Nomenclature
$(x,y,z)$ Cartesian coordinate system | |
$m$ shape factor | |
${c}_{p}$ specific heat | |
${\boldsymbol{V}}$ flow field | |
$T$ fluid temperature | |
Subscripts |
|
${T}_{0}$ lower temperature at upper wall | |
$f$ fluid phase | |
${T}_{h}$ higher temperature at lower wall | |
$s$ solid phase | |
$u,v,w$ velocity components | |
$N$ nanofluid | |
$k$ thermal conductivity | |
Greek symbols |
|
${U}_{w}$ velocity of stretching sheet | |
$\rho $ density | |
$N{u}_{x}$ local Nusselt number | |
$\mu $ dynamic viscosity | |
$p$ pressure | |
$\nu $ kinematic viscosity | |
${q}_{w}$ wall heat flux | |
$\alpha $ thermal diffusivity | |
$Pr$ Prandtl number | |
$\sigma $ electrical conductivity | |
${C}_{f}$ skin friction coefficient | |
${\rm{\Omega }}$ angular velocity | |
$a$ stretching rate | |
$\eta $ similarity variable | |
${B}_{0}$ uniform magnetic field | |
$\theta $ dimensionless temperature | |
$f^{\prime} ,g$ dimensionless velocity along $x,y$ direction | |
$\phi $ nanoparticle volume fraction |
1. Introduction
2. Problem formulation
Figure 1. Geometry of the problem. |
Table 1. Thermophysical features of water and magnetite [23]. |
Physical properties | Water | Magnetite |
---|---|---|
ρ (kg m−3) | 997 | 5180 |
${c}_{p}$ (J kg−1 K) | 4179 | 670 |
$k$ (W m−1 K) | 0.613 | 9.7 |
Σ (ω−1 m−1) | 0.05 | 25000 |
3. Methodology of solution
Figure 2. Computational flowchart for shooting method. |
Table 2. Comparison of ${C}_{f}R{e}_{x}^{1/2}$and $R{e}_{x}^{-1/2}N{u}_{x}$ with results of reference [46] for diverse values of $S$ and $\phi $ at lower/upper surface. |
${C}_{f}R{e}_{x}^{1/2}$ at $\eta =0$ | $R{e}_{x}^{-1/2}N{u}_{x}$ at $\eta =0$ | ${C}_{f}R{e}_{x}^{1/2}$ at $\eta =1$ | $R{e}_{x}^{-1/2}N{u}_{x}$ at $\eta =1$ | ||||||
---|---|---|---|---|---|---|---|---|---|
$S$ | $\phi $ | Present | Reference [46] | Present | Reference [46] | Present | Reference [46] | Present | Reference [46] |
−1 | 0 | −9.54155 | −9.54155 | 0.39001 | 8.85752 | 0.39001 | 8.85752 | 4.82296 | 4.82296 |
0 | 0 | −4.09595 | −4.09595 | 1.33610 | 1.33610 | 1.95289 | 1.95289 | 0.80249 | 0.80249 |
1 | 0 | 2.23565 | 2.23565 | 2.19498 | 2.19498 | −3.81099 | −3.81099 | 0.05776 | 0.05776 |
4. Results and discussion
Table 3. Values of ${C}_{f}R{e}_{x}^{1/2}$ at $\eta =0$ for diverse amounts of the emergent parameters at lower surface ($Pr=6.2,m=3$). |
${A}_{2}=0,{A}_{3}=0$ | ${A}_{2}=0,{A}_{3}=0$ | ${A}_{2}=0,{A}_{3}=1$ | ${A}_{2}=1,{A}_{3}=1$ | ||||||
---|---|---|---|---|---|---|---|---|---|
$S$ | $\phi \,$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ |
−1 | 0.0 | −9.73485 | −9.46810 | −9.79701 | −9.54155 | −9.85479 | −9.71457 | −9.91343 | −9.77916 |
.05 | −9.13191 | −8.86501 | −9.19873 | −8.94492 | −9.26257 | −9.13383 | −9.32502 | −9.20260 | |
0.1 | −8.90239 | −8.63544 | −8.97118 | −8.71810 | −9.04916 | −8.93734 | −9.11280 | −9.00691 | |
0 | 0.0 | −4.04282 | −4.08556 | −4.05316 | −4.09595 | −4.10870 | −4.21578 | −4.11862 | −4.22533 |
.05 | −3.80166 | −3.84440 | −3.81267 | −3.85545 | −3.87325 | −3.98571 | −3.88373 | −3.99574 | |
0.1 | −3.70986 | −3.75259 | −3.72115 | −3.76393 | −3.79018 | −3.91095 | −3.80085 | −3.92109 | |
1 | 0.0 | 2.10694 | 2.21327 | 2.13398 | 2.23565 | 2.08922 | 2.17591 | 2.11487 | 2.19632 |
.05 | 1.98635 | 2.09260 | 2.01478 | 2.11584 | 1.96702 | 2.05181 | 1.99379 | 2.07274 | |
0.1 | 1.94044 | 2.04667 | 1.96944 | 2.07024 | 1.91874 | 2.00092 | 1.94581 | 2.02185 |
Table 4. Values of ${C}_{f}R{e}_{x}^{1/2}$ at $\eta =1$ for diverse amounts of the emergent parameters at upper surface ($Pr=6.2,m=3$). |
${A}_{2}=0,{A}_{3}=0$ | ${A}_{2}=0,{A}_{3}=0$ | ${A}_{2}=0,{A}_{3}=1$ | ${A}_{2}=1,{A}_{3}=1$ | ||||||
---|---|---|---|---|---|---|---|---|---|
S | $\phi $ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ |
−1 | 0.0 | 8.38630 | 8.80893 | 8.43150 | 8.85752 | 8.41637 | 8.86134 | 8.45885 | 8.90381 |
.05 | 7.90512 | 8.33141 | 7.95344 | 8.38359 | 7.93761 | 8.38735 | 7.98257 | 8.43195 | |
0.1 | 7.72200 | 8.14982 | 7.77163 | 8.20351 | 7.75840 | 8.21225 | 7.80408 | 8.25710 | |
0 | 0.0 | 1.97639 | 1.95318 | 1.97585 | 1.95289 | 1.96054 | 1.92301 | 1.95995 | 1.92260 |
.05 | 1.85583 | 1.83265 | 1.85527 | 1.83238 | 1.83865 | 1.80013 | 1.83804 | 1.79970 | |
0.1 | 1.80993 | 1.78677 | 1.80937 | 1.78651 | 1.79071 | 1.75049 | 1.79008 | 1.75005 | |
1 | 0.0 | −3.88037 | −3.77309 | −3.92117 | −3.81099 | −3.94475 | −3.89718 | −3.98358 | −3.93178 |
.05 | −3.63962 | −3.53346 | −3.68284 | −3.57339 | −3.70947 | −3.66769 | −3.75030 | −3.70367 | |
0.1 | −3.54799 | −3.44230 | −3.59221 | −3.48305 | −3.62631 | −3.59253 | −3.66774 | −3.62872 |
Table 5. Values of $R{e}_{x}^{-1/2}N{u}_{x}\,$at $\eta =0$ for diverse amounts of the emergent parameters at lower surface ($Pr=6.2,m=3$). |
${A}_{2}=0,{A}_{3}=0$ | ${A}_{2}=0,{A}_{3}=0$ | ${A}_{2}=0,{A}_{3}=1$ | ${A}_{2}=1,{A}_{3}=1$ | ||||||
---|---|---|---|---|---|---|---|---|---|
$S$ | $\phi $ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ |
−1 | 0.0 | 0.67952 | 0.39074 | 0.67911 | 0.39001 | 0.67831 | 0.38720 | 0.67792 | 0.38656 |
0.05 | 0.72001 | 0.45850 | 0.71957 | 0.45763 | 0.71873 | 0.45436 | 0.71832 | 0.45362 | |
0.1 | 0.75438 | 0.51950 | 0.75396 | 0.51859 | 0.75304 | 0.51488 | 0.75265 | 0.51412 | |
0 | 0.0 | 1.16185 | 1.33661 | 1.16161 | 1.33610 | 1.16061 | 1.33143 | 1.16038 | 1.33096 |
0.05 | 1.14116 | 1.29207 | 1.14092 | 1.29157 | 1.13991 | 1.28691 | 1.13968 | 1.28646 | |
0.1 | 1.12372 | 1.25487 | 1.12350 | 1.25441 | 1.12247 | 1.24974 | 1.12226 | 1.24933 | |
1 | 0.0 | 1.62894 | 2.19513 | 1.62886 | 2.19498 | 1.62793 | 2.19164 | 1.62785 | 2.19150 |
0.05 | 1.55400 | 2.06685 | 1.55390 | 2.06665 | 1.55296 | 2.06314 | 1.55287 | 2.06295 | |
0.1 | 1.48916 | 1.95237 | 1.48907 | 1.95215 | 1.48810 | 1.94848 | 1.48801 | 1.94828 |
Table 6. Values of $R{e}_{x}^{-1/2}N{u}_{x}\,$at $\eta =1$ for diverse amounts of the emergent parameters at upper surface ($Pr=6.2,m=3$). |
${A}_{2}=0,{A}_{3}=0$ | ${A}_{2}=0,{A}_{3}=0$ | ${A}_{2}=0,{A}_{3}=1$ | ${A}_{2}=1,{A}_{3}=1$ | ||||||
---|---|---|---|---|---|---|---|---|---|
$S$ | $\phi $ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | $A{}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ | ${A}_{1}=0.5$ | ${A}_{1}=1$ |
−1 | 0.0 | 2.43038 | 4.82218 | 2.43055 | 4.82296 | 2.43193 | 4.83047 | 2.43209 | 4.83117 |
.05 | 2.20060 | 4.14128 | 2.20078 | 4.14220 | 2.20212 | 4.14988 | 2.20229 | 4.15068 | |
0.1 | 2.01768 | 3.60497 | 2.01785 | 3.60587 | 2.01917 | 3.61358 | 2.01933 | 3.61435 | |
0 | 0.0 | 0.89872 | 0.80217 | 0.89889 | 0.80249 | 0.89964 | 0.80560 | 0.89981 | 0.80590 |
.05 | 0.91101 | 0.82579 | 0.91118 | 0.82611 | 0.91195 | 0.82930 | 0.91212 | 0.82960 | |
0.1 | 0.92151 | 0.84603 | 0.92167 | 0.84634 | 0.92246 | 0.84961 | 0.92261 | 0.84989 | |
1 | 0.0 | 0.26551 | 0.05769 | 0.26564 | 0.05776 | 0.26592 | 0.05810 | 0.26605 | 0.05817 |
.05 | 0.31627 | 0.08475 | 0.31642 | 0.08485 | 0.31675 | 0.08535 | 0.31690 | 0.08544 | |
0.1 | 0.36629 | 0.11706 | 0.36646 | 0.11719 | 0.36685 | 0.11787 | 0.36700 | 0.11799 |
Figure 3. (a) $f^{\prime} \left(\eta \right),$ (b) $g\left(\eta \right),$ (c) $\theta \left(\eta \right)\,$ for nanoparticle concentration $\phi $ when ${A}_{1}={A}_{2}={A}_{3}=2,S=1.$ |
Figure 4. (a) $f^{\prime} \left(\eta \right),$ (b) $g\left(\eta \right),$ (c) $\theta \left(\eta \right)\,$ for magnetite-water nanofluid with parameter S when ${A}_{1}={A}_{2}={A}_{3}=0.5,\phi =0.2.$ |
Figure 5. (a) $f^{\prime} \left(\eta \right),$ (b) $g\left(\eta \right),$ (c) $\theta \left(\eta \right)\,$ for magnetite-water nanofluid with parameter S when ${A}_{2}=0.2,{A}_{3}=0,\phi =0.2,S=5.$ |
Figure 6. (a) $f^{\prime} \left(\eta \right),$ (b) $g\left(\eta \right),$ (c) $\theta \left(\eta \right)\,$ for magnetite-water nanofluid with parameter ${A}_{2}$ when ${A}_{1}=0.2,{A}_{3}=0,\phi =0.2,S=-3.$ |
Figure 7. (a) $f^{\prime} \left(\eta \right),$ (b) $g\left(\eta \right),$ (c) $\theta \left(\eta \right)\,$ for magnetite-water nanofluid with parameter ${A}_{3}$ when ${A}_{1}=2,{A}_{2}=2,\phi =0.2,S=-1.$ |
Figure 8. Change of $R{e}_{x}^{1/2}{C}_{f}$ at $\eta =0$ for diverse values of (a) S, (b) ${A}_{1},$ (c) ${A}_{2},$ (d) ${A}_{3}.$ |
Figure 9. Change of $R{e}_{x}^{1/2}{C}_{f}$ at $\eta =1$ for diverse values of (a) S, (b) ${A}_{1},$ (c) ${A}_{2},$ (d) ${A}_{3}.$ |
Figure 10. Change of $R{e}_{x}^{-1/2}N{u}_{x}$ at $\eta =0$ for various values of (a) S, (b) ${A}_{1},$ (c) ${A}_{2},$ (d) ${A}_{3}.$ |
Figure 11. Change of $R{e}_{x}^{-1/2}N{u}_{x}$ at $\eta =1$ for diverse values of (a) S, (b) ${A}_{1},$ (c) ${A}_{2},$ (d) ${A}_{3}.$ |
Figure 12. Change of streamlines for diverse S values. |
5. Conclusions
• | The concentration of nanoparticles increases the velocity, along with the temperature of the nanofluid. |
• | The flow field is emphatically impacted by the injection/suction parameter. |
• | A surge in the magnetic as well as rotation factor results in a boost in the thermal field. |
• | The rate of heat transfer boosts for the magnetite-water nanofluid when compared to that for the standard fluid against the concentration of nanoparticles. |
• | The magnetic parameter augments the Nusselt number, although it diminishes the coefficient of the drag force. |
• | Drag force coefficient is lower for the magnetite-water nanofluid as compared to that for the standard fluid against the nanoparticle concentration. |
• | The rotation factor tends to increase the drag coefficient, although it slightly diminishes the Nusselt number. |