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0Propulsion and Power Research 2016;5(4):267-278
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ELSEVIER
ORIGINAL ARTICLE
Stefan blowing effect on bioconvective flow of ^ nanofiuid over a solid rotating stretchable disk
CrossMark
N.A. LatiffM.J. Uddin, A.I. Md. Ismail
School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia
Received 4 August 2015; accepted 26 October 2015 Available online 2 December 2016
KEYWORDS
Stefan blowing; Bio-nanofluid; Stretching disk; Rotating disk; Heat transfer
Abstract A mathematical model for the unsteady forced convection over rotating stretchable disk in nanofluid containing micro-organisms and taking into account Stefan blowing effect is presented theoretically and numerically. Appropriate transformations are used to transform the governing boundary layer equations into non-linear ordinary differential equations, before being solved numerically using the Runge-Kutta-Fehlberg method. The effect of the governing parameters on the dimensionless velocities, temperature, nanoparticle volume fraction (concentration), density of motile microorganisms as well as on the local skin friction, local Nusselt, Sherwood number and motile microorganisms numbers are thoroughly examined via graphs. It is observed that the Stefan blowing increases the local skin friction and reduces the heat transfer, mass transfer and microorganism transfer rates. The numerical results are in good agreement with those obtained from previous literature. Physical quantities results from this investigation show that the effects of higher disk stretching strength and suction case provides a good medium to enhance the heat, mass and microorganisms transfer compared to blowing case. © 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Rotating flows have applications in many industries and natural phenomena. Studies on rotating disk problem rely on the solution to the governing equations. Von Kärmän [1]
"Corresponding author. Tel.: (+604) 6533284.
E-mail address: nuramalinalatif@gmail.com (N.A. Latiff).
Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.
was the first to study steady flow over a solid rotating disk by an integral method [2]. Cochran and Goldstein [3] improved Von Karman [1] solution by using patching two series solution. Accordingly, model was improved by Benton [4] who extended the hydrodynamic problem to flow starting impulsively from rest. A study by Bodonyi [5] has found the unsteadiness in rotating disk problem is caused by impulsively rotating disk. Pop [6] studied two-dimensional unsteady flow due to eccentric disk rotation for
http://dx.doi.org/10.1016/j.jppr.2016.11.002
2212-540X © 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd74.07).
Nomenclature
■f Cp
db Dn DT
f (n) fw
Le Lb mw Nb
Nd Nt Nur
chemotaxis constant (unit: dimensionless) nanoparticles volume fraction (unit: dimensionless) fluid nanoparticles volume fraction (unit: dimensionless) local skin friction coefficient in r-direction local skin friction coefficient in e-direction wall nanoparticles volume fraction (unit: dimensionless) ambient nanoparticle volume fraction (unit: dimensionless)
Brownian diffusion coefficient (unit: m2/s) microorganism diffusion coefficient (unit: m2/s) thermophoretic diffusion coefficient (unit: m2/s) dimensionless axial stream function (unit: dimensionless) Stefan blowing parameter fw — (Cf — CM ) /2 (1 — Cw ))
dimensionless circumferential stream function (unit:
dimensionless)
mass transfer coefficient
Lewis number [Le — a/DB)
bioconvection Schmidt number (Lb — a/Dm)
surface mass flux (unit: W/m2)
Brownian motion parameter (Nb — tDb AC/ a
(unit: dimensionless)
Biot number (Nd — (hm/DB) (unit: dimensionless)
thermophoresis parameter (unit: dimensionless) local Nusselt number (Nur — rqw /kf (Tw — Tw)) (unit: dimensionless)
number of motile microorganism (unit: dimensionless) wall motile microorganisms (unit: dimensionless) fluid pressure
bioconvection Péclet number (Pe — bWc/v) (unit: dimensionless)
Prandtl number (Pr — v/a) (unit: dimensionless surface microorganism flux
^(v (1 — fit)/a)
(Nt — tDt AT /T ^ a
qw surface heat flux (unit: W/m )
Qnf local wall motile microorganism number
{Qnf = rq„ /Dnnw)
f dimensional coordinate along the plate (unit: m)
Ref local Reynolds number lRef — Q~f2/v)
S unsteadiness parameter (S — P/Q)
Shf local Sherwood number (Shf — fmw/DB(Cw — CM))
t time (unit: s)
T nanofluid temperature (unit: K)
Tf fluid temperature (unit: K)
Tw wall temperature (unit: K) Tambient temperature (unit: K)
uf velocity components along the f-axis (unit: m/s)
uz velocity components along the z-axis (unit: m/s)
ug velocity components along the g-axis (unit: m/s)
Wc maximum cell swimming speed (unit: m/s)
z dimensional coordinate normal to the plate (unit: m)
Greek letters
effective thermal diffusivity (unit: m2/s) disk stretching parameter (unit: dimensionless) constant (unit: dimensionless)
dimensionless number of motile microorganism (unit: dimensionless)
dimensionless nanoparticles volume fraction (unit: dimensionless)
independent similarity variable (unit: dimensionless)
dynamic viscosity of the fluid (unit: kg/(m • s))
dimensionless temperature (unit: dimensionless)
nanofluid density (unit: kg/m3)
ratio of the effective heat capacity of the nanoparticle
material to the fluid heat capacity (unit:
dimensionless)
skin friction in f-direction (unit: Pa) skin friction in g-direction (unit: Pa) kinematic viscosity (unit: m2/s) stream function (unit: m2/s)
$ (n) n
Tr Te v V
which both the disk and fluid are at rest. Erdogan [7] next extended Pop [6] and suggested three-dimensional problem which considered the problem involving both disk and fluid at rest. He proposed that both the disk and fluid at infinity are initially rotating about z-axis with the same angular velocity. Recently, a series of studies rely on rotating disk was investigated by Turkyilmazoglu [8,9] and [10]. Tur-kyilmazoglu [8] and [9] first focuses on the three dimensional steady flow of an electrically conducting fluid over a rotating stretchable disk with a uniform magnetic field. Next, Turkyilmazoglu [10] extended his previous researches by using water-based nanofluids into the model. Asghar et al. [11] used the Lie group analysis to study three dimensional of viscous fluid on a stretchable rotating disk in radial direction. Sheikholeslami [12] investigated MHD flow and heat transfer with the combined effects of different types of nanofluids and injection/suction parameter between two parallel plates in a rotating system. Srinivas et al. [13] applied the method of HAM to study the unsteady MHD
viscous fluid flow between expanding/contracting rotating porous disks with viscous dissipation. They found that the percentage to increase in dimensionless temperature is much higher for the combined effect of expansion disk with injection compared to contraction disk with injection. Dandapat et al. [14] studied uniform transverse magnetic field on unsteady two-layer film flow over a rotating disk under planar interface assumption. Sheikholeslami et al. [15] considered the three dimensional steady flow of condensation film on inclined rotating disk to study the effect of nanofluid spraying for cooling process. Studies on three-dimensional unsteady rotating disk include those of Hobiny et al. [16], Watson and Wang [17], Rashidi et al. al [18], Sparrow and Cess [19] and Khan et al. [20].
Interest in the use of nanofluids to enhance the flow, heat and mass transfer has developed significantly among many researchers. The term 'nanofluids' was first suggested by Choi [21] to describe the pure fluids with suspended nanoparticles. He defined nanofluid as a fluid containing
nanoscale-sized particles between 1 and 100 nm in the base fluids (water, oils and ethylene glycol) [22]. Boungiorno [23] introduced the effects of Brownian diffusion and the thermophoresis into conservation equations to describe the behavior of nanofluids. Next, Nield and Kutnetsov [24] introduced the idea of Buongiorno's model and this idea was further developed by many researchers in various application involved nanofluids. Recently, Rohni et al. [25] studied the flow and heat transfer using Buongiorno's model for a nanofluid shrinking sheet with suction. Zaimi et al. [26] studied the unsteady flow in contacting cylinder over nanofluid. Ahmadi et al. [27] analyzed the flow and heat transfer with a stretching flat plate in a nanofluid and Ro§ca and Pop [28] investigated the flow characteristics from a moving surface in an external uniform free stream of nanofluid. Hajmohammadi et al. [29] studied the flow and heat transfer of nanofluids past a permeable surface with convective boundary condition and they found that the increases of nanoparticles volume fraction with injection surface enhance the heat transfer rate and skin friction. They concluded that in the case of permeable surface the types of nanoparticles play an important role to increase the heat transfer rate while in the case of impermeable surfaces the type of nanoparticles is insignificant.
In many industrial applications (extrusion of plastic sheet, paper manufacture, glass blowing etc.), information of flow, heat and mass transfer process occurred between stretching surface and fluid flows are required. In addition, we note that an interesting situation (called blowing effect) where extensive diffusion of mass from the stretching surface to the ambient (e.g. evaporation in paper drying process) can occur. This situation occurs due to temperature difference and water content of the wet paper sheet. The theory of blowing effect can be adopted from Stefan problem. Therefore, we consider Stefan blowing effect in this study which is different from permeable surface in mass injection or blowing due to transpiration [30]. Previously, many researchers have investigated various flow configurations involving the blowing effect [30-33] for regular fluids. We believe that there is no study on unsteady rotating stretchable disk with Stefan blowing effect for any kind of fluids and this motivates our present study.
Bioconvection is due to motile microorganism being generally heavier than water so that they are likely to swim in upward direction and cause an unstable top heavy density stratification. Thermo-bioconvection plays an important role in geophysical phenomena e.g. in hot springs colonized by motile micro-organism named thermophiles i.e. heat loving microorganisms [34]. Another application is in the field of microbial enhanced oil recovery where microorganisms and nutrients are inserted in oil-bearing layers to adjust permeability variation. Accordingly, Nield and Kuznetsov [35] suggested that microorganisms may contribute toward the improvement in biomicrosystems where they play an important role in mass transport enhancement and mixing. A series of analysis conducted by Kuznetsov [36], Kutnet-sov [37] and Kuznetsov [38] found that the insertion of
gyrotactic microorganisms into nanofluids tend to increase nanofluids stability as a suspension. Therefore, combining both advantages of nanofluid and microorganism (bio-nanofluids), the heat transfer rate is anticipated to improve even more extensively. Recently, many researchers investigated the problems which involved bioconvection in nanofluids. They include Kutnetsov [37], Tham et al. [39], Xu and Pop [40] and Xu [41].
To the authors' knowledge so far no studies have been reported which address the combined effects of unsteady rotating stretchable and impermeable disk by including the blowing effect. Motivated by previous investigations and applications, in this paper we propose such a study by extending the work of Fang and Tao [42] to investigate the flow, heat, mass and microorganism transfer. It important to note that due to the blowing effects from mass transfer, the momentum and concentration equations are coupled and should be solved simultaneously. The coupled nonlinear partial differential equations governing the flow, heat and mass transfer are reduced to a set of coupled nonlinear ordinary differential equations by using similarity transformations. The reduced equations are then solved numerically using the Runge-Kutta fourth-order integration scheme in association with the shooting method. The effect of various physical parameters on the velocity, temperature, concentration and microorganism fields is studied.
2. Mathematical formulations of the problem
Consider the unsteady motion of a bioconvection, three-dimensional, viscous incompressible nanofluid over a rotating impermeable disk. A schematic of the physical configuration and the cylindrical polar coordinate system r, 0 and z is shown in Figure 1. The flow is through a rotating impermeable disk where the Stefan blowing model is assumed. The surface of the plate is subjected to boundary conditions with mass convective and Stefan blowing effect in w-velocity conditions. Stefan blowing effect considered in this study deals with impermeable surface which is different from permeable surface in mass injection or blowing due to transpiration [30]. It is also assumed that the condition at the vicinity to the disk is Cf >Cw Therefore this situation is called mass
Figure 1 Physical diagram.
convective boundary condition when there exists different concentration from the vicinity to the disk concentration, wall concentration and ambient concentration which provides a mass transfer coefficient hm.
With the above mentioned assumptions, the boundary layer equations governing the conservation of mass, momentum in r, 0 and z direction, energy, nanoparticles volume friction and density of motile microorganisms in cylindrical coordinates reads [42,24]:
dû- uF duz
1F + = + ^ = 0; or r dz
d uF _ d uF _ d uF
—— + u- —— + uz ——
d t d r d z
1 d P p d -
(d2uf d2uF 1 d u
+v + -r-2 + —:
\o r2 d z2
dug _ dug _dug uF ug
— + u,- — + w—--—
dt df dz r
d2ug df"
1 dug r df
dûz _ ~df + u
+v ■
duz dû-z
--- + uz-
- df z dz
' d2uz d2uz 1 1
"df2 + 'dz2 4
1 du-z r dr
is the fluid density, a is the fluid thermal diffusivity, t is the ratio of heat capacity of nanoparticle and heat capacity of fluid, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, b is the chemotaxis constant, Wc is the maximum cell swimming speed and Dn is the microorganism diffusivity coefficient. In this study, we assumed the disk rotation speed has a form of ug (r, 0,0) = Q/(1 — fît) and the disk stretching velocity is proportional to the radius defined as uF (r, g, 0) = acQ -/(1 — fît). It is important to mention that such a disk rotation velocity was introduced by Watson and Wang [17]. Thus, the relevant boundary conditions are [42,43,33,44]:
u-f (r, g, 0)= fcQI, ug (r, g, 0)= ^,
uz (-, g, 0)= -
= — Djw ¡dC
§ , T (-, g, 0) = Tw
— Db f = hm (jf — Jw), n (-, g, 0) = nw,
Ur (r, 0,1) — 0, U0 (r, 0,1) — 0, T (r, 0,1) — Ti, C (r, 0,1) = C 1, n (r, 0,1) — ,
where Q is the angular velocity, ft is a constant (if ft — 0, the problem reduces to steady rotation disk case and if ft a 0, the problem reduces to unsteady rotation disk case), ac is the strength of disk stretching, the subscript f, w and 1 represent conditions at the fluid, wall and ambient fluid, respectively; and hm is the convective fluid concentration.
2.1. Similarity transformations
dT _ dT _ dT
— + u- — + uz — = a dt dr dz
£t 1 dT
df2 - df
dTdJ dTdJ\ DT
dr dr dz dz I Tm
—, 2 dT\
dJ _ dJ _ dJ
+ u---h uz — = DB
dt dr dz
1 dJ - df
d2J Hz2
DT id2 T
+Tm {d-2'
1 dT - dr
dn _ dn _ dn lb Wc
— + u---h uz--h ■=—
dt dr dz Jœ
d (ndJÇ)
dz\ dz
d2n 1 dn dr2 r df
d2n ~dz2
are the
Here, the velocity vector [V =[uF, ug, uz velocity components along -, g and z directions, respectively; t is the time, T, J and n are the temperature, nanoparticle volume fraction and microorganisms respectively; v is the kinematic viscosity, P is the fluid pressure, p
Following Fang and Tao [42], we adopt the following transformations:
_ Qr _ Qr
, "-F = y—fîtf (n), ug = fîtg (n),
— 2sfQv
V^—fît
f (nX g = T — T1 ,
Tw — T
J Ji * = J - J ,
X nw '
Substituting Eq. (9) into Eqs. (2)-(7) and boundary conditions (8), we finally obtained the following system of ordinary differential equations:
f" + 2ff" —f2 + g2 — s(f + 2 nf ^ = 0, (10)
g" + 2fg — 2gf — ^g + ^ ng') = 0,
g" + Nb4>'g' + Nté2 — - sp^g' + 2Pf& = 0, (12)
1 ^tV Nt g" — - sn* + 2f*' = 0,
LePr^ LePrNb 2
^X" - 1 Snx' + 2fX - Pe \x$" + $ x] = 0. (14)
The corresponding boundary conditions in Eq. (8) become:
f (0) = «, g (0)=1, f (0)= Lw $ (0); 0 (0)=1,
$ (0)=-Nd \1 - $(0)], x(1)= 1, f (1) = 0, g (1) = 0, 0 (1) = 0, $ (1) = 0, x (1) = 0. (15)
The dimensionless parameters are S = ^ (unsteadiness parameter), Pr = a (Prandtl number), Nb = (Brow-
nian motion), Nt = (thermophoresis), Le = (Lewis number), Lb = —- (bioconvection Lewis number), Pe =
—2 ' PfU
Cfg —
—2 ' Pfu0
Nu- — ,_ Jw_
kf (Tw — T i)
DB (Cw — CM)
where , Tg , qw , mw and qn represent the shear stress in r-direction, shear stress in g-direction, surface heat flux, surface mass flux and the surface motile microorganism flux respectively. These quantities are defined as follows:
du-f du-^ [dug du-
oz or — 0 \_dz or
\dT 1 \dCl
(bioconvection Péclet number), fw — (Cf—Cl)
T- — ^
qw — —k qn— —D
(Stefan
mw — — Db
z — 0
z — 0
z — 0
2 (1 - Cw )
blowing parameter), a (disk stretching parameter) and
Substitute Eqs. (9) and (17) in Eq. (16),
- hm , VL
fw represent for mass transfer from the rotating disk to the free stream (i.e. evaporation) and negative value of fw represent for mass transfer from the free stream to the rotating disk wall (Fang and Tao [42]). In short, for fw>0, it shows mass blowing at the rotating disk wall and for fw<0, it turn into mass suction. Meanwhile, the value of S>0 indicates the acceleration and S<0 shows the deceleration of the rotating disk. However, results presented in this study focused on the deceleration of the rotating disk.
2.2. Physical quantities
The important physical quantities from an engineering point of view which describe the fluid flow characteristics in r-direction, skin friction in g-direction, heat, mass and microorganism transfer rate are the local skin friction r-direction, local skin friction in g-direction, the local Nusselt number, local Sherwood number and the local wall motile microorganism number. These are, respectively, defined by:
(Biot number). Note that positive value Re1=2 f (1 - fit) 1/2 = f'"(0)
>- V2 _ rm,
Re1/2 Cf e (1 — ßt) —1/2 — g (0) , Re—1/2 Nu- (1 — ßt) —1/2 — — e (0) , Rer—1/2 Sh- (1 — ßt) —1/2 — — 4>' (0) , Re1/2 Qn- (1 — ßt) —1/2 — —X (0):
Here, Re- — is the local Reynolds number.
2.3. Numerical solutions and validation
The analytical solution from Eqs. (10)—(14) with boundary conditions Eq. (15) may not be readily obtained and hence we have to solve it numerically. The set of nonlinear ordinary differential have been solved numerically by employing the Runge-Kutta-Fehlberg fourth-fifth (RKF45) order numerical method available in the symbolic code, Maple 17, taking S, Pr, Nb, Nt, Le, Lb and Pe as controlling parameters. Note that, to have a true similarity solution, all parameters must be free from original independent variables. Therefore, in order to validate our result, the values of f" (0) and g 0 (0) were compared
Table 1 The calculated values of f" (0) and g (0) from the present model in the absence of energy, concentration and microorganism
equation.
S (a — 0.0) (a — 1.0)
f " (0) g (0) f "(0) g (0)
Shooting [42] RK45 (Present) Shooting [42] RK45 (Present) Shooting [42] RK45 (Present) Shooting [42] RK45 (Present)
— 0.1 0.5308 0.5287 — 0.5789 — 0.5779 —0.9189 —0.9191 — 1.4656 — 1.4656
— 0.2 0.5515 0.5496 — 0.5416 — 0.5407 — 0.8894 — 0.8896 — 1.4441 — 1.4442
— 0.5 0.6143 0.6132 — 0.4284 — 0.4280 — 0.8007 — 0.8008 — 1.3797 — 1.3797
—1.0 0.7198 0.7196 — 0.2366 — 0.2365 — 0.6520 — 0.6520 —1.2716 — 1.2716
— 2.0 0.9315 0.9315 0.1550 0.1550 —0.3517 —0.3517 — 1.0534 — 1.0534
— 5.0 1.5627 1.5628 1.3609 1.3609 0.5632 0.5632 — 0.3882 0.3882
-10.0 2.6008 2.6008 3.4139 3.4139 2.1153 2.1153 0.7424 0.7424
— 20.0 4.6464 4.6464 7.5796 7.5799 5.2706 5.2706 3.0464 3.0463
with the values of Fang and Tao [42] (Table 1). The table shows the comparison our result for values of f" (0) and g (0) with various disk stretching parameter values when the energy, concentration and micro-organism equation are neglected. The comparison results show excellent agreement among data. Thus the set up numerical code for current model is justify.
3. Results and discussions
In Figures 2-5, the flow characteristics under different controlling parameters are shown. Profiles of f(n), g (n), — 0(n), — $ (n) and — x (n) represent the radial velocity, circumferential velocity, temperature, nanoparticle volume
Figure 2 Effect of unsteadiness (S) and blowing/suction fw) parameters on the (a) radial velocity (b) circumferential velocity, (c) temperature, (d) nanoparticle volume fraction, and (e) microorganism profiles.
Figure 3 Effect of the strength of disk stretching (a) and blowing/suction (fw) parameters on the (a) radial velocity (b) circumferential velocity, (c) temperature, (d) nanoparticle volume fraction, and (e) microorganism profiles.
fraction and microorganism respectively. Figure 2(a)-(e) show the profiles of radial and circumferential velocity, temperature, nanoparticle volume fraction and microorganism respectively when all other parameters are kept constant for different values of the unsteadiness parameter S and suction/blowing parameter fw. It important to mention that the value of S> 0 indicates acceleration and S< 0 shows the
deceleration of the rotating disk. However, results presented in this study focus on the deceleration of the rotating disk. In Figure 2(a), the radial velocity near the surface disk increases in suction case as |S| increases. However, the radial velocity far away from surface disk decreases due to less fluids being pumped toward the disk. The circumferential velocity decreases with n in blowing case as |S|
Figure 4 Effect of strength of disk stretching (a) and blowing/suction fw) parameters on the (a) radial local skin friction, (b) circumferential local friction, (c) local Nusselt numbers, (d) Sherwood number, and (c) motile microorganisms number for different unsteadiness parameter.
decreases (Figure 2(b)). However, as |S| increasing (that is, fast deceleration), it causes the fluid circumferential velocity to overshoot and the circumferential fluid velocity near the surface disk to become higher than the disk rotational speed. The same trend of circumferential velocity was observed by Fang and Tao [42] and Watson and Wang [17]. In Figure 2(c), the thermal boundary layer thickness
increases in blowing case as | S| decreasing (slow deceleration). Next an interesting phenomenon can be observed in Figure 2(d) when the nanoparticle volume fraction near the surface disk decreases as the value of fw and |S| increasing. As fw increases, an nextensive diffusion of mass from the stretching surface to the ambient occurs. Hence, it affects the nanoparticle volume fraction to decrease near the
Figure 5 Effect of Biot number (Nd) and blowing/suction (fw) parameters on the (a) radial local skin friction, (b) circumferential local friction, (c) local Nusselt numbers, (d) Sherwood number, and (c) motile microorganisms number for different unsteadiness parameter.
surface and increase far away from the surface disk. Next, Figure 2(e) shows that the microorganism boundary layer thickness increases in the blowing case as |S| decreases. Therefore, it can be concluded that the combined effects of increasing blowing parameter and increasing value of | S| cause the radial and circumferential velocity profile to decrease. The combined effects of blowing parameter and
decreasing value of | S| will affect the temperature, nano-particle volume fraction near the surface disk and causes the microorganism profile to increase.
The profiles of f (n) , g (n) , — g (n) , — (n) and — x (n) under slow deceleration S — —0.5 for the different disk stretching parameters and suction/blowing parameters are shown in Figure 3(a)-(e). It can be observed in Figure 3
(a) that when the disk stretching is increasing, the radial velocity profile decreases and hence it decays to zero for both suction and blowing case. It is noted that when the disk start to stretch, more fluids are pumped into the disk and hence radial velocity increase. Also, the radial velocity boundary layer thickness near the surface disk is increased for the case fw > 0. In Figure 3(b), the circumferential velocity increases as the blowing parameter increases with no disk stretching effect (a — 0). This situation occurs due to fast deceleration. Next, the trends show in Figure 3(c)-(e) represented for temperature, nanoparticle volume fraction and microorganism which have common characteristics to those discussed in Figure 2(c)-(e). But the difference in this case is mainly due to stretching. Therefore, it can be concluded that the combined effects of increasing blowing parameter and increasing value of a, causes the radial velocity profile to increase. In comparison, Srinivas et al. [13] found that the percentage to increase in dimensionless temperature is much higher for the combined effect of expansion disk with injection compared to contraction disk with injection. However, in this study we investigated the Stefan blowing effects and found that the combined effects of blowing parameter with non-stretching disk affect the circumferential velocity, temperature, nanoparticle volume fraction near the surface disk and microorganism profile to increase.
The variation trend of the local skin friction in r-direc-tion, local skin friction in 0-direction, local Nusselt number, Sherwood number and number of motile microorganism with various effects of disk stretching strength, blowing/ suction and steady/unsteady cases are shown in Figure 4 (a)-(e). From Figure 4(a) and (b), it is seen that for a given S and fw parameter, both f" (0) and g (0) decreases with the increase of a. It is interesting to note that there exist f" (0) — 0 for both suction and blowing when a — 0 which leads to a frictionless stretching disk. The temperature and microorganism gradient at the wall is proportional to the heat transfer flux and microorganism transfer flux respectively. The — 0 (0) and — x (0) increase with the increase of a and blowing parameter (see Figure 4(c) and (e)). However in Figure 4(d), the steady case shows the values of — <p' (0) remain constant as a increases. From Figure 4(c)-(e), it is clearly shown the unsteady state provide a medium to enhance the heat, mass and microorganism transfer.
Next, Figure 5(a)-(e) show the effects of Biot number and blowing/suction parameter on the local skin friction in r direction, local skin friction in 0-direction, Nusselt number, Sherwood number and microorganism number respectively. From Figure 5(a) and (b), as Nd and suction increases, the local skin friction in both direction decreases. As Nd and blowing increases, the local skin friction tend to increase. However, as Nd increases, the value of skin friction at the wall remains constant when fw — 0 (no suction/blowing). The wall temperature gradient, as shown in Figure 5(c) is increased with an increase in the suction and Nd parameter. Also it seen that the wall temperature gradient always remains constant in the case of no suction/blowing. Next Figure 5(d)-(e) show the Nd parameter significantly affect
the mass and microorganism transfer. From the wall mass and microorganism gradient plots, it can be observed that the mass and the microorganism transfer increases as Nd increase. Also, noticed that increasing the blowing parameter reduces the mass and microorganism transfer.
4. Conclusions
Our focus in this study is to investigate an unsteady rotating stretchable and impermeable disk by including the Stefant blowing effect. Similarity transformation technique is used to convert the three dimensional of the momentum, temperature, nanoparticle volume fraction and microorganism equations into a set of ordinary differential equations which were solved numerically using the Runge-Kutta method. The effect of the governing parameters on the flow, temperature, nanoparticle concentration and density of motile microorganisms are described. The flow, heat and microorganism transfer are presented graphically to clarify the influence of the different physical parameters including suction/blowing parameter (fw), disk stretching strength parameter (a) and unsteadiness parameter (S<0). Based on the numerical results, the important phenomena observed are:
• Increasing of disk stretching strength (a) leads to a reduction in the local skin friction and enhance the heat transfer, mass and microorganism transfer.
• In the case of suction fw > 0, the local skin friction tends to decrease and the heat transfer, mass transfer and microorganism transfer tend to increase.
• In the case of blowing fw>0, the local skin friction tends to increases and the heat transfer, mass transfer and microorganism transfer tend to decrease.
For future development, this model can be extended by accounting for variable fluid properties and the nonlinear effects including chemical reaction, magnetohydrodynamic (MHD), viscous dissipation ([45,46]) and radiation. Future work can also be extended by using semi-analytical methods (Hajmohammadi [47], Khan et al. [48], Gul et al. [49], Hajmohammadi [50] and Hajmohammadi and Nourazar [51]) to solve ordinary differential equations with the boundary conditions. In addition, researchers also interested to investigate the effects of nanoparticles in non-regular convective configurations. Therefore, the different configurations include helical coiled tube with constant wall heat flux, various thickness of trapezoidal wall placed between a heat source and a cold fluid upon the hot spot temperature, flow in the pipe when the wall is heated, boundary layer flow over heated segments on plates and trapezoidal solid conducting wall intruded by an isothermal cavity are reported by Ko and Ting [52], Hajmohammadi et al. [53], Hajmohammadi et al. [54], Hajmohammadi et al. [55] and Pouzesh [56] respectively. It can be referred for future work to enhance the heat transfer, mass transfer and
microorganism transfer rate for different non-regular configurations with nanofluid application.
Acknowledgements
The authors acknowledge financial support from Universiti
Sains Malaysia, RU Grant 1001/PMATHS/81125.
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