Modeling and analysis of thin film flow of Fuzzified Johnson Segalman nanofluid using fuzzy extension of He-Laplace scheme

Figures & data

Figure 1. Geometry of the problem.

Table 1. Thermophysical properties of base fluid and nanoparticles.

Base fluid/Nanoparticles	$\rho (kg/m^3)$	$C_p(J/kgK)$
Copper	18933	385
Water	997	4179

Table 2. Fuzzy solutions and residual errors for the influence of $\varphi$ (lifting problem) when $r=0.7$ ${\mathbb{W}}_e$ = 0.01, ${\mathbb{S}}_t$ = 0.001 and $\lambda$ = 0.1.

		$\varphi =0.1$		$\varphi =0.5$		$\varphi =0.8$
$r=0.7$	$X$	Solution	Error	Solution	Error	Solution	Error
(lower bound)	$0.0$	$-0.3$	$-$6.57$\times {10}^{-11}$	$-0.3000000$	$-$8.97$\times {10}^{-11}$	$-0.3000000$	$-$1.07$\times {10}^{-10}$
	$0.2$	$-0.360180$	$-$5.26$\times {10}^{-11}$	$-0.3601800$	$-$7.17$\times {10}^{-11}$	$-0.360180$	$-$8.60$\times {10}^{-11}$
	$0.4$	$-0.4203200$	$-$3.94$\times {10}^{-11}$	$-0.4203200$	$-$5.37$\times {10}^{-11}$	$-0.4203200$	$-$6.45$\times {10}^{-11}$
	$0.6$	$-0.4804200$	−2.63$\times {10}^{-11}$	$-0.4804200$	$-$3.59$\times {10}^{-11}$	$-0.4804200$	$-$4.30$\times {10}^{-11}$
	$0.8$	$-0.5404800$	$-$1.31$\times {10}^{-11}$	$-0.5704950$	$-$1.80$\times {10}^{-13}$	$-0.5404800$	$-$2.17$\times {10}^{-13}$
	$1.0$	$-0.600500$	$-$9.60$\times {10}^{-14}$	$-0.0073508$	$-$1.79$\times {10}^{-13}$	$-0.6005000$	$-$2.57$\times {10}^{-13}$
(upper bound)	$0.0$	$0.3000000$	6.50$\times {10}^{-11}$	$0.3000000$	8.85$\times {10}^{-11}$	$0.3000000$	1.06$\times {10}^{-11}$
	$0.2$	$0.3598199$	5.19$\times {10}^{-11}$	$0.3598199$	7.08$\times {10}^{-11}$	$0.35981999$	8.50$\times {10}^{-11}$
	$0.4$	$0.4196799$	3.90$\times {10}^{-11}$	$0.4196799$	5.31$\times {10}^{-11}$	$0.4196799$	6.37$\times {10}^{-11}$
	$0.6$	$0.479579$	2.62$\times {10}^{-11}$	$0.479579$	3.54$\times {10}^{-11}$	$0.4795799$	4.25$\times {10}^{-11}$
	$0.8$	$0.5395199$	1.29$\times {10}^{-11}$	$0.5395199$	1.76$\times {10}^{-11}$	$0.5395199$	2.11$\times {10}^{-11}$
	$1.0$	$0.5994999$	$-$9.61$\times {10}^{-14}$	$0.5994999$	$-$1.79$\times {10}^{-13}$	$0.5994999$	$-$2.57$\times {10}^{-13}$

Table 3. Fuzzy solutions and residual errors for the influence of ${\mathbb{S}}_t$ (lifting problem) when ${\mathbb{W}}_e$ = 0.001, $\varphi$ = 0.3 and $\lambda =\frac{1}{8}$.

		${\mathbb{S}}_t=0.001$		${\mathbb{S}}_t=0.01$		${\mathbb{S}}_t=0.1$
$r=0.2$	$X$	Solution	Error	Solution	Error	Solution	Error
(lower bound)	$0.0$	$-0.8$	$-$2.05$\times {10}^{-12}$	$-0.8$	$-$2.08$\times {10}^{-10}$	$-0.8$	$-$2.37$\times {10}^{-8}$
	$0.2$	$-0.960180$	$-$1.64$\times {10}^{-12}$	$-0.961800$	$-$1.66$\times {10}^{-10}$	$-0.978000$	$-$1.84$\times {10}^{-8}$
	$0.4$	$-1.120320$	$-$1.23$\times {10}^{-12}$	$-1.123200$	$-$1.24$\times {10}^{-11}$	$-1.152000$	$-$1.34$\times {10}^{-9}$
	$0.6$	$-1.280420$	$-$8.20$\times {10}^{-13}$	$-1.284200$	$-$8.24$\times {10}^{-11}$	$-1.3220000$	$-$8.71$\times {10}^{-9}$
	$0.8$	$-1.440480$	$-$4.10$\times {10}^{-13}$	$-1.444800$	$-$4.11$\times {10}^{-11}$	$-1.488000$	$-$4.23$\times {10}^{-9}$
	$1.0$	$-1.600500$	$-$6.71$\times {10}^{-16}$	$-1.605000$	$-$6.71$\times {10}^{-15}$	$-1.650000$	$-$6.71$\times {10}^{-14}$
(upper bound)	$0.0$	$0.8$	2.04$\times {10}^{-12}$	$0.8$	2.01$\times {10}^{-10}$	$0.8$	1.72$\times {10}^{-8}$
	$0.2$	$0.9598199$	1.63$\times {10}^{-12}$	$0.9581999$	1.61$\times {10}^{-10}$	$0.941999$	1.43$\times {10}^{-8}$
	$0.4$	$1.11967999$	1.22$\times {10}^{-12}$	$1.1167999$	1.21$\times {10}^{-11}$	$1.0879999$	1.11$\times {10}^{-9}$
	$0.6$	$1.2795799$	8.17$\times {10}^{-13}$	$1.2757999$	8.13$\times {10}^{-11}$	$1.2379999$	7.67$\times {10}^{-9}$
	$0.8$	$1.4395199$	4.08$\times {10}^{-13}$	$1.4351999$	4.08$\times {10}^{-11}$	$1.3919999$	3.97$\times {10}^{-9}$
	$1.0$	$1.599499$	$-$6.71$\times {10}^{-16}$	$1.5949999$	$-$6.71$\times {10}^{-15}$	$1.549999$	$-$6.71$\times {10}^{-14}$

Table 4. Fuzzy solutions and residual errors of the membership function under the influence of $\lambda$ (lifting problem) when ${\mathbb{S}}_t$ = 0.001 ${\mathbb{W}}_e$ = 0.002 and $\varphi =0.03$.

		$\lambda =0.3$		$\lambda =0.6$		$\lambda =0.9$
$X=1.0$	$r$	Solution	Error	Solution	Error	Solution	Error
(lower bound)	$0.0$	$-2.0005$	$-$1.38$\times {10}^{-14}$	$-2.0005$	$-$6.81$\times {10}^{-15}$	$-2.0005$	$-$6.01$\times {10}^{-16}$
	$0.2$	$-1.6005$	$-$5.65$\times {10}^{-15}$	$-1.6005$	$-$2.79$\times {10}^{-15}$	$-1.6005$	$-$2.46$\times {10}^{-16}$
	$0.4$	$-1.2005$	$-$1.79$\times {10}^{-15}$	$-1.2005$	$-$8.84$\times {10}^{-15}$	$-1.2005$	$-$7.77$\times {10}^{-17}$
	$0.6$	$-0.8005$	$-$3.53$\times {10}^{-16}$	$-0.8005$	$-$1.75$\times {10}^{-16}$	$-0.8005$	$-$1.56$\times {10}^{-17}$
	$0.8$	$-0.4005$	$-$2.21$\times {10}^{-17}$	$-0.4005$	$-$1.09$\times {10}^{-17}$	$-0.4005$	$-$7.84$\times {10}^{-19}$
	$1.0$	$-0.0005$	$-$4.82$\times {10}^{-25}$	$-0.0005$	2.80$\times {10}^{-25}$	$-0.0005$	$-$2.17$\times {10}^{-25}$
(upper bound)	$0.0$	$1.995$	$-$1.38$\times {10}^{-14}$	$1.995$	$-$6.81$\times {10}^{-15}$	$1.995$	$-$6.01$\times {10}^{-16}$
	$0.2$	$1.5995$	$-$5.65$\times {10}^{-15}$	$1.5995$	$-$2.79$\times {10}^{-15}$	$1.5995$	$-$2.46$\times {10}^{-16}$
	$0.4$	$1.1995$	$-$1.79$\times {10}^{-15}$	$1.1995$	$-$8.84$\times {10}^{-15}$	$1.1995$	$-$7.77$\times {10}^{-17}$
	$0.6$	$0.7995$	$-$3.53$\times {10}^{-16}$	$0.7995$	$-$1.75$\times {10}^{-16}$	$0.7995$	$-$1.56$\times {10}^{-17}$
	$0.8$	$0.3995$	$-$2.21$\times {10}^{-17}$	$0.3995$	$-$1.09$\times {10}^{-17}$	$0.3995$	$-$7.84$\times {10}^{-19}$
	$1.0$	$0.0005$	$-$4.82$\times {10}^{-25}$	$0.0005$	2.80$\times {10}^{-25}$	$0.0005$	$-$2.17$\times {10}^{-25}$

Table 5. Fuzzy solutions and residual errors of the membership function under the influence of ${\mathbb{W}}_e$ (lifting problem) when ${\mathbb{S}}_t$ = 0.001 $\lambda$ = 0.99 and $\varphi =0.1$.

		${\mathbb{W}}_e=0.002$		${\mathbb{W}}_e=0.02$		${\mathbb{W}}_e=0.2$
$X=1.0$	$r$	Solution	Error	Solution	Error	Solution	Error
(lower bound)	$0.0$	$-2.0005$	$-$7.74$\times {10}^{-18}$	$-2.0005$	$-$7.67$\times {10}^{-14}$	$-2.0005$	$-$7.68$\times {10}^{-10}$
	$0.2$	$-1.6005$	$-$3.14$\times {10}^{-18}$	$-1.6005$	$-$3.14$\times {10}^{-14}$	$-1.6005$	$-$3.14$\times {10}^{-10}$
	$0.4$	$-1.2005$	$-$1.18$\times {10}^{-18}$	$-1.2005$	$-$9.94$\times {10}^{-15}$	$-1.2005$	$-$9.94$\times {10}^{-11}$
	$0.6$	$-0.8005$	$-$1.98$\times {10}^{-19}$	$-0.8005$	$-$1.96$\times {10}^{-15}$	$-0.8005$	$-$1.96$\times {10}^{-11}$
	$0.8$	$-0.4005$	1.45$\times {10}^{-19}$	$-0.4005$	$-$1.23$\times {10}^{-16}$	$-0.4005$	$-$1.22$\times {10}^{-12}$
	$1.0$	$-0.0005$	2.17$\times {10}^{-19}$	$-0.0005$	$-$2.17$\times {10}^{-19}$	$-0.0005$	$-$2.25$\times {10}^{-19}$
(upper bound)	$0.0$	$1.995$	$-$7.74$\times {10}^{-18}$	$1.995$	$-$7.67$\times {10}^{-14}$	$1.995$	$-$7.68$\times {10}^{-10}$
	$0.2$	$1.5995$	$-$3.14$\times {10}^{-18}$	$1.5995$	$-$3.14$\times {10}^{-14}$	$1.5995$	$-$3.14$\times {10}^{-10}$
	$0.4$	$1.1995$	$-$1.18$\times {10}^{-18}$	$1.1995$	$-$9.94$\times {10}^{-15}$	$1.1995$	$-$9.94$\times {10}^{-11}$
	$0.6$	$0.7995$	$-$1.98$\times {10}^{-19}$	$0.7995$	$-$1.96$\times {10}^{-15}$	$0.7995$	$-$1.96$\times {10}^{-11}$
	$0.8$	$0.3995$	1.45$\times {10}^{-19}$	$0.3995$	$-$1.23$\times {10}^{-16}$	$0.3995$	$-$1.22$\times {10}^{-12}$
	$1.0$	$0.0005$	2.17$\times {10}^{-19}$	$0.0005$	$-$2.17$\times {10}^{-19}$	$0.0005$	$-$2.25$\times {10}^{-19}$

Table 6. Fuzzy flow rate in lifting and drainage case when $\varphi =0.6$, ${\mathbb{W}}_e$ = 0.5, ${\mathbb{S}}_t$ = 0.3 and $\lambda$ = 0.1.

	Flow Rate in Lifting Case		Flow Rate in Draining Case
$r-cut$	${\mathcal{Q}}_L$	${\mathcal{Q}}_U$	${\mathcal{Q}}_L$	${\mathcal{Q}}_U$
$0.0$	$-1.73316$	$1.31009$	$-1.31009$	$1.73316$
$0.2$	$-1.38003$	$1.04767$	$-1.04767$	$1.38003$
$0.4$	$-1.04359$	$0.773674$	$-0.773674$	$1.04359$
$0.6$	$-0.719828$	$0.490183$	$-0.490183$	$0.719828$
$0.8$	$-0.405894$	$0.198695$	$-0.198695$	$0.405894$
$1.0$	$-0.1$	$-0.1$	$0.1$	$0.1$

Table 7. Fuzzy solutions and residual errors for the influence of $\varphi$ (drainage problem) when ${\mathbb{W}}_e$ = 0.01, ${\mathbb{S}}_t$ = 0.001 and $\lambda$ = 0.2.

		$\varphi =0.1$		$\varphi =0.5$		$\varphi =0.7$
$r=0.7$	$X$	Solution	Error	Solution	Error	Solution	Error
(lower bound)	$0.0$	$-0.3$	$-$6.30$\times {10}^{-11}$	$-0.3$	$-$8.59$\times {10}^{-11}$	$-0.3$	$-$9.73$\times {10}^{-11}$
	$0.2$	$-0.35982$	$-$5.04$\times {10}^{-11}$	$-0.35982$	$-$6.87$\times {10}^{-11}$	$-0.35982$	$-$7.79$\times {10}^{-11}$
	$0.4$	$-0.41968$	$-$3.78$\times {10}^{-10}$	$-0.41968$	$-$5.16$\times {10}^{-11}$	$-0.41968$	$-$5.84$\times {10}^{-11}$
	$0.6$	$-0.47958$	$-$2.52$\times {10}^{-11}$	$-0.47958$	$-$3.43$\times {10}^{-11}$	$-0.47958$	$-$3.89$\times {10}^{-11}$
	$0.8$	$-0.53952$	$-$1.26$\times {10}^{-11}$	$-0.53952$	$-$1.71$\times {10}^{-11}$	$-0.53952$	$-$1.94$\times {10}^{-11}$
	$1.0$	$-0.5995$	$9.03\times {10}^{-14}$	$-0.5995$	1.68$\times {10}^{-13}$	$-0.5995$	2.16$\times {10}^{-13}$
(upper bound)	$0.0$	$0.3$	6.38$\times {10}^{-11}$	$0.3$	8.69$\times {10}^{-11}$	$0.3$	9.85$\times {10}^{-11}$
	$0.2$	$0.36018$	5.10$\times {10}^{-11}$	$0.36018$	6.95$\times {10}^{-11}$	$0.36018$	7.88$\times {10}^{-11}$
	$0.4$	$0.42032$	3.82$\times {10}^{-11}$	$0.42032$	5.21$\times {10}^{-11}$	$0.42032$	5.91$\times {10}^{-11}$
	$0.6$	$0.48042$	2.55$\times {10}^{-11}$	$0.48042$	3.48$\times {10}^{-11}$	$0.48042$	3.94$\times {10}^{-11}$
	$0.8$	$0.54048$	1.28$\times {10}^{-11}$	$0.54048$	1.75$\times {10}^{-11}$	$0.54048$	1.98$\times {10}^{-11}$
	$1.0$	$0.6005$	9.03$\times {10}^{-14}$	$0.6005$	1.68$\times {10}^{-13}$	$0.6005$	2.16$\times {10}^{-13}$

Table 8. Fuzzy solutions and residual errors for the influence of ${\mathbb{S}}_t$ (draining problem) when ${\mathbb{W}}_e$ = 0.002, $\varphi$ = 0.3 and $\lambda =0.84$.

		${\mathbb{S}}_t=0.001$		${\mathbb{S}}_t=0.01$		${\mathbb{S}}_t=0.1$
$r=0.35$	$X$	Solution	Error	Solution	Error	Solution	Error
(lower bound)	$0.0$	$-0.65$	$-$7.85$\times {10}^{-13}$	$-0.65$	$-$7.71$\times {10}^{-11}$	$-0.65$	$-$6.33$\times {10}^{-9}$
	$0.2$	$-0.77982$	$-$6.28$\times {10}^{-13}$	$-0.7782$	$-$6.20$\times {10}^{-11}$	$-0.762$	$-$5.31$\times {10}^{-9}$
	$0.4$	$-0.90968$	$-$4.71$\times {10}^{-13}$	$-0.9068$	$-$4.67$\times {10}^{-11}$	$-0.878$	$-$4.17$\times {10}^{-9}$
	$0.6$	$-1.03958$	$-$3.14$\times {10}^{-13}$	$-1.0358$	$-$3.12$\times {10}^{-11}$	$-0.998$	$-$2.90$\times {10}^{-9}$
	$0.8$	$-1.16952$	$-$1.57$\times {10}^{-13}$	$-1.1652$	$-$1.57$\times {10}^{-11}$	$-1.122$	$-$1.51$\times {10}^{-9}$
	$1.0$	$-1.2995$	6.56$\times {10}^{-17}$	$-1.295$	6.54$\times {10}^{-16}$	$-1.25$	6.55$\times {10}^{-15}$
(upper bound)	$0.0$	$0.65$	7.88$\times {10}^{-13}$	$0.65$	8.02$\times {10}^{-11}$	$0.65$	9.41$\times {10}^{-9}$
	$0.2$	$0.78018$	6.31$\times {10}^{-13}$	$0.7818$	6.39$\times {10}^{-11}$	$0.798$	7.28$\times {10}^{-9}$
	$0.4$	$0.91032$	4.73$\times {10}^{-13}$	$0.9132$	4.78$\times {10}^{-11}$	$0.942$	5.27$\times {10}^{-9}$
	$0.6$	$1.04042$	3.15$\times {10}^{-13}$	$1.0442$	3.17$\times {10}^{-11}$	$1.082$	3.39$\times {10}^{-9}$
	$0.8$	$1.17048$	1.57$\times {10}^{-13}$	$1.1748$	1.58$\times {10}^{-11}$	$1.218$	1.63$\times {10}^{-9}$
	$1.0$	$1.3005$	6.56$\times {10}^{-17}$	$1.305$	6.54$\times {10}^{-16}$	$1.35$	6.54$\times {10}^{-15}$

Table 9. Fuzzy solutions and residual errors of the membership function under the influence of $\lambda$ (draining problem) when ${\mathbb{S}}_t$ = 0.001 ${\mathbb{W}}_e$ = 0.002 and $\varphi =0.03$.

		$\lambda =0.3$		$\lambda =0.6$		$\lambda =0.9$
$X=1.0$	$r$	Solution	Error	Solution	Error	Solution	Error
(lower bound)	$0.0$	– 1.995	1.41$\times {10}^{-14}$	– 1.995	6.95$\times {10}^{-15}$	– 1.995	6.13$\times {10}^{-16}$
	$0.2$	– 1.5995	3.76$\times {10}^{-15}$	– 1.5995	2.85$\times {10}^{-15}$	– 1.5995	2.51$\times {10}^{-16}$
	$0.4$	– 1.1995	1.82$\times {10}^{-15}$	– 1.1995	9.01$\times {10}^{-16}$	– 1.1995	7.93$\times {10}^{-17}$
	$0.6$	– 0.7995	3.59$\times {10}^{-16}$	– 0.7995	1.78$\times {10}^{-16}$	– 0.7995	1.59$\times {10}^{-17}$
	$0.8$	– 0.3995	2.26$\times {10}^{-17}$	– 0.3995	1.11$\times {10}^{-17}$	– 0.3995	8.03$\times {10}^{-19}$
	$1.0$	– 0.0005	4.79$\times {10}^{-25}$	– 0.0005	– 2.61$\times {10}^{-25}$	– 0.0005	2.17$\times {10}^{-19}$
(upper bound)	$0.0$	2.0005	1.41$\times {10}^{-14}$	2.0005	6.95$\times {10}^{-15}$	2.0005	6.13$\times {10}^{-16}$
	$0.2$	1.6005	3.76$\times {10}^{-15}$	1.6005	2.85$\times {10}^{-15}$	1.6005	2.51$\times {10}^{-16}$
	$0.4$	1.2005	1.82$\times {10}^{-15}$	1.2005	9.01$\times {10}^{-16}$	1.2005	7.93$\times {10}^{-17}$
	$0.6$	0.8005	3.59$\times {10}^{-16}$	0.8005	1.78$\times {10}^{-16}$	0.8005	1.59$\times {10}^{-17}$
	$0.8$	0.4005	2.26$\times {10}^{-17}$	0.4005	1.11$\times {10}^{-17}$	0.4005	8.03$\times {10}^{-19}$
	$1.0$	0.0005	4.79$\times {10}^{-25}$	0.0005	– 2.61$\times {10}^{-25}$	0.0005	2.17$\times {10}^{-19}$

Table 10. Fuzzy solutions and residual errors of the membership function under the influence of ${\mathbb{W}}_e$ (draining problem) when ${\mathbb{S}}_t$ = 0.001 $\lambda$ = 0.99 and $\varphi =0.11$.

		${\mathbb{W}}_e=0.002$		${\mathbb{W}}_e=0.02$		${\mathbb{W}}_e=0.2$
$X=1.0$	$r$	Solution	Error	Solution	Error	Solution	Error
(lower bound)	$0.0$	– 1.995	7.88$\times {10}^{-18}$	– 1.995	7.80$\times {10}^{-14}$	– 1.995	7.81$\times {10}^{-10}$
	$0.2$	– 1.5995	3.42$\times {10}^{-18}$	– 1.5995	3.19$\times {10}^{-14}$	– 1.5995	3.20$\times {10}^{-10}$
	$0.4$	– 1.1995	9.85$\times {10}^{-19}$	– 1.1995	1.01$\times {10}^{-15}$	– 1.1995	1.01$\times {10}^{-10}$
	$0.6$	– 0.7995	4.19$\times {10}^{-19}$	– 0.7995	1.99$\times {10}^{-15}$	– 0.7995	1.99$\times {10}^{-11}$
	$0.8$	– 0.3995	– 7.16$\times {10}^{-20}$	– 0.3995	1.25$\times {10}^{-16}$	– 0.3995	1.25$\times {10}^{-12}$
	$1.0$	– 0.0005	– 9.98$\times {10}^{-26}$	– 0.0005	2.17$\times {10}^{-19}$	– 0.0005	2.23$\times {10}^{-19}$
(upper bound)	$0.0$	2.0005	7.88$\times {10}^{-18}$	2.0005	7.80$\times {10}^{-14}$	2.0005	7.81$\times {10}^{-10}$
	$0.2$	1.6005	3.42$\times {10}^{-18}$	1.6005	3.19$\times {10}^{-14}$	1.6005	3.20$\times {10}^{-10}$
	$0.4$	1.2005	9.85$\times {10}^{-19}$	1.2005	1.01$\times {10}^{-15}$	1.2005	1.01$\times {10}^{-10}$
	$0.6$	0.8005	4.19$\times {10}^{-19}$	0.8005	1.99$\times {10}^{-15}$	0.8005	1.99$\times {10}^{-11}$
	$0.8$	0.4005	– 7.16$\times {10}^{-20}$	0.4005	1.25$\times {10}^{-16}$	0.4005	1.25$\times {10}^{-12}$
	$1.0$	0.0005	– 9.98$\times {10}^{-26}$	0.0005	2.17$\times {10}^{-19}$	0.0005	2.23$\times {10}^{-19}$

Table 11. Comparison of HLM with ADM.

$r=1$	$X$	HLM Sol.	ADM Sol.	Error HLPM	Error ADM [47]
(crisp)	$0.1$	$0.0095$	$0.0095$	$1.04\times {10}^{-8}$	$3.63\times {10}^{-8}$
	$0.3$	$0.0256$	$0.0256$	$1.72\times {10}^{-8}$	$1.33\times {10}^{-8}$
	$0.5$	$0.0377$	$0.0377$	– $1.82\times {10}^{-8}$	$3.46\times {10}^{-8}$
	$0.7$	$0.0457$	$0.0457$	– $1.82\times {10}^{-8}$	$4.48\times {10}^{-9}$
	$0.9$	$0.0497$	$0.0497$	– $1.82\times {10}^{-8}$	$5.53\times {10}^{-11}$

Figure 2. Influence of increasing ${\mathbb{W}}_e$ on flow rate (lifting case) when $\varphi$=8, $\lambda =1.45$, ${\mathbb{W}}_e$=0.3 and ${\mathbb{S}}_t$ =0.5.

Figure 7. Influence of increasing $\lambda$ on triangular membership function (lifting case) for $\varphi$=2, $X$=1, $W_e$=0.3 and $S_t$ =1.

Figure 8. Influence of $\varphi$ on triangular membership function (lifting case) for $\lambda$=0.1, $x$=0.5, $W_e$=1 and $S_t$=0.2.

Figure 9. Influence of increasing $W_e$ on flow rate (draining case) when $\varphi$=4, $\lambda =1.3$ and $S_t$=0.5.

Figure 10. Influence of $\lambda$ on fuzzy velocity profile (draining case) when $r=0.4$, $\varphi$ = 0.03 and $W_e$=0.4.

Figure 11. Influence of $S_t$ on fuzzy velocity profile (draining case) when $\lambda$=0.2,$r=0.4$, $\varphi$ = 0.2 and $W_e$=1.0.

Figure 12. Influence of $W_e$ on fuzzy velocity profile (draining case) when $\lambda$=0.14, $r=0.2$, $\varphi$ = 2 and $S_t$ =0.9.

Figure 13. Influence of increasing $\varphi$ on fuzzy velocity profile (draining case) for $r$=0.53, $\lambda =3$ $W_e$=0.98 and $S_t$ =0.65.

Figure 14. Influence of increasing $\lambda$ on triangular membership function (draining case) for $\varphi$=2, $x$=1, $W_e$=0.3 and $S_t$ =1.

Figure 3. Influence of $\lambda$ on fuzzy velocity profile (lifting case) when $r=0.75$, ${\mathbb{S}}_t=1$, $\varphi$ = 0.03 and ${\mathbb{W}}_e$ =0.4.

Figure 4. Influence of $\varphi$ on fuzzy velocity profile (lifting case) when $r=0.24$, ${\mathbb{S}}_t=0.2$, $\lambda$ = 0.01 and ${\mathbb{W}}_e$ =1.

Figure 5. Influence of ${\mathbb{S}}_t$ on fuzzy velocity profile (lifting case) when $\lambda$=0.2,$r$=0.51, $\varphi$ = 0.3 and ${\mathbb{W}}_e$ =1.0.

Figure 6. Influence of ${\mathbb{W}}_e$ on fuzzy velocity profile (lifting case) when $\lambda$=0.1,$r$=0.6, $\varphi$ = 4 and ${\mathbb{S}}_t$ =0.5.

M. Kamran Alam, M.T. Rahim, T. Haroon, S. Islam, and A.M. Siddiqui, Solution of steady thin film flow of johnson–segalman fluid on a vertical moving belt for lifting and drainage problems using adomian decomposition method, Appl. Math. Comput. 218 (21) (2012), pp. 10413–10428. doi:10.1016/j.amc.2012.03.095.

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