Unsteady hydromagnetic-free convection flow with suction/injection

Figures & data

Figure 1. Schematic of the problem.

Figure 2. Velocity profile showing the effect of $m_1$ for $\text{(a)}\lambda =0\text{and}\text{(b)}\lambda =1$ for impulsive motion $\left(Pr=1.0,Gr=1.0,s=0.5\right)$.

Figure 3. Velocity profile showing the effect of $m_1$ for $\text{(a)}\lambda =0\text{and}\text{(b)}\lambda =1$ for accelerated motion $(Pr=1.0,Gr=1.0)$.

Figure 4. Velocity profile showing the effect of $Gr$ for $\text{(a)}\lambda =0\text{and}\text{(b)}\lambda =1$ for impulsive motion $(Pr=1.0,t=0.5,m_1=1.0)$.

Figure 5. Velocity profile showing the effect of $Gr$ for $\text{(a)}\lambda =0\text{and}\text{(b)}\lambda =1$ for accelerated motion $(Pr=1.0,t=0.5,m_1=1.0)$.

Figure 6. Velocity profile showing the effect of suction/injection parameter $s$ for $\text{(a)}\lambda =0\text{and}\text{(b)}\lambda =1$ for impulsive motion $(Pr=1.0,t=0.5,m_1=1.0)$.

Figure 7. Velocity profile showing the effect of suction/injection parameter $s$ for $(\text{a)}\lambda =0\text{and}\text{(b})\lambda =1$ for accelerated motion $(Pr=1.0,m_1=1.0)$.

Figure 8. Variation of skin friction versus time for different values of $m_1$with $(\text{a)}\lambda =0\text{and}\text{(b})\lambda =1$for impulsive motion $(Gr=1.0,Pr=1.0)$.

Figure 9. Variation of skin friction versus time for different values of $m_1$with $(\text{a)}\lambda =0\text{and}\text{(b})\lambda =1$for accelerated motion $(Gr=1.0,Pr=1.0)$.

Figure 10. Variation of skin friction versus time for different values of $Gr$ for $(\text{a)}\lambda =0\text{and}\text{(b})\lambda =1$ for impulsive motion $(m_1=1.0,Pr=1.0,s=0.5)$.

Figure 11. Variation of skin friction versus time for different values of $Gr$ for $(\text{a)}\lambda =0\text{and}\text{(b})\lambda =1$ for accelerated motion ( $(m_1=1.0,Pr=1.0,s=0.5)$.

Table 1. Numerical comparison of velocity obtained using the Riemann-sum approximation method and those obtained using the convolution integrals method by Chandran et al. [13] at $Gr=1.0$, $t=0.1$, $s\to 0$ (impulsive case).

	Chandran et al. [13]		Present work
	$\lambda =0$
$y/m_1$	0.1	0.5	0.1	0.5
0.0	1.000	1.000	1.000	1.000
0.2	0.656	0.650	0.657	0.649
0.4	0.373	0.366	0.372	0.365
0.6	0.181	0.176	0.180	0.176
0.8	0.074	0.072	0.074	0.072
1.0	0.025	0.025	0.025	0.025

Table 2. Numerical comparison of velocity obtained using the Riemann-sum approximation method and those obtained using the convolution integrals method by Chandran et al. [13] at $Gr=1.0$, $t=0.1$, $s\to 0$ (impulsive case).

	Chandran et al. [13]		Present work
	$\lambda =1$
$y/m_1$	0.1	0.5	0.1	0.5
0.0	1.000	1.000	1.000	1.000
0.2	0.662	0.676	0.662	0.676
0.4	0.381	0.406	0.380	0.405
0.6	0.190	0.222	0.189	0.221
0.8	0.084	0.120	0.083	0.119
1.0	0.035	0.073	0.035	0.073

Table 3. Numerical comparison of velocity obtained using the Riemann-sum approximation method and those obtained using the convolution integrals method by Chandran et al. [13] at $Gr=1.0$, $t=0.1$, $s\to 0$ (accelerated case).

	Chandran et al. [13]		Present work
	$\lambda =0$
$y/m_1$	0.1	0.5	0.1	0.5
0.0	0.100	0.100	0.100	0.100
0.2	0.050	0.050	0.051	0.050
0.4	0.023	0.022	0.023	0.022
0.6	0.009	0.009	0.009	0.009
0.8	0.003	0.003	0.003	0.003
1.0	0.001	0.001	0.001	0.001

Table 4. Numerical comparison of velocity obtained using the Riemann-sum approximation method and those obtained using the convolution integrals method by Chandran et al. [13] at $Gr=1.0$, $t=0.1$, $s\to 0$ (accelerated case).

	Chandran et al. [13]		Present work
	$\lambda =1$
$y/m_1$	0.1	0.5	0.1	0.5
0.0	0.10	0.100	0.100	0.100
0.2	0.050	0.052	0.051	0.052
0.4	0.023	0.025	0.023	0.025
0.6	0.009	0.011	0.009	0.011
0.8	0.003	0.005	0.003	0.005
1.0	0.001	0.003	0.001	0.003

Table 5. Numerical comparison of skin friction obtained using the Riemann-sum approximation method and those obtained using the convolution integrals method by Chandran et al. [13] [impulsive] $Gr=1.0$, $t=0.1$, $s\to 0$.

			Chandran et al. [13]		Present work
$\mathit{G}\mathit{r}$	$\mathit{t}$	${\mathit{m}}_1$	$\lambda =0$	$\lambda =1$	$\lambda =0$	$\lambda =1$
0.5	0.1	0.1	1.7771	1.7416	0.4132	0.4489
		0.3	1.8126	1.7067	0.3777	0.4839
		0.5	1.8479	1.6724	0.3424	0.5182
	0.2	0.1	1.2370	1.1868	0.3103	0.3606
		0.3	1.2870	1.1386	0.2603	0.4090
		0.5	1.3364	1.0922	0.2109	0.4556
1.0	0.1	0.1	1.7523	1.7168	0.4382	0.4738
		0.3	1.7879	1.6819	0.4026	0.5088
		0.5	1.8232	1.6477	0.3672	0.5430
	0.2	0.1	1.1872	1.1371	0.3601	0.4104
		0.3	1.2375	1.0891	0.3098	0.4586
		0.5	1.2871	1.0430	0.2602	0.5049

Table 6. Numerical comparison of skin friction obtained using the Riemann-sum approximation method and those obtained using the convolution integrals method by Chandran et al. [13] [accelerated case] $Gr=1.0$, $t=0.1$, $s\to 0$.

			Chandran et al. [13]		Present work
$\mathit{G}\mathit{r}$	$\mathit{t}$	${\mathit{m}}_1$	$\lambda =0$	$\lambda =1$	$\lambda =0$	$\lambda =1$
0.5	0.1	0.1	0.3332	0.3308	0.3337	0.3313
		0.3	0.3356	0.3285	0.3361	0.3290
		0.5	0.3381	0.3263	0.3385	0.3268
	0.2	0.1	0.4582	0.4515	0.4594	0.4526
		0.3	0.4651	0.4452	0.4663	0.4463
		0.5	0.4720	0.4390	0.4731	0.4401
1.0	0.1	0.1	0.3084	0.3060	0.3087	0.3064
		0.3	0.3109	0.3038	0.3112	0.3041
		0.5	0.3134	0.3016	0.3137	0.3019
	0.2	0.1	0.4085	0.4018	0.4095	0.4028
		0.3	0.4156	0.3957	0.4167	0.3967
		0.5	0.4227	0.3898	0.4238	0.3908

Chandran P, Singh AK, Sacheti NC. A unified approach to analytical solution of a hydromagnetic free convection flow. Sci Math Jpn. 2001;53:467.

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