Exact solutions for unsteady free convection flow over an oscillating plate due to non-coaxial rotation.

Ahmad Qushairi Mohamad,Ilyas Khan,Zulkhibri Ismail,Sharidan Shafie

doi:10.1186/s40064-016-3748-2

Ahmad Qushairi Mohamad, Ilyas Khan + Show 2 more

Open Access

https://doi.org/10.1186/s40064-016-3748-2

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Abstract

BackgroundNon-coaxial rotation has wide applications in engineering devices, e.g. in food processing such as mixer machines and stirrers with a two-axis kneader, in cooling turbine blades, jet engines, pumps and vacuum cleaners, in designing thermal syphon tubes, and in geophysical flows. Therefore, this study aims to investigate unsteady free convection flow of viscous fluid due to non-coaxial rotation and fluid at infinity over an oscillating vertical plate with constant wall temperature.MethodsThe governing equations are modelled by a sudden coincidence of the axes of a disk and the fluid at infinity rotating with uniform angular velocity, together with initial and boundary conditions. Some suitable non-dimensional variables are introduced. The Laplace transform method is used to obtain the exact solutions of the corresponding non-dimensional momentum and energy equations with conditions. Solutions of the velocity for cosine and sine oscillations as well as for temperature fields are obtained and displayed graphically for different values of time (t ), the Grashof number (Gr), the Prandtl number (Pr ), and the phase angle (omega t). Skin friction and the Nusselt number are also evaluated.ResultsThe exact solutions are obtained and in limiting cases, the present solutions are found to be identical to the published results. Further, the obtained exact solutions also validated by comparing with results obtained by using Gaver–Stehfest algorithm.ConclusionThe interested physical property such as velocity, temperature, skin friction and Nusselt number are affected by the embedded parameters time (t), the Grashof number (Gr), the Prandtl number (Pr ), and the phase angle (omega t).

Highlights

Non-coaxial rotation has wide applications in engineering devices, e.g. in food processing such as mixer machines and stirrers with a two-axis kneader, in cooling turbine blades, jet engines, pumps and vacuum cleaners, in designing thermal syphon tubes, and in geophysical flows
In order to understand the physical aspects of the problem, the numerical results for velocity [Eqs. (53, 54)] and temperature [Eq (55)] are computed and plotted graphically for different values of time t, Grashof number Gr, Prandtl number Pr, phase angle ωt, and amplitude of the plate oscillations U0
All of these graphs are displayed for a real part of velocity and for an imaginary part of velocity

Summary

Introduction

Non-coaxial rotation has wide applications in engineering devices, e.g. in food processing such as mixer machines and stirrers with a two-axis kneader, in cooling turbine blades, jet engines, pumps and vacuum cleaners, in designing thermal syphon tubes, and in geophysical flows. The problems of Newtonian fluids are complicated due to the non-linearity of Navier Stokes equations. Studies of Newtonian fluids in the presence of heat transfer are scarce, when one is interested in exact solutions. The study of heat transfer in Newtonian fluids, especially due to convection, is important in many engineering applications, such as automatic control systems consisting of electrical and electronic components, regularly subjected to periodic heating and cooled by a free convection process (Manna et al 2007; Sajid et al 2008; Sahoo et al 2010; Chandran et al 2005; Chaudhary and Jain 2006; Deka and Das 2011; Narahari and Nayan 2011)

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