# Flow of Newtonian Incompressible Fluids in Square Media: Isogeometric vs. Standard Finite Element Method

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

_{0}denotes the value of some reference velocity magnitude, d

_{p}stands for the diameter of pellets, and $\mathsf{\gamma}$ is the dynamic viscosity of the fluid.

## 3. Introduction to B-Spline NURBS Method

## 4. The Approximate Formulation of the Navier–Stokes Equation

## 5. Numerical Results

^{2}and ${\mathrm{H}}^{1}$:

**i.****Numerical results on the unit square**

- Prescribed values: Dirichlet boundary conditions involve prescribing the values of certain flow variables (e.g., velocity, pressure, temperature) at the boundary. This allows the simulation to mimic real-world scenarios where specific boundary conditions are known or can be measured [18].
- Physical relevance: Dirichlet boundary conditions align with physical phenomena and constraints. They allow the simulation to capture the influence of real-world boundaries, such as walls or inlets/outlets, on the flow behavior. By imposing appropriate Dirichlet conditions, the flow near boundaries can be accurately represented, considering factors such as the no-slip condition on walls or prescribed inflow/outflow conditions at inlets and outlets.
- Ease of implementation: Dirichlet boundary conditions are easily implemented in numerical solvers. They do not require intricate mathematical formulations or additional equations compared to other boundary conditions.

**ii.****Benchmark NURBS and FEM for square example**

**Velocity fields**

**Pressure fields**

_{x}= 0.0526 and A

_{y}= 0.0526,

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**A square geometry, defined as the set [0, 1] × [0, 1], and the control mesh of its NURBS representation 16 × 16.

**Figure 3.**Convergence of relative L2 error norm for different approaches with shape functions of order p: p = 1 (

**a**), p = 2 (

**b**), and p = 3 (

**c**) at Re = 10.

**Figure 4.**Convergence of relative L2 error norm for different approaches with shape functions of order p: p = 1 (

**a**), p = 2 (

**b**), and p = 3 (

**c**) at Re = 100.

**Figure 5.**Convergence of relative L2 error norm for different approaches with shape functions of order p: p = 1 (

**a**), p = 2 (

**b**), and p = 3 (

**c**) at Re = 1000.

**Figure 6.**Velocity vector field representation: FEM approximate solution (

**a**), IGA approximate solution (

**b**), and exact solution (

**c**).

**Figure 7.**The variation of the pressure field of the point A: (

**a**) the computed solution (NURBS), and (

**b**) the exact solution.

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**MDPI and ACS Style**

Scutaru, M.L.; Guendaoui, S.; Koubaiti, O.; El Ouadefli, L.; El Akkad, A.; Elkhalfi, A.; Vlase, S.
Flow of Newtonian Incompressible Fluids in Square Media: Isogeometric vs. Standard Finite Element Method. *Mathematics* **2023**, *11*, 3702.
https://doi.org/10.3390/math11173702

**AMA Style**

Scutaru ML, Guendaoui S, Koubaiti O, El Ouadefli L, El Akkad A, Elkhalfi A, Vlase S.
Flow of Newtonian Incompressible Fluids in Square Media: Isogeometric vs. Standard Finite Element Method. *Mathematics*. 2023; 11(17):3702.
https://doi.org/10.3390/math11173702

**Chicago/Turabian Style**

Scutaru, Maria Luminița, Sohaib Guendaoui, Ouadie Koubaiti, Lahcen El Ouadefli, Abdeslam El Akkad, Ahmed Elkhalfi, and Sorin Vlase.
2023. "Flow of Newtonian Incompressible Fluids in Square Media: Isogeometric vs. Standard Finite Element Method" *Mathematics* 11, no. 17: 3702.
https://doi.org/10.3390/math11173702