Mathematical model of oxytactic bacteria’s role on MHD nanofluid flow across a circular cylinder: application of drug-carrier in hypoxic tumour

Figures & data

Figure 1. (a) Tumour microenvironment to the breast cancer [26], (b) Flow model containing oxytactic bacteria and nanoparticle with coordinate system, (c) Oxytactic bacteria injected intravenously into human.

Table 1. Comparison of $-{\theta }^{\mathrm{\prime }}\left(0\right)$ for different values of $\lambda$ for $N_r=N_t=N_b=R_c=0$, $B_{i1}\to \mathrm{\infty }$, and $P_r=1$.

	Merkin	Nazar	Rashad	Present
$\lambda$	[39]	et al. [36]	et al. [38]	outcomes
$-1.75$	$0.4199$	$0.4205$	$0.4202$	$0.4198$
$-1.5$	$0.4576$	$0.4601$	$0.4579$	$0.4573$
$-1.0$	$0.5067$	$0.5080$	$0.5068$	$0.5067$
$-0.5$	$0.5420$	$0.5430$	$0.5421$	$0.5421$
$0.0$	$0.5705$	$0.5710$	$0.5706$	$0.5705$
$0.5$	$0.5943$	$0.5949$	$0.5947$	$0.5945$
$0.88$	$0.6096$	$0.6112$	$0.6111$	$0.6111$
$0.89$	$0.6110$	$0.6116$	$0.6114$	$0.6113$
$1.0$	$0.6158$	$0.6160$	$0.6160$	$0.6159$
$2.0$	$0.6497$	$0.6518$	$0.6518$	$0.6518$
$5.0$	$0.7315$	$0.7320$	$0.7319$	$0.7315$

Figure 2. Effect of ${\mathit{P}}_e$ on velocity profile $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 3. Effect of ${\mathit{P}}_e$ on temperature profile $\mathit{\theta }$.

Figure 4. Effect of ${\mathit{P}}_e$ on motile bacterial density profile $\mathrm{\aleph }$.

Figure 5. Effect of ${\mathit{P}}_e$ on oxygen concentrations.

Figure 6. Effect of $\mathit{P}ϵ$ on mobile bacterial density profile $\mathrm{\aleph }$.

Figure 7. Effect of ${\mathit{B}}_{\mathrm{\Delta }\mathit{n}}$ on oxygen concentration profile $C$

Figure 8. Effect of ${\mathit{R}}_{\mathit{b}}$ on velocity profile $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 9. Effect of ${\mathit{R}}_{\mathit{b}}$ on velocity profile $\theta$.

Figure 10. Effect of ${\mathit{R}}_{\mathit{b}}$ on motile bacterial density profile $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 11. Effect of ${\mathit{R}}_{\mathit{b}}$ on oxygen concentration profile $\theta$.

Figure 12. Effect of ${\mathit{N}}_{\mathit{b}}$ on temperature profile $\theta$.

Figure 13. Effect of ${\mathit{N}}_{\mathit{t}}$ on temperature profile $\theta$.

Figure 14. Effect of ${\mathit{N}}_{\mathit{t}}$ on motile bacterial density profile $\mathrm{\aleph }$.

Figure 15. Effect of ${\mathit{N}}_{\mathit{t}}$ on oxygen concentration profile $C$.

Figure 16. Effect of ${\mathit{P}}_{\mathit{r}}$ on velocity profile $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 17. Effect of ${\mathit{P}}_{\mathit{r}}$ on motile bacterial density profile $\mathrm{\aleph }$.

Figure 18. Effect of $\lambda$ on velocity profile $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 19. Effect of $\lambda$ on motile bacterial density profile $\mathrm{\aleph }$.

Figure 20. Effect of $\mathbf{M}$ on motile bacterial density profile $f^{\mathrm{\prime }}\left(\eta \right)$.

Figure 21. Effect of $\mathbf{M}$ on motile bacterial density profile $\mathrm{\aleph }$.

Figure 22. Flow chart for describing the numerical approach.

Table 2. Behaviour of $f^{\mathrm{\prime }\mathrm{\prime }}{|}_{\eta =0}$, $-{\theta }^{\mathrm{\prime }}{|}_{\eta =0}$, $-{\mathrm{\aleph }}^{\mathrm{\prime }}{|}_{\eta =0}$, $-{\phi }^{\mathrm{\prime }}{|}_{\eta =0}$ and $-C^{\mathrm{\prime }}{|}_{\eta =0}$ for different physical parameter.

$\lambda$	$P_r$	$N_r$	$N_t$	$N_b$	$S_c$	$L_e$	$W$	$B_{\mathrm{\Delta }n}$	$f^{\mathrm{\prime }\mathrm{\prime }}{|}_{\eta =0}$	$-{\theta }^{\mathrm{\prime }}{|}_{\eta =0}$	$-{\mathrm{\aleph }}^{\mathrm{\prime }}{|}_{\eta =0}$	$-{\phi }^{\mathrm{\prime }}{|}_{\eta =0}$	$-C^{\mathrm{\prime }}{|}_{\eta =0}$
$0.5$									$0.7945$	$0.2098$	$1.1087$	$0.1299$	$0.1669$
$1$									$1.6780$	$0.2930$	$1.2450$	$0.1607$	$0.1933$
$1.5$									$2.4798$	$0.3450$	$1.3069$	$0.1815$	$0.2117$
$0.5$	$1$								$0.7945$	$0.2098$	$1.1087$	$0.1299$	$0.1669$
	$1.6$								$0.7854$	$0.2181$	$1.4403$	$0.1562$	$0.2053$
	$2$								$0.7778$	$0.2191$	$1.6785$	$0.1713$	$0.2272$
	$1$	$1.1$							$0.9171$	$0.2157$	$1.1252$	$0.1312$	$0.1682$
		$1.3$							$0.9570$	$0.2175$	$1.1301$	$0.1316$	$0.1685$
		$1.5$							$1.0160$	$0.2202$	$1.1372$	$0.1322$	$0.1691$
		$1.1$	$0.1$						$1.0865$	$0.1930$	$5.8746$	$0.1325$	$0.1696$
			$0.7$						$1.1903$	$0.1929$	$8.7134$	$0.1333$	$0.1704$
			$1.3$						$1.3099$	$0.1911$	$11.9561$	$0.13409$	$0.1713$
			$0.1$	$1$					$1.0865$	$0.19297$	$5.8746$	$0.1325$	$0.1696$
				$3$					$0.9393$	$0.1608$	$2.6507$	$0.1302$	$0.1674$
				$6$					$0.9062$	$0.1302$	$2.1154$	$0.1295$	$0.1668$
				$1$	$0.3$				$1.0840$	$0.1930$	$5.9691$	$0.1324$	$0.1695$
					$0.5$				$1.0826$	$0.1930$	$6.0312$	$0.1324$	$0.1695$
					$0.7$				$1.0804$	$0.1930$	$6.1232$	$0.1323$	$0.1694$
					$0.3$	$0.1$			$1.0840$	$0.1930$	$5.9691$	$0.1324$	$0.1695$
						$0.7$			$1.4243$	$0.2805$	$15.6685$	$0.3038$	$0.4228$
						$1.3$			$1.8529$	$0.3726$	$26.2805$	$0.4186$	$0.5933$
						$0.1$	$0.2$		$1.1618$	$0.1142$	$8.7645$	$0.1320$	$0.1693$
							$0.4$		$1.3610$	−$0.1909$	$15.8415$	$0.1305$	$0.1683$
							$0.6$		$1.7540$	−$1.2026$	$29.0220$	$0.1249$	$0.1640$
							$0.2$	$0.3$	$1.2116$	$0.1247$	$9.7818$	$0.1354$	$0.2030$
								$0.6$	$1.2870$	$0.1407$	$11.3507$	$0.1405$	$0.2533$
								$0.9$	$1.3636$	$0.1569$	$12.9736$	$0.1456$	$0.3033$

Figure 23. Scheme of the integration between experiments and mathematical models [54].

Table

Parameters	${\mathit{L}}_{\mathit{e}}$	${\mathit{N}}_{\mathit{r}}$	${\mathit{N}}_{\mathit{b}}$	${\mathit{S}}_{\mathit{c}}={\mathit{S}}_{\mathit{b}}$	${\mathit{N}}_{\mathit{t}}$	${\mathit{E}}_{\mathit{c}}$	$\lambda$	${\mathit{P}}_{\mathit{e}}$	${\mathit{P}}_{\mathit{r}}$
Values	$0.5$	$0.5$	$1$	$5$	$0.1$	$0.1$	$1$	0.7	0.7

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