Analysis of flow and heat transfer inside oscillatory squeezed thin films subject to a varying clearance
Introduction
Flow and heat transfer inside thin films have received a significant amount of attention in the recent years because they are widely utilized in many engineering applications such as in lubrication, heat pipes, microchannels and fluidic cells. External disturbances such as unbalances in rotating machines and increased noise levels from the surroundings can result in oscillatory relative motions between the plates surrounding the thin film. Even small oscillating motions can have a substantial impact because the thickness of thin films is very small. Accordingly, the dynamics and thermal characterization of thin films will be altered.
The chambers for chemical and biological detection systems such as fluidic cells for chemical or biological microcantilever probes (Lavrik et al. [1]) are an important example for applications of thin films. Small noise levels that may be present at the boundaries can produce flow instabilities inside the fluidic cells. These disturbances have large influence on the measurements of the detecting elements specially those utilizing microcantilevers. These detecting elements are very sensitive to flow conditions. Therefore, a special design for these fluidic cells such as considering converging or diverging thin films is needed in order to transport the target proteins to the probes with minimum effects of turbulence or thermal disturbances.
There are many studies that have investigated flow in hydrodynamic or squeezed thin films like Langlois [2] who solved analytically the momentum equations for hydrodynamic pressure in isothermal squeezed films with fluid density varying according to the pressure. Later on, towards the end of the 20th century, the interest in studying flow inside squeezed thin films increased. Hamza [3] and Bhattacharyya et al. [4] have studied the squeeze effects on the temperature distribution inside the thin film. However their works were concerned with flow between two parallel disks and were simplified to a one-dimensional flow case. Recently, Khaled and Vafai [5], [6] considered heat transfer in incompressible squeezed thin films with sinusoidal squeezing for two different models: low and large Reynolds number models. To the authors knowledge the literature lacks studies that are concerned with the effects of pressure squeezing on flow and heat transfer inside convergent or divergent thin films which is related to a number of application such as fluidic cells utilizing detection of biological agents.
In this work, the effects of external squeezing on flow and heat transfer inside thin films having its upper plate slightly inclined are studied. Thin films with positive inclination forms divergent thin films while they become convergent films when the inclination is negative. The governing equations for both low and large Reynolds number flow models are non-dimensionalized and then solved numerically and the influence of squeezing Reynolds number, dimensionless amplitude of the upper plate’s motion, squeezing number, thermal squeezing parameter and dimensionless slope for the upper plate are determined on both flow and heat transfer characteristics inside non-flat thin films. Finally, few correlations are generated for Nusselt numbers at the exit of the thin film as functions of both the dimensionless slope of the upper plate and the dimensionless amplitude of the upper plate’s motion for two thermal conditions: constant wall temperature (CWT) and uniform wall heat flux (UHF).
Section snippets
Problem formulation
Consider a two-dimensional thin film that has a small thickness h compared to its length B. The x-axis is taken in the direction of the length of the thin film, the horizontal direction, while y-axis is taken in the vertical direction along the thickness of the thin film as shown in Fig. 1. The lower plate of the thin film is horizontal and fixed while the upper plate is inclined and allowed to have sinusoidal vertical motion such that the thickness of the thin film can be expressed according
Numerical analysis
Eqs. , , are first transformed to the new computational domain (ξ,η,τ*) and then solved using the alternating direction implicit method (ADI), see Ref. [10]. After each half time step for Eqs. , , Eq. (27) was solved using the method of successive over relaxation SOR. The dimensionless velocities in Eq. (26) as well as the dimensionless vorticity at the plates of the thin film, seen in Eqs. (31c,d), were calculated initially at previous half time steps and then corrected by consecutive
Low Reynolds numbers flow model
Fig. 3, Fig. 4 illustrate the effect of dimensionless slope of the upper plate of the thin film κ on the Nusselt number for CWT and UHF conditions, respectively. It is noticed that all NuL and NuU for CWT condition and NuL for UHF condition have oscillatory behaviors. Note that NuU is zero for the UHF condition. Further, both NuL and NuU for CWT condition and NuL for UHF condition decrease as κ increases due to a decrease in the average axial velocity as κ increases. An interesting feature for
Conclusions
The effects of external squeezing of the upper plate of a thin film that has a linearly varying thickness with the horizontal distance have been considered on flow and heat transfer for a wide range of squeezing Reynolds numbers. In the present study, the governing equations have been non-dimensionalized and reduced to two categories: low and large Reynolds number flow models. Both categories were compared at a limiting case and were found to be in excellent agreement. It was found that flow
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