Brownian motion and thermophoresis influence in magnetized Maxwell upper-convected stagnation point fluid flow via a stretching porous surface

Figures & data

Figure 1. Physical model of the problem.

Figure 2. $M_p$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$.

Figure 3. $\alpha$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$.

Figure 4. $\lambda$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Figure 5. Darcy number $Da$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$.

Figure 6. Chemical reaction parameter $Kn$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Figure 7. $Nb$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Figure 8. $Nt$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Figure 9. Velocity slip $S_{\nu }$ and thermal slip $S_t$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\vartheta \left(\eta \right)$.

Figure 10. Solutal slip $S_o$ on $\vartheta \left(\eta \right)$ and $\varphi \left(\eta \right)$.

Table 1. $f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$, ${\vartheta }^{\mathrm{\prime }}\left(0\right)$ and ${\varphi }^{\mathrm{\prime }}\left(0\right)$ for constant values $Le=1$, $S=-0.5$, ${\text{β}}_n=0.01$, $\zeta =0.01$ and variation of other flow parameters.

$\lambda$	$Nt$	$Nb$	$Da$	$M_p$	$Pr$	$E_c$	$S_{\nu }$	$S_t$	$R_d$	$\alpha$	$f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$	$-{\vartheta }^{\mathrm{\prime }}\left(0\right)$	$-{\varphi }^{\mathrm{\prime }}\left(0\right)$
0.5	0.2	0.2	5	0.5	1.8	0.1	0.3	0.3	10	$\pi /2$	−0.463344	0.265015	0.614910
0.8											−0.201600	0.340889	0.645896
	0.5										−0.463344	0.223149	0.726435
		0.3									−0.463344	0.244782	0.605082
			10								−0.454898	0.266954	0.615199
				1.0							−0.501774	0.256543	0.613629
					2.2						−0.463344	0.251899	0.658834
						0.2					−0.463344	0.252772	0.623277
							1.0				−0.276511	0.245801	0.595464
								1.0			−0.463344	0.225353	0.606913
									5		−0.463344	0.268798	0.606923
										$\pi /6$	−0.430097	0.307495	0.724908

Table 2. Comparison of $f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$ and ${\varphi }^{\mathrm{\prime }}\left(0\right)$ with $Kn$ by setting $M_p=Pr=Le=1,$ ${\text{β}}_n=\lambda =0.2$ and $Da=Nt=\zeta =S_{\nu }=0$.

	Shahid [19]	Agunbiade et al. [36]	Present	Shahid [19]	Agunbiade et al. [36]	Present
$K_n$	$f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$	$f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$	$f^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)$	$-{\varphi }^{\mathrm{\prime }}\left(0\right)$	$-{\varphi }^{\mathrm{\prime }}\left(0\right)$	$-{\varphi }^{\mathrm{\prime }}\left(0\right)$
1.0	1.272470	1.272470	1.272470	1.16786	1.16788	1.16788
1.2	–	–	–	1.25227	1.25228	1.25228
1.5	–	–	–	1.36885	1.36886	1.36886

Table 3. Residual error analysis of the respective momentum, energy and concentration distribution for the define data $\lambda =0.5$, $M_p=0.5$, $R_d=10$, $Pr=6.8$, $Kn=0.3$, $E_c=0.1$, $Nt=0.2$, $Nb=0.2$, $Le=1$, $S=-0.5$, $S_{\nu }=0.3$, $S_t=0.3$, $S_o=0.3$, ${\text{β}}_n=0.2$, $\zeta =0.01$, $Da=10$, and $\alpha =\pi /3$.

Np	$\left|\text{Re}{\text{s}}_f\left(\eta \right)\right|$	$\left|\text{Re}{\text{s}}_{\vartheta }\left(\eta \right)\right|$	$\left|\text{Re}{\text{s}}_{\phi }\left(\eta \right)\right|$
$5$	$4.0518630417\times {10}^{-1}$	$1.6092651982\times {10}^{-2}$	$5.3660450658\times {10}^{-2}$
$8$	$1.5471094404\times {10}^{-2}$	$8.8612831793\times {10}^{-3}$	$6.1491649243\times {10}^{-3}$
$11$	$7.0553604843\times {10}^{-4}$	$7.8284204245\times {10}^{-4}$	$3.1576492279\times {10}^{-4}$
$14$	$3.1337779797\times {10}^{-5}$	$6.9267194511\times {10}^{-6}$	$3.1627430407\times {10}^{-5}$
$17$	$2.0307966870\times {10}^{-6}$	$3.8183103897\times {10}^{-6}$	$1.4977840037\times {10}^{-6}$
$20$	$1.6834982678\times {10}^{-8}$	$1.1788969374\times {10}^{-7}$	$1.1724493597\times {10}^{-8}$
$23$	$1.8282973616\times {10}^{-8}$	$3.4052651363\times {10}^{-8}$	$9.6155563438\times {10}^{-9}$
$26$	$8.4649453750\times {10}^{-9}$	$2.7721791457\times {10}^{-9}$	$2.0066340040\times {10}^{-9}$
$30$	$1.2158786195\times {10}^{-9}$	$5.0074415932\times {10}^{-9}$	$2.3271267156\times {10}^{-9}$

Shahid A. The effectiveness of mass transfer in the MHD upper-convected Maxwell fluid flow on a stretched porous sheet near stagnation point: A numerical investigation. Inventions. 2020;5:64. doi:10.3390/inventions5040064

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Agunbiade SA, Oyekunle TL, Akolade MT. Radiative and MHD Dissipative heat effects on upper-convected maxwell fluid flow and material time relaxation over a permeable stretched sheet. Comput Therm Sci. 2023;15(3):45–59. doi:10.1615/ComputThermalScien.2022043596

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