Abstract—The Marangoni convective instability in a horizontal fluid layer with the insoluble surfactant and nondeformable free surface is investigated. The surface tension at the free surface is linearly dependent on the temperature and concentration gradients. At the bottom surface, the temperature conditions of uniform temperature and uniform heat flux are considered. By linear stability theory, the exact analytical solutions for the steady Marangoni convection are derived and the marginal curves are plotted. The effects of surfactant or elasticity number, Lewis number and Biot number on the marginal Marangoni instability are assessed. The surfactant concentration gradients and the heat transfer mechanism at the free surface have stabilizing effects while the Lewis number destabilizes fluid system. The fluid system with uniform temperature condition at the bottom boundary is more stable than the fluid layer that is subjected to uniform heat flux at the bottom boundary. Keywords—Analytical solutions, Marangoni Instability, Nondeformable free surface, Surfactant. I. INTRODUCTION ONVECTION driven by surface tension effects is called as Marangoni-Bénard convection or simply Marangoni convection. The Marangoni convection can be observed in industrial and technological processes such as the crystal growth production, welding and semi-conductor manufacturing. The convective instability in a horizontal fluid layer heated from below and cooled from above was first studied experimentally by Bénard [1] and theoretically by Rayleigh [2] and Pearson [3]. Marangoni convection usually can affect the quality of the products due to striations, dendrites and bubbles that occur during the manufacturing process. The Marangoni instability problems due to temperature- dependent surface tension have been investigated for steady and oscillatory convection by [4] and [5]. Numerous studies on the effects of physical factors are considered such as the effect of feedback control [6]−[7], internal heat generation [8], variable viscosity [9] and porous layer [10]. Another important factor is the influence of surface-active agents on thermocapillary convection where the surface tension is dependent on the concentration gradients [11]. Mikishev and Nepomnyashchy [11] studied the long-wavelength Marangoni convection in a liquid layer with insoluble surfactant using perturbation method. Ainon Syazana Ab. Hamid, Seripah Awang Kechil and Ahmad Sukri Abd. Aziz are with the Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Shah Alam, 40450 Selangor, Malaysia (phone: +603- 55435429; fax: +603-55435501; e-mail: ainonsyazana@gmail.com; e-mail: seripah_awangkechil@salam.uitm.edu.my). In this paper, we shall investigate the Marangoni convection in a fluid layer with nondeformable surface in the presence of insoluble surfactant and subject to uniform temperature and uniform heat flux at the bottom boundary. We will find the exact analytical solutions for the steady Marangoni convection for the bottom conditions of uniform temperature and uniform heat flux. II. PROBLEM FORMULATION Consider a horizontal layer of fluid with thickness d, bounded below by a rigid wall plate and above by a flat free surface subject to a transverse temperature gradient. Two- dimensional Cartesian coordinates x and z are introduced with z = 0 coincides with the plate surface and z-axis is directed vertically upward. The two-dimensional consideration is sufficient for the development of the linear stability theory because of the rotational symmetry [11]. The surface tension σ, is assumed to depend linearly on both temperature T and surfactant concentration Γ, , 2 1 0 Γ − − = σ σ σ σ T (1) where σ 0 is reference value of surface tension, T ∂ −∂ = / 1 σ σ and Γ ∂ −∂ = / 2 σ σ . Heat is transmitted from the free surface to the atmosphere by Newton’s law of cooling , 0 = + ∂ ∂ qT T n λ (2) where λ is the fluid’s thermal conductivity, n is a normal unit vector to the surface and q is the rate of heat transfer at the free surface. The system is governed by the equations of conservation of mass, momentum and energy given by 0, = ⋅ ∇ v (3) , ) ( 2 v v v v ∇ + −∇ = ∇ ⋅ + ∂ ∂ ν ρ p t (4) , ) ( 2 T T t T ∇ = ∇ ⋅ + ∂ ∂ χ v (5) where ) , ( w u = v , T is the temperature, ρ density, p pressure, ν kinematic viscosity, χ thermal diffusivity, ∇ gradient vector and t is the time. By using the linear stability theory and the introduction of infinitesimal disturbances and scaling for length, time, velocity, temperature and pressure as d d d d β χ χ , , , 2 and Ainon Syazana Ab. Hamid, Seripah Awang Kechil and Ahmad Sukri Abd. Aziz Marangoni Instability in a Fluid Layer with Insoluble Surfactant C World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:5, No:10, 2011 1547 International Scholarly and Scientific Research & Innovation 5(10) 2011 scholar.waset.org/1999.7/10848 International Science Index, Mathematical and Computational Sciences Vol:5, No:10, 2011 waset.org/Publication/10848