Here local Reynolds number is\(\text{Re}_{s}=u_{w}\sqrt{\frac{2l\left(1-\alpha_{0}t\right)}{\text{νa}}}\).
Results and discussion
We formulated a mathematical model to analyze the unsteady flow of
micropolar fluid over a curved surface under slip effects.
Figs. 2-6 reveal the influence of unsteady parameter (\(A\)), magnetic
parameter (\(\beta\)), velocity slip parameter (\(\delta\)), curvature
parameter (\(K\)), micropolar parameter (\(K_{1}\)) on the velocity
profile for week concentration. Fig. 2 exposes the impacts of \(A\) on
velocity gradient. The boundary layer of the velocity profile reduced
when the values of \(A\) increases. Unsteady parameters \(A\) reduced
the velocity of flow when the values increase unsteady parameters
increases. Fig. 3 discloses the influence of \(\beta\) on the velocity
gradient. It is seen that velocity gradient slow down towards the
surface as \(\beta\) increases. Fig. 4 illustrates the influences of\(\delta\) on the velocity gradient. The velocity gradient increases as
the velocity slip parameters, \(\delta,\) also increases. This is
because velocity slip accelerates the flow which increases the fluid
velocity in our case. Fig. 5 exposes the effects of \(K\) on the
velocity gradient. Similar to before, the behaviour of the velocity
gradient hightens as \(K\) increases. The curvature parameter
accelerates the flows which increase the fluid velocity in our case.
Fig. 6 exposures the inspiration of \(\text{\ K}_{1}\) on the velocity
gradient. It is seen that velocity gradient slows down towards the
surface as \(\text{\ K}_{1}\) increases. Figs. 9-11 depict the impacts
of \(A\), \(K\) and \(K_{1}\) micropolar profile. Fig. 9 shows the
influence of \(A\) on the micropolar profile. It is seen that the
micropolar profile slows down when the values of \(A\) rise. Fig. 10
highlights the impacts of \(K\) on the micropolar profile. The
micropolar profile was found to be rise when values of \(K\) are
increased. The micropolar parameter \(K_{1}\) shows the influence on the
micropolar profile which is perceived in Fig. 11. As we approach higher
values of \(K_{1}\ \)the mircopolar profile increases, since as\(K_{1}\) increases, it reduces the micropolar profile thickness. The
encouragement of \(A\), Bi, \(K\) and \(\Pr\) on the
temperature profile is shown in Figs. 12-15. Fig. 12 reveals the
influence of \(A\) on the temperature profile. We see that the thermal
boundary layer thickness reduces for higher values of \(A\). Fig. 13
highlights the effect of Bi on the temperature profile.Bi rises which increases the temperature away from the
surface. Fig. 14 reveals the influence of \(K\) on the temperature
profile. The curvature parameter increases which enhances the
temperature near the surface. Fig. 15 reveals the influence of \(\Pr\)on temperature profile. The Prandtl number \(\Pr\) increases which in
turn reduces the temperature profile.
The impacts of the physical parameters on \(f^{\prime\prime}(0)\) are in Table 1. The
values of K rise as \(f^{\prime\prime}(0)\) increases. The impacts of\(A\) on \(f^{\prime\prime}(0)\) are revealed in Table 1. The opposite behaviours are
noted for A and \(f^{\prime\prime}(0)\). As the A enhanes,\(f^{\prime\prime}(0)\) is found to decline near the surface. The influence of the\(\beta\) on the \(f^{\prime\prime}(0)\) is found in Table 1. \(f^{\prime\prime}(0)\) reduced as\(\beta\) and \(K_{1}\) rise, whereas the opposing behaviour is noted
for δ and \(f^{\prime\prime}(0)\). As δ enhances ,\(f^{\prime\prime}(0)\)is found to decline near the surface. Table 2 shows the variation of\(K,\ \ A,\ \ Pr,\ \ Bi\) on the \(\theta^{\prime}(0)\). It is observed that the
curvature parameters increases as \(\theta^{\prime}(0)\) is reduced. This tells
us that heat transfer is reduced near the surface when the values of
curvature parameters rise. The influence of the unsteady parameters at
the heat transfer rate, show that the heat transfer rate becomes
augments when rising the value of the unsteady parameters. The Prandtl
number and heat transfer have the same behaviour of incremental increase
as is noted in the Table 2. The Biot number and heat transfer rate show
the same behaviour of enhancing as is set out in Table 2.