transient free convection flow past an infinite vertical cylinder with thermal stratification

Journal of Mechanical Science and Technology 26 (8) (2012) 2229~2237

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0602-5

Transient free convection flow past an infinite vertical cylinder with

thermal stratification† Rudra Kanta Deka and Ashish Paul*

Department of Mathematics, Gauhati University, Guwahati-781014, India

(Manuscript Received May 7, 2011; Revised January 9, 2012; Accepted March 12, 2012)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract This paper presents an analytical solution of one-dimensional free convection flow past an infinite vertical circular cylinder in a strati-

fied fluid medium. The dimensionless unsteady coupled linear governing equations are solved by Laplace transform technique for the case when the Prandtl number is unity. Due to the effect of thermal stratification, the velocity, temperature, skin-friction and Nusselt number shows oscillatory behavior at smaller times and then reaches steady state at larger times, while this behavior is not seen in the absence of stratification.

Keywords: Vertical cylinder; Heat transfer; Thermal stratification; Laplace transformation ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

In recent years, convective heat transfer problems about cy-lindrical bodies have attracted attention of researchers because of their wide applications in the field of science and technol-ogy. Cylinder, which is the geometry for heat generating fuel rods in a reactor, requires emergency cooling of the fuel ele-ment. In glass and polymer industries, hot filaments, which are considered as a vertical cylinder, are cooled as they pass through the surrounding environments. Many researchers under different physical conditions have investigated steady and unsteady free convective flow along vertical cylinders. Sparrow and Gregg [1] first studied the heat transfer from vertical cylinders. Subsequently, Goldstein and Briggs [2] presented an analysis of the transient free convective flow past vertical flat plate and circular cylinder by employing Laplace transform technique. Tetsu and Haruo [3] also studied the heat transfer from vertical cylinder. Nagendra et al. [4] presented a

boundary-layer analysis of free-convection heat transfer from a vertical cylinder with uniform heat flux at its surface. Min-kowycz and Ping [5] made an analysis for free convective flow about a vertical cylinder embedded in a saturated porous medium, where surface temperature of the cylinder varies as a power function of distance from the leading edge. Crane [6] studied the natural convection from a vertical cylinder for the limiting case of very large Prandtl number, keeping the Grashof number finite. Sokovishin and Shapiro [7] carried out

works to study the effect of radiation on free convective heat liberation from the surface of a vertical cylinder and found that the radiative component of thermal flux equalizes the surface temperature. Later on, Sokovishin and Shapiro [8] restudied the heat transfer from a vertical cylinder in the pres-ence of exponentially decaying heat flux. Aziz and Na [9] used perturbation method to solve the laminar natural convec-tion flow from an isothermal thin vertical cylinder and estab-lished the efficiency over the popular local non-similarity and finite difference methods. Yücel [10] considered free convec-tion about a vertical cylinder in a porous medium subject to constant wall temperature or constant wall heat flux. Gorla [11] solved the boundary layer flow on a continuous moving vertical cylinder numerically.

Velusamy and Garg [12] carried out a numerical study for the transient natural convection over heat generating vertical cylinders and obtained excellent agreement with previous experimental steady-state data as well as with one-dimensional theoretical results. Hossain and Alim [13] pre-sented a free convection flow of an optically dense viscous incompressible fluid along a vertical thin circular cylinder with effect of radiation, when the surface temperature is uni-form by finite difference method. Yih [14] presented numeri-cal solutions for the effect of radiation on natural convection about an isothermal vertical cylinder embedded in a saturated porous medium. Gori et al. [15] investigated theoretically the natural convection around a vertical thin cylinder with con-stant wall heat flux for fluids with Prandtl numbers in the range 0.7-730 and compared their findings with the experi-mental results of Minkowycz and Sparrow [16]. Reddy and

*Corresponding author. Tel.: +919864524239, Fax.: +913612570270 E-mail address: [email protected]

† Recommended by Associate Editor Jun Sang Park © KSME & Springer 2012

2230 R. K. Deka and A. Paul / Journal of Mechanical Science and Technology 26 (8) (2012) 2229~2237

Reddy [17] studied the interaction of free convection with thermal radiation of a viscous incompressible unsteady MHD flow past a vertical cylinder with variable surface temperature and concentration by finite-difference method.

In contrast, closed form of solutions on natural convection along a vertical cylinder is limited in literature, which is very much necessary for validating numerical models. The present study is mainly on the one-dimensional transient flow along a vertical cylinder by including the pressure work term in the thermodynamic energy equation and also the variation of am-bient temperature. As, we shall see, in the context of one-dimensional model studied by Goldstein and Briggs [2], the pressure work and vertical temperature advection are of the same form and both processes can be combined into a single advection term. This combined effect is termed as thermal stratification or stratification for distinction from the earlier studies. The analytical study pertaining to thermal stratifica-tion is limited to the works of Shapiro and Fedorovich [18], Shapiro and Fedorovich [19], Magyari et al. [20] for flow past vertical plates. Later on, Shapiro and Fedorovich [21] carried on study for vertical plates and circular cylinders seeking solu-tions in the form of harmonic oscillators. Ganesan and Loga-nathan [22] presented a numerical study of free convective flow past a moving, semi-infinite vertical cylinder with con-stant temperature and mass diffusion in a thermally stratified medium.

In our study, we have extended the works of Goldstein and Briggs [2] in the presence of thermal stratification. Solutions for velocity and temperature are obtained by the Laplace transform technique. Noteworthy to mention here that for derivation in closed form of the Laplace inverse, the computa-tions are restricted for Prandtl number of unity [19]. A com-parison of the solutions for the fluid with thermal stratification is made with the classical solutions, when there is no thermal stratification. Also, the impact of the stratification on reaching steady state is studied over the case of no stratification.

2. Mathematical analysis

We consider a vertical cylinder of radius 0r′ , which is situ-ated in an otherwise quiescent environment having tempera-ture ( )T z∞′ ′ , z′ -axis is being taken vertically upwards along the axis of the cylinder and the radial co-ordinate r′ normal to the cylinder. Initially, it is assumed that the cylinder and fluid are at the same environment temperature ( )T z∞′ ′ . It is also assumed that at ' 0t ≥ , the temperature near the cylinder raised to constant temperature wT ′ . Then the Boussinesq flow is governed by,

( )g ''

u uT T' rt r r r

νβ ∞

′ ′∂ ∂ ∂⎛ ⎞′= − + ⎜ ⎟′ ′ ′∂ ∂ ∂⎝ ⎠ (1)

' ''

T Tr ut r r r

α γ∂ ∂ ∂⎛ ⎞′ ′ ′= −⎜ ⎟′ ′ ′∂ ∂ ∂⎝ ⎠ (2)

with initial and boundary conditions,

0

0 : 0 0: 0 at .

0 asw

t' u , T' T' rt' > u , T' T' r r u , T' T' r

∞

∞

′ ′≤ = = ∀ ⎫⎪′ ′ ′= = = ⎬⎪′ ′→ → →∞⎭

(3)

The variables and physical quantities involved are men-

tioned in the nomenclature. However, the particular variable γ ′ , the main concern of our study, is the ambient stratification parameter [≡ ( ) /dT z dz∞′ ′ ′ for Boussinesq flow of liquids or gases, ≡ ( ) / / pdT z dz g C∞′ ′ ′ + for a perfect gas with pressure work term retained], z′ is the height. Thus γ ′ , is the combi-nation of thermal stratification and compression, where

( ) /dT z dz∞′ ′ ′ stands for thermal stratification and g/Cp for com-pression. It should be noted, however, that the compression work term is generally quite small (since g = 9.8 m s −2, Cp = 1004 J Kg−1 K−1), and that the main interest in our solutions will probably be in the effects of temperature stratification. We have retained the compression work term so that numeri-cal convection models can be developed for examining the validity of the solution. It will be seen during the study that the increase in the thermal stratification has a significant effect on the study. Also, it can be predicted that the compression work being the additive one to thermal stratification, also plays a similar role on the solution of temperature and vertical veloc-ity as well.

The term uγ ′ ′− appearing in Eq. (2) is due to the com-bined effects of compression and vertical temperature advec-tion. The appearance of this term in the energy equation is interpreted as: warm fluid rises, expands and cools relative to the environment, whereas cool fluid subsides, compresses and warms relative to the environment [19].

Introducing the following non-dimensional quantities,

( )

02

0 0

3 00 2

,

w

w

w

r u r t' T' T'R , U t , T r ν r T' T'ν T' T' γ rPr , Gr gβ r , Sα ν T' T'

ν ∞

∞

∞

∞

′ ′ ′ − ⎫= = = = ⎪′ ′ − ⎪⎬′ ′− ⎪′= = =⎪− ⎭

(4)

Eqs. (1) and (2) reduce to,

2

2

1U U U GrTt R R R

∂ ∂ ∂= + +

∂ ∂ ∂ (5)

2

2

1 1T T T SUt Pr R R R

⎛ ⎞∂ ∂ ∂= + −⎜ ⎟∂ ∂ ∂⎝ ⎠

(6)

with initial and boundary conditions as:

0: 0 0

>0: 0 1 1 . 0 0

t U , T Rt U , T at R U , T as R

≤ = = ∀ ⎫⎪= = = ⎬⎪→ → →∞⎭

(7)

Here, Gr is the Grashof number, Pr is the Prandtl number

R. K. Deka and A. Paul / Journal of Mechanical Science and Technology 26 (8) (2012) 2229~2237 2231

and S is the thermal stratification parameter. It is to be noted that the trivial case (S = 0) corresponds to the equations ob-tained by Goldstein and Briggs [2]. In our study, we have designated the trivial case as ‘Classical case’ as a distinction from present study.

3. Solutions with stratification (S ≠ 0)

To solve the unsteady Eqs. (5) and (6) subject to the initial and boundary conditions (7), we apply Laplace transform technique for the case of Prandtl number of unity, as for arbi-trary Prandtl number, the Laplace transform technique would lead to a difficult inverse transformation step (integrand of the Bromwich integral would be a complicated multivalued func-tion).

Laplace transform of Eqs. (5) and (6) (with Pr = 1) give rise to,

2

2

1 0d U dU pU GrTdR R dR

+ − + = (8)

2

2

1 0d T dT pT SUdR R dR

+ − − = (9)

where p is the parameter of Laplace transformation defined by,

for example, 0

{ ( , )} ( , ) ( , )ptL T R t e T R t dt T R p∞

−= =∫ . Eliminat-

ing T from Eqs. (8) and (9) and after mere rearrangement, we have,

2 22 2

2 2

1 1 0d d d da b UdR R dR dR R dR

⎧ ⎫⎧ ⎫+ − + − =⎨ ⎬⎨ ⎬

⎩ ⎭⎩ ⎭ (10)

where 2 2 2, and .a p iM b p iM M SGr= + = − = Now, assuming

22

2

1d d b U WdR R dR

⎧ ⎫+ − =⎨ ⎬

⎩ ⎭ (11)

and then using Eq. (10), we have solution for W(R) as,

( ) ( )1 0 2 0 W C I a R C K a R= + (12)

where 1C and 2C are arbitrary constants.

Using Eq. (12) in Eq. (11) we have,

( ) ( )2

21 0 2 02

1 .d d b U C I a R C K a RdR R dR

⎧ ⎫+ − = +⎨ ⎬

⎩ ⎭ (13)

Now, to solve the non-homogenous Eq. (13) by the method

of variation of parameter, we assume,

( ) ( )0 0 U AI b R BK b R= + (14)

as the complete solution of the Eq. (13), where A and B are functions of R such that,

( ) ( )1 0 1 0 0 .A I b R B K b R+ = (15)

Using Eq. (14) in Eq. (13) and solving we have,

( ) ( ){ } ( ) ( )1 1 1 1 1 0 2 0 .b A I b R B K b R C I a R C K a R− = + (16)

Solving Eqs. (15) and (16) for 1A and 1B , we have

( ) ( ) ( ) ( )1 1 0 0 2 0 0 dAA C R I a R K b R C R K a R K b RdR

= = +

(17)

( ) ( ) ( ) ( )1 1 0 0 2 0 0 .dBB C R I a R I b R C R K a R I b RdR

= = −⎡ + ⎤⎣ ⎦

(18) In determining 1A and 1B , we have the following proper-

ties on Bessel function (see Refs. [23, 24]), namely;

( ) ( ) ( ) ( )

0 1 0 1

0 1 1 0

( ) ( ), ( ) ( ),

1 .

d dI bR bI bR K bR bK bRdR dR

I bR K bR I bR K bRbR

= = −

+ =

Integrating Eqs. (17) and (18), we get,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

11 0 0 12 2

20 1 0 1 32 2

C RA a I a R K b R b I a R K b Ra b

C R b K a R K b R a K b R K a R Ca b

= ⎡ + ⎤⎣ ⎦−

+ ⎡ − ⎤ +⎣ ⎦−

(19)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

10 1 0 12 2

21 0 0 1 42 2

C RB a I b R I a R b I a R I b Ra b

C R b I b R K a R a I b R K a R Ca b

−= ⎡ − ⎤⎣ ⎦−

+ ⎡ + ⎤ +⎣ ⎦−

(20)

where 3C and 4C are integration constants. It is to mention here that to determine A and B above, we have derived and used the following identities,

( ) ( ) ( ) ( )

( ) ( )

0 0 0 12 2

0 1

RR I a R I b R dR a I b R I a Ra b

b I a R I b R

= ⎡⎣−− ⎤⎦

∫

( ) ( ) ( ) ( )

( ) ( )

0 0 1 02 2

0 1

RR I a R K b R dR a I a R K b Ra b

b I a R K b R

= ⎡⎣−+ ⎤⎦

∫

( ) ( ) ( ) ( )

( ) ( )

0 0 0 12 2

0 1

K

.

RR a R K b R dR b K a R K b Ra b

a K b R K a R

= ⎡⎣−− ⎤⎦

∫


Using Eqs. (19) and (20) in Eq. (14), after simplification, we have the Laplace transform of U as,

( ) ( ) ( ) ( )1 0 2 0

3 0 4 0

2 2

C I aR C K aRU C I bR C K bR

iM iM= + + + (21)

Now, since I0(aR) and I0(bR) are not bounded as R →∞, and

we are seeking for bounded solution for U, so we set C1 = C3 = 0 and accordingly, we have,

( ) ( )2 0

4 0

.

2C K aR

U C K bRiM

= + (22)

Using (22) in Eq. (8), we have

( ) ( )2 0 4 02 2 0 .GrT C K aR iMC K bR+ − = (23)

Now, taking Laplace transforms of the conditions, namely,

U = 0, T = 1 at R = 1, into account, we have

( )2

0

GrCpK p iM

−=

+ and

( )4

0

.2

GrCpiM K p iM

=−

Thus from Eqs. (22) and (23), using C2, C4 above, we get,

( )( )

( )( )

0 0

0 02

K R p iM K R p iMGrUiM pK p iM pK p iM

⎧ ⎫− +⎪ ⎪= −⎨ ⎬− +⎪ ⎪⎩ ⎭

(24)

and

( )( )

( )( )

0 0

0 0

1 .2

K R p iM K R p iMT

pK p iM pK p iM

⎧ ⎫− +⎪ ⎪= +⎨ ⎬− +⎪ ⎪⎩ ⎭

(25)

Inverse Laplace transforms of Eqs. (24) and (25), respec-

tively, gives the expressions for vertical velocity and tempera-ture as:

( )( )

( )( )

( ) ( ){ } ( )2

0 0

0 0

2

4 20

2

sin cos2 ,V t

K R iM K R iMGrUMi K iM K iM

e V Mt M MtGr Γ R V V dVπM V M

−∞

⎧ ⎫−⎪ ⎪= −⎨ ⎬−⎪ ⎪⎩ ⎭

⎧ ⎫+⎪ ⎪+ ⎨ ⎬+⎪ ⎪⎩ ⎭

∫

(26)

( )( )

( )( )

( ) ( ){ } ( )2

0 0

0 0

2

4 20

12

cos sin2 , V t

K R iM K R iMT

K iM K iM

e V Mt M MtΓ R V V dV

π V M

−∞

⎧ ⎫−⎪ ⎪= +⎨ ⎬−⎪ ⎪⎩ ⎭

⎧ ⎫−⎪ ⎪+ ⎨ ⎬+⎪ ⎪⎩ ⎭

∫

(27)

where ( ) ( ) ( ) ( ) ( )( ) ( )

0 0 0 02 20 0

,J RV Y V Y RV J V

R VJ V Y V

−Γ =

+.

From the expressions of velocity and temperature, the lo-cations of peak acceleration ( maxUR ) and peak rate of tem-perature change ( maxTR ) are respectively obtained from

2 / 0U R t∂ ∂ ∂ = and 2 / 0T R t∂ ∂ ∂ = and found to coincide with each other obtainable from the identity

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

2 1 0 1 0 22 2

0 0 0

0V t J RV Y V Y RV J Ve V dV

J V Y V

∞− −

=+∫ .

It is noteworthy to mention here that these locations depend on time specifically in coincidence with the observations made by Shapiro and Fedorovich [19] in case of flow past vertical plate. The peak rates of velocity and temperature change at these locations are,

( )( ) ( ) ( ) ( )

( ) ( )

max

2

at

0 0 0 0max max

2 20 0 0

2 sin

UR R

V t U U

Ut

J R V Y V Y R V J VGr Mt e VdVM J V Y Vπ

•

• •

=

∞−

∂∂

−= −

+∫

(28)

( )( ) ( ) ( ) ( )

( ) ( )

max

2

at

0 0 0 0max max

2 20 0 0

2 cos .

TR R

V t T T

Tt

J R V Y V Y R V J VMt e VdV

π J V Y V

•

• •

=

∞−

∂∂

−= −

+∫

(29) Now, from Eqs. (28) and (29), it is seen that the peak accel-

eration and peak rate of temperature change oscillate.

The non-dimensional skin friction 1

R

UR

τ=

∂= −

∂ and Nus-

selt number 1

R

TNuR =

∂= −

∂are obtained from Eqs. (26) and

(27) as:

( )( )

( )( )

( ) ( ){ }

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

2

1 1

0 0

2

4 20

1 0 1 0 22 20 0

2

sin cos2

V t

iM K iM iM K iMGrτMi K iM K iM

e V Mt M MtGrπM V M

J V Y V Y V J VV dV

J V Y V

−∞

⎧ ⎫− −⎪ ⎪= −⎨ ⎬−⎪ ⎪⎩ ⎭

⎧ +⎪+ ⎨+⎪⎩

⎫− ⎪× ⎬+ ⎪⎭

∫ (30)

( )( )

( )( )

( ) ( ){ }

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

2

1 1

0 0

2

4 20

1 0 1 0 22 20 0

12

cos sin2

.

V t

iM K iM iM K iMNu

K iM K iM

e V Mt M Mtπ V M

J V Y V Y V J VV dV

J V Y V

−∞

⎧ ⎫− −⎪ ⎪= +⎨ ⎬−⎪ ⎪⎩ ⎭

⎧ −⎪+ ⎨+⎪⎩

⎫− ⎪× ⎬+ ⎪⎭

∫ (31)


Thus, the plate heat flux is finite at time t = 0, while for flow past vertical plate it is infinite [19]. Ironically, it is followed from the expressions of velocity, temperature, skin-friction and Nusselt number that as time t approaches ∞, the expres-sions become time independent. This can be attributed to be the expressions for steady-state velocity, temperature, skin-friction and Nusselt number as can be seen in the following section.

3.1 Steady-state solutions

Steady-state equations are obtained by neglecting time de-rivative term from Eqs. (5) and (6). Solving these steady-state equations and using the boundary conditions (7) we obtain the expressions for velocity and temperature as:

( )( )

( )( )

0 0

0 02S

K R iM K R iMGrUMi K iM K iM

⎧ ⎫−⎪ ⎪= −⎨ ⎬−⎪ ⎪⎩ ⎭

(32)

( )( )

( )( )

0 0

0 0

1 .2S

K R iM K R iMT

K iM K iM

⎧ ⎫−⎪ ⎪= +⎨ ⎬−⎪ ⎪⎩ ⎭

(33)

Also, the corresponding steady-state skin friction,

1

SS

R

UR

τ=

∂= −

∂ and Nusselt number,

1

SS

R

TNuR =

∂= −

∂ are

obtained from Eqs. (32) and (33) as:

( )( )

( )( )

1 1

0 02S

iM K iM iM K iMGrτMi K iM K iM

⎧ ⎫− −⎪ ⎪= −⎨ ⎬− −⎪ ⎪⎩ ⎭

(34)

( )( )

( )( )

1 1

0 0

1 .2S

iM K iM iM K iMNu

K iM K iM

⎧ ⎫− −⎪ ⎪= +⎨ ⎬−⎪ ⎪⎩ ⎭

(35)

Thus, the expressions (26), (27), (30) and (31) approach to

(32), (33), (34) and (35) respectively, when t →∞ as pre-dicted. Therefore, the mathematical findings reveal that the transient velocity, temperature, skin-friction and Nusselt num-ber reaches steady state for larger times.

Now, from the expressions for velocity, temperature, skin-friction and Nusselt number in the case of both unsteady and steady state, it is clear that before reaching steady state for larger times, the behavior is oscillatory one at smaller times.

4. Solution without stratification (S = 0)

In order to have a comparative study, we have also obtained classical solutions for velocity (UC), temperature (TC), skin friction (τC) and Nusselt number (NuC) for the flow without stratification. Here, equations are non-dimensionalized by the same set of non-dimensional quantities (4) and hence the non-dimensional equations are same as Eqs. (5) and (6) with S = 0. The Laplace transformed quantities CU and CT can readily be obtained as:

( )( )

( ) ( )( )

1 1 0

20 02C

K R p K p K R pGrU Rp p K p K p

⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭

(36)

( )( )

0

0

.C

K R pT

p K p= (37)

Inverse Laplace transforms of , C CU T give rise to the ex-

pressions for classical velocity and temperature as:

( ) ( )2

1 20

1 ,V tC

Gr dVU e Γ R Vπ V

∞−= −∫ (38)

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

2 0 0 0 02 2

0 0 0

21 V tC

J RV Y V Y RV J V dVT eπ J V Y V V

∞− −

= ++∫ (39)

where

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

( ) ( ) ( ) ( ){ } ( ) ( ){ }( ) ( ){ }

( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( ){ }

1 0 1 01 2 2

0 0

2 21 0 0 1 0 0

22 20 0

0 0 1 0 1 022 2

0 0

2 .

J RV Y V Y RV J VΓ R,V R

J V Y V

Y V J RV Y RV J V J V Y V

J V Y V

J V Y V J V J RV Y V Y RV

J V Y V

−=

+

+ −+

+

−−

+

The non-dimensional classical skin-friction 1

Cc

R

UR

τ=

∂= −

∂

and Nusselt number 1

CC

R

TNuR =

∂= −

∂are obtained from Eqs.

(38) and (39) as:

( )

( ) ( ) ( ) ( ){ } ( ) ( ) ( ) ( ){ }( ) ( ){ }

2

0

1 0 1 0 0 2 0 2

2 20 0

1

V tc

Grτ eπ

J V )Y V Y V J V V J V Y V Y V J VJ V Y V

∞−= − ×

⎧− − + −⎪×⎨+⎪⎩

∫

( ) ( ) ( ) ( ){ } ( ) ( ) ( ) ( ){ }( ){ }

2 2 2 21 1 0 0 0 0 1 1

2 22 20 0

2

J V Y V J V Y V Y V J V J V Y V dVVVJ (V) Y V

⎫− − − ⎪+ ⎬+ ⎪⎭

(40) ( ) ( ) ( ) ( )

( ) ( )2 1 0 1 0

2 20 0 0

2 .V tC

J V Y V Y V J VNu e dV

π J V Y V

∞−

⎧ ⎫−⎪ ⎪= ⎨ ⎬+⎪ ⎪⎩ ⎭

∫ (41)

5. Results and discussion

In order to get a physical insight into the problem, computa-tions for velocity, temperature, skin friction and Nusselt num-ber are performed for different values of Grashof number, stratification parameter and time. The computed results are presented in Figs. 1-12. Solutions with thermal stratification are compared with the classical case, when there is no thermal stratification.


Velocity profiles represented by Fig. 1 shows the effect of stratification parameter S for fixed values of Gr = 5 and t = 1.5. Fig. 2 shows the effect of Gr for S = 0.2 and t = 1.5. It is ob-served that velocity decreases with increasing S and increases with increasing Gr. Fig. 3 and Fig. 4 respectively show the effects of S and Gr on velocity profiles against time at a non-dimensional distance R =1.5 from the surface of the cylinder. It is interesting to observe that classical velocity increases unboundedly with time. In contrast, due to the presence of stratification, velocity increases with oscillatory behavior for

smaller time, but for larger time it becomes steady. Also, the time required to reach the steady state increases with increas-ing S and decreases with increasing the values of Gr.

Fig. 5 shows the effect of thermal stratification on tempera-ture profiles at Gr = 5 and t = 1.5. It is observed from the fig-ure that temperature decreases with increasing thermal stratifi-cation. Also temperature becomes negative in presence of high thermal stratification. This is because the fluid near the cylinder can have temperature lower than the ambient.

Fig. 6 represents the effect of Grashof number on tempera-

1 2 3 4 5 60.0

0.4

0.8

1.2

U

R

Gr=5, t=1.5S=0S=0.1S=0.3S=0.5S=1

Fig. 1. Effect of stratification on velocity profiles.

1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

U

R

t=1.5, S=0.2Gr=3Gr=5Gr=10

Fig. 2. Effect of Grashof number on velocity profiles.

0 2 4 6 8 100

1

2

3

4

U

t

R=1.5, Gr=5S=0S=0.1S=0.2S=0.5S=1

Fig. 3. Velocity profiles with respect to time.

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

U

t

S=0.2, R=1.5Gr=5Gr=10Gr=15

Fig. 4. Velocity profiles with respect to time.

1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

T

R

t=1.5, Gr=5S=0S=0.1S=0.3S=0.5S=1

Fig. 5. Effect of stratification on temperature profiles.

1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

T

R

t=1.5, S=0.2Gr=1Gr=5Gr=10

Fig. 6. Effect of Grashof number on temperature.


ture profiles at S = 0.2 and t = 1.5. It is observed that tem-perature decreases with increasing Grashof number. Fig. 7 and Fig. 8 respectively shows the effects of S and Gr on tempera-ture profile against time at a non-dimensional distance R = 1.5. From these figures we observe that initially temperature in-creases with oscillating character but for larger time it be-comes steady. Also, time required to reach the steady state increases with increasing S or Gr.

The values of skin friction represented by Fig. 9 and Fig. 10

show the effects of thermal stratification and Grashof number respectively. It is observed from these figures that classical skin friction decreases monotonically with time, but in pres-ence of thermal stratification though it decreases initially, but becomes steady after certain time. Skin friction increases with increasing S but decreases with increasing Grashof number.

Fig. 11 and Fig. 12 respectively shows the effects of thermal stratification and Grashof number on Nusselt number i.e. rate of heat transfer. It is observed that initially, Nusselt number

0 2 4 6 8 100.3

0.4

0.5

0.6

0.7

0.8

R=1.5, Gr=5

T

t

S=0S=0.1S=0.2S=0.3S=0.5

Fig. 7. Temperature profiles with respect to time.

0 2 4 6 8 100.3

0.4

0.5

0.6

T

t

R=1.5, S=0.2Gr=5Gr=10Gr=15

Fig. 8. Temperature profiles with respect to time.

0 2 4 6 8-9

-8

-6

-4

-2

τ

t

Gr=5S=0S=0.1S=0.2S=0.5

Fig. 9. Effect of stratification on skin friction.

0 2 4 6 8-8

-6

-4

-2

0

τ

t

S=0.2Gr=5Gr=7Gr=10

Fig. 10. Effects of Gr on skin friction.

0 2 4 6 8 100.5

0.8

1.2

1.6 Gr=5

Nu

t

S=0 S=0.1S=0.3S=0.5

Fig. 11. Effect of stratification on Nusselt number.

0 2 4 6 8 100.9

1.0

1.2

1.4

1.6

1.8

Nu

t

S=0.2Gr=3Gr=5Gr=10

Fig. 12. Effects of Gr on Nusselt number.


decreases but after certain time it becomes steady. Nusselt number increases with increasing S, but decreases with in-creasing Grashof number.

Thus on the basis of the above observations, the following conclusions are made:

(1) Solutions for velocity and temperature obtained for un-steady state approaches to the solutions of steady state as t tends to infinity.

(2) The time required reaching steady state increases with increasing Gr or decreasing S.

(3) Velocity increases with increasing Gr but decreases with increasing S.

(4) Temperature decreases with increasing S or Gr. In pres-ence of high stratification non-dimensional temperature be-comes negative.

(5) Skin friction decreases with increasing Gr but increases with increasing S.

(6) Rate of heat transfer increases as Gr or S increases.

Nomenclature------------------------------------------------------------------------

pC : Specific heat at constant pressure Gr : Grashof number g : Acceleration due to gravity

0J : Bessel function of first kind and order zero 1J : Bessel function of first kind and order one 0K : Modified Bessel function of second kind and order

zero 1K : Modified Bessel function of second kind and order

one L : Laplace operator Nu : Nusselt number Pr : Prandtl number r : Radial coordinate

0r : Radius of the cylinder R : Dimensionless radial coordinate S : Dimensionless stratification parameter 't : Time

t : Dimensionless time 'T : Temperature

T : Dimensionless temperature T : Laplace transformation of T u : z′ -component of velocity U : Dimensionless velocity U : Laplace transformation of U V : Dummy real variable used in integral

0Y : Bessel function of second kind and order zero 1Y : Bessel function of second kind and order one α : Thermal diffusivity of fluid ν : Kinematic viscosity β : Coefficient of thermal expansion of fluid

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Rudra Kanta Deka obtained his M.Sc., Ph.D., D.Sc. degrees from Gauhati Uni-versity in 1986, 1998 and 2012. Dr. Deka was recipient of research award given by University Grants Commission, New Delhi in 2002. Dr. Deka has pub-lished 78 papers in journals and confer-ence proceedings.

Ashish Paul obtained his M.Sc. and Ph.D in Mathematics from Gauhati Uni-versity. Dr. Paul has published 15 papers in journals and conference proceedings.

transient free convection flow past an infinite vertical cylinder with thermal stratification

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