Influence of slip over an exponentially moving vertical plate with Caputo-time fractional derivative

Abstract

In this article, the impact of a MHD is analyzed on a VF accompanying double convection, because of the transfer inducted by temperature and concentration gradients along with the slip at the boundary. Furthermore, impacts of chemical reaction and heat generation are also taken into account. The concept of non-integer Caputo time fractional derivative is utilized for a generalized VF model comprising three PDEs of momentum, heat and mass transfer accompanying initial and boundary constraints. The LT technique and Stf.A and Tzu.A are acquired to utilize the desirable outcomes of velocity, temperature and concentration. The influence of physical parameters and flow is graphically analyzed via computational software (MathCad). The outcomes attained as specific cases are also marvelously agree with the published results from the literature. Finally, it has been seen that the rising values of the slip coefficient reduces the fluid velocity. This represents the impact of slip at the boundary on the fluid flow.

Appendix

$$\begin{aligned}&\phi (\beta ,-\sigma :z)=\Sigma _{n=0}^{\infty }\frac{z^{\text{n}}}{n!\Gamma (\beta -\sigma n)}\\&L^{-1}\left[ \frac{1}{q^{2}}\exp \left( -a\sqrt{q^{\upalpha }+b}\right) \right] =\\&\int _{0}^{\infty }\int _{0}^{t}erfc\left( \frac{a}{2\sqrt{x}}\right) \frac{\exp (-bx)}{\tau }\phi (0,\alpha ;-x\tau ^{-\upalpha })\nonumber \\&\qquad \left[ \frac{(t-\tau )^{1-\upalpha }}{\Gamma (2-\alpha )}+ b(t-\tau )\right] \mathrm{d}\tau \mathrm{d}x\\&L^{-1}\left[ \frac{\exp (-y\sqrt{q+\alpha })}{q-\beta }\right] =e^{\upbeta {\text{t}}}\Phi (y,t;\alpha +\beta )\\&L^{-1}\left[ \frac{\exp (-y\sqrt{q+\alpha })}{q^{2}}\right] =\\&\frac{1}{2}\left[ (t-\frac{y}{2\sqrt{\alpha }})\exp (-y\sqrt{\alpha }) erfc(\frac{a}{2\sqrt{t}}-\sqrt{bt})\right] \nonumber \\&\qquad +\left[ (t+\frac{a}{2\sqrt{b}})\exp (a\sqrt{b})erfc(\frac{a}{2\sqrt{t}}+\sqrt{bt})\right] \\&L^{-1}\{F(y+\beta )\}=\exp (-yt)f(t) where L^{-1}\{F(q)\}=f(t)\\&L^{-1}\left[ \frac{q^{\upalpha -\upbeta }}{q^{\upalpha }+y}\right] \nonumber \\&\quad =t^{\upbeta -1}E_{\alpha ,\beta }(-yt^{\upalpha }); E_{\alpha ,\beta }(z)=\sum _{k=0}^{\infty }\frac{z^{\text{k}}}{\Gamma (\alpha k+\beta )};\quad \alpha>0,\beta >0\\&L^{-1}\left[ \frac{\exp (-y\sqrt{s^{\upalpha }+\beta })}{s^{\upalpha }+b}\right] =\\&\int _{0}^{\infty }erfc(\frac{y}{2\sqrt{u}})\frac{\exp (-\beta u)}{t}\phi (0,-\alpha ;-ut^{-\upalpha })\mathrm{d}u\\&\Psi (y,t;\alpha +\beta )=\\&\frac{1}{2\sqrt{\alpha +\beta }}\left[ \exp (-y\sqrt{\alpha +\beta })erfc\left( \frac{y}{2\sqrt{t}}\right. \right. \\&\qquad \left. \left. -\sqrt{(\alpha +\beta )t}\right) -\exp (y\sqrt{\alpha +\beta })erfc\left( \frac{y}{2\sqrt{t}}+\sqrt{(\alpha +\beta )t}\right) \right] \\&\Phi (y,t;\alpha _{1}+\beta _{1})=\\&\frac{1}{2}\left[ \exp (-y\sqrt{\alpha _{1}+\beta _{1}})erfc\left( \frac{y}{2\sqrt{t}}-\sqrt{(\alpha _{1}+\beta _{1})t}\right) \right. \\&\left. +\exp (y\sqrt{\alpha _{1}+\beta _{1}})erfc\left( \frac{y}{2\sqrt{t}}+\sqrt{(\alpha _{1}+\beta _{1})t}\right) \right] \\&\Phi (y,t;\alpha +\beta )=\\&\frac{1}{2}\left[ \exp (-y\sqrt{\alpha +\beta })erfc\left( \frac{y}{2\sqrt{t}}-\sqrt{(\alpha +\beta )t}\right) \right. \\&\left. +\exp (y\sqrt{\alpha +\beta })erfc\left( \frac{y}{2\sqrt{t}}+\sqrt{(\alpha +\beta )t}\right) \right] \\ \end{aligned}$$