Research Article Analytical Solutions of Heat Transfer and Film … · 2019. 7. 31. · Research...

Research ArticleAnalytical Solutions of Heat Transfer and Film Thickness withSlip Condition Effect in Thin-Film Evaporation for Two-PhaseFlow in Microchannel

Ahmed Jassim Shkarah,1,2 Mohd Yusoff Bin Sulaiman,1 and Md. Razali bin Hj Ayob1

1Faculty of Mechanical Engineering, Universiti Teknikal Malaysia Melaka (UTeM), Melaka, Malaysia2Department of Mechanical Engineering, Thi-Qar University, 64001 Nassiriya, Iraq

Correspondence should be addressed to Ahmed Jassim Shkarah; [email protected]

Received 20 June 2014; Revised 24 November 2014; Accepted 25 November 2014

Academic Editor: Salvatore Alfonzetti

Copyright © 2015 Ahmed Jassim Shkarah et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Physical and mathematical model has been developed to predict the two-phase flow and heat transfer in a microchannel withevaporative heat transfer. Sample solutions to the model were obtained for both analytical analysis and numerical analysis. It isassumed that the capillary pressure is neglected (Morris, 2003). Results are provided for liquid film thickness, total heat flux, andevaporating heat flux distribution. In addition to the sample calculations that were used to illustrate the transport characteristics,computations based on the current model were performed to generate results for comparisons with the analytical results ofWang etal. (2008) andWayner Jr. et al. (1976).The calculated results from the current model match closely with those of analytical results ofWang et al. (2008) andWayner Jr. et al. (1976).This work will lead to a better understanding of heat transfer and fluid flow occurringin the evaporating film region and develop an analytical equation for evaporating liquid film thickness.

1. Introduction

Over the last decade, micromachining technology has beenincreasingly used to develop highly efficient heat sink coolingdevices due to advantages such as lower coolant demandsand smaller machinable dimensions. One of the most impor-tant micromachining technologies is the ability to fabricatemicrochannels. Hence, the studies of fluid flow and heattransfer in microchannels which are two essential parts ofsuch devices have attracted attention due to their broadpotential for solving both engineering andmedical problems.Heat sinks are classified as either single-phase or two-phase according to whether liquid boiling occurs inside themicrochannels. The primary parameters that determine thesingle-phase and two-phase operating regimes are the heatflux through the channel wall and the coolant flow rate. For afixed heat flux (heat load), the coolantmaymaintain its liquidstate throughout the microchannels. For a lower flow rate,the liquid coolant flowing inside themicrochannel may reachits boiling point, causing flow boiling to occur, resulting in

a two-phase heat sink [1–3]. Since the initial conception ofmicroheat pipes in 1984 by Cotter [4], a number of analyticaland experimental investigations have been conducted. Mostof these investigations have concentrated on the capillary heattransport capability. With rapid advancements in microelec-tronic devices, their required total power and power densityare significantly increasing. Thin-film evaporation plays animportant role in these modern highly efficient heat transferdevices. When thin-film evaporation occurs in the thin-filmregion, most of the heat transfers through a narrow areabetween the nonevaporation region and intrinsic meniscusregion as shown in Figure 1. The flow resistance of the vaporphase during thin-film evaporation is very small comparedwith the vapor flow in the liquid phase in a typical nucleateboiling heat transfer configuration. In addition, the superheatneeded for the phase change in the thin-film region is muchsmaller than that for a bubble growth in a typical nucleateboiling, in particular, at the initial stage of the bubble growth.The heat transfer efficiency of thin-film evaporation is muchhigher than the nucleate boiling heat transfer. Because

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 369581, 15 pageshttp://dx.doi.org/10.1155/2015/369581

2 Mathematical Problems in Engineering

thin-film evaporation occurs in a small region, increasingthin-film regions andmaintaining its stability are very impor-tant for heat transfer enhancement. It is known that thethermodynamic properties of the liquid thin-film region arevery different from those of the macroregion. The effect ofthe intermolecular forces between the liquid thin film andwall can be characterized by the disjoining pressure, whichcontrols thewettability and stability of liquid thin film formedon the wall. A better understanding of the evaporationmechanisms governed by the disjoining pressure in the thin-film region, especially for high heat flux, is very importantin the development of highly efficient heat transfer devices.As early as 1972, Potash Jr. and Wayner Jr. [5] expanded theDerjaguin-Landau-Verwey-Overbeek (DLVO) theory [6] todescribe evaporation and fluid flow from an extendedmenis-cus. Following this work, extensive investigations have beenconducted to further understand mechanisms of fluid flowcoupled with evaporating heat transfer in thin-film region.Stephan and Busse [7] developed a mathematical modelbased on the theoretical analysis presented by Wayner Jr. etal. [5, 8] to investigate the heat transfer coefficient occurringin small triangular grooves and found that the interfacetemperature variation plays an important role in the thin-film evaporation. Schonberg andWayner Jr. [9] developed ananalytical model by ignoring capillary pressure and found ananalytical solution for the maximum heat evaporation fromthe thin film. Ma and Peterson [10] studied the thin-filmprofile, heat transfer coefficient, and temperature variationalong the axial direction of a triangular groove. Hanlon andMa [11] found that when particles become smaller, thin-filmregion can be significantly increased. More recently, Shaoand Zhang [12] considered the effect of thin-film evaporationon the heat transfer performance in an oscillating heat pipe.Wang et al. [13, 14] established a simplified model basedon the Young-Laplace equation and obtained an analyticalsolution for the total heat transfer in the thin-film region. Inthe current investigation, amathematicalmodel is establishedand its analytical solutions are obtained to evaluate the heatflux, total heat transport per unit length along the thin-film profile, thin-film thickness, and location for the maxi-mum heat evaporation in the thin-film region andmaximumheat transfer rate per unit length by thin-film evaporation.

2. Theoretical Analysis

Figure 1 illustrates a schematic of an evaporating thin filmformed on a wall. For the current investigation, it is assumedthat fluid flow in the thin-film region is two-dimensional andpressure in the liquid film is a function of the 𝑥-coordinateonly. The wall temperature, 𝑇

𝑤, is greater than the vapor

temperature, 𝑇V. The momentum equation governing thefluid flow in the thin film can be found by

𝑑𝑝𝑙

𝑑𝑥

− 𝜇𝑙

𝜕2

𝑢

𝜕𝑦2= 0, (1)

where

Liquid bulk

NonevaporatingEvaporating

Intrinsic meniscus

y

x

𝛿𝛿0

ṁe

ṁx

thin film

T�

Tw

Figure 1: Schematic of an evaporating thin film.

𝑢 = 𝑢wall = 𝛽𝑑𝑢

𝑑𝑥

wall

𝑦 = 0,

𝛿𝑢

𝛿𝑦

= 0 𝑦 = 𝛿.

(2)

Solving (1) with boundary condition (2), we get

𝑢 = (

1

𝜇𝑙

𝑑𝑝𝑙

𝑑𝑥

)

𝑦2

2

−

𝛿

𝜇𝑙

𝑑𝑝𝑙

𝑑𝑥 1

𝑦 + 𝑢wall,

𝑢wall = 𝛽𝑑𝑢

𝑑𝑥

wall

,

(3)

where 𝛽 is the slip coefficient. If 𝛽 is equal to zero, then noslip boundary condition is obtained.

So we can find

𝑢 =

1

𝜇𝑙

𝑑𝑝𝑙

𝑑𝑥

(

𝑦2

2

− 𝛿𝑦 − 𝛽𝛿) , (4)

where

𝛽 = 𝛽0(

1

√1 − 𝛾/𝛾𝑐

) , (5)

where 𝛽0is the limiting slip length and 𝛾

𝑐represents the

critical value of the shear rate. The slip coefficient underthe condition of this study turns out to be approximately1 ∗ 10

−9m. From (4), the mass flow rate at a given location 𝑥can be found as

�̇�𝑥= ∫

𝛿

0

𝜌𝑙𝑢𝑙𝑑𝑦 =

𝜌𝑙

𝜇𝑙

𝑑𝑝𝑙

𝑑𝑥

(

−2𝛿3

− 6𝛽𝛿2

6

) . (6)

Taking a derivative of (6),

𝑑�̇�𝑥

𝑑𝑥

=

𝑑

𝑑𝑥

[

𝜌𝑙

𝜇𝑙

𝑑𝑝𝑙

𝑑𝑥

(

−2𝛿3

− 6𝛽𝛿2

6

)] . (7)

But

�̇�𝑒= −

𝑑�̇�𝑥

𝑑𝑥

(8)

so the net evaporative mass transfer can be obtained as

�̇�𝑒= −

𝑑�̇�𝑥

𝑑𝑥

=

𝑑

𝑑𝑥

[

𝜌𝑙

𝜇𝑙

𝑑𝑝𝑙

𝑑𝑥

(

2𝛿3

+ 6𝛽𝛿2

6

)] . (9)

Mathematical Problems in Engineering 3

The heat transfer rate by evaporation occurring at the liquid-vapor interface in the thin-film region can be determined by

𝑞

= �̇�𝑒ℎ𝑓𝑔= ℎ𝑓𝑔

𝑑

𝑑𝑥

[

𝜌𝑙

𝜇𝑙

𝑑𝑝𝑙

𝑑𝑥

(

2𝛿3

+ 6𝛽𝛿2

6

)] . (10)

But at the same time 𝑞 in (10) is equal to the heat transferrate through the liquid thin film; that is,

𝑞

= 𝑘𝑙

𝑇wall − 𝑇𝛿𝛿

. (11)

From expanding the Clausius-Clapeyron equation, we canfind

(

𝑑𝑝

𝑑𝑇

)

sat=

ℎ𝑓𝑔

𝑇V (1

𝜌V−

1

𝜌𝑙

)

, (12)

𝑇𝛿= 𝑇V (1 +

Δ𝑝

𝜌Vℎ𝑓𝑔) . (13)

The pressure difference between vapor and liquid, Δ𝑝, at theliquid-vapor interface is due to both the capillary pressureand disjoining pressure and is expressed using the augmentedYoung-Laplace equation:

Δ𝑝 = 𝑝V − 𝑝𝑙 = 𝑝𝑐 + 𝑝𝑑. (14)

The disjoining pressure for a nonpolar liquid is expressed as

𝑝𝑑=

𝐴

𝛿3, (15)

where𝐴 is the dispersion constant and 𝛿 is the film thickness.The capillary pressure is the product of interfacial curvature𝐾 and surface tension coefficient 𝜎:

𝑝𝑐= 𝜎𝐾, (16)

𝐾 = (

𝑑2

𝛿

𝑑𝑥2)[1 + (

𝑑𝛿

𝑑𝑥

)

2

]

3/2

. (17)

Substituting (13) into (11) the heat flux can be rewritten as

𝑞

=

𝑇wall − 𝑇V (1 + Δ𝑝/𝜌Vℎ𝑓𝑔)

𝛿/𝑘𝑙

. (18)

Substituting (14) into (16) the heat flux can be rewritten as

𝑞

=

𝑇wall − 𝑇V (1 + (𝑝𝑐 + 𝑝𝑑) /𝜌Vℎ𝑓𝑔)

𝛿/𝑘𝑙

. (19)

Substituting (19) into (10) gives


𝛿/𝑘𝑙

= ℎ𝑓𝑔

𝑑

𝑑𝑥

[

𝜌𝑙

𝜇𝑙

𝑑𝑝𝑙

𝑑𝑥

(

2𝛿3

+ 6𝛽𝛿2

6

)] .

(20)

We can find 𝑑𝑝𝑙/𝑑𝑥 by differentiating (14) with respect to 𝑥,

and assume uniform vapor pressure, 𝑝V, along the meniscus:

𝑑𝑝𝑙

𝑑𝑥

= −(

𝑑 (𝑝𝑐+ 𝑝𝑑)

𝑑𝑥

) . (21)

Substituting (21) into (20) yields


𝛿/𝑘𝑙

= −ℎ𝑓𝑔

𝑑

𝑑𝑥

[

𝜌𝑙

𝜇𝑙

(

𝑑 (𝑝𝑐+ 𝑝𝑑)

𝑑𝑥

)(

2𝛿3

+ 6𝛽𝛿2

6

)] .

(22)

For the evaporating thin-film region, the disjoining pressureis one dominant parameter, which governs the fluid flow inthe evaporating thin-film region. And in the evaporating thinfilm region, the absolute disjoining pressure is much largerthan the capillary pressure especially when the curvaturevariation along the meniscus is very small. In order to findthe primary factor affecting the thin-film evaporation in theevaporating thin-film region, it is assumed that the capillarypressure, 𝑝

𝑐, is neglected.

Then, (22) becomes

𝑇wall − 𝑇V (1 + 𝑝𝑑/𝜌Vℎ𝑓𝑔)

𝛿/𝑘𝑙

= −ℎ𝑓𝑔

𝑑

𝑑𝑥

[

𝜌𝑙

𝜇𝑙

(

𝑑𝑝𝑑

𝑑𝑥

)(

2𝛿3

+ 6𝛽𝛿2

6

)] .

(23)

From (15) we find 𝑝𝑑and 𝑑𝑝

𝑑/𝑑𝑥;

𝑑𝑝𝑑

𝑑𝑥

=

−3𝐴

𝛿4

𝑑𝛿

𝑑𝑥

. (24)

Substituting (15) and (24) into (23) yields

𝑇wall − 𝑇V (1 + (𝐴/𝛿3

) /𝜌Vℎ𝑓𝑔)

𝛿/𝑘𝑙

= −ℎ𝑓𝑔

𝑑

𝑑𝑥

[

𝜌𝑙

𝜇𝑙

(

−3𝐴

𝛿4

𝑑𝛿

𝑑𝑥

)(

2𝛿3

+ 6𝛽𝛿2

6

)] .

(25)

We can rewrite the heat flux as

𝑞

= ℎ𝑓𝑔

𝑑

𝑑𝑥

[

𝜌𝑙𝐴

𝜇𝑙

𝑑𝛿

𝑑𝑥

(

1

𝛿

+

3𝛽

𝛿2)] . (26)

Simplifying (25) gives

1

ℎ𝑓𝑔

(

𝑘𝑙𝑇wall𝛿

−

𝑘𝑙𝑇V

𝛿

−

𝑘𝑙𝑇V𝐴

𝜌Vℎ𝑓𝑔𝛿4)

=

𝑑

𝑑𝑥

[

𝜌𝑙𝐴

𝜇𝑙

𝑑𝛿

𝑑𝑥

(

1

𝛿

+

3𝛽

𝛿2)] .

(27)

We need to solve (27) to find 𝑑𝛿/𝑑𝑥.


1

0.8

0.6

0.4

0.2

0

0 4 8 12 16 20

qt

(W/m

)

Analytical solution of Wang et al. [13]Numerical solution of Schonberg and WaynerCurrent full numerical solutionCurrent analytical solution

Twall − T�

Figure 2: Comparison of the total heat transfer rate through thinfilm region with results presented byWang et al. [13] andWayner Jr.et al. [8].

2

1.8

1.6

1.4

1.2

1

0 2 4 6

𝛿/𝛿

0

x/𝛿0

Numerical solutionAnalytical solution

Tw − T� = 0.1K

Figure 3: Dimensionless evaporative film thickness profile at asuperheat of 0.1 K.

𝛿/𝛿

0

x/𝛿0

8

6

4

2

0

0 2 4 6

Tw − T� = 0.5K


Figure 4: Dimensionless evaporative film thickness profile at asuperheat of 0.5 K.

By multiplying 2 sides by 2 ∗ [(𝑑𝛿/𝑑𝑥)(1/𝛿 + 3𝛽/𝛿2)] andintegrating two sides with respect to 𝑥 we get

[

𝑑𝛿

𝑑𝑥

(

1

𝛿

+

3𝛽

𝛿2)]

2

+ 𝐷

=

2𝜇𝑙

ℎ𝑓𝑔𝐴𝜌𝑙

[

𝑘𝑙

𝛿

(𝑇V − 𝑇wall) +3𝑘𝑙𝛽

2𝛿2(𝑇V − 𝑇wall)

+

𝑘𝑙𝑇V𝐴

4𝜌Vℎ𝑓𝑔𝛿4+

3𝛽𝑘𝑙𝑇V𝐴

5𝜌Vℎ𝑓𝑔𝛿5] + 𝐶.

(28)

𝐷, 𝐶 are integration constants.So

[

𝑑𝛿

𝑑𝑥

]

2

=

1

[1/𝛿 + 3𝛽/𝛿2]2

× (

2𝜇𝑙


[

𝑘𝑙

𝛿

(𝑇V − 𝑇wall)

+

3𝑘𝑙𝛽


+

𝑘𝑙𝑇V𝐴



5𝜌Vℎ𝑓𝑔𝛿5] + (𝐶 − 𝐷)) .

(29)

Mathematical Problems in Engineering 5𝛿/𝛿

0

x/𝛿0

20

16

12

8

4

0

0 2 4 6

Tw − T� = 1K


Figure 5: Dimensionless evaporative film thickness profile at asuperheat of 1 K.

Let

𝐵 = 𝐶 − 𝐷, (30)

𝑑𝛿

𝑑𝑥

= (

1

[1/𝛿 + 3𝛽/𝛿2]2

× (

2𝜇𝑙


× [

𝑘𝑙

𝛿



+

𝑘𝑙𝑇V𝐴



5𝜌Vℎ𝑓𝑔𝛿5] + 𝐵))

1/2

.

(31)

To find constant 𝐵 we apply the boundary condition

𝑑𝛿

𝑑𝑥

= 0 𝛿 = 𝛿𝑜. (32)

So we get

𝐵 = −

2𝜇𝑙


× [

𝑘𝑙

𝛿𝑜


2𝛿2

𝑜

(𝑇V − 𝑇wall)

+

𝑘𝑙𝑇V𝐴

4𝜌Vℎ𝑓𝑔𝛿4

𝑜

+


5𝜌Vℎ𝑓𝑔𝛿5

𝑜

] .

(33)

𝛿/𝛿

0

x/𝛿0

60

40

20

0

0 2 4 6

Tw − T� = 2K



From (26) and (27),

𝑞

= ℎ𝑓𝑔

𝑑

𝑑𝑥

[

𝜌𝑙𝐴

𝜇𝑙

𝑑𝛿

𝑑𝑥

(

1

𝛿

+

3𝛽

𝛿2)]

= (

𝑘𝑙𝑇wall𝛿

−

𝑘𝑙𝑇V

𝛿

−

𝑘𝑙𝑇V𝐴

𝜌Vℎ𝑓𝑔𝛿4) .

(34)

So

(𝜇𝑙𝑘𝑙/ℎ𝑓𝑔𝜌𝑙𝐴) (𝑇

𝑤− 𝑇V) − (𝜇𝑙𝑘𝑙𝑇V/ℎ

2

𝑓𝑔𝜌𝑙𝜌V𝐴) (1/𝛿

3

)

𝛿

=

𝑑

𝑑𝑥

[

𝑑𝛿

𝑑𝑥

(

1

𝛿

+

3𝛽

𝛿2)] .

(35)

Then

𝑑

𝑑𝑥

[

𝑑𝛿

𝑑𝑥

(

1

𝛿

+

3𝛽

𝛿2)]

=

[(𝑘𝑙𝜐𝑙/ℎ𝑓𝑔𝐴) (𝑇

𝑤− 𝑇V)] − (𝑘𝑙𝜐𝑙𝑇V/ℎ

2

𝑓𝑔𝜌V) 𝛿−3

𝛿

.

(36)

Note that

𝜐𝑙=

𝜇𝑙

𝜌𝑙

. (37)

6 Mathematical Problems in Engineering𝛿/𝛿

0

x/𝛿0

800

600

400

200

0

0 2 4 6

Tw − T� = 5K



The optimum thickness of the evaporating thin film, 𝛿optimum,where the heat flux reaches its maximum, can be found bytaking a derivative of 𝑥 for heat flux equation (34):

𝑑𝑞

𝑑𝑥

= 0

=

𝑑 (𝑘𝑙𝑇wall/𝛿 − 𝑘𝑙𝑇V/𝛿 − 𝑘𝑙𝑇V𝐴/𝜌Vℎ𝑓𝑔𝛿

4

)𝛿=𝛿optimum

𝑑𝑥

.

(38)

So

(−

𝑇wall

(𝛿optimum)2+

𝑇V

(𝛿optimum)2+

4𝑇V𝐴

𝜌Vℎ𝑓𝑔 (𝛿optimum)5) = 0.

(39)

Then

𝛿optimum =3√

4𝐴𝑇V

𝜌Vℎ𝑓𝑔 (𝑇𝑤 − 𝑇V). (40)

For the nonevaporating film region, the heat flux is zero.Clearly, the interface temperature is equal to the wall tem-perature. The equilibrium thickness 𝛿

𝑜can be readily found

by

𝑞

= (

𝑘𝑙𝑇wall𝛿𝑜

−

𝑘𝑙𝑇V

𝛿𝑜

−

𝑘𝑙𝑇V𝐴

𝜌Vℎ𝑓𝑔𝛿4

𝑜

) = 0. (41)

𝛿/𝛿

0

x/𝛿0

60000

40000

20000

0

0 2 4 6

Tw − T� = 10K



So

𝛿0≥3√

𝐴𝑇V

𝜌Vℎ𝑓𝑔 (𝑇𝑤 − 𝑇V),

𝛿min0

=3√

𝐴𝑇V

𝜌Vℎ𝑓𝑔 (𝑇𝑤 − 𝑇V).

(42)

From (31), (33), and (34) we get

𝛿optimum =3√4𝛿𝑜. (43)

The total heat transfer rate per unit width along themeniscus,𝑞𝑡, can be calculated by

𝑞𝑡= ∫

𝑥

0

𝑞

𝑑𝑥. (44)

From (34)

𝑞𝑡= ∫

𝑥

0

ℎ𝑓𝑔

𝑑

𝑑𝑥

[

𝜌𝑙𝐴

𝜇𝑙

𝑑𝛿

𝑑𝑥

(

1

𝛿

+

3𝛽

𝛿2)]𝑑𝑥. (45)

So

𝑞𝑡=

ℎ𝑓𝑔𝜌𝑙𝐴

𝜇𝑙

𝑑𝛿

𝑑𝑥

(

1

𝛿

+

3𝛽

𝛿2) . (46)

Dimensionless thickness of the evaporating region can bedefined as

̂𝛿 =

𝛿

𝛿𝑜

, (47)

where 𝛿𝑜is the thickness of the nonevaporating region.


x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 4 8 12 16

Tw − T� = 0.3K


Figure 9: Dimensionless heat flux profile at a superheat of 0.3 K.

x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 4 8 12 16

Tw − T� = 0.7K


Figure 10: Dimensionless heat flux profile at a superheat of 0.7 K.

A dimensionless position also can be defined as

𝜓 =

𝑥

𝛿𝑜

. (48)

x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 2 4 6

Tw − T� = 1K


Figure 11: Dimensionless heat flux profile at a superheat of 1 K.

x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 2 4 6

Tw − T� = 2K



For dimensionless heat flux,

𝜙 =

𝑞

𝑞

𝑜

, (49)


x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 2 4 6

Tw − T� = 5K



x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 2 4 6

Tw − T� = 10K



where

𝑞

𝑜=

𝑘𝑙(𝑇𝑤− 𝑇V)

𝛿𝑜

(50)

is the heat flux at the interface temperature equal to the vaportemperature.

x/𝛿0

𝜙

𝛽 = 0 (no slip)

0.8

0.6

0.4

0.2

0

0 2 4 6


Figure 15: Dimensionless heat flux profile at a 𝛽 = 0 nm.

Substituting (36), (34), and (50) in (49) we found

𝜙 =


𝜇𝑙

×

([(𝑘𝑙𝜐𝑙/ℎ𝑓𝑔𝐴) (𝑇

𝑤− 𝑇V)] − (𝑘𝑙𝜐𝑙𝑇V/ℎ

2

𝑓𝑔𝜌V) 𝛿−3

) /𝛿

(𝑘𝑙(𝑇𝑤− 𝑇V) /𝛿𝑜)

=

𝛿𝑜

𝛿

−

𝛿4

𝑜

𝛿4,

(51)

or

𝜙 =

1

̂𝛿

(1 −

1

̂𝛿3

) . (52)

By considering 𝑑𝜙/𝑑̂𝛿 = 0, we can determine the maximumdimensionless heat flux 𝜙max. The local heat flux through theevaporating thin film reaches its maximum when ̂𝛿 = 41/3.Letting ̂𝛿 = 41/3 in (52), the maximum dimensionless heatflux, ̂𝛿max, can be found as

̂𝛿max =

3

44/3

≈ 0.473. (53)

Equation (53) indicates that the maximum heat flux occur-ring in the evaporating thin-film region is not greater than0.473 times of the characteristic flux heat.


𝛽 = 0.5 ∗ 10−9 m

x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 2 4 6


Figure 16: Dimensionless heat flux profile at a 𝛽 = 0.5 nm.

Consider the heat transfer coefficient of

𝑞

= ℎ (𝑇𝑠− 𝑇V) . (54)

To solve (31) we have two methods:

(1) exact solution or analytical solution,

(2) numerical solution.

We are going to solve (31) by using both methods

2.1. Analytical Solution. We can rewrite (31) in the form

𝑑𝛿

𝑑𝑥

= (Σ𝜎 (𝛿) (1 −

𝛿0

𝛿

) + Ψ𝜎 (𝛿) (1 −

𝛿2

0

𝛿2)

+ Ω𝜎 (𝛿) (

𝛿4

0

𝛿4− 1) + Π𝜎 (𝛿) (

𝛿5

0

𝛿5− 1))

1/2

,

(55)

where

𝜎 (𝛿) =

𝛿2

0

(𝛿0/𝛿 + (3𝛽/𝛿

0) (𝛿2

0/𝛿2))2,

Σ =

2𝜇𝑙𝑘𝑙(𝑇𝑤− 𝑇V)


1

𝛿0

,

𝛽 = 1 ∗ 10−9 m

x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 2 4 6



Ψ =

3𝜇𝑙𝛽 (𝑇𝑤− 𝑇V)


(

1

𝛿0

)

2

,

Ω =

𝜇𝑙𝑘𝑙𝑇V

2ℎ2

𝑓𝑔𝜌V(

1

𝛿0

)

4

,

Π =

6𝜇𝑙𝑘𝑙𝛽𝑇V

5ℎ2

𝑓𝑔𝜌V

(

1

𝛿0

)

5

,

Λ =

3𝛽

𝛿0

.

(56)

Since Σ ≫ Ψ,Π,Ω so we can rewrite (55) as

𝑑𝛿

𝑑𝑥

= √Σ(

𝛿2

0

(𝛿0/𝛿 + 3 (𝛽𝛿

2

0/𝛿0𝛿2))

× [(1 −

𝛿0

𝛿

) +

Ψ

Σ

(1 −

𝛿2

0

𝛿2)

+

Ω

Σ

(

𝛿4

0

𝛿4− 1) +

Π

Σ

(

𝛿5

0

𝛿5− 1)])

1/2

.

(57)


𝛽 = 3 ∗ 10−9 m

x/𝛿0

𝜙

0.8

0.6

0.4

0.2

0

0 2 4 6



We introduce ̂𝛿 = 𝛿/𝛿0and 𝑥 = 𝑥√Σ to get

𝑑̂𝛿

𝑑𝑥

= (

̂𝛿2

(1 + Λ (1/̂𝛿))

2

× [(1 −

1

̂𝛿

) +

Ψ

Σ

(1 −

1

̂𝛿2

)

+

Ω

Σ

(

1

̂𝛿4

− 1) +

Π

Σ

(

1

̂𝛿5

− 1)])

1/2

.

(58)

Since 3𝛽/𝛿0≪ 1, we can make the following approximation:

1

(1 + Λ (1/̂𝛿))

2= 1 − 2

Λ

̂𝛿

+ (

Λ

̂𝛿

)

2

. (59)

To get

𝑑̂𝛿

𝑑𝑥

= (̂𝛿2

[1 − 2

Λ

̂𝛿

+ 3(

Λ

̂𝛿

)

2

]

× [(1 −

1

̂𝛿

) +

Ψ

Σ

(1 −

1

̂𝛿2

)

+

Ω

Σ

(

1

̂𝛿4

− 1) +

Π

Σ

(

1

̂𝛿5

− 1)])

1/2

,

x/𝛿0

𝛽 = 0 (no slip)

0 2 4 6

𝛿/𝛿

0

4000

3000

2000

1000

0


Figure 19: Dimensionless evaporative film thickness profile at a 𝛽 =0 nm.

𝑑̂𝛿

𝑑𝑥

= ([1 − 2

Λ

̂𝛿

+ 3(

Λ

̂𝛿

)

2

]

× [ (̂𝛿2

−̂𝛿) +

Ψ

Σ

(̂𝛿2

− 1)

+

Ω

Σ

(

1

̂𝛿2

−̂𝛿2

) +

Π

Σ

(

1

̂𝛿3

−̂𝛿2

)])

1/2

(60)

we have

𝑑̂𝛿

𝑑𝑥

= (̂𝛿2

[1 +

Ψ

Σ

−

Ω

Σ

−

Π

Σ

]

−̂𝛿 [1 + 2Λ [1 +

Ψ

Σ

−

Ω

Σ

−

Π

Σ

]]

− [

Ψ

Σ

− 2Λ] + ⋅ ⋅ ⋅

1

̂𝛿

+ ⋅ ⋅ ⋅

1

̂𝛿2

+ ⋅ ⋅ ⋅ )

1/2

.

(61)

Let

Γ = 1 +

Ψ

Σ

−

Ω

Σ

−

Π

Σ

. (62)

Because ̂𝛿 ≥ 1, we neglect 1/̂𝛿, 1/̂𝛿2, . . . . Therefore, thesolution is accurate for ̂𝛿 ≫ 1 as

𝑑̂𝛿

𝑑𝑥

=√̂𝛿2Γ −

̂𝛿 [1 + 2ΛΓ] − [Γ − 1 − 2ΛΓ]. (63)

The general solution of (63) by using the initial condition

̂𝛿 = 1 𝑥 = 0 (64)


4000

3000

2000

1000

0

𝛽 = 0.5 ∗ 10−9 m

x/𝛿0

0 2 4 6

𝛿/𝛿

0


Figure 20: Dimensionless evaporative film thickness profile at a 𝛽 =0.5 nm.

is

̂𝛿 =

1

4 × (Γ/ (1 + 2ΛΓ))

×

1

2 × (Γ/ (1 + 2ΛΓ)) − 1

× 𝑒−𝑥√Γ

{4

Γ

1 + 2ΛΓ

(

Γ

1 + 2ΛΓ

− 1)

+ [1 + (2

Γ

1 + 2ΛΓ

− 1) 𝑒−𝑥√Γ

]

2

} .

(65)

And from (65) we can evaluate an analytical solution for𝑞𝑡(𝛿 → ∞) as

𝑞𝑡=


𝜇𝑙

√Σ. (66)

2.2. Numerical Solution. The solution can be readily obtainedusing the fourth order Runge-Kutta method for the evaporat-ing thin film profile. The governing equation (31) is solvedwith the use of a Runge-Kutta (4) method; the solutionprocedure is iterative. As a first guess 𝛿

0, the values from

the previous step are used, and the calculated values arereturned from the Runge-Kutta solver and compared to theguess values. A comparison of the guess and calculated valuesis performed and looped until reaching convergence criteriafor both film thicknesses. The numerical solver is coded inMATLAB. With these initial conditions, we have

𝛿 = 𝛿0

𝑥 = 0,

𝑑𝛿

𝑑𝑥

= 0 𝑥 = 0.

(67)

𝛽 = 1 ∗ 10−9 m

4000

3000

2000

1000

0

x/𝛿0

0 2 4 6

𝛿/𝛿

0Numerical solutionAnalytical solution


Table 1: Liquid properties and operating conditions.

Liquid Water𝐴 10−20 J𝑇V 353K

∘

𝜌V 0.083 kg/m3

𝜐𝑙

0.4996 ∗ 10−6m2/sℎ𝑓𝑔

2382700 J/kg𝑘𝑙

0.65w/m⋅K

3. Results and Discussion

3.1. Comparison of Analytical Solution and Full Model. Aspresented above, a mathematical model for predicting evap-oration and fluid flow in thin-film region is developed.Utilizing dimensionless analysis, analytical and numericalsolutions are obtained for the heat flux distribution, totalheat transfer rate per unit length, location of the maximumheat flux, and ratio of the conduction to convection thermalresistance in the evaporating film region. In order to verifythe analytical solution derived herein, results predicted byWang et al. [13] and numerical solution by Wayner Jr. et al.[8] are used. Figure 2 shows the comparison of analyticaland numerical results of the total heat transfer rate throughthin-film region with results presented by Wang et al. [13]and Wayner Jr. et al. [8]. Total heat flux is presented infunction of the superheat temperature. Our numerical andanalytical solutions are compared to the ones of Wangand Schonberg and Wayner. It shows good agreement formoderate superheat temperatures; however, our analyticalsolution tends to underestimate total heat flux at large


Table 2: Comparison of previous studies on evaporating extended meniscus.

Authors Numerical solution analytical solution Finding analytical equation for 𝛿 Slip conditionPotash and Wayner [5] o x x xMoosman and Homsy [15] o o x xSchonberg and Wayner [9] o o x xStephan and Busse [7] o x x xSchonberg et al. [16] o o x xShikhmurzaev [17] x o x xMa and Peterson [10] o x x xPismen and Pomeau [18] x o x xCatton and Stroes [19] o o x xChoi et al. [20] o x x oQu and Ma [21] o o x xChoi et al. [22] o x x oPark and Lee [23] o x x xBy Morris [24] x o x xDemsky and Ma [25] o x x xJiao et al. [26] x o x xNa et al. [27] o o x xSultan et al. [28] x o x xWang et al. [14] o x x xMa et al. [29] o x x xWang et al. [13] o o x xZhao et al. [30] o x x oZhao et al. [31] o x x oBenselama et al. [32] x o x xBiswal et al. [33] o x x oLiu et al. [34] x o x xBai et al. [35] o o x xBiswal et al. [36] o x x xThokchom et al. [37] o x x xYang et al. [38] o x x xCurrent study o o o o

superheat temperatures. In addition, the model presentedherein can be used to predict analytically and numericallyall of the equilibrium film thickness, heat flux distribution,film thickness variation of evaporating film region,maximumtotal heat transfer rate through the evaporating film region,and ratio of the conduction to convection thermal resistance.The following calculations and predictions are based onthe thermal properties and operating conditions shown inTable 1.

3.2. Comparison of Analytic Equation for 𝛿 with PreviousStudies. It was seen from Table 2 that, in many studies on awide range of time, we did not find any researcher who foundthe analytical equation for 𝛿 even only approximately, but wefound that they have numerical studies. So according to this,we can say that (65) is the first analytical equation for 𝛿 or atleast approximately.

3.3. Distribution of Evaporative Film Thickness. Figures 3,4, 5, 6, 7, and 8 compare our numerical and analyticalpredictions for the dimensionless film thickness as functionof the dimensionless position for 0.1 K, 0.5 K, 1 K, 2 K, 5 K, and10K superheat temperatures.They clearly show that the errorof the analytical approximation is decreasing with increasingsuperheat temperature.We can see that for large positions thesolution is dominated by exponential growth.

Figures 9, 10, 11, 12, 13, and 14 compare our numericaland analytical predictions for the dimensionless heat flux asfunction of the dimensionless position for 0.3 K, 0.7 K, 1 K,2 K, 5 K, and 10K superheat temperatures. Similar propertycan be seen; that is, the error of the analytical approximationis decreasing with increasing superheat temperature. Themaximum of the heat flux is predicted correctly for allvalues of the superheat temperature analytically; however, thelocation of themaximum is predictedwith significant error atlow superheat temperatures.


𝛽 = 3 ∗ 10−9 m

4000

3000

2000

1000

0

x/𝛿0

0 2 4 6

𝛿/𝛿

0



Figures 15, 16, 17, and 18 show that nondimensional heatflux is presented as function of the nondimensional positionfor different values of 𝛽: 0, 0.5, 1 and 3×10−9. As 𝛽 gets larger,the error in the analytical heat flux approximation is gettinglarger.

Figures 19, 20, 21, and 22 compare our numerical andanalytical predictions for the dimensionless film thickness asfunction of the dimensionless position for different values ofthe slip coefficient, 𝛽: 0, 0.5, 1 and 3×10−9. Comparing the fig-ures indicates similar conclusions as previously mentioned:smaller slip coefficient yields smaller error in the analyticalapproximation.

4. Conclusions

This paper presents a mathematical model for predictingevaporation and fluid flow in thin-film region. The thin-film region of the extended meniscus is delineated. Utilizinganalytical solutions were obtained for heat flux distribution,total heat transfer, and liquid film thickness in the evapo-rating film region. The mathematical model also developsan analytic equation for 𝛿. So according to this, we can saythat (65) is the first analytical equation for 𝛿 or at leastapproximately. Also we can conclude that there is small effectof slip condition𝛽 on the thin-film evaporation for two-phaseflow in microchannel. The dimensionless heat flux throughthin-film region is a function of dimensionless thickness.Also the results showed the assumption that neglecting thecapillary pressure is acceptable.

Highlights

(i) Analytical two-phase flow formicrochannel heat sinkis studied.

(ii) Numerical two-phase flow formicrochannel heat sinkis studied.

(iii) New evaporating film thickness equation is devel-oped.

Nomenclature

𝐴: Dispersion constant (J)ℎ𝑓𝑔: Heat of vaporization (J/kg)

𝑘: Conductivity (W/m⋅K)�̇�𝑒: Interface net evaporative mass transfer (kg/(m2s))

�̇�𝑥: Mass flow rate (kg/ms)

𝑝𝑐: Capillary pressure (N/m2)

𝑝𝑙: Liquid pressure (N/m2)

𝑝V: Vapor pressure (N/m2)

Δ𝑝: 𝑝1− 𝑝2(N/m)

𝑞: Heat flux (w/m2)𝑞

𝑜: Characteristic heat flux (w/m2)

𝜙: Dimensionless heat flux𝜙max: Maximum dimensionless heat flux𝑞tot: Total heat transfer rate per unit width (W/m)𝑇: Temperature (K)𝑢: Velocity along 𝑥-axis (m/s)𝑥: 𝑥-coordinate (m)𝜓: Dimensionless 𝜓-coordinate𝑦: 𝑦-coordinate (m).

Greek Symbols

𝛿: Film thickness (m)𝛿0: Equilibrium film thickness or

characteristic thickness (m)̂𝛿: Dimensionless film thickness𝛿optimum: Optimum thickness corresponding to the

maximum heat flux (m)𝜇: Dynamic viscosity (N s/m2)𝜐: Kinematic viscosity (m2/s)𝜌: Density (kg/m3)𝜎: Surface tension (N/m)𝛽: The slip coefficient (m).

Subscripts

⋅: Time rate of change𝑙: Liquidmax: Maximum quantitytot: TotalV: Vapor𝑤: Wall.


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is supported by Universiti Teknikal MalaysiaMelaka (UTeM) andThi Qar University.

References

[1] A. J. Shkarah,M. Y. B. Sulaiman, and R. B. H. Ayob, “Two-phaseflow inmicro-channel heat sink review paper,”The InternationalReview ofMechanical Engineering, vol. 7, no. 1, pp. 231–237, 2013.

[2] A. J. Shkarah, M. Y. B. Sulaiman, M. R. B. H. Ayob, and H.Togun, “A 3D numerical study of heat transfer in a single-phasemicro-channel heat sink using graphene, aluminum and siliconas substrates,” International Communications in Heat and MassTransfer, vol. 48, pp. 108–115, 2013.

[3] A. J. Shkarah, M. Y. B. Sulaiman, and M. R. B. Hj, “Boiling twophase flow inmicrochannels: a review,” Indian Journal of Scienceand Technology, vol. 6, no. 11, pp. 5013–5018, 2013.

[4] T. Cotter, “Principles and prospects formicro heat pipes,” NASASTI/Recon Technical Report 84, 1984.

[5] M. Potash Jr. and P. C. Wayner Jr., “Evaporation from a two-dimensional extended meniscus,” International Journal of Heatand Mass Transfer, vol. 15, no. 10, pp. 1851–1863, 1972.

[6] B. V. Derjaguin and Z. M. Zorin, “Optical study of theabsorption and surface condensation of vapors in the vicinityof saturation on a smooth surface,” in Proceedings of the 2ndInternational Congress on Surface Activity, vol. 2, pp. 145–152,London, UK, 1956.

[7] P. C. Stephan and C. A. Busse, “Analysis of the heat transfercoefficient of grooved heat pipe evaporator walls,” InternationalJournal of Heat and Mass Transfer, vol. 35, no. 2, pp. 383–391,1992.

[8] P. C. Wayner Jr., Y. K. Kao, and L. V. LaCroix, “The interlineheat-transfer coefficient of an evaporating wetting film,” Inter-national Journal of Heat and Mass Transfer, vol. 19, no. 5, pp.487–492, 1976.

[9] J. A. Schonberg and P. C. Wayner Jr., “Analytical solutionfor the integral contact line evaporative heat sink,” Journal ofThermophysics and Heat Transfer, vol. 6, no. 1, pp. 128–134, 1992.

[10] H. B. Ma and G. P. Peterson, “Temperature variation and heattransfer in triangular grooves with an evaporating film,” JournalofThermophysics andHeat Transfer, vol. 11, no. 1, pp. 90–97, 1997.

[11] M. A. Hanlon and H. B. Ma, “Evaporation heat transfer insintered porous media,” Journal of Heat Transfer, vol. 125, no.4, pp. 644–652, 2003.

[12] W. Shao and Y. Zhang, “Effects of film evaporation and conden-sation on oscillatory flow and heat transfer in an oscillating heatpipe,” Journal of Heat Transfer, vol. 133, no. 4, Article ID 042901,2011.

[13] H. Wang, S. V. Garimella, and J. Y. Murthy, “An analyticalsolution for the total heat transfer in the thin-film region of anevaporating meniscus,” International Journal of Heat and MassTransfer, vol. 51, no. 25-26, pp. 6317–6322, 2008.

[14] H.Wang, S.V.Garimella, and J. Y.Murthy, “Characteristics of anevaporating thin film in a microchannel,” International Journal

of Heat and Mass Transfer, vol. 50, no. 19-20, pp. 3933–3942,2007.

[15] S.Moosman andG.M.Homsy, “Evaporatingmenisci of wettingfluids,” Journal of Colloid and Interface Science, vol. 73, no. 1, pp.212–223, 1980.

[16] J. A. Schonberg, S. DasGupta, and P. C. Wayner Jr., “Anaugmented Young-Laplace model of an evaporating meniscusin a microchannel with high heat flux,” Experimental Thermaland Fluid Science, vol. 10, no. 2, pp. 163–170, 1995.

[17] Y. D. Shikhmurzaev, “Moving contact lines in liquid/liquid/solid systems,” Journal of Fluid Mechanics, vol. 334, pp. 211–249,1997.

[18] L. M. Pismen and Y. Pomeau, “Disjoining potential and spread-ing of thin liquid layers in the diffuse-interface model coupledto hydrodynamics,”Physical ReviewE: Statistical, Nonlinear, andSoft Matter Physics, vol. 62, no. 2, pp. 2480–2492, 2000.

[19] I. Catton and G. R. Stroes, “A semi-analytical model to pre-dict the capillary limit of heated inclined triangular capillarygrooves,” Journal of Heat Transfer, vol. 124, no. 1, pp. 162–168,2002.

[20] C.-H. Choi, K. J. A. Westin, and K. S. Breuer, “To slip or not toslip—water flows in hydrophilic and hydrophobic microchan-nels,” in Proceedings of the ASME International MechanicalEngineering Congress and Exposition (IMECE ’02), pp. 557–564,American Society of Mechanical Engineers, New Orleans, La,USA, November 2002.

[21] W. Qu and T. Ma, “Effects of the polarity of working fluids onvapor-liquid flowandheat transfer characteristics in a capillary,”Microscale Thermophysical Engineering, vol. 6, no. 3, pp. 175–190, 2002.

[22] C.-H. Choi, K. J. A. Westin, and K. S. Breuer, “Apparent slipflows in hydrophilic and hydrophobic microchannels,” Physicsof Fluids, vol. 15, no. 10, pp. 2897–2902, 2003.

[23] K. Park and K.-S. Lee, “Flow and heat transfer characteristicsof the evaporating extended meniscus in a micro-capillarychannel,” International Journal of Heat and Mass Transfer, vol.46, no. 24, pp. 4587–4594, 2003.

[24] S. J. S. Morris, “The evaporating meniscus in a channel,” Journalof Fluid Mechanics, vol. 494, pp. 297–317, 2003.

[25] S. Demsky and H. Ma, “Thin film evaporation on a curvedsurface,” Microscale Thermophysical Engineering, vol. 8, no. 3,pp. 285–299, 2004.

[26] A. J. Jiao, R. Riegler, H. B. Ma, and G. P. Peterson, “Thin filmevaporation effect on heat transport capability in a grooved heatpipe,” Microfluidics and Nanofluidics, vol. 1, no. 3, pp. 227–233,2005.

[27] Y. W. Na, J. Chung, and F. Forster, “Numerical modelingof pressure drop and pump design for high power densitymicrochannel heat sinks with boiling,” in Proceedings of theInternational Mechanical Engineering Congress and Exposition(ASME ’05), pp. 349–357, American Society of MechanicalEngineers, Orlando, Fla, USA, November 2005.

[28] E. Sultan, A. Boudaoud, and M. Ben Amar, “Evaporation of athin film: diffusion of the vapour and Marangoni instabilities,”Journal of Fluid Mechanics, vol. 543, pp. 183–202, 2005.

[29] H. B. Ma, P. Cheng, B. Borgmeyer, and Y. X. Wang, “Fluidflow and heat transfer in the evaporating thin film region,”Microfluidics and Nanofluidics, vol. 4, no. 3, pp. 237–243, 2008.

[30] J. J. Zhao, X. F. Peng, and Y. Y. Duan, “Scale effects andslip microflow characteristics of evaporating thin films ina microchannel,” in Proceedings of the ASME 2nd Interna-tional Conference on Micro/Nanoscale Heat and Mass Transfer


(MNHMT ’09), vol. 2, pp. 61–70, American Society of Mechan-ical Engineers, Shanghai, China, December 2009.

[31] J. J. Zhao, X. F. Peng, and Y. Y. Duan, “Slip and micro flowcharacteristics near a wall of evaporating thin films in a microchannel,”Heat Transfer—Asian Research, vol. 39, no. 7, pp. 460–474, 2010.

[32] A. M. Benselama, S. Harmand, and K. Sefiane, “A perturbationmethod for solving the micro-region heat transfer problem,”Physics of Fluids, vol. 23, no. 10, Article ID 102103, 1994.

[33] L. Biswal, S. K. Som, and S. Chakraborty, “Thin film evapora-tion in microchannels with interfacial slip,” Microfluidics andNanofluidics, vol. 10, no. 1, pp. 155–163, 2011.

[34] X. Liu, D. Guo, G. Xie, S. Liu, and J. Luo, “Boiling in the waterevaporating meniscus induced by Marangoni flow,” AppliedPhysics Letters, vol. 101, no. 21, Article ID 211602, 2012.

[35] L. Bai, G. Lin, and G. P. Peterson, “Evaporative heat transferanalysis of a heat pipe with hybrid axial groove,” Journal of HeatTransfer, vol. 135, no. 3, Article ID 31503, 2013.

[36] L. Biswal, S. K. Som, and S. Chakraborty, “Thin film evaporationinmicrochannels with slope- and curvature-dependent disjoin-ing pressure,” International Journal of Heat and Mass Transfer,vol. 57, no. 1, pp. 402–410, 2013.

[37] A. K. Thokchom, A. Gupta, P. J. Jaijus, and A. Singh, “Analysisof fluid flow and particle transport in evaporating dropletsexposed to infrared heating,” International Journal of Heat andMass Transfer, vol. 68, pp. 67–77, 2014.

[38] F. Yang, X. Dai, Y. Peles, P. Cheng, J. Khan, and C. Li,“Flow boiling phenomena in a single annular flow regimein microchannels (I): characterization of flow boiling heattransfer,” International Journal of Heat and Mass Transfer, vol.68, pp. 703–715, 2014.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Research Article Analytical Solutions of Heat Transfer and Film … · 2019. 7. 31. · Research...

Documents

Transcript of Research Article Analytical Solutions of Heat Transfer and Film … · 2019. 7. 31. · Research...