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Table 11 – Heat transfer coefficient −2′(0) at different [i, m] orders of the ISHAM compared
with the numerical solutions for different values of M , Pr and Ec when K = 1.
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116 ON A LINEARISATION METHOD FOR REINER-RIVLIN SWIRLING FLOW
Figures 1-2 show the effect of M and K on the radial and axial velocity pro-
files respectively. Increasing M reduces the radial component of the velocity
while increasing K enhances F . The axial velocity H(η) increases with the
magnetic parameter but decreases when K is increased. There is excellent agree-
ment between the second order ISHAM solutions for F(η) and H(η) and the
numerical result.
The tangential velocity component and the temperature profiles are show in
Figures 3-4 for different values of M and K respectively. An increase in M
reduces G(η) while enhancing the 2(η). When K values are increased both
G(η) and 2(η) decrease.
In Figure 5 temperature profiles are presented for varying values of Pr
and Ec. The temperature decreases with increasing Prandtl numbers while an
increase in the Eckert number enhances the temperature.
6 Conclusion
A novel approach for accelerating the convergence of the spectral homotopy
analysis method that is used to solve nonlinear equations in science and engi-
neering has been proposed and applied successfully to the nonlinear system of
equations governing the Reiner-Rivlin fluid in with Joule heating and viscous
dissipation. The primary objective of the algorithm is to improve the initial
approximate solution. The improved approximations are then used in the algo-
rithm of the spectral-homotopy analysis method to reduce the number of itera-
tions required to achieve convergence and better accuracy. The shear stresses in
the radial and azimuthal directions were computed and the corresponding abso-
lute errors determined. Convergence to the numerical solutions of the ISHAM
approximate solutions was achieved at the 2nd orders for all flow parameters
while the SHAM converged at the 8th order for some of the flow parameters.
The effects of flow parameters was also investigated for the radial and tan-
gential shear stresses for both the Newtonian (K = 0) and non-Newtonian
(K 6= 0) cases. For the Newtonian case, increasing M reduces F ′(0) and
enhances −G ′(0) while in the non-Newtonian case increasing M enhances
both F ′(0) and −G ′(0).
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ZODWA G. MAKUKULA, PRECIOUS SIBANDA and SANDILE S. MOTSA 117
0 1 2 3 4 5 6 7 80
0.05
0.1
0.15
0.2
0.25
η
F(η)
M = 0M = 0.5M = 1M = 1.5
0 1 2 3 4 5 60
0.05
0.1
0.15
0.2
0.25
0.3
η
F(η)
K = 0K = 2K = 4K = 6
Figure 1 – On the comparison between the 2nd order ISHAM solution and the numer-
ical solution (solid line) for F(η) at different values of M (K = 2) and K (M = 1)
when Pr = 1, Ec = 0.3.
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118 ON A LINEARISATION METHOD FOR REINER-RIVLIN SWIRLING FLOW
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
η
−H(η)
M = 0M = 0.5
M = 1M = 1.5
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
η
−H(η)
K = 0K = 2
K = 4K = 6
Figure 2 – On the comparison between the 2nd order ISHAM solution and the numer-
ical solution (solid line) for −H(η) at different values of M (K = 2) and K (M = 1)
when Pr = 1, Ec = 0.3.
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ZODWA G. MAKUKULA, PRECIOUS SIBANDA and SANDILE S. MOTSA 119
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η
G(η)
M = 0M = 0.5M = 1M = 1.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η
G(η)
K = 2K = 4K = 6K = 8
Figure 3 – On the comparison between the 2nd order ISHAM solution (figures) and the
bvp4c numerical solution (solid line) for G(η) at different values of M (K = 2) and
K (M = 1) when Pr = 1, Ec = 0.3.
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120 ON A LINEARISATION METHOD FOR REINER-RIVLIN SWIRLING FLOW
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η
Θ(η)
M = 0M = 0.5M = 1M = 1.5
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η
Θ(η)
K = 0K = 2K = 4K = 6
Figure 4 – On the comparison between the 2nd order ISHAM solution (figures) and the
bvp4c numerical solution (solid line) for 2(η) at different values of M (K = 2) and
K (M = 0.1) when Pr = 1, Ec = 0.3.
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ZODWA G. MAKUKULA, PRECIOUS SIBANDA and SANDILE S. MOTSA 121
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
η
Θ(η)
Pr = 1Pr = 3Pr = 5Pr = 7
0 1 2 3 4 5 6 7 80
0.5
1
1.5
2
2.5
η
Θ(η)
Ec = 0Ec = 3
Ec = 6Ec = 9
Figure 5 – On the comparison between the 2nd order ISHAM solution (figures) and the
bvp4c numerical solution (solid line) for 2(η) at different values of Pr (Ec = 0.3) and
Ec (Pr = 1) when M = 0.1, K = 1, L = 30, N = 150.
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122 ON A LINEARISATION METHOD FOR REINER-RIVLIN SWIRLING FLOW
The effect of K and M was determined and it was observed that an increase
in K results in an increase in F(η), and a decrease in H(η), G(η) and 2(η)
while increasing M increases H(η) and 2(η) while both F(η) and G(η) de-
creases. 2(η) decreased with an increase in the Ec and decreased with an
increase in Pr . The success of the ISHAM in solving the non-linear equations
governing the von Kármán flow of an electrically conducting non-Newtonian
Reiner-Rivlin fluid in the presence of viscous dissipation, Joule heating and heat
transfer proves that the ISHAM fits as a newly improved method of solution that
can be used to solve non-linear problems arising in science and engineering.
Acknowledgements. The authors wish to acknowledge financial support from
the University of KwaZulu-Natal and the National Research Foundation (NRF).
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