Thermophoresis and Brownian motion effects on peristaltic nanofluid flow for drug delivery applications

Accepted Manuscript Thermophoresis and Brownian motion effects on peristaltic nanofluid flow for drug delivery applications

Seiyed E. Ghasemi PII: DOI: Reference:

S0167-7322(17)30921-2 doi: 10.1016/j.molliq.2017.04.067 MOLLIQ 7223

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

2 March 2017 11 April 2017 15 April 2017

Please cite this article as: Seiyed E. Ghasemi , Thermophoresis and Brownian motion effects on peristaltic nanofluid flow for drug delivery applications. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/j.molliq.2017.04.067

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ACCEPTED MANUSCRIPT Thermophoresis and Brownian motion effects on peristaltic nanofluid flow for drug delivery applications Seiyed E. Ghasemi  Young Researchers and Elite Club, Sari Branch, Islamic Azad University, Sari, Iran

Abstract

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In this study a simple and highly accurate semi-analytical method called the Differential

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Transformation Method (DTM), is used for solving the governing equations of peristaltic

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nanofluid flow in drug delivery systems. The effects of thermophoresis and Brownian motion parameters on temperature and velocity fields are discussed in details. The validity of the results

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of DTM solution are verified via comparison with numerical results obtained using fourth order Runge-Kutta method. The results show that by increasing the Brownian motion and

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thermophoresis parameters the temperature profile increases. Also, results reveal that DTM is very effective and convenient. Furthermore, it is found that this method can be easily extended to other strongly nonlinear heat transfer equations and can be found widely applicable in

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engineering and science.

Method (DTM), Drug delivery.

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1. Introduction

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Keywords: Nanoparticles, Thermophoresis, Brownian motion, Differential Transformation

Peristalsis is one of major transport mechanism used for fluid transportation in physiology. This

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mechanism comprises of waves of area contraction and relaxation propagating along the channel walls. Applications of peristalsis in physiology can be seen in food transport through esophagus,

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chyme movement through intestine, urine transport from kidneys to the bladder, fluid mechanics in perivascular space of brain, sperm transport in male reproductive tract etc. Main benefit of peristalsis is that the fluid transported through this mechanism avoids contamination caused due to direct contact with external environment. This advantage convinced engineers to apply this mechanism in designing several industrial devices e.g. roller/finger pumps, hose pumps, heart– lung and dialysis machines and pumps used in transport of several corrosive and sensitive fluids



Corresponding author: E-mail: [email protected]

ACCEPTED MANUSCRIPT Nomenclature Half width of the channel

b

Wave amplitude

c

Velocity of the wave

DB

Brownian diffusion coefficient

DT

Thermophoretic diffusion coefficient

F

Nanoparticle volume fraction

GrF

Species Grashof number

GrT

Thermal Grashof number

Nb

Brownian motion parameter

Nt

Thermophoresis parameter

k

Thermal conductivity

Re

Reynlods number

Pr

Prandtl number

g

Gravitational acceleration

u, v

Axial velocity and transverse velocity



 

IP CR

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CE

Volumetric volume expansion coefficient Amplitude ratio Dynamic viscosity of the fluid

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

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M

ED

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Greek symbols



T

a

Kinetic viscosity of the fluid Wave number

f

Fluid density

p

Density of nanoparticle



Dimensionless nanoparticle volume fraction



wavelength

ACCEPTED MANUSCRIPT in nuclear industry. Peristaltic transport of nanofluids is of considerable importance in biomedical engineering particularly in modern drug delivery systems, in hyperthermia and cryosurgery as means to destroy the undesired tissues in cancer therapy [1–3]. Maiti and Misra [4] studied of the peristaltic motion of blood in the micro-circulatory system, by taking into account the non-Newtonian nature of blood and the non-uniform geometry of the micro-vessels, e.g. arterioles and venules. Influence of applied magnetic field on the peristaltic

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transport of copper–water nanofluid in presence of Halland Ohmic heating effects was studied by

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Abbasi et al.[5]. Their Results depicted that addition of copper nanoparticles reduces the velocity

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and temperature of fluid. Peristaltic transport of silver-water nanofluid in the presence of viscous dissipation, heat generation/absorption, Ohmic heating and slip effects was developed by Abbasi

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et al.[6]. Results show that addition of 5% silver nanoparticles reduces the velocity of base fluid by almost 10% and its temperature by 16%. Shehzad et al.[7] examined mixed convective

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peristaltic transport of water based nanofluids with viscous dissipation and heat generation/absorption using two different models of the effective thermal conductivity of

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nanofluids. The two cases of Maxwell's and Hamilton–Crosser's thermal conductivity models were used in their analysis. Their results indicated that when the thermal conductivity of

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discontinuous phase (nanoparticles) is relatively higher than that of the continuous phase (water) then the results predicted by the Maxwell's and the Hamilton–Crosser's model differ by a large

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amount. Most differential equations of engineering problems do not have exact analytic solutions so approximation and numerical methods must be used. A great deal of effort has been expended

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in attempting to find robust and stable numerical and analytical methods for solving differential equations of physical interest. The analytical and numerical methods have been introduced to

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solve these equations, such as the Homotopy Perturbation Method (HPM) [8,9], Adomian Decomposition Method (ADM) [10], Variational Iteration Method (VIM) [11], Modified Homotopy Perturbation Method (MHPM) [12], Parameterized Perturbation Method (PPM) [13], Least Square Method (LSM) [14-16], Collocation Method (CM) [17], Optimal Homotopy Asymptotic Method (OHAM) [18,19], Keller-Box Method (KBM) [20-22], Differential Quadrature Method (DQM) [23,24] and Exp-function method [25]. The effect of nonlinear matter flow on the dynamics of electrified jets in electrospinning process was examined numerically by Valipour and Ghasemi [26].

ACCEPTED MANUSCRIPT One of the semi-exact methods which do not need small parameters is the Differential Transformation Method. The concept of differential transformation method (DTM) was first introduced by Zhou [27] in 1986 and it was used to solve both linear and nonlinear initial value problems in electric circuit analysis. This method can be applied directly for linear and nonlinear differential equation without requiring linearization, discretization, or perturbation and this is the main benefit of this method. Ghafoori et al. [28] used the DTM for solving the nonlinear

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oscillation equation. Abdel-Halim Hassan [29] has applied the DTM for different systems of

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differential equations and he has discussed the convergency of this method in several examples

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of linear and non-linear systems of differential equations. R. Abazari and M. Abazari [30] have applied the DTM and RDTM (reduced differential transformation method) for solving the

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generalized Hirota–Satsuma coupled KdV equation. They compared the results with exact solution and they found that RDTM is more accurate than the classical DTM. Rashidi et al. [31]

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solved the problem of mixed convection about an inclined flat plate embedded in a porous Medium by DTM; they applied the Pade approximant to increase the convergence of the

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solution. Abbasov et al. [32] employed DTM to obtain approximate solutions of the linear and non-linear equations related to engineering problems and they showed that the numerical results

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are in good agreement with the analytical solutions. Balkaya et al. [33] applied the DTM to analyze the vibration of an elastic beam supported on elastic soil. Borhanifar et al. [34] employed

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DTM on some PDEs and their coupled versions. Moradi and Ahmadikia [35] applied the DTM to solve the energy equation for a temperature-dependent thermal conductivity fin with three

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different profiles. Moradi [36] applied DTM for thermal characteristics of straight rectangular fin for all type of heat transfer (convection and radiation) and compared it results by ADM and

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numerical method with fourth order Rang- Kutta method using shooting method. Also recently Ghasemi et al. [37,38] applied the Differential Transformation Method to analyze the temperature distribution in solid and porous fins with temperature-dependent heat generation and thermal conductivity. Recently the peristaltic flow of nanofluids in drug delivery systems was studied analytically and numerically by Ghasemi et al. [39]. They applied Least Squares Method (LSM), Galerkin Method (GM) and fourth-order Runge–Kutta numerical method to solve the problem. Prime goal of this study is to model and theoretically analyze the drug delivery system with peristaltic flow of nanofluids. To solve the governing equations of the problem, Differential

ACCEPTED MANUSCRIPT Transformation Method (DTM) is used. Also the effects of Brownian motion and thermophoresis parameters on temperature, velocity and nanoparticle fraction profiles are investigated. 2. Mathematical model Fig. 1 shows the schematic of the geometry of peristaltic fluid flow in drug delivery system studied in the present work. Consider the peristaltic pumping of a conducting fluid through a

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channel. A longitudinal train of progressive sinusoidal waves takes place on the upper and lower

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walls of the channel. We have restricted our problem to the half-width of the channel as shown in

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Fig 1.

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Fig. 1. Geometry of peristaltic pumping in drug delivery system.

2



(  ct )

)1(

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h( , t )  a  b sin

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The geometry of the channel wall is given by [40] :

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Where t , h and  represent time, transverse vibration of the wall and axial coordinate respectively. Utilizing the Oberbeck-Boussinesq approximation, the conservation of total mass,

.v  0

f (

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momentum, thermal energy, and nanoparticle fraction are as follow:

v  v.v)  p  2 v   F  p  (1  F )  f (1   (T  T0 )) g t

)2(

)3(

ACCEPTED MANUSCRIPT

(  c) f (

  D T  v.T )  k 2T  (  c) p  DB.T  ( T )T .T  t T0  

)4(

D F  v.F  DB  2  ( T ) 2T t T0

)5(

T  T1 , F  F1

at   h

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at   0

)6(

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T  T0 , F  F0

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The boundary conditions are taken to be:

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Considering the nanoparticle concentration to be dilute, and selecting a suitable choice for the reference pressure, the momentum equation can be rewritten as:

v  v.v)  p   2 v  (  p   f 0 )( F  F0 )  (1  F0 )  f 0  (T  T0 )  g t

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f (

)7(

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be presented as follows [41,42]:

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Now by utilizing standard boundary-layer approximation, the governing equations (2) to (5) may

)8(

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u v  0  

   p  2u  2u u  v )u     ( 2  2 )  g (1  F0 )  f 0  (T  T0 )  (    f 0 )( F  F0 )  )9( t     

f (

   p  2v  2v u  v )v     ( 2  2 )  g (1  F0 )  f 0  (T  T0 )  (    f 0 )( F  F0 )  )11( t     

(

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(  c) f (

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f (

 D T     2T  2T F T F T T  u  v )T  k ( 2  2 )  (  c) p  DB (  )  T (( ) 2  ( ) 2 )  t   T0          

    2 F  2 F D  2T  2T u  v ) F  DB ( 2  2 )  T ( 2  2 ) t   T0    

Introducing the following non-dimensional quantities:

)11(

)12(

ACCEPTED MANUSCRIPT  u v a ct pa 2 h  ,  ,t , u , v , p , h   1   sin(2 ) ,   , a c c a    c   f ca b   ga3 (1  F0 )(T1  T0 ) T  T0 F  F0  , v ,  ,  , Re  , GrT  , f0 a v2  T1  T0 F1  F0 

GrF 

 f 0v 2

, Pr 

v(  c) f k

, Nb 

(  c) p DB ( F1  F0 ) k

)13(

,

T

(  c) p DT (T1  T0 ) kT0

IP

Nt 

ga3 (    f 0 )( F1  F0 )

(8)-(12) lead to the following non-dimensional equations:

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u v  0  

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Using the low Reynolds number and large wavelength approximations, the governing Equation

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p  2u   GrT  GrF    2

ED

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p 0 

)14(

)15(

)16(

)17(

 2 Nt  2  0  2 Nb  2

)18(

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 2     Nb  Nt ( ) 2  0 2    

With the boundary conditions as the form:

u 0 

AC

  0,   0,

  1,   1, u  0

at   0

)19( at   h

3. Fundamentals of Differential transformation Method (DTM) In this section the fundamental basic of the Differential Transformation Method (DTM) is introduced. For understanding method’s concept, suppose that x(t ) is an analytic function in domain D, and t  ti represents any point in the domain. The function x(t ) is then represented by

ACCEPTED MANUSCRIPT one power series whose center is located at ti . The Taylor series expansion function of x(t ) is in form of:

(t  t i )k k! k 0 

x (t )  

 d k x (t )   dt k    t ti

t  D

(20)

The Maclaurin series of x(t ) can be obtained by taking ti  0 in Eq. (20) expressed as: t  D

(21)

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T

t k  d k x (t )  x (t )     k k  0 k !  dt  

t 0

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As explained in [27] the differential transformation of the function x(t ) is defined as follows: H k  d k x (t )    k k  0 k !  dt  t 0 

(22)

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X (k )  

Where X (k ) represents the transformed function and x(t ) is the original function. The

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differential spectrum of X (k ) is confined within the interval t  0, H  , where H is a constant



x (t )   (

t k ) X (k ) H

(23)

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k 0

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value. The differential inverse transform of X (k ) is defined as follows:

It is clear that the concept of differential transformation is based upon the Taylor series

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expansion. The values of function X (k ) at values of argument k are referred to as discrete, i.e. X (0) is known as the zero discrete, X (1) as the first discrete, etc. The more discrete available,

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the more precise it is possible to restore the unknown function. The function x(t ) consists of the T-function X (k ) , and its value is given by the sum of the T-function with (t / H )k as its

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coefficient. In real applications, at the right choice of constant H, the larger values of argument k the discrete of spectrum reduce rapidly. The function x(t ) is expressed by a finite series and Eq. (23) can be written as: n

x (t )   ( k 0

t k ) X (k ) H

(24)

Some important mathematical operations performed by differential transform method are listed in Table 1.

Table1. Some fundamental operations of the differential transform method

ACCEPTED MANUSCRIPT Origin function

Transformed function

x (t )   f (x )   g (t )

X (k )   F (k )  G (k )

d m f (t ) x (t )  dt m

X (k ) 

x (t )  f (t ) g (t )

X (k )   F (l )G (k  l )

(k  m )!F (k  m ) k! k

l 0

1 k!

X (k ) 

X (k ) 

x (t )  cos(t   )

sin(

k ) 2

k!

k

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k!

cos(

k ) 2

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4. Application of DTM for Problem

X (k ) 

k

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x (t )  sin(t   )

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x (t )  exp(t )

if k = m, if k  m.

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1, X (k )   (k  m )   0,

T

x (t )  t m

k

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Now we apply DTM from table 1 into Eqs. (17) and (18) to find  ( ) and  ( ) .

(k  2)(k  1)(k  2)  N b  (r  1)(r  1)(k  1  r )(k  1  r )  r 0

k

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Nt  (r  1)(r  1)(k  1  r )(k  1  r )  0

(25)

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r 0

Nt (k  2)(k  1)(k  2)  0 Nb

(26)

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(k  2)(k  1)(k  2) 

Where  and  represent the DTM transformed form of  and  , respectively. Rearranging Eqs. (25) and (26), a simple recurrence set of equations is obtained as follows:

(k  2)  k k 1   N ( r  1)  ( r  1)( k  1  r )  ( k  1  r )  N b t  ( r  1)( r  1)( k  1  r )( k  1  r )   (k  2)(k  1)  r 0 r 0  (27)



ACCEPTED MANUSCRIPT (k  2)  

 Nt  1 (k  2)(k  1)(k  2)   (k  2)(k  1)  Nb 

(28)

Similarly, the transformed form of boundary conditions can be written as: (0)  0, (1)  A (0)  0, (1)  B

(29)

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Using transformed boundary conditions and Eqs. (27) and (28) (when Nt = 2 and Nb=1) :

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1 (2)   AB  A2 2 (2)  AB  2 A2

(31)

... Where A and B are unknown coefficients.

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1 1 1 4 1 (3)   A( AB  2 A2 )  ( AB  A2 ) B  A( AB  A2 ) 3 3 2 3 2 2 2 1 8 1 (3)  A( AB  2 A2 )  ( AB  A2 ) B  A( AB  A2 ) 3 3 2 3 2

(30)

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After specifying  ( ) and  ( ) , two equations based on A and B will be obtained for  ( ) and

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 ( ) .

By applying boundary condition at   1 (when h=1) into  ( ) and  ( ) (Eq. 19), two equations

PT

with two unknown coefficients will be appeared and A and B can be determined easily by solving them:

394240 413951 ,B   124843 124843

CE

A

(32)

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Finally,  ( ) and  ( ) when Nt = 2 and Nb=1 can be defined as:

394240 591360 2 591360 3 443520 4 266112 5 133056 6            124843 124843 124843 124843 124843 124843 57024 7 21384 8 7128 9 10692 10 2916 11 729 12            124843 124843 124843 624215 624215 624215

(33)

413951 1182720 2 1182720 3 887040 4 532224 5 266112 6            124843 124843 124843 124843 124843 124843 114048 7 42768 8 14256 9 21384 10 5832 11 1458 12            124843 124843 124843 624215 624215 624215

(34)

 ( ) 

 ( )  

ACCEPTED MANUSCRIPT By substituting Eqs. (33) and (34) in Eq. (15) , when GrF  GrT  1 and

p  0 , u ( ) will be 

obtained as:

20433282 269397 3 147840 4 88704 5 44352 6 19008 7           56803565 249686 124843 124843 124843 124843 7128 8 2376 9 3564 10 972 11 243 12 729             13  124843 124843 624215 624215 624215 8114795 2187  14 113607130

u ( ) 

CR

IP

T

(35)

5. Numerical Method

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It is obvious that the type of the current problem is boundary value problem (BVP) and the appropriate numerical method needs to be selected. The numerical solution is performed using

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Fourth order Rung-Kutta method. This method is a method of numerically integrating ordinary differential equations using a trial step at the midpoint of an interval to cancel out lower-order error terms [43]. The fourth-order formula is:

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k4  hf ( xn  h, yn  k3 )

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1 1 k3  hf ( xn  h, yn  k2 ) 2 2

ED

1 1 k2  hf ( xn  h, yn  k1 ) 2 2

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k1  hf ( xn , yn )

AC

1 1 1 1 yn 1  yn  k1  k2  k3  k4  O(h5 ) 6 3 3 6

6. Results and discussion The governing equations for peristaltic nanofluid flow through a channel are solved by DTM. In order to validate the present solution of the problem and find the accuracy, the comparison between analytical solution and numerical method is done. The numerical solution applied to solve the present case is the fourth order Runge–Kutta approach. A good agreement between the present method and numerical solution is observed in Fig.2 for temperature, velocity and nanoparticle volume fraction profiles, which confirms the validity of the proposed solution.

US

CR

IP

T

ACCEPTED MANUSCRIPT

(b)

(c)

AC

CE

PT

ED

M

AN

(a)

Fig. 2. Comparison between DTM solutions with numerical results for: a) ?? , b) u and c) ??

Fig.3a, 3b and 3c are depicted for showing the effect of Brownian motion parameter ( N b ) on temperature, velocity and nanoparticle volume fraction profiles, respectively. As seen in Fig.3a by increasing N b temperature profile increases that is the regime is heated. Fig.3b confirms that increasing N b lead to a decrease in magnitude of velocity values. As Fig.3c shows by increasing Brownian motion parameter, the species diffusion is increased. The nanofluid is a two-phase

ACCEPTED MANUSCRIPT fluid in which casual motion of the suspended nanoparticles enhances the rate of energy

US

CR

IP

T

exchange and concentration in the low regime.

(b)

AC

CE

PT

ED

M

AN

(a)

(c)

Fig. 3. Effect of Brownian motion parameter on: a) ?? , b) u and c) ??

The effect of thermophoresis parameter ( N t ) on temperature, velocity and nanoparticle volume fraction profiles is investigated through Fig.4 (a, b and c). Fig.4a confirms that increasing N t lead to a increase in temperature profile. As seen in Fig.4b by increasing N t , the magnitude of

ACCEPTED MANUSCRIPT velocity values decreases. Fig.4c shows that with an increase in thermophoresis parameter, the

AN

US

CR

IP

T

nanoparticle volume fraction profile decreases.

(b)

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(a)

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Fig. 4. Effect of thermophoresis parameter on: a) ?? , b) u and c) ??

7. Conclusion In this paper, Differential Transformation Method (DTM) has been successfully applied to analyze the effect of nanoparticles on peristaltic fluid flow for drug delivery application. Also,

complementary numerical solutions were obtained via the fourth order Runge-Kutta. It was

ACCEPTED MANUSCRIPT shown that very excellent agreement achieved between numerical solutions and DTM. The results indicated that the temperature distribution is strongly depending on Brownian motion and thermophoresis parameters. Also, it was illustrated that the magnitude of velocity is decreased with increasing the Brownian motion and thermophoresis parameters. Furthermore, with an increase in Brownian motion parameter, the nanoparticle fraction was increased while increasing

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thermophoresis parameter reduced nanoparticle fraction profile.

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ACCEPTED MANUSCRIPT Highlights Thermophoresis and Brownian motion effects on peristaltic nanofluid flow is studied.  The magnitude of velocity is decreased with increasing the thermophoresis parameter.

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With an increase in Brownian motion parameter, the nanoparticle fraction is increased.