ZAMM  Z. Angew. Math. Mech. 81 (2001) 10, 699 –– 709 Hossain, M. A., Hussain, S., Rees, D. A. S. Influence of Fluctuating Surface Temperature and Concentration on Natural Convection Flow from a Vertical Flat Plate A linearized theory is used to investigate how a free double-diffusive boundary layer flow is affected by small amplitude temporal variations in the surface temperature and species concentration. The mean temperature and the mean species concentration are assumed to vary as a power n of the distance measured from the leading edge. Three distinct methods, namely, a perturbation method for low frequencies, an asymptotic series expansion for high frequencies, and a finite difference method for intermediate frequencies, are used. Calculations have been carried out for a wide range of param- eters in order to examine the results obtained from the three methods. Comparisons are made in terms of the amplitudes and phases of the surface heat transfer and surface mass transfer. It has been found that the amplitudes and phase angles predicted by perturbation theory and the asymptotic method are in good agreement with the finite difference computa- tions. MSC (2000): 76D99, 80A20 1. Introduction The study of laminar boundary layers in the presence of an oscillatory flow with a steady mean component was first undertaken by Lighthill [1]. He considered the effects of small fluctuations in the free stream velocity on the skin- friction and heat transfer for plates and cylinders by employing the Karman-Pohlhausen approximate integral method. Nanda and Sharma [2] and Eshghy et al. [3] later extended Lighthill’s theory for free convective flows. Muhuri and Maiti [4] and Singh et al. [5] investigated the free-convection flow and heat transfer along a semi-infinite horizontal plate with a small amplitude surface temperature oscillation about a non-zero mean by using the approximate integral method. On the other hand, Kelleher and Yang [6] investigated the heat transfer response of laminar free convec- tion boundary layers along vertical heated plates to surface-temperature oscillations. Solutions of the governing nonsi- milar equations for the fluctuating flow and heat transfer were obtained for both small and large frequencies using suitable perturbation techniques. With an oscillating surface heat flux, the problem of natural convective flow from a vertical surface has been investigated by Hossain et al. [7]. The additional effects of a transverse magnetic field on the problem posed by Kelleher and Yang, has more recently been investigated by Hossain et al. [8]. In all the above studies solutions were obtained for both small and large frequencies by using suitable perturbation techniques. For intermediate frequencies the equations governing the fluctuating flow and heat transfer were obtained by using both an implicit finite difference method and the local non-similarity method of Minkowycz and Sparrow [9]. In these inves- tigations involving small-amplitude oscillations, it has been found that in the low-frequency range, the oscillating com- ponent of the skin-friction always lags behind the plate temperature oscillations while the rate of heat transfer has a phase lead. In the high-frequency range, the velocity and temperature in the boundary layer are of ‘shear-wave’ type, predicting a phase lead of 45  in the rate of heat transfer fluctuation and an equivalent phase lag in the skin-friction fluctuation. There are many transport processes in industry and in the environment in which buoyancy forces arise for both thermal and mass diffusion as a result of the co-existence of thermal and concentration gradients. In free convection these may either hinder or aid one another. Somers [10], Wilcox [11], and Gill et al. [12] studied the effects of mass transfer on free convection. Similarity analysis of the natural convection flows adjacent to both vertical and horizontal surfaces which result from the combined buoyancy effects of thermal and mass diffusion was first investigated by Gebhart and Pera [13] and Pera and Gebhart [14]. The combined effect of buoyancy forces from thermal and mass diffusion on forced convection was studied by Chen et al. [15]. In this latter paper attention is directed to forced convection along vertical and inclined plates for which the plate is either maintained at a uniform temperature and concentration or subjected to a uniform surface heat-flux and mass-flux. Both local nonsimilarity and finite difference methods for the fluid having Prandtl numbers 0.7 and 7.0 gave solutions of the transformed conservation equations. Hossain [16] also investigated the effect of transpiration along with the combined effect of buoyancy forces from thermal and mass diffusion on forced convective heat and mass transfer from a vertical plate. But all the above studies are confined to steady flows. Very much less attention has been given to the effects of combined heat and mass trans- fer on unsteady flow. Among those few investigators Hossain and Begum [17] studied the combined effect of heat and mass transfer to unsteady convection flow from a doubly infinite vertical permeable surface with small-amplitude fluc- tuations in the free stream, the surface temperature, and the surface species concentration. Recently, Hossain [18] investigated the effect of a fluctuating surface temperature on the unsteady two dimensional natural convection flow with uniform species concentration at the surface by the use of the Karman-Pohlhausen approximate integral method. Hossain, M. A. et al.: Fluctuating Surface Temperature and Natural Convection Flow 699 # WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2001 0044-2267/01/1010-0699 $ 17.50þ.50/0