MHD mixed convection heat transfer over a non-linear slender elastic sheet with variable fluid properties

Comparison of wall temperature gradient θ′ (0) for different values of Prandtl number and magnetic parameter when Mn = 0_0, ε = 0_0, λ = 0_ and m = 1_0_

Pr	Present results	Gmbka & Bobba [32]	Chen [33]	Ali [34]
0.01	-0.01017936	-0.0099	0.0091	-
0.72	-0.4631462	-0.4631	-0.46315	-0.4617
1.0	-0.5826707	-0.5820	-0.58199	-0.5 801
3.0	-1.16517091	-1.1652	-1.16523	-1.1599
5.0	-1.56800866	-	-	-
10.0	-2.308029	-2.3080	-2.30796	-2.2960
100.0	-7.769667	-7.7657	-	-

Variation of skin fricrion f″(0) and uall-tempeiarure gradient θ′(0) for different values of pliyskal parameters_

m	Mn	θr	Pr	ε	λ	α = 0.0		α = 0.1		α = 0.2
						f″(0)	θ′(0)	f″(0)	θ′(0)	f″(0)	θ′(0)
2.0	0.1	-5.0	1.0	0.1	-5.0	-1.545068	-1.080793	-1.522204	-1.068792	-1.499637	-1.057054
					0.0	-1335278	-1.146790	-1.331519	-1.134678	-1.310604	-1.122755
					0.5	-1.185840	-1.189561	-1.165799	-1.177318	-1.146117	-1.165239
					1.0	-1.031486	-1.222858	-1.012449	-1-210512	-0.993795	-1.198313
5.0	0.5	-5.0	1.0	0.0	0.1	-1353883	-1.530281	-1.322420	-1.502237	-1.291802	-1.474429
				0.1		-1352389	-1380212	-1320975	-1356161	-1290407	-1332783
				0.2		-1.351008	-1.262758	-1.319643	-1.242086	-1.289123	-1.222008
5.0	0.5	-5.0	1.0	0.1	0.1	-1.353871	-1.383738	-1.320976	-1.135616	-1.290407	-1.332783
			2.0			-1.361864	-2.092309	-1.330033	-2.037640	-1.290407	-1.984734
			5.0			-1.371232	-3.507749	-1.334456	-3.362246	-1.307821	-3.223160
			10.0			-1.376316	-5.101436	-1.344081	-4.806734	-1.312650	-4.529388
λ	Mn	θr	Pr	ε	α	m = – 0.2		m = 0.0		m = 5.0
						f″(0)	θ′(0)	f″(0)	θ′(0)	f″(0)	θ′(0)
0.1	0.0	-5.0	1.0	0.1	0.0	-0.227799	5.478141	-0.631212	0.880672	-1.376995	-1.399540
					0.1	-0.409797	3.568688	-0.718014	0.679241	-1.336629	-1.376940
					0.2	-0.593180	2.335793	-0 808435	0.50008	-1.297614	-1.354864
					0.3	-0.775887	1.545657	-0.902260	0.337709	-1259952	-1333293
					0.5	-1.141195	0.598557	-1.099277	0.049424	-1.188558	-1.291576
					0.75	-1.603216	-0.121641	-1.360758	-0.266291	-1.106376	-1.241904
λ	α	θr	Pr	ε	m	Mn = 0.0			Mn = 0.25
						f″(0)		θ′(0)	f″(0)		θ′(0)
0.1	0.0	-5.0	1.0	0.1	-0.2	-0.248745		4.692819	-0.548033		3.405338
					-0.1	-0.484413		1.895248	-0.725686		1.517615
					0.0	-0.640512		0.806492	-0.849481		0.688922
					1.0	-1.134166		-0.889315	-1.275859		-0.858419
					5.0	-1.375488		-1.399645	-1.496541		-1.367532
					10.0	-1.426228		-1.494909	-1.543140		-1.464027
λ	α	m	Pr	ε	θr	α = 0.0			α = 0.5
						f″(0)		θ′(0)	f″(0)		θ′(0)
0.1	0.0	2.0	1.0	0.1	-10.0	-1212481		-1.169201	-1.123290		-1.109093
					-5.0	-1.312789		-1.152769	-1.211471		-1.094139
					-3.0	-1.435761		-1.132209	-1.317998		-1.075621
					-2.0	-1.576301		-1.108233	-1.437581		-1.054299

Comparison of skin friction –f″(0) when for different values of m and α with θr → ∞, m = 0, ε = 0_0, Mn = 0_0, λ = 0_0_

α	m	Fang et al. [23] By shooting method	Khader and Megahed [24] when Stip velocity parameter λ = 0 By Chebyshev spcctral method	Present Results
0.5	10	1.0603	1.0603	1.060309
	9	1.0589	1.0588	1.058812
	7	1.0550	1.0551	1.055122
	5	1.0486	1.0486	1.048615
	3	1.0359	1.0358	1.035805
	2	1.0234	1.0234	1.023424
	1	1.0	1.0	1.0
	0.5	0.9799	0.9798	0.979848
	0	0.9576	0.9577	0.957727
	-1/2	1.1667	1.1666	1.166666
0.25	10	1.1433	1.1433	1.143309
	9	1.1404	1.1404	1.140481
	7	1.1323	1.1323	1.132332
	5	1.1186	1.1186	1.118662
	3	1.0905	1.0904	1.090409
	1	1.0	1.0	1.0
	0.5	0.9338	0.9337	0.933770
	0	0.7843	0.7843	0.784309
	-1/3	0.5000	0.5000	0.500000
	-1/2	0.0833	0.08322	0.0832227