The mathematical model consists of time-dependent Navier-Stokes and energy equations in a time-dependent (moving) domain. Further, the Marangoni effect and a temperature-dependent dynamic contact angle will also be incorporated in the model. The domain will be handled by the arbitrary Lagrangian-Eulerian approach. Moreover, the PDEs will be discretized by finite elements in space and by a fractional step-theta scheme in time. Since the density of the alloy is very high, the PDEs become convection dominated, and therefore a local projection stabilization will be used.

The developed numerical scheme will be used to get an insight into the effects of Marangoni convection on the dislocation and accumulation of the liquidized alloy coating. Further, parametric studies will be performed to study the dislocation phenomenon. we believe that the findings of this numerical study will help to develop technologies to use high heating rates to increase ultra-high-strength steel production in HFDQ process.

In addition to the targeted application, the considered type of problems initiate new scientific questions in fundamental areas of computational mathematics. Moreover, the findings will have a huge impact in automobile industries.

The free surface/two-phase flow will be described by the time-dependent Navier-Stokes equations in a moving domain, whereas the rigid body (ship) motion will be described by the equations of variation of linear and angular momentum. Apart from the other challenges associated with the solution of the coupled Navier-Stokes and momentum equations, tracking and/or capturing the moving free surface/interface makes the computation more complicate. In this project, finite element schemes based on the arbitrary Lagrangian-Eulerian (ALE) will be developed for the coupled equations.

The developed scheme can be used to solve the ship resistance problem, that is, to predict the resistance of the ship moving at low speed through still water. In practice, a very high accuracy of resistance prediction is often required to determine the hull form of design, and the proposed ALE-FEM will be well suited for these models, and will be capable of simulating free surface/two-phase flows around ships very accurately.

The mathematical model consists of time-dependent Navier-Stokes and viscoelastic constitutive equations in a time-dependent (moving) domain. Moreover, the partial differential equations are discretized by finite elements in space and by backward Euler scheme in time. Since the PDEs are convection dominated, a local projection stabilization (LPS) is used to obtained a stable finite element scheme.

The developed numerical scheme can be used to get an insight into the effects of viscoelasticity on the flow dynamics of a droplet spreading upon impact on flat substrate and a bubble rising in a viscoelastic fluid due to buoyancy. Further, parametric studies will be performed to study the flow dynamics.

Findings of this numerical study can be used to develop technologies related to polymer additives in controlled droplet deposition and enhanced oil recovery mechanisms.

The mathematical model consists of Magnetohydrodynamics (MHD) equations that involves time-dependent Navier-stokes equations and Maxwell’s equations of electromagnetism. To calculate heat transfer convection-diffusion equation is coupled with MHD equations. The PDEs are discretized by finite elements in space and by a fraction step-theta scheme in time. Further, for simulating the flow with high Reynolds number, a projection based variational multiscale scheme (VMS) is used. The VMS allows the separation of entire range of scales in the flow field into three groups enabling different numerical treatment for different groups.

The developed scheme can be used to study the effect on heat transfer in MHD flows. Further, effects on heat transfer due to the introduction of obstacle of various shapes and sizes, inside the duct, can be investigated to efficiently design a nuclear reactor.

Due to the presence of multitude of scales in turbulent flows, the standard Galerkin approach has severe limitations. Variational multiscale method (VMS) is relatively a new technique that can be used to solve the Navier-Stokes equations accurately for turbulent flows. Much like LES (large eddy simulation), VMS separates flow scales into resolved and unresolved scales, and the effects of the unresolved scales are incorporated into the resolved scales by a turbulence model.