\documentclass[11pt,twoside]{article}\makeatletter

\IfFileExists{xcolor.sty}%
  {\RequirePackage{xcolor}}%
  {\RequirePackage{color}}
\usepackage{colortbl}
\usepackage{wrapfig}
\usepackage{ifxetex}
\ifxetex
  \usepackage{fontspec}
  \usepackage{xunicode}
  \catcode`⃥=\active \def⃥{\textbackslash}
  \catcode`❴=\active \def❴{\{}
  \catcode`❵=\active \def❵{\}}
  \def\textJapanese{\fontspec{Noto Sans CJK JP}}
  \def\textChinese{\fontspec{Noto Sans CJK SC}}
  \def\textKorean{\fontspec{Noto Sans CJK KR}}
  \setmonofont{DejaVu Sans Mono}
  
\else
  \IfFileExists{utf8x.def}%
   {\usepackage[utf8x]{inputenc}
      \PrerenderUnicode{–}
    }%
   {\usepackage[utf8]{inputenc}}
  \usepackage[english]{babel}
  \usepackage[T1]{fontenc}
  \usepackage{float}
  \usepackage[]{ucs}
  \uc@dclc{8421}{default}{\textbackslash }
  \uc@dclc{10100}{default}{\{}
  \uc@dclc{10101}{default}{\}}
  \uc@dclc{8491}{default}{\AA{}}
  \uc@dclc{8239}{default}{\,}
  \uc@dclc{20154}{default}{ }
  \uc@dclc{10148}{default}{>}
  \def\textschwa{\rotatebox{-90}{e}}
  \def\textJapanese{}
  \def\textChinese{}
  \IfFileExists{tipa.sty}{\usepackage{tipa}}{}
\fi
\def\exampleFont{\ttfamily\small}
\DeclareTextSymbol{\textpi}{OML}{25}
\usepackage{relsize}
\RequirePackage{array}
\def\@testpach{\@chclass
 \ifnum \@lastchclass=6 \@ne \@chnum \@ne \else
  \ifnum \@lastchclass=7 5 \else
   \ifnum \@lastchclass=8 \tw@ \else
    \ifnum \@lastchclass=9 \thr@@
   \else \z@
   \ifnum \@lastchclass = 10 \else
   \edef\@nextchar{\expandafter\string\@nextchar}%
   \@chnum
   \if \@nextchar c\z@ \else
    \if \@nextchar l\@ne \else
     \if \@nextchar r\tw@ \else
   \z@ \@chclass
   \if\@nextchar |\@ne \else
    \if \@nextchar !6 \else
     \if \@nextchar @7 \else
      \if \@nextchar (8 \else
       \if \@nextchar )9 \else
  10
  \@chnum
  \if \@nextchar m\thr@@\else
   \if \@nextchar p4 \else
    \if \@nextchar b5 \else
   \z@ \@chclass \z@ \@preamerr \z@ \fi \fi \fi \fi
   \fi \fi  \fi  \fi  \fi  \fi  \fi \fi \fi \fi \fi \fi}
\gdef\arraybackslash{\let\\=\@arraycr}
\def\@textsubscript#1{{\m@th\ensuremath{_{\mbox{\fontsize\sf@size\z@#1}}}}}
\def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}}
\def\abbr{}
\def\corr{}
\def\expan{}
\def\gap{}
\def\orig{}
\def\reg{}
\def\ref{}
\def\sic{}
\def\persName{}\def\name{}
\def\placeName{}
\def\orgName{}
\def\textcal#1{{\fontspec{Lucida Calligraphy}#1}}
\def\textgothic#1{{\fontspec{Lucida Blackletter}#1}}
\def\textlarge#1{{\large #1}}
\def\textoverbar#1{\ensuremath{\overline{#1}}}
\def\textquoted#1{‘#1’}
\def\textsmall#1{{\small #1}}
\def\textsubscript#1{\@textsubscript{\selectfont#1}}
\def\textxi{\ensuremath{\xi}}
\def\titlem{\itshape}
\newenvironment{biblfree}{}{\ifvmode\par\fi }
\newenvironment{bibl}{}{}
\newenvironment{byline}{\vskip6pt\itshape\fontsize{16pt}{18pt}\selectfont}{\par }
\newenvironment{citbibl}{}{\ifvmode\par\fi }
\newenvironment{docAuthor}{\ifvmode\vskip4pt\fontsize{16pt}{18pt}\selectfont\fi\itshape}{\ifvmode\par\fi }
\newenvironment{docDate}{}{\ifvmode\par\fi }
\newenvironment{docImprint}{\vskip 6pt}{\ifvmode\par\fi }
\newenvironment{docTitle}{\vskip6pt\bfseries\fontsize{22pt}{25pt}\selectfont}{\par }
\newenvironment{msHead}{\vskip 6pt}{\par}
\newenvironment{msItem}{\vskip 6pt}{\par}
\newenvironment{rubric}{}{}
\newenvironment{titlePart}{}{\par }

\newcolumntype{L}[1]{){\raggedright\arraybackslash}p{#1}}
\newcolumntype{C}[1]{){\centering\arraybackslash}p{#1}}
\newcolumntype{R}[1]{){\raggedleft\arraybackslash}p{#1}}
\newcolumntype{P}[1]{){\arraybackslash}p{#1}}
\newcolumntype{B}[1]{){\arraybackslash}b{#1}}
\newcolumntype{M}[1]{){\arraybackslash}m{#1}}
\definecolor{label}{gray}{0.75}
\def\unusedattribute#1{\sout{\textcolor{label}{#1}}}
\DeclareRobustCommand*{\xref}{\hyper@normalise\xref@}
\def\xref@#1#2{\hyper@linkurl{#2}{#1}}
\begingroup
\catcode`\_=\active
\gdef_#1{\ensuremath{\sb{\mathrm{#1}}}}
\endgroup
\mathcode`\_=\string"8000
\catcode`\_=12\relax

\usepackage[a4paper,twoside,lmargin=1in,rmargin=1in,tmargin=1in,bmargin=1in,marginparwidth=0.75in]{geometry}
\usepackage{framed}

\definecolor{shadecolor}{gray}{0.95}
\usepackage{longtable}
\usepackage[normalem]{ulem}
\usepackage{fancyvrb}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{marginnote}

\renewcommand{\@cite}[1]{#1}


\renewcommand*{\marginfont}{\itshape\footnotesize}

\def\Gin@extensions{.pdf,.png,.jpg,.mps,.tif}

  \pagestyle{fancy}

\usepackage[pdftitle={Analytical Study on Motion Behavior of Non-Spherical Particles in Incompressible Fluid with Presence of Electrostatic Force},
 pdfauthor={}]{hyperref}
\hyperbaseurl{}

	 \paperwidth210mm
	 \paperheight297mm
              
\def\@pnumwidth{1.55em}
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5}
\setcounter{tocdepth}{3}
\clubpenalty=8000
\emergencystretch 3em
\hbadness=4000
\hyphenpenalty=400
\pretolerance=750
\tolerance=2000
\vbadness=4000
\widowpenalty=10000

\renewcommand\section{\@startsection {section}{1}{\z@}%
     {-1.75ex \@plus -0.5ex \@minus -.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\Large\bfseries}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
     {-1.75ex\@plus -0.5ex \@minus- .2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\Large}}
\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
     {-1.5ex\@plus -0.35ex \@minus -.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\large}}
\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
     {-1ex \@plus-0.35ex \@minus -0.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\normalsize}}
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
     {1.5ex \@plus1ex \@minus .2ex}%
     {-1em}%
     {\reset@font\normalsize\bfseries}}


\def\l@section#1#2{\addpenalty{\@secpenalty} \addvspace{1.0em plus 1pt}
 \@tempdima 1.5em \begingroup
 \parindent \z@ \rightskip \@pnumwidth 
 \parfillskip -\@pnumwidth 
 \bfseries \leavevmode #1\hfil \hbox to\@pnumwidth{\hss #2}\par
 \endgroup}
\def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\def\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\@ifundefined{c@section}{\newcounter{section}}{}
\@ifundefined{c@chapter}{\newcounter{chapter}}{}
\newif\if@mainmatter 
\@mainmattertrue
\def\chaptername{Chapter}
\def\frontmatter{%
  \pagenumbering{roman}
  \def\thechapter{\@roman\c@chapter}
  \def\theHchapter{\roman{chapter}}
  \def\thesection{\@roman\c@section}
  \def\theHsection{\roman{section}}
  \def\@chapapp{}%
}
\def\mainmatter{%
  \cleardoublepage
  \def\thechapter{\@arabic\c@chapter}
  \setcounter{chapter}{0}
  \setcounter{section}{0}
  \pagenumbering{arabic}
  \setcounter{secnumdepth}{6}
  \def\@chapapp{\chaptername}%
  \def\theHchapter{\arabic{chapter}}
  \def\thesection{\@arabic\c@section}
  \def\theHsection{\arabic{section}}
}
\def\backmatter{%
  \cleardoublepage
  \setcounter{chapter}{0}
  \setcounter{section}{0}
  \setcounter{secnumdepth}{2}
  \def\@chapapp{\appendixname}%
  \def\thechapter{\@Alph\c@chapter}
  \def\theHchapter{\Alph{chapter}}
  \appendix
}
\newenvironment{bibitemlist}[1]{%
   \list{\@biblabel{\@arabic\c@enumiv}}%
       {\settowidth\labelwidth{\@biblabel{#1}}%
        \leftmargin\labelwidth
        \advance\leftmargin\labelsep
        \@openbib@code
        \usecounter{enumiv}%
        \let\p@enumiv\@empty
        \renewcommand\theenumiv{\@arabic\c@enumiv}%
	}%
  \sloppy
  \clubpenalty4000
  \@clubpenalty \clubpenalty
  \widowpenalty4000%
  \sfcode`\.\@m}%
  {\def\@noitemerr
    {\@latex@warning{Empty `bibitemlist' environment}}%
    \endlist}

\def\tableofcontents{\section*{\contentsname}\@starttoc{toc}}
\parskip0pt
\parindent1em
\def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}}
\newenvironment{reflist}{%
  \begin{raggedright}\begin{list}{}
  {%
   \setlength{\topsep}{0pt}%
   \setlength{\rightmargin}{0.25in}%
   \setlength{\itemsep}{0pt}%
   \setlength{\itemindent}{0pt}%
   \setlength{\parskip}{0pt}%
   \setlength{\parsep}{2pt}%
   \def\makelabel##1{\itshape ##1}}%
  }
  {\end{list}\end{raggedright}}
\newenvironment{sansreflist}{%
  \begin{raggedright}\begin{list}{}
  {%
   \setlength{\topsep}{0pt}%
   \setlength{\rightmargin}{0.25in}%
   \setlength{\itemindent}{0pt}%
   \setlength{\parskip}{0pt}%
   \setlength{\itemsep}{0pt}%
   \setlength{\parsep}{2pt}%
   \def\makelabel##1{\upshape ##1}}%
  }
  {\end{list}\end{raggedright}}
\newenvironment{specHead}[2]%
 {\vspace{20pt}\hrule\vspace{10pt}%
  \phantomsection\label{#1}\markright{#2}%

  \pdfbookmark[2]{#2}{#1}%
  \hspace{-0.75in}{\bfseries\fontsize{16pt}{18pt}\selectfont#2}%
  }{}
      \def\TheFullDate{2015-05-15 (revised: 15 May 2015)}
\def\TheID{\makeatother }
\def\TheDate{2015-05-15}
\title{Analytical Study on Motion Behavior of Non-Spherical Particles in Incompressible Fluid with Presence of Electrostatic Force}
\author{}\makeatletter 
\makeatletter
\newcommand*{\cleartoleftpage}{%
  \clearpage
    \if@twoside
    \ifodd\c@page
      \hbox{}\newpage
      \if@twocolumn
        \hbox{}\newpage
      \fi
    \fi
  \fi
}
\makeatother
\makeatletter
\thispagestyle{empty}
\markright{\@title}\markboth{\@title}{\@author}
\renewcommand\small{\@setfontsize\small{9pt}{11pt}\abovedisplayskip 8.5\p@ plus3\p@ minus4\p@
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus2\p@
\belowdisplayshortskip 4\p@ plus2\p@ minus2\p@
\def\@listi{\leftmargin\leftmargini
               \topsep 2\p@ plus1\p@ minus1\p@
               \parsep 2\p@ plus\p@ minus\p@
               \itemsep 1pt}
}
\makeatother
\fvset{frame=single,numberblanklines=false,xleftmargin=5mm,xrightmargin=5mm}
\fancyhf{} 
\setlength{\headheight}{14pt}
\fancyhead[LE]{\bfseries\leftmark} 
\fancyhead[RO]{\bfseries\rightmark} 
\fancyfoot[RO]{}
\fancyfoot[CO]{\thepage}
\fancyfoot[LO]{\TheID}
\fancyfoot[LE]{}
\fancyfoot[CE]{\thepage}
\fancyfoot[RE]{\TheID}
\hypersetup{citebordercolor=0.75 0.75 0.75,linkbordercolor=0.75 0.75 0.75,urlbordercolor=0.75 0.75 0.75,bookmarksnumbered=true}
\fancypagestyle{plain}{\fancyhead{}\renewcommand{\headrulewidth}{0pt}}

\date{}
\usepackage{authblk}

\providecommand{\keywords}[1]
{
\footnotesize
  \textbf{\textit{Index terms---}} #1
}

\usepackage{graphicx,xcolor}
\definecolor{GJBlue}{HTML}{273B81}
\definecolor{GJLightBlue}{HTML}{0A9DD9}
\definecolor{GJMediumGrey}{HTML}{6D6E70}
\definecolor{GJLightGrey}{HTML}{929497} 

\renewenvironment{abstract}{%
   \setlength{\parindent}{0pt}\raggedright
   \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}}
   \vskip16pt
   \textcolor{GJBlue}{\large\bfseries\abstractname\space}
}{%   
   \vskip8pt
   \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}}
   \vskip16pt
}

\usepackage[absolute,overlay]{textpos}

\makeatother 
      \usepackage{lineno}
      \linenumbers
      
\begin{document}

             \author[1]{A. R.  Noorpoor}

             \affil[1]{  University of Tehran}

\renewcommand\Authands{ and }

\date{\small \em Received: 6 April 2015 Accepted: 30 April 2015 Published: 15 May 2015}

\maketitle


\begin{abstract}
        


In this paper, the accelerated motion of non-spherical particles in an incompressible fluid in both the presence and the absence of electrostatic force was investigated. Differential transformation method (DTM) and a FORTRAN code was used to calculate the instantaneous velocity of particles. Regarding particles? instantaneous velocity in the absence of electrostatic force, DTM approach was resulted in a proper accordance with previous studies which utilized variational iteration method (VIM). In addition, a good agreement between DTM and VMT was seen as sphericity of particles was varied from 0.5 to 0.9. The results showed that falling velocity increased with increasing sphericity. Moreover, the presence of electrostatic force (by assuming the electrical load equal to 1 micro colon) was compared to the one with no electrostatic force. The results showed that the falling velocity was decreased by 23.33% under the effect of an additional resistant force.

\end{abstract}


\keywords{electrostatic, particles, acceleration motion, fluid, liquid.}

\begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) 
\textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark}
\end{textblock*}

\begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) 
\textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}}
\end{textblock*}


\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
edimentation of solid or liquid particles in fluids occurs in the different natural and artificial phenomena. Different researches were studied the behavior of dispersed particles in an incompressible media, in which most of the mutilized Eulerian-Eulerian or Eulerian-Lagrangian approaches. When using an Eulerian-Lagrangian approach for two-phases fluids, the continuity and momentum balance equations must be derived from the hydrodynamics of the continuous phase. However, each encapsulated particle is considered as a main mass for which velocity and state are derived from Newton's second law as follows?? ?? ???? ?? ???? = ?? ?? , ???? ?? ???? = ?? ?? (\textbf{1})\par
where?? ?? is particle mass,??is velocity, ??is time, ?? is original analytical function, and?? ?? is the resultant of forces that are appliedtothe particle, including gravity, buoyancy, drag, virtual mass force, the Basset force, and lift force.\par
Equation \hyperref[b0]{(1)} has been solved numerically in different studies by various methods, for example the finite difference method \hyperref[b0]{[1]}. Some analytical methods which were applied for analysis of the acceleration of spherical and non-spherical particles motion in Newtonian fluids was addressed in \hyperref[b1]{[2]}\hyperref[b2]{[3]}\hyperref[b3]{[4]}\hyperref[b4]{[5]}\hyperref[b5]{[6]}\hyperref[b6]{[7]}\hyperref[b7]{[8]}\hyperref[b8]{[9]}\hyperref[b9]{[10]}. Jalaal and Ganji \hyperref[b2]{[3]}studied spherical and non-spherical particles motion in unsteady state at Newtonian media. They used friction coefficient governed by Chhabra and Ferreira's equations \hyperref[b10]{[11]}for a range of Reynolds numbers by using the homotopy perturbation method (HPM). Jalaal et al. \hyperref[b5]{[6]}studied non-spherical particles motion at the Newtonian media using the VIM and friction equations derived by Chien \hyperref[b11]{[12]}. In another researches, Jalaal et al. \hyperref[b3]{[4]} and Jalaal and Ganji \hyperref[b4]{[5]}studied non-spherical particles by using the HPM.\par
Stokes \hyperref[b12]{[13]}assigned following equation for drag coefficient of a sedimenting particle. The equation is derived for a flow field that is totally dominated by viscous diffusion as belowC D = 24 Re (2)\par
where C D is drag coefficient and Re is Reynolds number. This equation denied the effect of inertia and is accepted for???? < 0.4. Therefore, Oseen \hyperref[b13]{[14]}, assuming the effect of inertia, completed the Stokes's equation as follow?? ?? = 24 ???? ?1 + 3 16 ?????\textbf{(3)}\par
Most of previous researches on spherical and non-spherical particles carried out experimentally and only a few of them were analytically investigated the solution of motion equations. In this regard, Proudman and Pearson \hyperref[b14]{[15]}proposed???? 2 ×ln (????) parameter to consider the behavior of drag coefficient for spherical particles and then Sano \hyperref[b15]{[16]}completed aforementioned equation. Lovalenti and Brady \hyperref[b16]{[17]} used the Kim and Karilla's equations \hyperref[b17]{[18]}to solve directly the behavior of applied forces on particles and derived different parameters of time reduction, Including?? ?1 , ?? ?5/2 ,?? ??? , ?? ?2 . ?? ??? , ?? ?1/2 , and ?? ?2 , depending on initial conditions. Ferreira and Chhabra \hyperref[b10]{[11]}achieved the following equation for 0 ? Re ?10 5    
\section[{S}]{S} 
\section[{Global Journal of Researches in Engineering}]{Global Journal of Researches in Engineering}?? ?? = 24 ???? ?1 + 1 48 ?????\textbf{(4)}\par
Equation ( \hyperref[formula_4]{4}) showed a suitable accordance with results of experimental researches. Also, in many cases, a linear equation can describe the drag force very well. In some cases, Reynolds number has an average value and the liner and exponential terms are used in \hyperref[b10]{[11,}\hyperref[b11]{12,}\hyperref[b18]{[19]} {\ref [20]} {\ref [21]} {\ref [22]}.\par
Regarding non-spherical particles, Haider and Levenspiel  {\ref [23]}proposed an accurate equation in order to contemplate drag force taking into account of Reynolds number and sphericity. Rewriting the force balance and assuming ?<<? ?? , the equation ( \hyperref[formula_0]{1}) can be implemented as \hyperref[b4]{(5)} where?? is acceleration due to gravity, ?? is fluid density, ?? ?? is particle density and ?? is particle's equivalent diameterThe main difficulty in solving equation (  {\ref 5}) is the non-linear part appeared to express drag force.By insertingequation (4) in equation (  {\ref 5}) and rewriting the equation (  {\ref 5}), the equation ( \hyperref[formula_6]{6}) is derived as followm ???? ???? = ???? ?1 ? ?? ?? ?? ? ? 1 8 ???? 2 ???? ?? ?? 2 ? 1 12 ???? 3 ?? ???? ?????? ???? ???? + ???? + ???? 2 ? ?? = 0 , ??(0) = 0\textbf{(6)}\par
Where ??, ??, ??, and ?? are constants and depend on physical condition of the system.\par
Different methods were proposed in previous studies to solve equation ( \hyperref[formula_6]{6}). Among those, VIM, a technique based on repeated integration proposed by He  {\ref [24]}, was used successfully to solve linear and nonlinear equations by different authors. This method is based on the true function by a general Lagrangian multiplier. DTM is another method to solve the linear and non-linear terms proposed by Yaghoobi and Torabi \hyperref[b7]{[8]}. Noorpoor and Nazari [25] used DTM to calculate the falling velocity of a particle in the acoustic field. 
\section[{II.}]{II.} 
\section[{Materials and Methods}]{Materials and Methods}\par
In this paper, equation (  {\ref 5}) was solved by using the Yaghoobi and Torabi's equations \hyperref[b7]{[8]}. The solving procedure included applying DTM by using a FORTRAN code.\par
By combining equation (  {\ref 5}), Reynolds number (???? = ?????? ?? , ?? is dynamic viscosity), and drag coefficient equation, the final equation of a falling acceleration of non-spherical particle motion in an incompressible fluid was written as follow??? + 1 12 ???? 3 ??? ???? ???? + 3.75???????? + 67.289?? (?5.03?) 8 ???? 2 ???? 2 ? ???? ?1 ? ?? ?? ?? ? = 0\textbf{(7)}\par
where ? is sphericity of particle. For simplicity, this equation's coefficients were assigned as a, b, c and d in further equations. First term of equation ( \hyperref[formula_7]{7})is to describe the added mass to a falling non-spherical particle in an incompressible fluid. The second and third term show the resistant force of particle's linear and non-linear motion respectively, and the fourth shows the applied gravity and buoyancy forces on the falling nonspherical particle in an incompressible fluid.?? = (?? + 1 12 ???? 3 ??)\textbf{(8)}?? = 3.75??????\textbf{(9)}?? = 67.289?? (?5.03?) 8 ???? 2 ?? (10) ?? = ???? ?1 ? ?? ?? ?? ?\textbf{(11)}\par
So, equation \hyperref[b6]{(7)} was rewritten as follow?? ???? ???? + ???? + ???? 2 ? ?? = 0, ??(0) = 0\textbf{(12)}\par
According to DTM conversion functions, equation ( \hyperref[formula_11]{12}) is provided as following transformed equation??(?? + 1)??(?? + 1) + ????(??) + ???? ??(??)??(?? ? 1) ?? ??=0 ? ? ?? × ??(??) = 0\textbf{(13)}\par
The value of ??(0) is equal to zero. Other value of ??(??) for k=1, 2, 3 ? are governed by above equation and can be calculated as bellow.??(1) = ?? ?? (14) ??(2) = ? 1 2 × ???? ?? 2 (15) ??(3) = 1 3! × ??(?? 2 ?2???? ) ?? 3 (16) ??(4) = ? 1 4! × ???? (?? 2 ?8???? ) ?? 4\textbf{(17)}??(5) = 1 5! × ??(?? 4 ?22?? 2 ???? +16?? 2 ?? 2 ) ?? 5 (18) ??(6) = ? 1 6! × ???? (?? 4 ?52?? 2 ???? +136?? 2 ?? 2 ) ?? 6\textbf{(19)}??(7) = 1 7! × ??(?? 6 ?114?? 4 ???? +720?? 2 ?? 2 ?? 2 ?272?? 3 ?? 3 ) ?? 7 (20) ??(8) = ? 1 8! × ???? (?? 6 ?240?? 4 ???? +3072?? 2 ?? 2 ?? 2 ?3968?? 3 ?? 3 ) ?? 8\textbf{(21)}\par
To calculate the velocity at each moment following equation is using?? ?? (??) = ? ? ?? ?? ?? ? ?? ?? ?? (??), 0 ? ?? ? ?? ??\par
spherical particles in an incompressible fluid is derived as?? ?? (??) = ??(1)?? + ??(2)?? 2 + ??(3)?? 3 + ??(4)?? 4 + ? (23)\par
In order to study the impact of electrostatic force, Coulomb's lawis used as follow?? = ?? 2 4×??×?? 0 ×?? 2 , ?? 0 = 8.85 × 10 ?12 (\textbf{24})\par
Where ?? is electrostatic force, ?? is thesigned magnitude of the charges, ?? is the distance between the charges, and ?? 0 is vacuum permittivity coefficient.m ???? ???? = ???? ?1 ? ?? ?? ?? ? ? 1 8 ???? 2 ???? ?? ?? 2 ? 1 12 ???? 3 ?? ???? ???? + ?? 2 4×??×?? 0 ×?? 2 (25)\par
Assuming that all varieties of equation ( \hyperref[formula_18]{24}) are independent of time, following equation can be written by rewriting terms of equation (  {\ref 25}) and inserting equation ( \hyperref[formula_7]{7}) in \hyperref[b10]{(11)}, and a non-linear deferential equation is formed:?? ???? ???? + ???? + ???? 2 ? ?? ? ?? = 0 , ??(0) = 0 (26)\par
By applying the DTM one quation (26) the following equation is derived. 
\section[{??(?? + 1)??(?? + 1) + ????(??) + ???? ??(??)??(?? ? 1)}]{??(?? + 1)??(?? + 1) + ????(??) + ???? ??(??)??(?? ? 1)}?? ??=0 ? ? (?? + ??) × ??(??) = 0\textbf{(27)}\par
By considering equations numbers from ( \hyperref[formula_12]{13}) to (27),the term (d+e) was replaced to (d) to calculate U(1) to U \hyperref[b7]{(8)}. As a result:??(1) = (??+??) ?? (28) ??(2) = ? 1 2 × ??(??+??) ?? 2 (29) ??(3) = 1 3! × (??+??)(?? 2 ?2??(??+??)) ?? 3 (30) ??(4) = ? 1 4! × ??(??+??)(?? 2 ?8??(??+??)) ?? 4 (31) ??(5) = 1 5! × (??+??)(?? 4 ?22?? 2 ??(??+??)+16?? 2 (??+??) 2 ) ?? 5 (32) ??(6) = ? 1 6! × ??(??+??)(?? 4 ?52?? 2 ??(??+??)+136?? 2 (??+??) 2 ) ?? 6 (33) ??(7) = 1 7! × (??+??)(?? 6 ?114?? 4 ??(??+??)+720?? 2 ?? 2 (??+??) 2 ?272?? 3 (??+??) 3 ) ?? 7 (34) ??(8) = ? 1 8! × ??(??+??)(?? 6 ?240?? 4 ??(??+??)+3072?? 2 ?? 2 (??+??) 2 ?3968?? 3 (??+??) 3 ) ?? 8\textbf{(35)}\par
Finally, using final derived equations described above, the equation (23)was solved as the general equation using a FORTRAN code. 
\section[{III.}]{III.} 
\section[{Results and Discussion}]{Results and Discussion}\par
The FORTRAN code is used for calculating instantaneous velocity,?? ?? (??), by assuming sphericity, density and equivalent diameterof particle equal to 0.9, 2100 ???? ?? 3 and 3mm respectively.\par
Figure  {\ref 1} illustrated the changes in the instantaneous velocity for sphericity of 0.9 when the fluid was water and no electrical load was applied. The falling velocity increased with increasing time. This increasing approach followed the linear behavior at the beginning, until the velocity passed 0.1 m/sec point. However, as it can be seen, the value of velocity reached to a constant value of 0.16 m/sec after 0.07 sec. Figure  {\ref 1} Figure 2 and 3 compares instantaneous velocities in water and ethylene glycol as their densities differ. The same approach can be seen regarding the increasing of velocity with time. However, the rate of increase in values of velocity was higher for water as its density is lower than that of ethylene glycol. In addition, the rate in ethylene glycol's representative diagram was increased in nonlinear approach in contrast with the one of water's.\par
Figure 4 represents the changes of particle's velocity when its sphericity changes. As it can be seen, the increase in sphericity resulted in an increase in falling velocity. The trend of increasing velocity was approximately the same before the velocity hit 0.04 m/sec independent of particles' sphericity. After this velocity, however, this rate was higher for particles with higher sphericity. Figure 4 Falling velocity of particles with lower sphericity reached a constant value in shorter times comparing the ones with lower sphericity. These results are consistent with Yaghoobi's and Torabi's \hyperref[b7]{[8]} and Jalaal's and Ganji's \hyperref[b2]{[3]}. An error analyzing was carried out in order to compare the results obtained in this study and Yaghoobi and Torabi's \hyperref[b7]{[8]}; for particles with sphericity of 0.5 and in absence of electrostatic force assuming the water as the incompressible fluid used for both studies, the value of falling velocity differ 6.7\%. Table \hyperref[tab_1]{1} lists important points which can be taken form Figure 4 for falling velocities of particles with different sphericities and the trend they increased. 
\section[{Table 1}]{Table 1}\par
In order to investigate the effects an electrostatic force on particles' sedimentation By inserting the above equation in equation (  {\ref 5}), the equation (  {\ref 25}) is derived. micro colon applied to a non-spherical particle with sphericity of 0.9 in water. The effects of electrostatic force in the motion behavior of particles in different distances between electric charges (0.5 m and 1.0 m) are shows in Figure 5. It can be found from this figure that applying electrostatic force reduced sedimentation velocity by 23.33\% due to the formation of a resistant force between particles. This reduction was more pronounced for shorter distance between the charges compared to the one with longer distance. Also, it can be seen from Figure 5 that the effect of electrostatic force was approximately constant with time especially after 0.04 sec.\par
IV. 
\section[{Conclusion}]{Conclusion}\par
The differential transformation method (DTM) was used for analytical investigation of non-spherical particles' falling velocity in an incompressible fluid in both the presence and the absence of electrostatic force. Results of this study indicated that, in general, the increasing curve of the sedimentation of non-spherical particles in incompressible fluids mostly behave nonlinear until it reaches the constant value. In addition, the effects of changing fluid's density in sedimentation behavior of particles was investigated. The results showed the falling velocity in water (the fluid with lower density) was higher than the one for ethylene glycol (the fluid with lower density).Moreover, when the sphericity increased, the sedimentation rate increased even though this rate differed for each sphericity. As results showed, particles with sphericity of 0.9 experienced the highest falling velocity and increasing rate followed by those with sphericity of 0.7 and 0.5 respectively. After applying an electric load of 1 micro colon in with distant between charges of 0.5 m and 1.0 m, the falling velocity decreased as a resistant force was created. This decrease was more noticeable for the distance of 0.5 with decreasing about 23\% in value of falling velocity.  
\section[{Global}]{Global} 
\section[{Tables}]{Tables}\par
Figure ( \hyperref[formula_0]{1}) -The changes of instantaneous velocity for sphericity of 0.9 with time in water and in absence of electrostatic force.     \begin{figure}[htbp]
\noindent\textbf{23}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Figure 2 Figure 3}\end{figure}
     \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.33280423280423277\textwidth}P{0.06746031746031746\textwidth}P{0.09444444444444444\textwidth}P{0.17989417989417988\textwidth}P{0.17539682539682538\textwidth}}
\multicolumn{3}{l}{Sphericity Time of linear velocity Maximum linear velocity}\tabcellsep Start of constant\tabcellsep Instantaneous\\
\tabcellsep \tabcellsep \tabcellsep velocity\tabcellsep velocity\\
\tabcellsep sec\tabcellsep m/sec\tabcellsep sec\tabcellsep m/sec\\
?=0.5\tabcellsep 0.01\tabcellsep 0.0306\tabcellsep 0.03\tabcellsep 0.06\\
?= 0.7\tabcellsep 0.03\tabcellsep 0.079\tabcellsep 0.04\tabcellsep 0.096\\
?=0.9\tabcellsep 0.06\tabcellsep 0.146\tabcellsep 0.08\tabcellsep 0.16\end{longtable} \par
  {\small\itshape [Note: FiguresGlobal Journals Inc. (US) Guidelines Handbook 2015 www.GlobalJournals.org]} 
\caption{\label{tab_1}Table 1 :}\end{figure}
 			\footnote{© 2015 Global Journals Inc. (US)} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Ganji ()]{b0}\label{b0} 	 		‘A semi-Analytical technique for nonlinear settling particle equation of Motion’.  		 			D D Ganji 		.  	 	 		\textit{Journal of Hydro-environment Research}  		2012. 6 p. .  	 
\bibitem[Jalaal et al. ()]{b6}\label{b6} 	 		‘A series exact solution for one-dimensional non-linear particle equation of motion’.  		 			M Jalaal 		,  		 			H Bararnia 		,  		 			G Domairry 		.  	 	 		\textit{Powder Technology}  		2011. 207 p. .  	 
\bibitem[Ferreira and Chhabra]{b18}\label{b18} 	 		\textit{Accelerating motion of a vertically falling sphere in incompressible},  		 			J M Ferreira 		,  		 			R P Chhabra 		.  		 	 
\bibitem[Chhabra and Ferreira ()]{b10}\label{b10} 	 		‘An analytical study of the motion of a sphere rolling down a smooth inclined plane in an incompressible Newtonian fluid’.  		 			R P Chhabra 		,  		 			J M Ferreira 		.  	 	 		\textit{Powder technology}  		1999. 104 p. .  	 
\bibitem[Jalaal and Ganji ()]{b1}\label{b1} 	 		‘An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid’.  		 			M Jalaal 		,  		 			D D Ganji 		.  	 	 		\textit{Powder Technology}  		2010. 198 p. .  	 
\bibitem[Jalaal et al. ()]{b2}\label{b2} 	 		‘An analytical study on settling of non-spherical particles’.  		 			M Jalaal 		,  		 			D D Ganji 		,  		 			G Ahmadi 		.  	 	 		\textit{Asia-Pacific Journal of Chemical Engineering}  		2012. 7 p. .  	 
\bibitem[Jalaal et al. ()]{b3}\label{b3} 	 		‘Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media’.  		 			M Jalaal 		,  		 			D D Ganji 		,  		 			G Ahmadi 		.  	 	 		\textit{Advanced Powder Technology}  		2010. 21 p. .  	 
\bibitem[Yaghoobi and Torabi ()]{b7}\label{b7} 	 		‘Analytical solution for settling of non-spherical particles in incompressible Newtonian media’.  		 			H Yaghoobi 		,  		 			M Torabi 		.  	 	 		\textit{Powder Technology}  		2012. p. .  	 
\bibitem[Proudman and Pearson ()]{b14}\label{b14} 	 		‘Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder’.  		 			I Proudman 		,  		 			J R A Pearson 		.  	 	 		\textit{Journal of Fluid Mechanics}  		1957. 2 p. .  	 
\bibitem[Jalaal et al. ()]{b5}\label{b5} 	 		‘Homotopy perturbation method for motion of a spherical solid particle in plane couette fluid flow’.  		 			\textbackslash  M Jalaal 		,  		 			M Nejad 		,  		 			P Jalili 		,  		 			M Esmaeilpour 		,  		 			H Bararnia 		,  		 			E Ghasemi 		,  		 			S Soleimani 		,  		 			D Ganji 		,  		 			S Moghimi 		.  	 	 		\textit{Computers \& Mathematics with Applications}  		2011. p. .  	 
\bibitem[Kim and Karrila ()]{b17}\label{b17} 	 		‘Microhydrodynamics: principles and selected applications’.  		 			S Kim 		,  		 			S J Karrila 		.  	 	 		\textit{Bubbles, drops and particles in non-Newtonian fluids},  				2007. 85 p. 251.  	 	 (CANADIAN JOURNAL OF CHEMICAL ENGINEERING) 
\bibitem[Oseen ()]{b13}\label{b13} 	 		\textit{Neuere methoden und ergebnisse in der hydrodynamik},  		 			C W Oseen 		.  		1927. Akad. Verlag-Ges.  	 
\bibitem[Yaghoobi and Torabi ()]{b9}\label{b9} 	 		‘Novel solution for acceleration motion of a vertically falling nonspherical particle by VIM-Padé approximant’.  		 			H Yaghoobi 		,  		 			M Torabi 		.  	 	 		\textit{Powder Technology}  		2012. 215 p. .  	 
\bibitem[Torabi and Yaghoobi ()]{b8}\label{b8} 	 		‘Novel solution for acceleration motion of a vertically falling spherical particle by HPM-Padé approximant’.  		 			M Torabi 		,  		 			H Yaghoobi 		.  	 	 		\textit{Advanced Powder Technology}  		2011. 22 p. .  	 
\bibitem[Stokes ()]{b12}\label{b12} 	 		\textit{On the effect of the internal friction of fluids on the motion of pendulums},  		 			G G Stokes 		.  		Press1851. Pitt.  	 
\bibitem[Jalaal and Ganji ()]{b4}\label{b4} 	 		‘On unsteady rolling motion of spheres in inclined tubes filled with incompressible Newtonian fluids’.  		 			M Jalaal 		,  		 			D D Ganji 		.  	 	 		\textit{Advanced Powder Technology}  		2011. 22 p. .  	 
\bibitem[Chien ()]{b11}\label{b11} 	 		‘Settling velocity of irregularly shaped particles’.  		 			S F Chien 		.  	 	 		\textit{SPE Drilling \& Completion}  		1994. 9 p. .  	 
\bibitem[Lovalenti and Brady ()]{b16}\label{b16} 	 		‘The hydrodynamic force on a rigid particle undergoing arbitrary timedependent motion at small Reynolds number’.  		 			P M Lovalenti 		,  		 			J F Brady 		.  	 	 		\textit{Journal of Fluid Mechanics}  		1993. 256 p. .  	 
\bibitem[Sano ()]{b15}\label{b15} 	 		‘Unsteady flow past a sphere at low Reynolds number’.  		 			T Sano 		.  	 	 		\textit{Journal of Fluid Mechanics}  		1981. 112 p. .  	 
\end{bibitemlist}
 			 		 	 
\end{document}
