## Latest News on Stokes Equations Research: Nov – 2019

**A stable finite element for the stokes equations**

We gift during this paper a brand new velocity-pressure finite component for the computation of Stokes flow. we have a tendency to discretize the speed field with continuous piecewise linear functions enriched by bubble functions, and also the pressure by piecewise linear functions. we have a tendency to show that this component satisfies the standard inf-sup condition and converges with initial order for each velocities and pressure. Finally we have a tendency to relate this component to families of higer order parts and to the favored Taylor-Hood component. **[1]**

**Application of a fractional-step method to incompressible Navier-Stokes equations**

A numerical methodology for computing three-dimensional, time-dependent incompressible flows is bestowed. the tactic is predicated on a fractional-step, or time-splitting, theme in conjunction with the approximate-factorization technique. it’s shown that the utilization of speed boundary conditions for the intermediate velocity field will result in inconsistent numerical solutions. applicable boundary conditions for the intermediate speed field are derived and tested. Numerical solutions for flows within a driven cavity and over a backward-facing step are bestowed and compared with experimental information and alternative numerical results. **[2]**

**Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method**

It is known that the Frisch-Hasslacher-Pomeau lattice-gas automaton model and connected models possess some rather unphysical effects. These ar (1) a non-Galilean invariableness caused by a density-dependent constant within the convection term, and (2) a velocity-dependent equation of state. during this paper, we tend to show that each of those effects will be eliminated precisely in an exceedingly lattice Boltzmann-equation model. **[3]**

**Scaling Relations and Self-Similarity of 3-Dimensional Reynolds-Averaged Navier-Stokes Equations**

Scaling conditions to attain self-similar solutions of three-dimensional (3D) Reynolds-Averaged Navier-Stokes Equations, as associate initial and boundary price downside, are obtained by utilizing Lie cluster of purpose Scaling Transformations. By suggests that of associate ASCII text file Navier-Stokes problem solver and therefore the derived self-similarity conditions, we tend to incontestable self-similarity inside the time variation of flow dynamics for a rigid-lid cavity downside underneath each up-scaled and down-scaled domains. The strength of the planned approach lies in its ability to think about the underlying flow dynamics through not solely from the governing equations into account however additionally from the initial and boundary conditions, thus permitting to get excellent self-similarity in numerous time and area scales. **[4]**

**The First Integral Method for the Two-dimensional Incompressible Navier-Stokes Equations**

In this paper, we tend to modify the primary integral methodology to search out actual solutions for The Two-Dimensional incompressible Navier-Stokes equations. This methodology is associate degree algebraical direct methodology used division theorem to search out the primary integral through polynomial and use undulation resolution to remodel the partial equation into the standard equation. we tend to get totally different actual solutions through the utilization of this methodology and these solutions square measure either of the formula of exponential, hyperbolic or pure mathematics functions. **[5]**

**Reference**

**[1]** Arnold, D.N., Brezzi, F. and Fortin, M., 1984. A stable finite element for the Stokes equations. Calcolo, 21(4), (Web Link)

**[2]** Kim, J. and Moin, P., 1985. Application of a fractional-step method to incompressible Navier-Stokes equations. Journal of computational physics, 59(2), (Web Link)

**[3]** Chen, H., Chen, S. and Matthaeus, W.H., 1992. Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. Physical Review A, 45(8), (Web Link)

**[4]** Scaling Relations and Self-Similarity of 3-Dimensional Reynolds-Averaged Navier-Stokes Equations

Ali Ercan & M. Levent Kavvas

Scientific Reports volume 7, Article number: 6416 (2017) (Web Link)

**[5]** Al-Hussein, A. (2018) “The First Integral Method for the Two-dimensional Incompressible Navier-Stokes Equations”, Journal of Advances in Mathematics and Computer Science, 26(5), (Web Link)