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MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature
Alexandria Engineering Journal (2017) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature Basant K. Jha, Bello J. Gwandu * Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria Received 27 February 2017; revised 16 August 2017; accepted 30 August 2017
KEYWORDS MHD; Free convection; Superhydrophobic slip; Temperature jump
Abstract This work treats convective heat transfer, for an incompressible electrically-conducting fluid, moving vertically through an isothermally heated parallel plate micro-channel, within a transverse magnetic field. One surface exhibited super-hydrophobic slip and temperature jump, while the other did not. The study aimed at discovering the possible effects of magnetism on the velocity, volume flow rate and Nusselt Number when either wall is heated by constant wall temperature. It was noted that it reduces velocity and flow rate in both cases. It, also, leads to decline in Nusselt Number when the super-hydrophobic side is heated, while heating the other plate showed a negligible magnetic effect, for low temperature jump coefficients. At the critical value of the temperature jump coefficient, c; the flow rates in both cases are equal. Before the critical value ðccrit: Þ is reached, heating the super-hydrophobic plate yields higher flow rate. After passing it, the reverse is the case. The effect of magnetism on ccrit: was, also, negative. The study will have applications in the design and maintenance of both mini- and micro-devices as well as in nano-science and nano-technology. Ó 2017 Faculty of Engineering, Alexandria University. 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 Advancement in technology in line with the needs of people for smaller equipment and lighter appliances turns the attentions * Corresponding author. Permanent Address: Department of Mathematics, Federal University Birnin Kebbi, P.M.B. 1157, Birnin Kebbi, Kebbi State, Nigeria. E-mail addresses: [email protected] (B.K. Jha), Mbj_gwn@ yahoo.com (B.J. Gwandu). Peer review under responsibility of Faculty of Engineering, Alexandria University.
of scientists, engineers and technologists towards researches, experimentations and practical studies in mini-technology, then micro-technology, and later nano-technology. This is what caused researchers in Fluid Mechanics to shift from studying flows in macro-channels to mini-channels, microchannels and nano-channels. Wang and Ng [1] investigated a similar flow to the present study, but for an electrically non-conducting fluid and outside a magnetic field. Other researchers like Chen and Weng [2], Avci and Aydin [3], Panda and Moulic [4], Weng and Chen [5], Bounomo and Manca [6,7] considered convection flows of rarefied gases through some channels, and, kept the two
http://dx.doi.org/10.1016/j.aej.2017.08.022 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. 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/). Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
2
B.K. Jha, B.J. Gwandu
Nomenclature u dimensional upward velocity of the flow (m/s) uslip,wall dimensional slip and wall velocities, respectively (m/s) y0 dimensional horizontal position between the plates (m) g acceleration due to gravity (m/s2) T dimensional temperature of the heated surface (K) Tjump,wall dimensional temperature jump and at wall, respectively (K) T0 dimensional temperature of the surroundings (K) Th dimensional temperature higher than T0 (K) ! E electric field force (N/C or V/m) ! magnetic Lorentz force (T) F ! B magnetic field vector (T) B0 constant magnetic field applied (T) ! J electric current density (A/m2) ! V 3-dimensional velocity vector (m/s) Y dimensionless horizontal position (–) U dimensionless upward velocity (–) Uslip,wall dimensionless slip and wall velocities, respectively (–)
plates under the same jump conditions. This is because the gas slips and temperature jumps depend on coefficients that are the properties of the fluids, thermal conditions of the flows, type of material of the channel and its quality. Authors [6,7] applied numerical methods while the rest used analytical methods. These studies quoted above, are all devoid of magnetic field application. On the other hand, many MHD studies were undertaken, such as Singh et al. [8], Makinde et al. [9], Ellahi et al. [10] and many others. These studies dealt with different flows under different conditions, but through ringed gaps (annuli) or cylindrical pipes (whether horizontal or vertical). There are, also, some studies in MHD Couette flows (where one wall moves relative to the other) or flows in cylindrical channels in which the two concentric cylinders rotate. They include studies like Makinde and Chinyoka [11], Subotic and Lai [12], and Jha and Apere [13]. Though the above studies included the application of magnetic field, attention was not paid to superhydrophobic nature of wall surfaces. Diverse researches were made bordering various kinds of slips, different dispositions of MHD, several forms of heat transfer, dissimilar types of flow, involving special kinds of fluid and with applications of several different methods. The trend continues, especially in recent years. Uddin et. al. [14] used a group theoretic method to find the effects of relevant parameters on the temperature, velocity and nanoparticle volume fraction of a nanofluid in a free convection involving a homogeneous isothermal irreversible chemical reaction. Slip effects were studied in relation to Biology under a special motion of microorganisms in fluid (called bioconvection) by Uddin et al. [15]. The flow was in a porous medium and the microorganisms moved by a combination of taxis and gyration. Uddin et al. [16] incorporated radiation in a nanofluid convection past a shrinking/stretching sheet. A combination
L M Q Nu ^ i
last point of displacement/channel width (m) magnetic component of the system (–) volume flow rate (m3/s) Nusselt number (–) the unit vector in the x (vertical) direction
Greek letters b thermal expansion coefficient (1/K) c dimensionless temperature jump coefficient (–) c0 dimensional temperature jump coefficient (m) m kinematic viscosity of the fluid (m2/s) q density of the fluid (kg/m3) r electrical conductivity of the fluid (A2s3/kg.m3 or S/m) h dimensionless temperature (–) hjump;wall dimensionless temperature jump and at wall, respectively (–) hb bulk temperature (K) k dimensionless slip length (–) k0 dimensional slip length (m)
of algebraic transformation and numerical methods were used to solve the Partial Differential Equations (PDEs). A similar situation, without radiation, was studied by Latiff et al. [17] but applied to Biology. The stretching/shrinking yielded slip effects. They found out that skin friction and Nusselt Number are decreased while microorganisms’ number is increased as the respective relevant slip parameters are increased. Uddin et al. [18] used Lie group algebraic theory in the solution of the Ordinary Differential Equations (ODEs) in their study of multiple slips convection through a Darcian porous medium. They discovered that increase in the magnetic parameter hinders velocity and increasing the permeability parameter favours temperature rise. Magnetohydrodynamic (MHD) slips in velocity and momentum were, also, involved in the heat and mass transfer study of nanofluids by Majeed et al. [19]. Different forms of heat transfer were combined together in the study by Uddin et al. [20]. A combination of algebraic transformation and numerical methods were, also, utilized. In addition to heat transfer, heat generation was, also, considered by Uddim et al. [21]. The stretching of the sheet produced slips. Magnetism was involved, and a numerical method followed the algebraic transformation method. In a related development, Nawaz et al. [22] used Homotopy Analysis Method (HAM) in solving the nonlinear problems that arose when they were studying heat transfer in a Jeffrey fluid flowing on a stretching circular membrane. A comparable work was done by Zeeshan and Majeed [23] with porosity and consideration of dipole effect. Sheikholeslami and Zeeshan [24] considered a constant heat flux in a similar study and used the Control Volume Finite Element Method (CVFEM) to solve the problem. MHD mixed convection was investigated by Zeeshan and Majeed [25] through a non Darcy stretching sheet. Numerical solutions were provided. Another bio-inclined study was performed by Bhatti et al. [26]. It involved peristalsis
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
MHD free convection flow in a vertical slit and numerical integration. A short and useful collection of communicative notes on micro-channel natural convection was released by Cheng and Weng [27]. Jha et al. [28] used an exact solution method in studying slip effects, fluid-wall interaction parameter, wall-ambient temperature difference ratio and rarefaction parameter in an MHD micro-channel natural convection. The basic necessary conditions for the rusting of iron are the presence of air and moisture. Even in mini- and microappliances that do not contain water or some other liquids, heat generation through friction and other means, leads to production of steam, which moistens the surfaces it touches. This made the engineers to think of introducing hydrophobic and even ultra-hydrophobic parts in their constructions and manufacturing. This study, therefore, makes an attempt to apply a transverse magnetic field to the natural convective flow in a super-hydrophobic micro-channel for investigating the outcome of the new combination. The findings will improve services in microchips production factories, oil and gas industries, mini-equipment assembling outfits, hospital instruments sterilization factories and other similar places. 2. Mathematical analysis Let us consider an electrically-conducting fluid moving steadily upward through a vertical parallel plate micro-channel by convection through heating one of the plates at a time. One of the surfaces is extremely difficult to wet (super-hydrophobic) as a result of a special micro-engineering treatment. The other side (no-slip surface) was not tempered with. The super-hydrophobic surface is kept at a position y0 ¼ 0 while the no-slip surface is at y0 ¼ L as shown by Fig. 1. It could be seen in literature such as [24], that the magnetic Lorentz force is described as the vector product of the current ! ! density, J and the magnetic field vector B . That is, ! ! ! F ¼ J B ð1Þ
3 Similarly, [14] defined the current density as ! ! ! ! J ¼ r½ E þ V B
ð2Þ
! where r is the electrical conductivity of the fluid, E is the ! electric field and V is the velocity vector. Combining Eqs. (1) and (2), we have ! ! ! ! ! F ¼ r½ E þ V B B ð3Þ Neglecting the electric field and taking one component each ! ! ! ! of both V and B ; that is, E ¼ 0, V ¼ ðu; 0; 0Þ and ! B ¼ ð0; B0 ; 0Þ; we have ! i ð4Þ F ¼ rB20 u^ The directions of B0 and u are indicated in Fig. 1. Using the ideas of [27,28], the slip velocity and temperature jump are, respectively, defined as follows: uslip ¼ k0
du dU ) Uslip ¼ k dY dy0
Tjump ¼ c0
dT dh ) hjump ¼ c dY dy0
ð5Þ ð6Þ
For velocity slip, as in [15–19], the particle velocity near the boundary is given as: u ¼ uwall þ uslip ) U ¼ Uwall þ Uslip
ð7Þ
Similarly, the temperature near the boundary, like in [15,16,18], is given as: T ¼ Twall þ Tjump ) h ¼ hwall þ hjump
ð8Þ
Here uwall and Twall are, respectively, the dimensional velocity and temperature at the wall. All related dimensionless velocity and temperature are, similarly defined. The Boussinesq buoyancy approximation and boundary conditions used by [1] are considered. Since the fluid considered is viscous, electrically conducting, and in the presence of magnetic field, the MHD approximations are attached to finally get the governing equations for the system (see Section 2.1). The Ordinary Differential Model Equations were solved by exact solution method. The graphs were plotted and interpretations given. Different temperature jump and slip conditions were used for the different plates because the main interest is in the super-hydrophobicity of a surface, not the characteristics of the fluid or the flow. This is in contrast with emphasis on rarefaction of gases which is the nature of the fluid itself. Micro-engineered structures like grooves, notches, holes and cones trap gases in their spaces and become filled, initially. This prevents the liquid from filling the spaces until the capillary pressure is exceeded. The situation causes a hydrodynamic slip leading to mixed boundary conditions. Gases conduct heat much slower than liquids. This leads to temperature jump between the two interfaces (liquid-solid and liquid-gas interfaces). The study was divided into two cases. 2.1. Case I (the super-hydrophobic surface being heated)
Fig. 1 Free convective flow of an electrically conducting fluid through a vertical micro-channel.
The governing equations of the system are as follows [1,14– 19,24,27,28]:
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
4 m
B.K. Jha, B.J. Gwandu d2 u rB2 u þ gbðT T0 Þ 0 ¼ 0 02 q dy
d2 T ¼0 dy02
ð9Þ
ð10Þ
subject to the following boundary conditions: ) du uðy0 Þ ¼ k0 dy 0 at y0 ¼ 0 dT Tðy0 Þ ¼ Th þ c0 dy 0 0
uðy Þ ¼ 0 Tðy0 Þ ¼ T0
Q¼
MSinhðMÞ þ ðkM3 2k 2ÞCoshðMÞ þ 2k þ 2 ð2c þ 2Þ½M3 SinhðMÞ þ kM4 CoshðMÞ
ð24Þ
hb ¼
ðE8 þ 6ÞQ 6E1 E2 E0 Q
ð25Þ
ð11Þ Nu ¼
at y0 ¼ L
Using Eqs. (18)–(20), (22) and (23), the following results were obtained:
ðE8 þ 6ÞQ 6E1 E2 6Q
ð26Þ
ð12Þ 2.2. Case II (the no-slip surface being heated)
Using the following dimensionless variables: ðY; c; kÞ ¼ ðy0 ; c0 ; k0 Þ=L; M2 ¼ rB20 L2 =qm;
h ¼ ðT T0 Þ=ðTh T0 Þ 1 U ¼ um½gbL2 ðTh T0 Þ
ð13Þ
Eqs. (9) and (10), respectively, become: d2 U þ h M2 U ¼ 0 dY2 dh ¼0 dY2
ð14Þ
2
ð15Þ
with the following boundary conditions: ) dh hðYÞ ¼ 1 þ c dY at Y ¼ 0 UðYÞ ¼ k dU dY hðYÞ ¼ 0 UðYÞ ¼ 0
ð16Þ
ð17Þ
The volume flow rate and bulk temperature are, respectively, given by the following equations: Z 1 UðYÞdY ¼ 0 ð18Þ Q¼ 0
R1 hb ¼
0
hðYÞUðYÞdY R1 UðYÞdY 0
ð19Þ
Eqs. (18) and (19) are used in both cases of heating. The Nusselt Numbers at the two walls are given by Eqs. (20) and (21) below. Nu ¼ Nu ¼
dh=dY jY ¼ 0 hb
dh=dY jY ¼ 1 hb
uðy0 Þ ¼ k0
du dU ) UðYÞ ¼ k dYY¼0 dy0 y0 ¼0
ð28Þ
Tðy0 Þ ¼ Th jy0 ¼L ) hðYÞ ¼ 1jY¼1
ð29Þ
uðy0 Þ ¼ 0jy0 ¼L ) UðYÞ ¼ 0jY¼1
ð30Þ
This gives the values of temperature and velocity to be, respectively, as below:
at Y ¼ 1
Eqs. (9) and (10) are the governing equations. Eq. (13) was used to change them into Eqs. (14) and (15). We, now, solve Eqs. (14) and (15), using the following boundary conditions: dT dh ) hðYÞ ¼ c ð27Þ Tðy0 Þ ¼ T0 þ c0 0 dY Y¼0 dy y0 ¼0
ð20Þ ð21Þ
Eq. (20) is used when the super-hydrophobic surface is heated, while Eq. (21) is used when the no-slip surface is heated. The following results for the temperature and velocity were obtained after solving Eqs. (14) and (15) with conditions (16) and (17).
hðYÞ ¼ ðY þ cÞ=ð1 þ cÞ UðYÞ ¼
ð31Þ
E10 Sinh½MðY 1Þ þ E11 ½ðY þ cÞCoshðMÞ ð1 þ cÞCoshðMYÞ E12 ð32Þ
Substituting these values appropriately into Eqs. (18), (19) and (21); the respective values of flow rate, bulk temperature and Nusselt Number at the no-slip wall are as follows: Q¼
E13 E14 E15 þ E16 E17 E18
ð33Þ
hb ¼
cQ þ E4 ðE6 E7 Þ þ E8 E9 ð1 þ cÞQ
ð34Þ
Nu ¼
Q cQ þ E4 ðE6 E7 Þ þ E8 E9
ð35Þ
The results obtained in both cases were considered as the value of magnetic component, M approaches zero. The limiting values tend to converge to the solutions obtained by Wang and Ng [1] (see Table 4). 3. Results and discussions 3.1. Case I
1Y hðYÞ ¼ 1þc UðYÞ ¼
ð22Þ
ð1 þ kÞ½SinhðMðY 1ÞÞ ðY 1Þ½SinhðMÞ þ kMCoshðMÞ ð1 þ cÞ½M2 SinhðMÞ þ kM3 CoshðMÞ ð23Þ
The fluctuation pattern of the velocity, as shown in Fig. 2, is similar to what was found in previous studies by other researchers. This shows that as particles move away from the super-hydrophobic surface they tend to have higher velocities for some time, then, the velocity falls before reaching the
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
MHD free convection flow in a vertical slit centre. But if both the surfaces are not super-hydrophobic, then, the velocity does not rise much and will not drop until around the centre of the channel. Each of the respective curves in Fig. 2 was considered under variable magnetic field strength. They exhibited similar fluctuations. So, one of them (when both k and c are each equal to 1) was used to represent them (Fig. 3). The pattern shows reduction in upward velocity (especially the peak velocity) with increase in magnetic field strength. Fig. 4 shows that the flow rate of Case I rises for values of slip length and temperature jump below 0.2. It reaches its peak value when these values range between 0.3 and 0.6 (depending on the strength of the magnetism applied). Then, it goes down. It was generally observed that this flow rate decreases with increase in magnetic strength. The shape of the graph shows that near the super-hydrophobic surface, there is a faster flow of fluid volume, then after a short distance, the flow falls continually. This is as a result of partial thermal insulation at the solid-gas-liquid interface, initially, and, later, due to the distance from the heated wall. Fig. 5 indicates that the Nusselt Number for Case I decreases (sharply for small values and mildly for bigger values) with increase in k or c. This is in conformity with what was found in previous studies. In addition to that, the Nusselt Number decreases with increase in magnetism. However, with magnetic component slightly greater than 2.0, the Nusselt Number would be kept at a barest minimum but constant value (irrespective of the slip length and temperature jump). The behaviour of the graph implies that both the distance from the super-hydrophobic surface and the strength of magnetism reduces the convective heat transfer in relation to the conductive heat transfer, since Nusselt Number is the ratio of convective heat transfer to conductive heat transfer.
5
Fig. 3 Variation of upward velocity for Case I with the magnetic component, M.
3.2. Case II Fig. 6 reveals a pattern of upward velocity fluctuation as expected, that is, in conformity with the results displayed in earlier researches. Here, as particles move away from the super-hydrophobic surface they tend to have higher velocities for a while, then, the velocity falls before reaching the centre.
Fig. 2 Variation of upward velocity for Case I with the displacement, y.
Fig. 4
Effect of magnetism on the flow rate in Case I.
But if both the surfaces are not super-hydrophobic, then, the velocity does not rise much, at the beginning, from zero and will not drop until after passing the centre of the channel, then, gradually, falls to Zero, again, at the other plate. It was, also, discovered that velocity in this Case reduces with increase in magnetism. This is shown by Fig. 7. The volume flow rate increases with increase in the value of c (the temperature jump coefficient) as portrayed by Fig. 8. However, it decreases with increase in magnetism. The infinite rise of the flow rate is due to the absence of super-hydrophobic thermal insulation at the heated surface. Fig. 9 illustrates that the Nusselt Number remains very close to Zero as the slip length or temperature jump increases for values of the magnetic factor from 0.5 to 1.5. The trend continues with M = 2.0 up to when slip or jump reaches 1.4. Then the Nusselt Number began to go down up to when the jump or slip is 1.8–2.0. At this interval, it began to rise sharply towards the peak value. Lack of trapped gases at the no-slip boundary reduces heat transfer by convection and negates the direction of conduction-convection flow in relation to the heating point.
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
6
Fig. 5
B.K. Jha, B.J. Gwandu
Effect of magnetism on the Nusselt Number in Case I.
Fig. 8
Effect of magnetism on the flow rate in Case II.
Fig. 6 Variation of upward velocity for Case II with the displacement, y.
Fig. 9
Fig. 7 Variation of upward velocity for Case II with the magnetic component, M.
Fig. 10 Comparison of flow rates in the two cases and existence of critical temperature jump coefficient (M = 4, k ¼ 1:0).
Effect of magnetism on the Nusselt Number in Case II.
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
MHD free convection flow in a vertical slit
7
Table 1 Variation of critical temperature jump coefficient (ccrit.) with the magnetic component (M) for the two cases combined together. k = 0.5 k = 1.0 k = 1.5 k = 2.0
M = 0.001
M=2
0.1676 0.2008 0.2150 0.2230
0.1535 0.1820 0.1933 0.1997
M=4 (8.4%) (9.4%) (10.1%) (10.4%)
0.1280 0.1470 0.1550 0.1588
M=6 (16.6%) (19.2%) (19.8%) (20.5%)
0.1056 0.1180 0.1230 0.1253
M=8 (17.5%) (19.7%) (20.6%) (21.1%)
0.0880 0.0970 0.1000 0.1017
(16.7%) (17.8%) (18.7%) (18.8%)
Note: The percentage indicated above was computed between two successive values from left to right. It does not hold from top to bottom.
Table 2 Variation of critical temperature jump coefficient (ccrit.) with the super-hydrophobic slip length (k) for the two cases combined together.
M = 0.001 M=2 M=4 M=6 M=8
k = 0.5
k = 1.0
k = 1.5
k = 2.0
0.1676 0.1535 0.1280 0.1056 0.0880
0.2008 0.1820 0.1470 0.1180 0.0970
0.2150 0.1933 0.1550 0.1230 0.1000
0.2230 0.1997 0.1588 0.1253 0.1017
(19.8%) (18.6%) (14.8%) (11.7%) (10.2%)
(7.1%) (6.2%) (5.4%) (4.2%) (3.1%)
(3.7%) (3.3%) (2.5%) (1.9%) (1.7%)
Note: The percentage indicated above was computed between two successive values from left to right. It does not hold from top to bottom.
Table 3
Comparison of velocities, flow rates and nusselt numbers for the two cases.
k
c
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
M
1 1 1 1 2 2 2 2
Case I
Case II
U (Peak)
Q
Nu
U (Peak)
Q
Nu
0.0583 0.0291 0.1466 0.0733 0.0460 0.0230 0.0935 0.0467
0.0379 0.0189 0.1004 0.0502 0.0298 0.0149 0.0631 0.0315
5.1020 5.0187 2.4503 2.3670 1.9824 1.9616 1.3308 1.3100
0.0583 0.0849 0.0854 0.1534 0.0460 0.0660 0.0558 0.0951
0.0379 0.0568 0.0677 0.1179 0.0298 0.0447 0.0437 0.0752
0.0586 0.0760 0.1063 0.1768 0.1980 0.3178 0.2984 0.7438
In addition to that, the flow rates in both cases were compared. It was found out that at a particular value of the temperature jump coefficient (critical value), the flow rates are the same (irrespective of which of the surfaces is heated). Before that value is reached, heating the super-hydrophobic surface gives a higher rate of fluid flow through the micro-channel. After that value is exceeded, heating the
Table 4 Comparison of Results of Both Cases in the Present Study with a Previous Work. (The velocities, here, are at the boundaries).
No-Slip surface gives the higher flow rate. This is depicted by Fig. 10. Generally, flow rate in Case I is in an inverse proportion with c, while in Case II it is a direct one. However, the graphs are not straight lines since the variation is monotone. Using the program leading to Fig. 10, it was observed that this critical value drops with increase in magnetism and rises with increase in the normalized super-hydrophobic slip length. As magnetism (M) approaches zero, it has a negligible effect on the Critical Temperature jump Coefficient. For possible experimental verification, the numerical values and their percentage variations are given in Tables 1 and 2. Tables 3 and 4 supply some numerical values of selected parameters for additional comparison.
Present work k¼c¼M¼1
Wang and Ng [1] k¼c¼1
Present work as M ! 0, k ¼ c ¼ 1
Case U Q Nu
I 0.0677 0.0502 2.3670
0.0833 0.0625 1.6071
0.0833 0.0625 1.6071
From the findings of this study, it could be concluded that the following situations hold:
Case U Q Nu
II 0.0000 0.1179 0.1768
0.0000 0.1458 0.7095
0.0000 0.1458 0.7095
(1) The value of peak upward velocity achievable through heating the super-hydrophobic surface is lower than that achieved by heating the no-slip surface when there is temperature jump and no super-hydrophobic slip or
4. Conclusions
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
8
B.K. Jha, B.J. Gwandu
(2)
(3)
(4)
(5)
(6)
(7)
there are both. If there is super-hydrophobic slip and no temperature jump, the reverse is the case. When there are neither, the peak velocities are equal. The bulk fluid flow rate observable when the superhydrophobic surface is being heated will be lower than the one to be observed when heating the no-slip surface, after a critical temperature jump coefficient (ccrit.) is exceeded. Before that value is reached, the reverse is the case. The critical value is when the two flow rates are equal, irrespective of which side is heated. Generally, the flow rate varies directly as slip length and/ or temperature jump coefficient, when the no-slip surface is heated; and inversely if the super-hydrophobic surface is heated. The peak ratio of the convective heat transfer to the conductive heat transfer, which is measured by the size of Nusselt Number, is higher when heating the superhydrophobic surface than the one attained by heating the no-slip surface. The ratio moves from higher values to lower ones in the first case. However, it remains almost unchanged in the second case, just to, unexpectedly, rise fast after the critical jump value of 1.8 when M = 2.0. In both cases, the existence and size of the magnetic factor, negatively, affects the peak upward velocity, the bulk fluid flow rate and the Nusselt Number (with the exception of the special case of Nusselt Number in case II). The percentage increase in the value of ccrit. decreases continuously with increase in k, especially at exposure to stronger magnetism. The percentage decrease in the value of ccrit. with increase in magnetism obeys the Agricultural ‘‘Law of Diminishing Returns” as it rises steadily until M = 6, then falls similarly. This suggests that even the application of magnetism follows that law.
The negative effects of magnetism (where applicable) in these processes could be useful, depending on the situation and application. In industries where poisonous electrically-conducting gases are emitted through vertical micro-channels, the results of this research advise that all magnets and magnetically-inducing materials be moved away from the channel for fast evacuation of the gas. This is, also, the case if it is feared that a sterilizing gas could corrode the walls of a channel or leave unwanted deposits on instruments being sterilized, by over-stay in the sterilization chamber. However, if it is a weak or lowconcentrated gas, magnetism could be added to make it stay longer. In the industrial production of gases in commercial quantities, high flow rate is more profitable. So, magnetism is to be moved afar. In renewable energy applications, if an electrically-conducting bio-gas is used, its life-span as a source of energy could be elongated by appropriate application of magnetism. The study, also, guides the industrialists in choosing their fluids. In all the above situations, provision of a superhydrophobic surface in the channel matters a lot. Also, the choice of which wall to heat and what value of magnetism to use are important.
Acknowledgement The corresponding author is grateful to the Federal University Birnin Kebbi, Nigeria, for financial assistance. Appendix A Constants used in the ð1þkÞ , E1 ¼ M2 ð1þcÞ½SinhðMÞþkMCoshðMÞ E3 ¼
work E0 ¼ 6ð1 þ cÞ, E2 ¼ M12 ½M SinhðMÞ,
E4 ¼ MSinhðMÞ , E5 ¼ SinhðMÞ CoshðMÞ1 , M M2 M2 ck 1 E6 ¼ ð1þcÞ½M2 SinhðMÞþkM3 CoshðMÞ, E7 ¼ ½M2 SinhðMÞCoshðMÞþkM 3 Cosh2 ðMÞ, 2þ3c E5 E8 ¼ 6M2 ð1þcÞ, E9 ¼ M2 CoshðMÞ ; E10 ¼ ðc kÞCoshðMÞ ð1 þ cÞ, E11 ¼ SinhðMÞ þ kMCoshðMÞ, E12 ¼ ð1 þ cÞM2 CoshðMÞ ½SinhðMÞ þ kMCoshðMÞ, E13 ¼ ½2kcM2 þ kM2 2k þ 4c þ 2 Cosh2 ðMÞ, E14 ¼ 2ð1 þ cÞSinh2 ðMÞ, E15 ¼ ð1 þ cÞ2kMSinh 1 , M2 ð1þcÞ
ðMÞCoshðMÞ, E16 ¼ ð1 þ 2cÞMSinhðMÞCoshðMÞ, E17 ¼ 2ð1 þ cÞCoshðMÞ, E18 ¼ 2ð1 þ cÞM3 CoshðMÞ½SinhðMÞþ kMCoshðMÞ. References
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H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature Basant K. Jha, Bello J. Gwandu * Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria Received 27 February 2017; revised 16 August 2017; accepted 30 August 2017
KEYWORDS MHD; Free convection; Superhydrophobic slip; Temperature jump
Abstract This work treats convective heat transfer, for an incompressible electrically-conducting fluid, moving vertically through an isothermally heated parallel plate micro-channel, within a transverse magnetic field. One surface exhibited super-hydrophobic slip and temperature jump, while the other did not. The study aimed at discovering the possible effects of magnetism on the velocity, volume flow rate and Nusselt Number when either wall is heated by constant wall temperature. It was noted that it reduces velocity and flow rate in both cases. It, also, leads to decline in Nusselt Number when the super-hydrophobic side is heated, while heating the other plate showed a negligible magnetic effect, for low temperature jump coefficients. At the critical value of the temperature jump coefficient, c; the flow rates in both cases are equal. Before the critical value ðccrit: Þ is reached, heating the super-hydrophobic plate yields higher flow rate. After passing it, the reverse is the case. The effect of magnetism on ccrit: was, also, negative. The study will have applications in the design and maintenance of both mini- and micro-devices as well as in nano-science and nano-technology. Ó 2017 Faculty of Engineering, Alexandria University. 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 Advancement in technology in line with the needs of people for smaller equipment and lighter appliances turns the attentions * Corresponding author. Permanent Address: Department of Mathematics, Federal University Birnin Kebbi, P.M.B. 1157, Birnin Kebbi, Kebbi State, Nigeria. E-mail addresses: [email protected] (B.K. Jha), Mbj_gwn@ yahoo.com (B.J. Gwandu). Peer review under responsibility of Faculty of Engineering, Alexandria University.
of scientists, engineers and technologists towards researches, experimentations and practical studies in mini-technology, then micro-technology, and later nano-technology. This is what caused researchers in Fluid Mechanics to shift from studying flows in macro-channels to mini-channels, microchannels and nano-channels. Wang and Ng [1] investigated a similar flow to the present study, but for an electrically non-conducting fluid and outside a magnetic field. Other researchers like Chen and Weng [2], Avci and Aydin [3], Panda and Moulic [4], Weng and Chen [5], Bounomo and Manca [6,7] considered convection flows of rarefied gases through some channels, and, kept the two
http://dx.doi.org/10.1016/j.aej.2017.08.022 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. 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/). Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
2
B.K. Jha, B.J. Gwandu
Nomenclature u dimensional upward velocity of the flow (m/s) uslip,wall dimensional slip and wall velocities, respectively (m/s) y0 dimensional horizontal position between the plates (m) g acceleration due to gravity (m/s2) T dimensional temperature of the heated surface (K) Tjump,wall dimensional temperature jump and at wall, respectively (K) T0 dimensional temperature of the surroundings (K) Th dimensional temperature higher than T0 (K) ! E electric field force (N/C or V/m) ! magnetic Lorentz force (T) F ! B magnetic field vector (T) B0 constant magnetic field applied (T) ! J electric current density (A/m2) ! V 3-dimensional velocity vector (m/s) Y dimensionless horizontal position (–) U dimensionless upward velocity (–) Uslip,wall dimensionless slip and wall velocities, respectively (–)
plates under the same jump conditions. This is because the gas slips and temperature jumps depend on coefficients that are the properties of the fluids, thermal conditions of the flows, type of material of the channel and its quality. Authors [6,7] applied numerical methods while the rest used analytical methods. These studies quoted above, are all devoid of magnetic field application. On the other hand, many MHD studies were undertaken, such as Singh et al. [8], Makinde et al. [9], Ellahi et al. [10] and many others. These studies dealt with different flows under different conditions, but through ringed gaps (annuli) or cylindrical pipes (whether horizontal or vertical). There are, also, some studies in MHD Couette flows (where one wall moves relative to the other) or flows in cylindrical channels in which the two concentric cylinders rotate. They include studies like Makinde and Chinyoka [11], Subotic and Lai [12], and Jha and Apere [13]. Though the above studies included the application of magnetic field, attention was not paid to superhydrophobic nature of wall surfaces. Diverse researches were made bordering various kinds of slips, different dispositions of MHD, several forms of heat transfer, dissimilar types of flow, involving special kinds of fluid and with applications of several different methods. The trend continues, especially in recent years. Uddin et. al. [14] used a group theoretic method to find the effects of relevant parameters on the temperature, velocity and nanoparticle volume fraction of a nanofluid in a free convection involving a homogeneous isothermal irreversible chemical reaction. Slip effects were studied in relation to Biology under a special motion of microorganisms in fluid (called bioconvection) by Uddin et al. [15]. The flow was in a porous medium and the microorganisms moved by a combination of taxis and gyration. Uddin et al. [16] incorporated radiation in a nanofluid convection past a shrinking/stretching sheet. A combination
L M Q Nu ^ i
last point of displacement/channel width (m) magnetic component of the system (–) volume flow rate (m3/s) Nusselt number (–) the unit vector in the x (vertical) direction
Greek letters b thermal expansion coefficient (1/K) c dimensionless temperature jump coefficient (–) c0 dimensional temperature jump coefficient (m) m kinematic viscosity of the fluid (m2/s) q density of the fluid (kg/m3) r electrical conductivity of the fluid (A2s3/kg.m3 or S/m) h dimensionless temperature (–) hjump;wall dimensionless temperature jump and at wall, respectively (–) hb bulk temperature (K) k dimensionless slip length (–) k0 dimensional slip length (m)
of algebraic transformation and numerical methods were used to solve the Partial Differential Equations (PDEs). A similar situation, without radiation, was studied by Latiff et al. [17] but applied to Biology. The stretching/shrinking yielded slip effects. They found out that skin friction and Nusselt Number are decreased while microorganisms’ number is increased as the respective relevant slip parameters are increased. Uddin et al. [18] used Lie group algebraic theory in the solution of the Ordinary Differential Equations (ODEs) in their study of multiple slips convection through a Darcian porous medium. They discovered that increase in the magnetic parameter hinders velocity and increasing the permeability parameter favours temperature rise. Magnetohydrodynamic (MHD) slips in velocity and momentum were, also, involved in the heat and mass transfer study of nanofluids by Majeed et al. [19]. Different forms of heat transfer were combined together in the study by Uddin et al. [20]. A combination of algebraic transformation and numerical methods were, also, utilized. In addition to heat transfer, heat generation was, also, considered by Uddim et al. [21]. The stretching of the sheet produced slips. Magnetism was involved, and a numerical method followed the algebraic transformation method. In a related development, Nawaz et al. [22] used Homotopy Analysis Method (HAM) in solving the nonlinear problems that arose when they were studying heat transfer in a Jeffrey fluid flowing on a stretching circular membrane. A comparable work was done by Zeeshan and Majeed [23] with porosity and consideration of dipole effect. Sheikholeslami and Zeeshan [24] considered a constant heat flux in a similar study and used the Control Volume Finite Element Method (CVFEM) to solve the problem. MHD mixed convection was investigated by Zeeshan and Majeed [25] through a non Darcy stretching sheet. Numerical solutions were provided. Another bio-inclined study was performed by Bhatti et al. [26]. It involved peristalsis
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
MHD free convection flow in a vertical slit and numerical integration. A short and useful collection of communicative notes on micro-channel natural convection was released by Cheng and Weng [27]. Jha et al. [28] used an exact solution method in studying slip effects, fluid-wall interaction parameter, wall-ambient temperature difference ratio and rarefaction parameter in an MHD micro-channel natural convection. The basic necessary conditions for the rusting of iron are the presence of air and moisture. Even in mini- and microappliances that do not contain water or some other liquids, heat generation through friction and other means, leads to production of steam, which moistens the surfaces it touches. This made the engineers to think of introducing hydrophobic and even ultra-hydrophobic parts in their constructions and manufacturing. This study, therefore, makes an attempt to apply a transverse magnetic field to the natural convective flow in a super-hydrophobic micro-channel for investigating the outcome of the new combination. The findings will improve services in microchips production factories, oil and gas industries, mini-equipment assembling outfits, hospital instruments sterilization factories and other similar places. 2. Mathematical analysis Let us consider an electrically-conducting fluid moving steadily upward through a vertical parallel plate micro-channel by convection through heating one of the plates at a time. One of the surfaces is extremely difficult to wet (super-hydrophobic) as a result of a special micro-engineering treatment. The other side (no-slip surface) was not tempered with. The super-hydrophobic surface is kept at a position y0 ¼ 0 while the no-slip surface is at y0 ¼ L as shown by Fig. 1. It could be seen in literature such as [24], that the magnetic Lorentz force is described as the vector product of the current ! ! density, J and the magnetic field vector B . That is, ! ! ! F ¼ J B ð1Þ
3 Similarly, [14] defined the current density as ! ! ! ! J ¼ r½ E þ V B
ð2Þ
! where r is the electrical conductivity of the fluid, E is the ! electric field and V is the velocity vector. Combining Eqs. (1) and (2), we have ! ! ! ! ! F ¼ r½ E þ V B B ð3Þ Neglecting the electric field and taking one component each ! ! ! ! of both V and B ; that is, E ¼ 0, V ¼ ðu; 0; 0Þ and ! B ¼ ð0; B0 ; 0Þ; we have ! i ð4Þ F ¼ rB20 u^ The directions of B0 and u are indicated in Fig. 1. Using the ideas of [27,28], the slip velocity and temperature jump are, respectively, defined as follows: uslip ¼ k0
du dU ) Uslip ¼ k dY dy0
Tjump ¼ c0
dT dh ) hjump ¼ c dY dy0
ð5Þ ð6Þ
For velocity slip, as in [15–19], the particle velocity near the boundary is given as: u ¼ uwall þ uslip ) U ¼ Uwall þ Uslip
ð7Þ
Similarly, the temperature near the boundary, like in [15,16,18], is given as: T ¼ Twall þ Tjump ) h ¼ hwall þ hjump
ð8Þ
Here uwall and Twall are, respectively, the dimensional velocity and temperature at the wall. All related dimensionless velocity and temperature are, similarly defined. The Boussinesq buoyancy approximation and boundary conditions used by [1] are considered. Since the fluid considered is viscous, electrically conducting, and in the presence of magnetic field, the MHD approximations are attached to finally get the governing equations for the system (see Section 2.1). The Ordinary Differential Model Equations were solved by exact solution method. The graphs were plotted and interpretations given. Different temperature jump and slip conditions were used for the different plates because the main interest is in the super-hydrophobicity of a surface, not the characteristics of the fluid or the flow. This is in contrast with emphasis on rarefaction of gases which is the nature of the fluid itself. Micro-engineered structures like grooves, notches, holes and cones trap gases in their spaces and become filled, initially. This prevents the liquid from filling the spaces until the capillary pressure is exceeded. The situation causes a hydrodynamic slip leading to mixed boundary conditions. Gases conduct heat much slower than liquids. This leads to temperature jump between the two interfaces (liquid-solid and liquid-gas interfaces). The study was divided into two cases. 2.1. Case I (the super-hydrophobic surface being heated)
Fig. 1 Free convective flow of an electrically conducting fluid through a vertical micro-channel.
The governing equations of the system are as follows [1,14– 19,24,27,28]:
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
4 m
B.K. Jha, B.J. Gwandu d2 u rB2 u þ gbðT T0 Þ 0 ¼ 0 02 q dy
d2 T ¼0 dy02
ð9Þ
ð10Þ
subject to the following boundary conditions: ) du uðy0 Þ ¼ k0 dy 0 at y0 ¼ 0 dT Tðy0 Þ ¼ Th þ c0 dy 0 0
uðy Þ ¼ 0 Tðy0 Þ ¼ T0
Q¼
MSinhðMÞ þ ðkM3 2k 2ÞCoshðMÞ þ 2k þ 2 ð2c þ 2Þ½M3 SinhðMÞ þ kM4 CoshðMÞ
ð24Þ
hb ¼
ðE8 þ 6ÞQ 6E1 E2 E0 Q
ð25Þ
ð11Þ Nu ¼
at y0 ¼ L
Using Eqs. (18)–(20), (22) and (23), the following results were obtained:
ðE8 þ 6ÞQ 6E1 E2 6Q
ð26Þ
ð12Þ 2.2. Case II (the no-slip surface being heated)
Using the following dimensionless variables: ðY; c; kÞ ¼ ðy0 ; c0 ; k0 Þ=L; M2 ¼ rB20 L2 =qm;
h ¼ ðT T0 Þ=ðTh T0 Þ 1 U ¼ um½gbL2 ðTh T0 Þ
ð13Þ
Eqs. (9) and (10), respectively, become: d2 U þ h M2 U ¼ 0 dY2 dh ¼0 dY2
ð14Þ
2
ð15Þ
with the following boundary conditions: ) dh hðYÞ ¼ 1 þ c dY at Y ¼ 0 UðYÞ ¼ k dU dY hðYÞ ¼ 0 UðYÞ ¼ 0
ð16Þ
ð17Þ
The volume flow rate and bulk temperature are, respectively, given by the following equations: Z 1 UðYÞdY ¼ 0 ð18Þ Q¼ 0
R1 hb ¼
0
hðYÞUðYÞdY R1 UðYÞdY 0
ð19Þ
Eqs. (18) and (19) are used in both cases of heating. The Nusselt Numbers at the two walls are given by Eqs. (20) and (21) below. Nu ¼ Nu ¼
dh=dY jY ¼ 0 hb
dh=dY jY ¼ 1 hb
uðy0 Þ ¼ k0
du dU ) UðYÞ ¼ k dYY¼0 dy0 y0 ¼0
ð28Þ
Tðy0 Þ ¼ Th jy0 ¼L ) hðYÞ ¼ 1jY¼1
ð29Þ
uðy0 Þ ¼ 0jy0 ¼L ) UðYÞ ¼ 0jY¼1
ð30Þ
This gives the values of temperature and velocity to be, respectively, as below:
at Y ¼ 1
Eqs. (9) and (10) are the governing equations. Eq. (13) was used to change them into Eqs. (14) and (15). We, now, solve Eqs. (14) and (15), using the following boundary conditions: dT dh ) hðYÞ ¼ c ð27Þ Tðy0 Þ ¼ T0 þ c0 0 dY Y¼0 dy y0 ¼0
ð20Þ ð21Þ
Eq. (20) is used when the super-hydrophobic surface is heated, while Eq. (21) is used when the no-slip surface is heated. The following results for the temperature and velocity were obtained after solving Eqs. (14) and (15) with conditions (16) and (17).
hðYÞ ¼ ðY þ cÞ=ð1 þ cÞ UðYÞ ¼
ð31Þ
E10 Sinh½MðY 1Þ þ E11 ½ðY þ cÞCoshðMÞ ð1 þ cÞCoshðMYÞ E12 ð32Þ
Substituting these values appropriately into Eqs. (18), (19) and (21); the respective values of flow rate, bulk temperature and Nusselt Number at the no-slip wall are as follows: Q¼
E13 E14 E15 þ E16 E17 E18
ð33Þ
hb ¼
cQ þ E4 ðE6 E7 Þ þ E8 E9 ð1 þ cÞQ
ð34Þ
Nu ¼
Q cQ þ E4 ðE6 E7 Þ þ E8 E9
ð35Þ
The results obtained in both cases were considered as the value of magnetic component, M approaches zero. The limiting values tend to converge to the solutions obtained by Wang and Ng [1] (see Table 4). 3. Results and discussions 3.1. Case I
1Y hðYÞ ¼ 1þc UðYÞ ¼
ð22Þ
ð1 þ kÞ½SinhðMðY 1ÞÞ ðY 1Þ½SinhðMÞ þ kMCoshðMÞ ð1 þ cÞ½M2 SinhðMÞ þ kM3 CoshðMÞ ð23Þ
The fluctuation pattern of the velocity, as shown in Fig. 2, is similar to what was found in previous studies by other researchers. This shows that as particles move away from the super-hydrophobic surface they tend to have higher velocities for some time, then, the velocity falls before reaching the
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
MHD free convection flow in a vertical slit centre. But if both the surfaces are not super-hydrophobic, then, the velocity does not rise much and will not drop until around the centre of the channel. Each of the respective curves in Fig. 2 was considered under variable magnetic field strength. They exhibited similar fluctuations. So, one of them (when both k and c are each equal to 1) was used to represent them (Fig. 3). The pattern shows reduction in upward velocity (especially the peak velocity) with increase in magnetic field strength. Fig. 4 shows that the flow rate of Case I rises for values of slip length and temperature jump below 0.2. It reaches its peak value when these values range between 0.3 and 0.6 (depending on the strength of the magnetism applied). Then, it goes down. It was generally observed that this flow rate decreases with increase in magnetic strength. The shape of the graph shows that near the super-hydrophobic surface, there is a faster flow of fluid volume, then after a short distance, the flow falls continually. This is as a result of partial thermal insulation at the solid-gas-liquid interface, initially, and, later, due to the distance from the heated wall. Fig. 5 indicates that the Nusselt Number for Case I decreases (sharply for small values and mildly for bigger values) with increase in k or c. This is in conformity with what was found in previous studies. In addition to that, the Nusselt Number decreases with increase in magnetism. However, with magnetic component slightly greater than 2.0, the Nusselt Number would be kept at a barest minimum but constant value (irrespective of the slip length and temperature jump). The behaviour of the graph implies that both the distance from the super-hydrophobic surface and the strength of magnetism reduces the convective heat transfer in relation to the conductive heat transfer, since Nusselt Number is the ratio of convective heat transfer to conductive heat transfer.
5
Fig. 3 Variation of upward velocity for Case I with the magnetic component, M.
3.2. Case II Fig. 6 reveals a pattern of upward velocity fluctuation as expected, that is, in conformity with the results displayed in earlier researches. Here, as particles move away from the super-hydrophobic surface they tend to have higher velocities for a while, then, the velocity falls before reaching the centre.
Fig. 2 Variation of upward velocity for Case I with the displacement, y.
Fig. 4
Effect of magnetism on the flow rate in Case I.
But if both the surfaces are not super-hydrophobic, then, the velocity does not rise much, at the beginning, from zero and will not drop until after passing the centre of the channel, then, gradually, falls to Zero, again, at the other plate. It was, also, discovered that velocity in this Case reduces with increase in magnetism. This is shown by Fig. 7. The volume flow rate increases with increase in the value of c (the temperature jump coefficient) as portrayed by Fig. 8. However, it decreases with increase in magnetism. The infinite rise of the flow rate is due to the absence of super-hydrophobic thermal insulation at the heated surface. Fig. 9 illustrates that the Nusselt Number remains very close to Zero as the slip length or temperature jump increases for values of the magnetic factor from 0.5 to 1.5. The trend continues with M = 2.0 up to when slip or jump reaches 1.4. Then the Nusselt Number began to go down up to when the jump or slip is 1.8–2.0. At this interval, it began to rise sharply towards the peak value. Lack of trapped gases at the no-slip boundary reduces heat transfer by convection and negates the direction of conduction-convection flow in relation to the heating point.
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Fig. 5
B.K. Jha, B.J. Gwandu
Effect of magnetism on the Nusselt Number in Case I.
Fig. 8
Effect of magnetism on the flow rate in Case II.
Fig. 6 Variation of upward velocity for Case II with the displacement, y.
Fig. 9
Fig. 7 Variation of upward velocity for Case II with the magnetic component, M.
Fig. 10 Comparison of flow rates in the two cases and existence of critical temperature jump coefficient (M = 4, k ¼ 1:0).
Effect of magnetism on the Nusselt Number in Case II.
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
MHD free convection flow in a vertical slit
7
Table 1 Variation of critical temperature jump coefficient (ccrit.) with the magnetic component (M) for the two cases combined together. k = 0.5 k = 1.0 k = 1.5 k = 2.0
M = 0.001
M=2
0.1676 0.2008 0.2150 0.2230
0.1535 0.1820 0.1933 0.1997
M=4 (8.4%) (9.4%) (10.1%) (10.4%)
0.1280 0.1470 0.1550 0.1588
M=6 (16.6%) (19.2%) (19.8%) (20.5%)
0.1056 0.1180 0.1230 0.1253
M=8 (17.5%) (19.7%) (20.6%) (21.1%)
0.0880 0.0970 0.1000 0.1017
(16.7%) (17.8%) (18.7%) (18.8%)
Note: The percentage indicated above was computed between two successive values from left to right. It does not hold from top to bottom.
Table 2 Variation of critical temperature jump coefficient (ccrit.) with the super-hydrophobic slip length (k) for the two cases combined together.
M = 0.001 M=2 M=4 M=6 M=8
k = 0.5
k = 1.0
k = 1.5
k = 2.0
0.1676 0.1535 0.1280 0.1056 0.0880
0.2008 0.1820 0.1470 0.1180 0.0970
0.2150 0.1933 0.1550 0.1230 0.1000
0.2230 0.1997 0.1588 0.1253 0.1017
(19.8%) (18.6%) (14.8%) (11.7%) (10.2%)
(7.1%) (6.2%) (5.4%) (4.2%) (3.1%)
(3.7%) (3.3%) (2.5%) (1.9%) (1.7%)
Note: The percentage indicated above was computed between two successive values from left to right. It does not hold from top to bottom.
Table 3
Comparison of velocities, flow rates and nusselt numbers for the two cases.
k
c
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
M
1 1 1 1 2 2 2 2
Case I
Case II
U (Peak)
Q
Nu
U (Peak)
Q
Nu
0.0583 0.0291 0.1466 0.0733 0.0460 0.0230 0.0935 0.0467
0.0379 0.0189 0.1004 0.0502 0.0298 0.0149 0.0631 0.0315
5.1020 5.0187 2.4503 2.3670 1.9824 1.9616 1.3308 1.3100
0.0583 0.0849 0.0854 0.1534 0.0460 0.0660 0.0558 0.0951
0.0379 0.0568 0.0677 0.1179 0.0298 0.0447 0.0437 0.0752
0.0586 0.0760 0.1063 0.1768 0.1980 0.3178 0.2984 0.7438
In addition to that, the flow rates in both cases were compared. It was found out that at a particular value of the temperature jump coefficient (critical value), the flow rates are the same (irrespective of which of the surfaces is heated). Before that value is reached, heating the super-hydrophobic surface gives a higher rate of fluid flow through the micro-channel. After that value is exceeded, heating the
Table 4 Comparison of Results of Both Cases in the Present Study with a Previous Work. (The velocities, here, are at the boundaries).
No-Slip surface gives the higher flow rate. This is depicted by Fig. 10. Generally, flow rate in Case I is in an inverse proportion with c, while in Case II it is a direct one. However, the graphs are not straight lines since the variation is monotone. Using the program leading to Fig. 10, it was observed that this critical value drops with increase in magnetism and rises with increase in the normalized super-hydrophobic slip length. As magnetism (M) approaches zero, it has a negligible effect on the Critical Temperature jump Coefficient. For possible experimental verification, the numerical values and their percentage variations are given in Tables 1 and 2. Tables 3 and 4 supply some numerical values of selected parameters for additional comparison.
Present work k¼c¼M¼1
Wang and Ng [1] k¼c¼1
Present work as M ! 0, k ¼ c ¼ 1
Case U Q Nu
I 0.0677 0.0502 2.3670
0.0833 0.0625 1.6071
0.0833 0.0625 1.6071
From the findings of this study, it could be concluded that the following situations hold:
Case U Q Nu
II 0.0000 0.1179 0.1768
0.0000 0.1458 0.7095
0.0000 0.1458 0.7095
(1) The value of peak upward velocity achievable through heating the super-hydrophobic surface is lower than that achieved by heating the no-slip surface when there is temperature jump and no super-hydrophobic slip or
4. Conclusions
Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022
8
B.K. Jha, B.J. Gwandu
(2)
(3)
(4)
(5)
(6)
(7)
there are both. If there is super-hydrophobic slip and no temperature jump, the reverse is the case. When there are neither, the peak velocities are equal. The bulk fluid flow rate observable when the superhydrophobic surface is being heated will be lower than the one to be observed when heating the no-slip surface, after a critical temperature jump coefficient (ccrit.) is exceeded. Before that value is reached, the reverse is the case. The critical value is when the two flow rates are equal, irrespective of which side is heated. Generally, the flow rate varies directly as slip length and/ or temperature jump coefficient, when the no-slip surface is heated; and inversely if the super-hydrophobic surface is heated. The peak ratio of the convective heat transfer to the conductive heat transfer, which is measured by the size of Nusselt Number, is higher when heating the superhydrophobic surface than the one attained by heating the no-slip surface. The ratio moves from higher values to lower ones in the first case. However, it remains almost unchanged in the second case, just to, unexpectedly, rise fast after the critical jump value of 1.8 when M = 2.0. In both cases, the existence and size of the magnetic factor, negatively, affects the peak upward velocity, the bulk fluid flow rate and the Nusselt Number (with the exception of the special case of Nusselt Number in case II). The percentage increase in the value of ccrit. decreases continuously with increase in k, especially at exposure to stronger magnetism. The percentage decrease in the value of ccrit. with increase in magnetism obeys the Agricultural ‘‘Law of Diminishing Returns” as it rises steadily until M = 6, then falls similarly. This suggests that even the application of magnetism follows that law.
The negative effects of magnetism (where applicable) in these processes could be useful, depending on the situation and application. In industries where poisonous electrically-conducting gases are emitted through vertical micro-channels, the results of this research advise that all magnets and magnetically-inducing materials be moved away from the channel for fast evacuation of the gas. This is, also, the case if it is feared that a sterilizing gas could corrode the walls of a channel or leave unwanted deposits on instruments being sterilized, by over-stay in the sterilization chamber. However, if it is a weak or lowconcentrated gas, magnetism could be added to make it stay longer. In the industrial production of gases in commercial quantities, high flow rate is more profitable. So, magnetism is to be moved afar. In renewable energy applications, if an electrically-conducting bio-gas is used, its life-span as a source of energy could be elongated by appropriate application of magnetism. The study, also, guides the industrialists in choosing their fluids. In all the above situations, provision of a superhydrophobic surface in the channel matters a lot. Also, the choice of which wall to heat and what value of magnetism to use are important.
Acknowledgement The corresponding author is grateful to the Federal University Birnin Kebbi, Nigeria, for financial assistance. Appendix A Constants used in the ð1þkÞ , E1 ¼ M2 ð1þcÞ½SinhðMÞþkMCoshðMÞ E3 ¼
work E0 ¼ 6ð1 þ cÞ, E2 ¼ M12 ½M SinhðMÞ,
E4 ¼ MSinhðMÞ , E5 ¼ SinhðMÞ CoshðMÞ1 , M M2 M2 ck 1 E6 ¼ ð1þcÞ½M2 SinhðMÞþkM3 CoshðMÞ, E7 ¼ ½M2 SinhðMÞCoshðMÞþkM 3 Cosh2 ðMÞ, 2þ3c E5 E8 ¼ 6M2 ð1þcÞ, E9 ¼ M2 CoshðMÞ ; E10 ¼ ðc kÞCoshðMÞ ð1 þ cÞ, E11 ¼ SinhðMÞ þ kMCoshðMÞ, E12 ¼ ð1 þ cÞM2 CoshðMÞ ½SinhðMÞ þ kMCoshðMÞ, E13 ¼ ½2kcM2 þ kM2 2k þ 4c þ 2 Cosh2 ðMÞ, E14 ¼ 2ð1 þ cÞSinh2 ðMÞ, E15 ¼ ð1 þ cÞ2kMSinh 1 , M2 ð1þcÞ
ðMÞCoshðMÞ, E16 ¼ ð1 þ 2cÞMSinhðMÞCoshðMÞ, E17 ¼ 2ð1 þ cÞCoshðMÞ, E18 ¼ 2ð1 þ cÞM3 CoshðMÞ½SinhðMÞþ kMCoshðMÞ. References
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Please cite this article in press as: B.K. Jha, B.J. Gwandu, MHD free convection flow in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: Heating by constant wall temperature, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.022