Convergence of the homotopy analysis method

The homotopy analysis method is studied in the present paper. The question of convergence of the homotopy analysis method is resolved. It is proven that under a special constraint the homotopy analysis method does converge to the exact solution of the sought solution of nonlinear ordinary or partial differential equations. An optimal value of the convergence control parameter is given through the square residual error. An error estimate is also provided. Examples, including the Blasius flow, clearly demonstrate why and on what interval the corresponding homotopy series generated by the homotopy analysis method will converge to the exact solution.Comment: 12 pages, 4 combined figure

Heat and Mass Transfer on the MHD Fluid Flow Due to a Porous Rotating Disk With Hall Current and Variable Properties

The steady magnetohydrodynamics (MHD

Do peaked solitary water waves indeed exist?

Many models of shallow water waves admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa-Holm equation. Besides, it is proved that Kelvin's theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, the phase speed of the peaked solitary waves has nothing to do with wave height. In addition, the kinetic energy of the peaked solitary waves either increases or almost keeps the same from free surface to bottom. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.Comment: 53 pages, 13 figures, 7 tables. Accepted by Communications in Nonlinear Science and Numerical Simulatio

An experimental study of airfoil instability tonal noise with trailing edge serrations

This paper presents an experimental study of the effect of trailing edge serrations on airfoil instability noise. Detailed aeroacoustic measurements are presented of the noise radiated by an NACA-0012 airfoil with trailing edge serrations in a low to moderate speed flow under acoustical free field conditions. The existence of a separated boundary layer near the trailing edge of the airfoil at an angle of attack of 4.2 degree has been experimentally identified by a surface mounted hot-film arrays technique. Hot-wire results have shown that the saw-tooth surface can trigger a bypass transition and prevent the boundary layer from becoming separated. Without the separated boundary layer to act as an amplifier for the incoming Tollmien-Schlichting waves, the intensity and spectral characteristic of the radiated tonal noise can be affected depending upon the serration geometry. Particle Imaging Velocimetry (PIV) measurements of the airfoil wakes for a straight and serrated trailing edge are also reported in this paper. These measurements show that localized normal-component velocity fluctuations that are present in a small region of the wake from the laminar airfoil become weakened once serrations are introduced. Owing to the above unique characteristics of the serrated trailing edges, we are able to further investigate the mechanisms of airfoil instability tonal noise with special emphasis on the assessment of the wake and non-wake based aeroacoustic feedback model. It has been shown that the instability tonal noise generated at an angle of attack below approximately one degree could involve several complex mechanisms. On the other hand, the non-wake based aeroacoustic feedback mechanism alone is sufficient to predict all discrete tone frequencies accurately when the airfoil is at a moderate angle of attack

Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.We investigate the effects of compressibility and wall cooling on the stationary, viscous (Type II) instability mode within the 3D boundary layer over rotating cones with half-angle greater than 40∘ 40∘ . The stationary mode is characterised by zero shear stress at the wall and a triple-deck solution is presented in the isothermal case. Asymptotic solutions are obtained which describe the structure of the wavenumber and the orientation of this mode as a function of local Mach number. It is found that a stationary mode is possible only over a finite range of local Mach number. Our conclusions are entirely consistent with the results of Seddougui 1990 , A nonlinear investigation of the stability models of instability of the trhee-dimensional Compresible boundary layer due to a rotating disc Q. J. Mech. Appl. Math. , 43, pt. 4. It is suggested that wall cooling has a significant stabilising effect, while reducing the half-angle is marginally destabilising. Solutions are presented for air

Linear stability analysis of MHD flow of micropolar fluid with thermal radiation and convective boundary condition: Exact solution

Magnetohydrodynamic (MHD) flow of micropolar fluid by including the thermal radiation and convective condition on a shrinking surface in the presence of mass suction effects has been investigated. The momentum, angular momentum and energy equations, and the solutions of these equations are valid for whole Navier stokes, and microrotational and energy equations have been solved exactly. We obtain the solution in the form of an incomplete γ function for the energy equation. The results reveal that dual solutions exist for certain domains of different physical parameters. Furthermore, high suction produces the high effect of drag force, and as a result, coefficient of skin friction increases in the first solution. Stability analysis has been performed and determined that the first solution is more stable

The cross-flow instability of the boundary layer on a rotating cone

Experimental studies have shown that the boundary-layer flow over a rotating cone is susceptible to cross-flow and centrifugal instability modes of spiral nature, depending on the cone sharpness. For half-angles (ψ) ranging from propeller nose cones to rotating disks (ψ ≥ 40°), the instability triggers co-rotating vortices, whereas for sharp spinning missiles (ψ 40°. Below this half-angle we suggest that an alternative instability mechanism is at work, which is not amenable to investigation using the formulation presented here

Numerical study of magneto-convective heat and mass transfer from inclined surface with Soret diffusion and heat generation effects : a model for ocean magnetohydrodynamics energy generator fluid dynamics

A mathematical model is developed for steady state magnetohydrodynamic (MHD) heat and mass transfer flow along an inclined surface in an ocean MHD energy generator device with heat generation and thermo-diffusive (Soret) effects. The governing equations are transformed into nonlinear ordinary differential equations with appropriate similarity variables. The emerging two-point boundary value problem is shown to depend on six dimensionless thermophysical parameters - magnetic parameter, Grashof number, Prandtl number, modified Prandtl number, heat source parameter and Soret number in addition to plate inclination. Numerical solutions are obtained for the nonlinear coupled ordinary differential equations for momentum, energy and salinity (species) conservation, numerically, using the Nachtsheim-Swigert shooting iteration technique in conjunction with the Runge- Kutta sixth order iteration scheme. Validation is achieved with Nakamura’s implicit finite difference method. Further verification is obtained via the semi-numerical Homotopy analysis method (HAM). With an increase in magnetic parameter, skin friction is depressed whereas it generally increases with heat source parameter. Salinity magnitudes are significantly reduced with increasing heat source parameter. Temperature gradient is decreased with Prandtl number and salinity gradient (mass transfer rate) is also reduced with modified Prandtl number. Furthermore, the flow is decelerated with increasing plate inclinations and temperature also depressed with increasing thermal Grashof number

Fluid flow and radiative nonlinear heat transfer over a stretching sheet

In the present paper, we endeavor to perform a numerical analysis in connection with the boundary layer flow induced in a quiescent fluid by a continuous sheet stretching with velocity uw (x) ∼x1/3 with heat transfer. The effects of thermal radiation using the nonlinear Rosseland approximation are investigated. We search for similarity solutions and reduce the problem to a couple of ordinary differential equations containing three dimensionless parameters: the radiation parameter NR, the temperature ratio parameter θw and the Prandtl number Pr. The computational results for velocity, temperature and heat transfer characteristics are presented in both graphical and tabular forms.Cortell Bataller, R. (2014). Fluid flow and radiative nonlinear heat transfer over a stretching sheet. Journal of King Saud University - Science. 26(2):161-167. doi:10.1016/j.jksus.2013.08.004S16116726