important in the study of breaking of a cylindrical jet of non-Newtonian fluid, electrohydrodynamic stability, and magnetohydrodynamic stability. Hence "Viscous correction for Potential Flow Analysis of Capillary and Kelvin-Helmholtz instability" is studied in this thesis. The chapter wise summary of the thesis is as follows: Chapter 1 consists of introduction to the general stability problems, capillary instability, Kelvin-Helmholtz instability, instability of liquid jet, viscous potential theory, viscous corrections for the viscous potential flow, some definitions and basic equations related to the stability analysis. A brief account of the related studies made by various authors in the field and a summary of the thesis are presented. In Chapter 2, viscous correction for viscous potential flow analysis of capillary instability when there is heat and mass transfer across the interface has been carried out. A dispersion relation that accounts for the growth of axisymmetric waves is derived and stability is discussed theoretically as well as numerically. Stability criterion is given in terms of critical value of wave number. It has been observed that heat transfer and irrotational shear stresses both have stabilizing effect on the stability of the system. In Chapter 3, viscous correction for the potential flow analysis of Kelvin-Helmholtz instability of the cylindrical interface has been studied when there is heat and mass transfer across the interface. Both asymmetric and axisymmetric disturbances are considered. Stability criterion is given in terms of critical value of relative velocity. It has been observed that the heat and mass transfer has destabilizing effect on the stability of the system. In Chapter 4, viscoelastic correction for viscoelastic potential flow analysis of a cylindrical jet with axisymmetric and asymmetric disturbances moving in an infinite viscous fluid has been studied. The cause of the instability in the liquid jet is Kelvin-Helmholtz instability due to the velocity difference and capillary instability due to surface tension. The stability analysis shows that viscoelastic liquid jets are less unstable than inviscid jets and more unstable than viscous liquid jets for both axisymmetric and asymmetric disturbances. In Chapter 5, viscous correction for the potential flow analysis of Kelvin-Helmholtz instability has been carried out when there is heat and mass transfer across the interface. Stability criterion is given in terms of critical value of relative velocity. It has been observed that the inclusion of irrotational shearing stresses have stabilizing effect on the stability of the system. In Chapter 6, viscous correction for the potential flow analysis of Kelvin-Helmholtz instability with heat and mass transfer in the presence of a horizontal electric field has been carried out. Stability criterion is given by a critical value of relative velocity of two fluids as well as critical value of the applied electric field. Various graphs with respect to physical parameters, such as wave number, viscosity ratio, ratio of dielectric constants of two fluids, heat transfer coefficients have been drawn and effect of various parameters have been described. It has been observed that relative velocity has destabilizing effect while electric field has stabilizing effect. In Chapter 7, the effect of irrotational shear stresses in the viscous potential flow analysis of Kelvin-Helmholtz instability with heat and mass transfer in the presence of a horizontal magnetic field has been studied. A critical value of magnetic field as well as relative velocity is obtained. A detailed analysis in terms of all physical parameters has been made. It has been observed that inclusion of irrotational shearing stresses have stabilizing effect on the stability of the system. Finally, conclusions are drawn and future research work is suggested in Chapter 8. ii