A finite element analysis of the distribution velocity in viscous incompressible fluids using the Lagrange interpolation function

In this work we use the finite element method to analyze the distribution of velocity in a viscous incompressible fluid flow using Lagrange interpolation function. The results obtained are highly accurate and converge fast to the exact solution as the number of elements increase. @JASEM A fluid is basically any substance that deforms continuously. Its motion is governed by the Stokes equations also known as Navier-Stokes equations. The fluid may be compressible or incompressible. In this study, we consider viscous incompressible fluids. It should be noted that stokes equation can also be used to model turbulent flow, in this case, the fluid parameters are treated as time average values (Lions, 1998 and Efunda, 2005). Finite element method is well treated in many standard texts (Burnett, 1987; Bickford, 1989; and Fagan, 1992). The determination of an accurate characterization of fluid flow has been the subject of much research. Some of these researchers include Stokes, Reynolds, Navier, Bernoulli, Petroff, Sommerfeld etc. Their works formed the fundamental principles of fluid dynamics. (Fox; McDonald,1996; Massey; Wardsmith, 2001) Also, Demkowicz et. al. (1990) developed a new finite element method for solving compressible Navier-Stokes equations. Their method was based on a version of Strang's operator splitting and an h-p adaptive finite element approximation in space. Wu and Jin (2005) proposed a numerical method for solving Stokes equation with corner singularity. First, they solved a simple eigenvalue problem, which was one dimension less than the original problem, to obtain the discrete expansion of the singularity near the corner. And then combined the approximation of the singularity and standard finite element basis functions to construct special finite element spaces, and solve the original problem in the special spaces on a conventional mesh. In this work we use the finite element method to analyze the distribution of velocity in a viscous incompressible fluid flow using Lagrange interpolation function. MATHEMATICAL FORMULATION Consider the relation: ( ) ( ) 0 p u u t x y ρ ρ ∂ ∂ ∂ + + = ∂ ∂ ∂ (1) At steady state, 0 t ∂ = ∂ (2) Thus, we have: ( ) u x ρ ∂ ∂ + ( ) u y ρ ∂ ∂ (3) for incompressible fluids, constant ρ = ( is density ρ ), this implies that: 0 u v x y ρ   ∂ ∂ + =   ∂ ∂   (4) from the continuity expression, 0 u v x y   ∂ ∂ + =   ∂ ∂   (5) A finite element analysis... AKPOBI, J A; AKPOBI, E D 32 from the conservation of linear momentum, for incompressible fluid flow we have: u f u u t σ ρ ∂   ∇ + = + ⋅∇   ∂   (6) xy x x u u u f u v x y t t y σ σ ρ ∂   ∂ ∂ ∂ ∂ + = + +   ∂ ∂ ∂ ∂ ∂   (7) y xy y v v u f u v x y t x x σ σ ρ ∂ ∂ ∂ ∂ ∂   + + = + +   ∂ ∂ ∂ ∂ ∂   (8)

A fluid is basically any substance that deforms continuously.Its motion is governed by the Stokes equations also known as Navier-Stokes equations.The fluid may be compressible or incompressible.In this study, we consider viscous incompressible fluids.It should be noted that stokes equation can also be used to model turbulent flow, in this case, the fluid parameters are treated as time average values (Lions, 1998 andEfunda, 2005).Finite element method is well treated in many standard texts (Burnett, 1987;Bickford, 1989;and Fagan, 1992).The determination of an accurate characterization of fluid flow has been the subject of much research.Some of these researchers include Stokes, Reynolds, Navier, Bernoulli, Petroff, Sommerfeld etc. Their works formed the fundamental principles of fluid dynamics.(Fox;McDonald,1996;Massey;Wardsmith, 2001) Also, Demkowicz et. al. (1990) developed a new finite element method for solving compressible Navier-Stokes equations.Their method was based on a version of Strang's operator splitting and an h-p adaptive finite element approximation in space.
Wu and Jin (2005) proposed a numerical method for solving Stokes equation with corner singularity.First, they solved a simple eigenvalue problem, which was one dimension less than the original problem, to obtain the discrete expansion of the singularity near the corner.And then combined the approximation of the singularity and standard finite element basis functions to construct special finite element spaces, and solve the original problem in the special spaces on a conventional mesh.
In this work we use the finite element method to analyze the distribution of velocity in a viscous incompressible fluid flow using Lagrange interpolation function.

MATHEMATICAL FORMULATION
Consider the relation: Thus, we have: from the conservation of linear momentum, for incompressible fluid flow we have: by substitution, ( ) ( ) Restricting the Stoke's equation for viscous, incompressible fluid flow, we have: let 0, ( ), ( ), 0 this is the reduced Stoke's model for viscous, incompressible, steady flow.Now considering equation.14 For steady state condition thus the stoke's model for viscous, incompressible, steady flow is

Developing the Velocity Distribution Model for Boundary Value Flow Problems
Since the boundary value problem are described by: 0 for 0 d du a cu q x l dy dy comparing this with the developed expression for steady state, viscous incompressible fluid flow ( ) ( ) rearranging and comparing: Generating a finite element model for boundary value problems: Integrating over the element domain: a cu q dy dy dy obtaining the primary and secondary variable from the weak form: du a dy secondary variable u primary variable boundary condition assumption: thus the variational form is: cwu wq dy wa y Q wa y Q dy dy For an arbitrary degree interpolation thus obtaining a general term, this could be given as:   In Example 2, the Fig. 2 shows a convergence of the numerical solution of velocity distribution to the exact solution.For the fluid with viscosity 0.92 kg(ms) -1 and pressure gradient 12N/m 3 , optimal velocity (16504.34m/s) is noticed midway between the stationary and moving plates.The fluid particles in contact with the moving plates assumes its velocity (400m/s), While those in contact with the stationary plates are at 'zero velocity'.It must be noted that the numerical technique described above gives the same velocity as the 'exact value' at nodal points.
In Example 3, Fig. 3 shows the numerical solution of velocity distribution of a viscous incompressible fluid of viscosity 0.4kg (ms) -1 and pressure gradient 20 N/m 3 at different location between the two fixed plates and the exact solution, for different number of finite elements.The results, as shown in the Figure using the technique converge fast towards the exact solution.The velocity of fluid flow assumes a maximum value of 22500m/s midway between both plates, and a 'zero' value at the location of the plates.
Table 1, shows the percentage difference of values between analytical solutions and solutions from our model for all the three Examples.A maximum percentage difference of 1.44% and a minimum difference of 0.0% between analytical solutions and the model were obtained.These results show that the solutions obtained using this model are highly accurate.

Conclusion:
In this work we have used the finite element method (with the Lagrange interpolation function) to analyze the distribution of velocity, in viscous incompressible fluids.The model developed is highly accurate, as shown by the results obtained.

Fig. 1 :Fig. 2 :
Fig.1: A graph of velocity against displacement for 2, 4, 6, 8, 10, 12 elements and the exact.Example 2: A viscous, incompressible fluid flows between two flat and parallel plates which are 100m apart and at a constant pressure gradient of 12N/m3 and he fluid is of viscosity of 0.92 kg/m/s.determine its velocity distribution U(y) if the velocity at the wall of the plates are U= 0 m/s; U b = 400 m/s Solution: Interpolation method: Lagrange interpolation technique.Element type: Linear.The solutions obtained using the model are shown in Fig. 2.
Fig.3represents the solutions to Example 3.In Example 1, Fig.1compares the numerical solution of velocity distribution for a viscous incompressible flow at different location between two faced plates and the exact solution.The result from this numerical technique converges towards the exact solution.The rate of converges increases fast as the number of elements increases.At the stationary plates, the fluid of viscosity 0.8 kg(ms) -1 has zero velocity (0 m/s), the flow velocity of the fluid is maximum (5000m/s), midway between the plates.