**Author details**

**Theorem 2**: Sequence of the solution of Eq. (75) converges to the solution of Eq. (74).

approach of FPM, the following problem is achieved:

3

å

1

*i*

*i*

( ) ( )

394 Numerical Simulation - From Brain Imaging to Turbulent Flows

1

2

2

ï

ï

ï

Then set the value of target function in *In*.

*S t*

Step 1: Read *ε*, and set *n* =1.

function in *In*.

**6. Conclusions**

3

2

*i*

The steps of using the above equations are as follows:

Step 2: Solve Eq. (73) or (76) in the FPM or the TLM, respectively.

Step 5: If | *In* − *In*−<sup>1</sup> | >*ε*1 then go to step 4, else *In* is the final answer.

**Proof**: The method of proof of this theorem is similar to the previous theorem. In the same

( )( )

×= + + - - + - +

1 1

*n ii i*

() ( ) ( )

11 11 11 1

<sup>ì</sup> é + + + + + ù= <sup>ï</sup> ë û <sup>ï</sup>

*<sup>h</sup> E x E x h E x mh*

*n n n*

() ( ) ( )

22 22 22 2

... <sup>0</sup> . <sup>2</sup>

<sup>ï</sup> é + + + + + ù= ë û <sup>=</sup> <sup>í</sup>

*<sup>h</sup> E x E x h E x mh*

*<sup>h</sup> E x E x h E x mh*

*n n n*

*n n n*

*C in*

î ³ ==

Step 4: Set *n*=*n*+1, and solve Eq. (73) or (76) in the FPM or the TLM, respectively.

33 33 33 3

<sup>ï</sup> é + + + + + ù= ë û <sup>ï</sup>

0 1,2,3. 1,2,...

() ( ) ( )

Step 3: If the previous step is infeasible, set *n*=*n*+1, and go to step 2, else set the value of target

In this chapter volume of fluid (VOF) scheme was introduced. This is one of the most effective methods employed in the simulation of two fluid flows interfaces with dramatic changes in density and viscosity. . These interfaces are represented implicitly by the values of a color function which is the fluid volume fraction. The advantage of the method is its ability to deal with arbitrarily shaped interfaces and to cope with large deformations, as well as interface rupture and coalescence in a natural way. In VOF the mass is rigorously conserved, provid‐ ed the discretization is conservative. However, advecting the interface without diffusing, dispersing, or wrinkling is a big issue. This can either be performed algebraically, in schemes

inf 2 ( 1) ( 1) <sup>2</sup>

*n ni ii*

*<sup>h</sup> I P Px m h Lx m h P x mh L x mh* <sup>=</sup>

1 1

*ni n i*

() () ( )( ) ( ) ( )

æ ö - + +- + + ç ÷

+- + è ø

... 0

... 0

2 ...

(76)

11 1 1

*Px Lx Px h Lx h*

Mohammad Javad Ketabdari

Address all correspondence to: ketabdar@aut.ac.ir

Faculty of Marine Technology, Amirkabir University of Technology, Tehran, Iran
