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 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 and \(f^{\prime\prime}(0)\). As the 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.