Boundary Layer Flow of a Nanofluid past a Stretching Sheet

7/25/2019 Boundary Layer Flow of a Nanofluid past a Stretching Sheet

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Ali Alhamaly

ME 536 Report 1

Boundary Layer Flow of a Nanofluid past aStretching Sheet

1. Introduction

In this report, the fluid flow over linearly stretching flat sheet of a nanofluid is investigated.

The report is mainly reproduction of the analysis and results found in [1]. The main objective is to

investigate the fluid flow, heat transfer, and mass transfer in the boundary layer of a fluid with

nanoparticles impeded in it.

The flow situation is considered as two dimensional boundary layer flow of a nanofluid

over a linearly stretching flat plat. The velocity of the flat plate is = where is a parameterthat signifies the rate of stretching of the flat plate. is the streamwise coordinate measured fromthe center of the plate where its velocity is zero. Figure 1 shows a schematic of the geometry

considered.

Figure 1. Geometry of the problem taken from Fig. 1 In Ref [1].


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The stretching flat plate is at temperature which is not known a priori. Instead, is theresult of a convective heat transfer process from below. This convection process is characterized

by a fluid temperature and a heat transfer coefficient . The temperature of the fluid far awayfrom the plate is

. The nanoparticle volume fraction at the flat plate is

while at far vertical

distant form the plate the volume fraction is .2. Mathematical Model

The governing equations for the problem described in the last sections are: continuity,

momentum, energy, and concentration. These equations are reduced for the case of two

dimensional laminar boundary layer to give [1]:

where

and

are the velocity components along the

and

directions, respectively,

is the

kinematic viscosity of the base fluid, is the thermal diffusivity of the base fluid, =/ is the ratio of nanoparticle heat capacity to the base fluid heat capacity, is theBrownian diffusion coefficient, is the thermophoretic diffusion, is the fluid temperature,and is the nanopartilces volume fraction.

+ = 0 (1)

+ = 2 2 (2)

+ = 22 + + 2 (3)

+ = 22 + 22 (4)


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The boundary conditions for the equations 1-4 is given by:

The system of equations 1-4 with the boundary conditions 5, 6 permit similarity

transformation. Exploiting this fact we define the following dimensionless quantities:

Using the similarities variables in Eq.7, Equations 1-4 becomes:

The boundary conditions in Equations 5 and 6 can be transformed to:

The dimensionless parameters appearing in Equations 9-11 are given by:

@ = 0 , = , = 0,

= , = (5)

, = 0, = 0, = , = (6)

= , = , = , = (7)

+ 2 = 0 (8) + + +2 = 0 (9)

+ + = 0 (10)

@ = 0 , = 0, = 0, = [1 ], = 1 (11)

, = 0, = 0, = 0 (12)


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Where is the Prandtl number,is the Lewis number, is Brownian motion parameter, is thermophoresis parameter, and is the Biot number.3. Solution Method

Equations 8-10 subjected to the boundary conditions given in Equations 11,12 were solvednumerically using MATLAB built in function bvp5c. It is worth to notice that Eq. 8 with

corresponding boundary conditions has an analytical solution, hence Eq.8 need not to be solved

numerically. The analytical solution is given by:

bvp5c does not accept second order differential equations, hence the Equations 9,10 need

to be converted to system of first order differential equations. To do this, we define new variables

as:

Using the definitions in Eq. 15 Equations 9 and 10 becomes:

= , = , = ,

= , =

2

(13)

= 1 (14)

= , 2 = , 3 = , 4 = (15)

(

2

3

4 ) = 2[1 2 +24 +22]

4

[1 4 +/ {[1 2 +24 +22]}] (16)


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The boundary conditions in Equations 11 and 12 can be transformed to:

4. Result Comparison

In this section, results obtained using MATLAB bvp5c function is compared with the

results given in [1].

Tables 1 through 4 are reproduction of Tables 1 through 5 in [1]. Table 3 here is equivalent to

Tables 3 and 4 in [1].

@ = 0 , 2 =1 , 3 = 1 (17)

, = 0 , 3 = 0 (18)

Ref [1] bvp5c0.07 0.0656 0.06557

0.20 0.1691 0.16908

0.70 0.4539 0.45391

2.00 0.9114 0.91135

7.00 1.8954 1.89540

20.00 3.3539 3.35390

70.00 6.4622 6.46219

0[1] 0bvp5c 0[1] 0bvp5c0.1 0.1 0.9524 0.95232 2.1294 2.12939

0.2 0.1 0.5056 0.50556 2.3819 2.381860.3 0.1 0.2522 0.25215 2.4100 2.41002

0.4 0.1 0.1194 0.11941 2.3997 2.39965

0.5 0.1 0.0543 0.05425 2.3836 2.38357

0.1 0.2 0.6932 0.69298 2.2740 2.27371

0.1 0.3 0.5201 0.52003 2.5286 2.52822

0.1 0.4 0.4026 0.40257 2.7952 2.79476

0.1 0.5 0.3211 0.32106 3.0351 3.03479

Table 1. Comparison of results for 0, = = = 0, =

Table 2. Comparison of results for 0and 0 , = 10, = 10 =


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0[1]

0bvp5c

0[1]

0bvp5c

0.1 0.1 0.0929 0.09291 2.2774 2.27742

0.2 0.1 0.0927 0.09273 2.2490 2.24896

0.3 0.1 0.0925 0.09255 2.2228 2.22282

0.4 0.1 0.0923 0.09234 2.1992 2.19920

0.5 0.1 0.0921 0.09213 2.1783 2.17835

0.1 0.2 0.0873 0.08733 2.3109 2.31094

0.2 0.2 0.0868 0.08676 2.3168 2.31682

0.3 0.2 0.0861 0.08612 2.3261 2.32607

0.4 0.2 0.0854 0.08539 2.3392 2.33924

0.5 0.2 0.0845 0.08454 2.3570 2.35704

0.1 0.3 0.0769 0.07688 2.3299 2.32995

0.2 0.3 0.0751 0.07508 2.3569 2.35693

0.3 0.3 0.0729 0.07292 2.3900 2.38997

0.4 0.3 0.0703 0.07027 2.4303 2.43031

0.5 0.3 0.0700 0.06697 2.4792 2.47923

0.1 0.4 0.0597 0.05966 2.3458 2.34583

0.2 0.4 0.0553 0.05535 2.3903 2.39025

0.3 0.4 0.0503 0.05027 2.4411 2.44114

0.4 0.4 0.0445 0.04455 2.4967 2.49667

0.5 0.4 0.0386 0.03860 2.5529 2.55287

0.1 0.5 0.0383 0.03833 2.3560 2.35604

0.2 0.5 0.0325 0.03250 2.4071 2.40711

0.3 0.5 0.0269 0.02690 2.4576 2.45763

0.4 0.5 0.0220 0.02199 2.5039 2.50386

0.5 0.5 0.0180 0.01801 2.5435 2.54351

0[1] 0bvp5c 0[1] 0bvp5c 0[1] 0bvp5c1 0.1 5 0.0789 0.07893 1.5477 1.54768 0.2107 0.21069

2 0.1 5 0.0806 0.08062 1.5554 1.55543 0.1938 0.19382

5 0.1 5 0.0735 0.07345 1.5983 1.59832 0.2655 0.26549

10 0.1 5 0.0387 0.03868 1.7293 1.72929 0.6132 0.61319

5 1.0 5 0.1476 0.14756 1.6914 1.69138 0.8524 0.85244

5 10.0 5 0.1550 0.15499 1.7122 1.71223 0.9845 0.98450

5 100 5 0.1557 0.15565 1.7144 1.71438 0.9984 0.99844

5 5 0.1557 0.15572 1.7146 1.71462 1.0000 0.999985 0.1 10 0.0647 0.06469 2.3920 2.39198 0.3531 0.35313

5 0.1 15 0.0600 0.05999 2.9899 2.98994 0.4001 0.40009

5 0.1 20 0.0570 0.05704 3.4881 3.48815 0.4296 0.42959

Table 4. Comparison of results for 0, 0, and 0, = = . 5

Table 3. Comparison of results for 0and 0 , = 10, = 10 = .1


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Table 1 shows results pertain to circumstances when the stretching sheet is at a constant

temperature and the Brownian and the thermophoresis effects are absent. Hence, table 1 represents

limiting cases that is used to test the capability of the numerical solver to handle singular boundary

conditions. As can be seen from Table 1, the results obtained matches Ref [1] to four digits. Notice

also that the wide range of Prandtl number considered in Table1 test the solver capability to solve

for different thermal boundary layer thicknesses. This requires different integration limits

depending on the Prandtl number. Table 2 shows another limiting case for the temperature

boundary condition while keeping the mass transfer physics intact. Again the comparison shows

great agreement. Notice that for some combinations of and the degree of agreement is onlyto three digits instead of four. Tables 3 and 4 represents results for regular parameter values. The

cases considered have wide range of variation for all the parameters affecting the solution. The

comparison shows great agreement between the current results and Ref [1].

Figures 2 and 3 are reproduction of Figures 3 and 7 in Ref [1] respectively. These figures

show the variation of the dimensionless temperature and nanoparticle concentration within the

boundary layer with Lewis number as a varying parameter. Figures 2 and 3 show exact behavior

and trend as their counterparts in Ref [1].

Figures 4 and 5 are reproduction of Figures 4 and 8 in Ref [1] respectively. These figures

show the variation of the dimensionless temperature and nanoparticle concentration within the

boundary layer with Biot number as a varying parameter.

Works Cited

[1] A. Aziz and O. D. Makinde, "Boundary layer flow of a nanofluid past a stretching sheet with aconvective boundary condition,"Int. J. Therm. Sci., vol. 50, pp. 1326-1332, 2011.


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Figure 2.Effect of on temperature profiles. = = . 1 , = 5 , = . 1. Reproduction ofFigure 3 In Ref [1].

Figure 3.Effect of on concentration profiles. = = . 1 , = 5 , = . 1. Reproduction ofFigure 7 In Ref [1].


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Figure 4.Effect of on temperature profiles. = = . 1 , = 5 , = 5 . Reproduction ofFigure 4 In Ref [1].

Figure 5.Effect of on concentration profiles. = = . 1 , = 5 , = 5 . Reproduction ofFigure 8 In Ref [1].

Boundary Layer Flow of a Nanofluid past a Stretching Sheet

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