Symmetry Analysis and Analytic Solutions for Unsteady Flows on a Stretching Surface in the Presence of Magnetic Field, Internal Heat Source and Thermocapillary Effects

Abstract

This study employs Lie point symmetry analysis to investigate the unsteady flow on a stretching surface. Such flows are common in various manufacturing processes such as extrusion, melt-spinning, and coating. Three cases have been discussed i.e. unsteady flow on a stretching surface in the presence of a variable magnetic field, its 1-dimensional optimal system, and unsteady flow on a stretching surface in the presence of thermocapillarity, an internal source or sink, and a variable magnetic field. For unsteady flow on a stretching surface in the presence of a variable magnetic field, a general linear combination of all admitted translational and scaling Lie point symmetries has been used to obtain the system invariants and general forms of the velocity, temperature, and concentration at the stretching surface. The deduced invariants provide a new generalized class of similarity transformations that convert the governing boundary layer equations into a system of non-linear ODEs. Analytic series solutions have been obtained for the resulting system of ODEs using Homotopy Analysis Method (HAM) and the effect of different parameters such as unsteadiness, magnetic parameter, Prandtl number, Schmidt number, and coefficients of the Lie point symmetries has been depicted graphically. It has been found that coefficients of the translational symmetries do not play any role in the solution while coefficients of the scaling symmetries can control the temperature and concentration fields. Secondly, a 1-dimensional optimal system of this flow is obtained which provides 22 new classes of similarity transformations that reduce the governing boundary layer equations into 22 news classes of ODEs, thus providing multiple new solutions of heat and mass transfer. Similarly, the hydrodynamics and thermal characteristics of the flow induced by the unsteady stretching of a sheet in the presence of thermocapillarity, internal heat source or sink, and variable magnetic field are investigated using Lie point analysis. The linear combinations of Lie point symmetries is again a Lie point symmetry. It is admitted by all boundary conditions while leave the stretching sheet velocity and temperature as a function of both distance and time. We utilize such a linear combination to develop Lie transformations that reduce the governing momentum and energy equations into a system of coupled non-linear ODEs. The resulting five-parameter problem namely, unsteadiness term 𝑆, magnetic parameter 𝑀𝑎, Prandtl number 𝑃𝑟, temperature-dependent v heat source or sink term 𝐺∗ and the thermocapillarity parameter 𝑀, is solved using Homotopy Perturbation Method (HPM). It has been found that thermocapillary forces drag the free surface of the fluid in the direction of the stretching sheet, due to which a local velocity minimum forms in the fluid. Thermocapillarity thickens the fluid film resulting in the increase of free surface velocity, temperature, and heat flux from the sheet while reducing the friction between the sheet and the fluid film. The temperature-dependent heat source or sink term and the magnetic parameter greatly affect the variation of the temperature across the fluid and can be useful in speeding up the cooling or heating of the fluid