A new family of unsteady boundary layers over a stretching surface

Abstract

In this paper, a new family of unsteady boundary layers over a stretching flat surface was proposed and studied. This new class of unsteady boundary layers involves the flows over a constant speed stretching surface from a slot, and the slot is moving at a certain speed. Depending on the slot moving parameter, the flow can be treated as a stretching sheet problem or a shrinking sheet problem. Both the momentum and thermal boundary layers were studied. Under special conditions, the solutions reduce to the unsteady Rayleigh problem and the steady Sakiadis stretching sheet problem. Solutions only exist for a certain range of the slot moving parameter, α. Two solutions are found for −53.55° < α < −45°. There are also two solution branches for the thermal boundary layers at any given Prandtl number in this range. Compared with the upper solution branch, the lower solution branch leads to simultaneous reduction in wall drag and heat transfer rate. The results also show that the motion of the slot greatly affects the wall drag and heat transfer characteristics near the wall and the temperature and velocity distributions in the fluids.

Introduction

Unsteady boundary layers are different from steady state ones due to extra time dependent terms in the governing equations, which also influence the fluid motion pattern and boundary layer separation [1], [2], [3]. Typical examples of unsteady boundary layers in the history of fluid mechanics are the Rayleigh problem and the Stokes oscillating plate [2], [3]. Two dimensional unsteady boundary layers have been studies by many researchers for different unsteady free stream velocity and flow configuration as discussed in the references and the references therein [3], [4], [5], [6]. An interesting unsteady state boundary layer problem was proposed and studied by Todd for a constant speed free stream passing a fixed semi-infinite flat plate [7]. The flow actually involves a moving leading edge with a certain rate of accretion or ablation. The momentum boundary layer was solved and discussed. Most recently, the thermal boundary layer has been studied and further discussions on the momentum boundary layer have been presented [8].

The flow induced by a stretching boundary is important in engineering applications [9], [10], [11]. There are many studies about this flow configuration in the literature [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. The pioneering work in this area was carried out by Sakiadis [12], [13]. His work was further verified by Tsou et al. [14] experimentally. The velocity of the surface was generalized to be a function of distance from the slot, where the surface was stretched out [15], [16], [22]. The heat transfer characteristics were extensively studied [18], [19], [21], [22], [23].Thermal boundary conditions included a power-law surface temperature or a power-law surface heat flux. Mass transfer such as fluid suction and injection was considered on the stretching surface. The stretching sheet problem for a rotating flow configuration was investigated [20]. Exponentially stretching velocity and rapidly decreasing velocity conditions were also discussed [24], [25], [26]. A new solution branch for both impermeable and permeable stretching sheets was found by Liao [28], [29], which indicates that multiple solutions for the stretching surfaces are possible under certain conditions. Three dimensional stretching wall problems were also investigated [17], [33], [36]. The effects of temperature-dependent viscosity on the boundary layers over a stretching surface were investigated [27], [34]. The flow and heat transfer of the boundary layers over a stretching sheet under the effect of a uniform shear free stream was studied recently [37]. These previous studies, however, focused on steady state flows over a stretching sheet. Unsteady boundary layer flows due to a stretching surface were also investigated by many researchers [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]. Among these studies, the stretching surface is moving with a speed changing with time. The heat transfer analysis of an unsteady boundary layer was performed, too. Under certain conditions, the unsteady boundary layer flows are exact solutions of the three-dimensional Navier–Stokes equations [38], [40], [41]. In the current work, inspired by the new family of unsteady boundary layers studied by Todd [7] and Fang [8], a new class of unsteady boundary layer over a stretching surface is proposed and investigated. For these newly proposed unsteady boundary layers, the stretching surface speed does not vary with time; rather, the slot, from which the sheet is stretched out, is moving at a certain speed. Some new characteristics of the momentum and thermal boundary layers of this flow configuration will be addressed and analyzed.