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Fractional Analysis of Unsteady Slip Flow of Viscous Fluid Confined to the Boundaries of an Annulus Driven by Exponentially Decaying/Growing Time-Dependent Pressure Gradient

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Abstract

A theoretical investigation highlighting the effect of fractional-order parameter and slip boundaries on the motion of a viscous fluid in an annular domain induced by exponential time-dependent pressure gradient has been reported. The action of decaying/growing exponential time-dependent pressure gradient in the circumferential direction and fractional-order on the fractionalized governing momentum equation based on the Caputo-Fabrizio fractional model has been carried out. For proper insight into the physical problem, the fractionalized model was first transformed and solved in the Laplace domain. Afterwards, a numerical Laplace inversing scheme based on Tzou’s algorithm has been utilized to comparatively analyze the solution for the flow field, shear stresses and flow vortex for the model under scrutiny. It was found that the fluid flow can be made faster by applying a growing pressure gradient, time and wall slippage on both walls. The effect of the fractional-order parameter is to decrease the flow and shear stresses. The instability of the flow can be controlled by enhancing the memory effect.

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The present paper has no data available.

Abbreviations

\({\mathrm{r}}_{1}\) :

Radius of the inner cylinder \((\mathrm{m})\)

\({\mathrm{r}}_{2}\) :

Radius of the outer cylinder \((\mathrm{m})\)

\(\mathrm{P}\) :

Static pressure \((\mathrm{Kg}/{\mathrm{ms}}^{2})\)

\(\mathrm{R}\) :

Dimensionless radius

\(\mathrm{s}\) :

Laplace parameter

\({\mathrm{t}}^{\prime}\) :

Dimensional time

\(\mathrm{r}^{\prime}\) :

Dimensional radial distance

\(\mathrm{t}\) :

Dimensionless time \((\mathrm{s})\)

\({\mathrm{U}}_{0}\) :

Reference velocity \((\mathrm{m}/\mathrm{s})\)

\({\mathrm{u}}_{\mathrm{r}^{\prime}}\) :

Radial velocity \((\mathrm{m}/\mathrm{s})\)

\(\mathrm{u}^{\prime}\) :

Circumferential velocity \((\mathrm{m}/\mathrm{s})\)

\(\mathrm{U}\) :

Dimensionless velocity

\(\upalpha \) :

Fractional-order parameter

\({\upbeta }_{1}\) :

Slip coefficient of inner cylinder

\({\upbeta }_{2}\) :

Slip coefficient of outer cylinder

\(\updelta \) :

Decaying/growing parameter of time-dependent pressure gradient

\({\upvarphi }\) :

Circumferential direction

\(\mathcal{L}\) :

Laplace parameter

\(\uplambda \) :

Radii ratio \(({\mathrm{r}}_{2}/{\mathrm{r}}_{1})\)

\(\uprho \) :

Fluid density \((\mathrm{Kg}/{\mathrm{m}}^{3})\)

\(\upomega \) :

Dean vortices

\(\uptau \) :

Skin friction

\(\upupsilon \) :

Dynamic viscosity of the fluid \((\mathrm{Kg}/\mathrm{ms})\)

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Funding

We declare that the authors received no funding in the course of this research.

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Authors and Affiliations

  1. Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

    Basant K. Jha

  2. Department of Mathematics, Iowa State University, Ames, USA

    Dauda Gambo

  3. Department of Mathematics, Federal University Dutse, Dutse, Nigeria

    Umar M. Adam

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  1. Basant K. Jha
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  2. Dauda Gambo
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  3. Umar M. Adam
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Correspondence to Dauda Gambo.

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Appendix

Appendix

$${\mathrm{A}}_{1}=\uplambda {\mathrm{K}}_{1}\left(\upgamma \right)-{\upbeta }_{2}{\mathrm{K}}_{1}\left(\upgamma \right)-{\uplambda }^{2}{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)+{\upbeta }_{2}\uplambda {\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)-{\upbeta }_{1}{\upbeta }_{2}{\mathrm{K}}_{1}\left(\upgamma \right)+{\upbeta }_{1}\uplambda {\mathrm{K}}_{1}\left(\upgamma \right)-\upbeta {\uplambda }^{2}{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right);$$
$${\mathrm{A}}_{2}={\upbeta }_{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\upgamma \right)-\mathrm{\lambda \alpha }{\mathrm{K}}_{1}\left(\upgamma \right)+{\uplambda }^{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)+{\upbeta }_{1}{\upbeta }_{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\upgamma \right)-{\upbeta }_{1}{\upbeta }_{2}\upgamma {\mathrm{K}}_{0}\left(\upgamma \right)-{\upbeta }_{1}\mathrm{\lambda \alpha }{\mathrm{K}}_{1}\left(\upgamma \right);$$
$${\mathrm{A}}_{3}={\upbeta }_{1}\mathrm{\lambda \gamma }{\mathrm{K}}_{0}\left(\upgamma \right)+{\upbeta }_{1}{\uplambda }^{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)+{\upbeta }_{2}{\uplambda }^{2}\upgamma {\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right);$$
$${\mathrm{A}}_{4}=\left(\frac{{\upbeta }_{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\upgamma \right)}{\mathrm{s}}+\frac{\mathrm{\lambda \alpha }{\mathrm{K}}_{1}\left(\upgamma \right)}{\mathrm{s}}\right);$$
$${\mathrm{A}}_{5}=\left(\frac{{\uplambda }^{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right);$$
$${\mathrm{A}}_{6}={\upbeta }_{1}{\upbeta }_{2}\uplambda {\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)-{\upbeta }_{2}\mathrm{\lambda \alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)-{\upbeta }_{1}{\upbeta }_{2}\mathrm{\lambda \alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)+\left(\frac{{\upbeta }_{2}\mathrm{\lambda \alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)+{\upbeta }_{1}{\upbeta }_{2}{\uplambda }^{2}\upgamma {\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right);$$
$${\mathrm{A}}_{7}={\upbeta }_{1}{\upbeta }_{2}\mathrm{\alpha \gamma }{\mathrm{K}}_{0}\left(\upgamma \right)-{\upbeta }_{2}{\uplambda }^{2}\mathrm{\alpha \gamma }{\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right)-\left(\frac{{\upbeta }_{1}{\upbeta }_{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\upgamma \right)}{\mathrm{s}}\right);$$
$${\mathrm{A}}_{8}=\left(\frac{{\upbeta }_{1}\mathrm{\lambda \alpha }{\mathrm{K}}_{1}\left(\upgamma \right)}{\mathrm{s}}\right)-\left(\frac{{\upbeta }_{1}{\uplambda }^{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)-\left(\frac{{\upbeta }_{1}{\upbeta }_{2}\mathrm{\alpha \gamma }{\mathrm{K}}_{0}\left(\upgamma \right)}{\mathrm{s}}\right)+\left(\frac{{\upbeta }_{1}\mathrm{\lambda \alpha \gamma }{\mathrm{K}}_{0}\left(\upgamma \right)}{\mathrm{s}}\right);$$
$${\mathrm{A}}_{9}=\left(\frac{{\upbeta }_{2}{\uplambda }^{2}\mathrm{\alpha \gamma }{\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)+\left(\frac{{\upbeta }_{1}{\upbeta }_{2}\mathrm{\lambda \alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)-{\upbeta }_{1}{\upbeta }_{2}{\uplambda }^{2}\mathrm{\alpha \gamma }{\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right)+\left(\frac{{\upbeta }_{1}{\upbeta }_{2}{\uplambda }^{2}\mathrm{\alpha \gamma }{\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right);$$
$${\mathrm{A}}_{10}={\upbeta }_{2}{\mathrm{I}}_{1}\left(\upgamma \right)-\uplambda {\mathrm{I}}_{1}\left(\upgamma \right)+{\uplambda }^{2}{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)-{\upbeta }_{2}\uplambda {\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)+{\upbeta }_{1}{\upbeta }_{2}{\mathrm{I}}_{1}\left(\upgamma \right)-{\upbeta }_{1}\uplambda {\mathrm{I}}_{1}\left(\upgamma \right)+\upbeta {\uplambda }^{2}{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)-{\upbeta }_{2}\mathrm{\alpha }{\mathrm{I}}_{1}\left(\upgamma \right);$$
$${\mathrm{A}}_{11}=\mathrm{\lambda \alpha }{\mathrm{I}}_{1}\left(\upgamma \right)-{\uplambda }^{2}\mathrm{\alpha }{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)-{\upbeta }_{1}{\upbeta }_{2}\mathrm{\alpha }{\mathrm{I}}_{1}\left(\upgamma \right)-{\upbeta }_{1}{\upbeta }_{2}\upgamma {\mathrm{I}}_{0}\left(\upgamma \right)+{\upbeta }_{1}\mathrm{\lambda \alpha }{\mathrm{I}}_{1}\left(\upgamma \right)+{\upbeta }_{1}\mathrm{\lambda \gamma }{\mathrm{I}}_{0}\left(\upgamma \right)-{\upbeta }_{1}{\uplambda }^{2}\mathrm{\alpha }{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right);$$
$${\mathrm{A}}_{12}={\upbeta }_{2}{\uplambda }^{2}\upgamma {\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right)+\left(\frac{{\upbeta }_{2}\mathrm{\alpha }{\mathrm{I}}_{1}\left(\upgamma \right)}{\mathrm{s}}-\frac{\mathrm{\lambda \alpha }{\mathrm{I}}_{1}\left(\upgamma \right)}{\mathrm{s}}\right)+\left(\frac{{\uplambda }^{2}\mathrm{\alpha }{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)-{\upbeta }_{1}{\upbeta }_{2}\uplambda {\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)+{\upbeta }_{2}\mathrm{\lambda \alpha }{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right);$$
$${\mathrm{A}}_{13}={\upbeta }_{1}{\upbeta }_{2}\mathrm{\lambda \alpha }{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)-\left(\frac{{\upbeta }_{2}\mathrm{\lambda \alpha }{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)+{\upbeta }_{1}{\upbeta }_{2}{\uplambda }^{2}\upgamma {\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right)+{\upbeta }_{1}{\upbeta }_{2}\mathrm{\alpha \gamma }{\mathrm{I}}_{0}\left(\upgamma \right)-{\upbeta }_{2}{\uplambda }^{2}\mathrm{\alpha \gamma }{\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right);$$
$${\mathrm{A}}_{14}=\left(\frac{{\upbeta }_{1}{\upbeta }_{2}\mathrm{\alpha }{\mathrm{I}}_{1}\left(\upgamma \right)}{\mathrm{s}}\right)-\left(\frac{{\upbeta }_{1}\mathrm{\lambda \alpha }{\mathrm{I}}_{1}\left(\upgamma \right)}{\mathrm{s}}\right)+\left(\frac{{\upbeta }_{1}{\uplambda }^{2}\mathrm{\alpha }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)-\left(\frac{{\upbeta }_{1}{\upbeta }_{2}\mathrm{\alpha \gamma }{\mathrm{I}}_{0}\left(\upgamma \right)}{\mathrm{s}}\right);$$
$${\mathrm{A}}_{15}=\left(\frac{{\upbeta }_{1}\mathrm{\lambda \alpha \gamma }{\mathrm{I}}_{0}\left(\upgamma \right)}{\mathrm{s}}\right)+\left(\frac{{\upbeta }_{2}{\uplambda }^{2}\mathrm{\alpha \gamma }{\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)-\left(\frac{{\upbeta }_{1}{\upbeta }_{2}\mathrm{\lambda \alpha }{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right)-{\upbeta }_{1}{\upbeta }_{2}{\uplambda }^{2}\mathrm{\alpha \gamma }{\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right)+\left(\frac{{\upbeta }_{1}{\upbeta }_{2}{\uplambda }^{2}\mathrm{\alpha \gamma }{\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right)}{\mathrm{s}}\right);$$
$${\mathrm{A}}_{16}=\mathrm{\lambda s}\left(\updelta +\mathrm{s}\right);$$
$${\mathrm{A}}_{17}={\upbeta }_{2}{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{1}\left(\upgamma \right)-{\upbeta }_{2}{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{1}\left(\upgamma \right)+\uplambda {\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{1}\left(\upgamma \right)-\uplambda {\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{1}\left(\upgamma \right)-{\upbeta }_{1}{\upbeta }_{2}{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{1}\left(\upgamma \right);$$
$${\mathrm{A}}_{18}={\upbeta }_{1}{\upbeta }_{2}{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{1}\left(\upgamma \right)+{\upbeta }_{1}\uplambda {\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{1}\left(\upgamma \right)-{\upbeta }_{1}\uplambda {\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{1}\left(\upgamma \right)-{\upbeta }_{1}{\upbeta }_{2}\upgamma {\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{0}\left(\upgamma \right)-{\upbeta }_{1}{\upbeta }_{2}\upgamma {\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right);$$
$${\mathrm{A}}_{19}={\upbeta }_{1}\mathrm{\lambda \gamma }{\mathrm{I}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{0}\left(\upgamma \right)+{\upbeta }_{1}\mathrm{\lambda \gamma }{\mathrm{K}}_{1}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{0}\left(\upgamma \right)+{\upbeta }_{2}\mathrm{\lambda \gamma }{\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{1}\left(\upgamma \right)+{\upbeta }_{2}\mathrm{\lambda \gamma }{\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{1}\left(\upgamma \right);$$
$${\mathrm{A}}_{20}={\upbeta }_{1}{\upbeta }_{2}\uplambda {\upgamma }^{2}{\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{0}\left(\upgamma \right)-{\upbeta }_{1}{\upbeta }_{2}\uplambda {\upgamma }^{2}{\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{0}\left(\upgamma \right)+{\upbeta }_{1}{\upbeta }_{2}\mathrm{\lambda \gamma }{\mathrm{I}}_{0}\left(\mathrm{\lambda \gamma }\right){\mathrm{K}}_{1}\left(\upgamma \right)+{\upbeta }_{1}{\upbeta }_{2}\mathrm{\lambda \gamma }{\mathrm{K}}_{0}\left(\mathrm{\lambda \gamma }\right){\mathrm{I}}_{1}\left(\upgamma \right);$$
$${\mathrm{C}}_{1}=\frac{{-(\mathrm{A}}_{1}+{\mathrm{A}}_{2}+{\mathrm{A}}_{3}-{\mathrm{A}}_{4}-{\mathrm{A}}_{5}+{\mathrm{A}}_{6}+{\mathrm{A}}_{7}+{\mathrm{A}}_{8}+{\mathrm{A}}_{9})}{{\mathrm{A}}_{16}({\mathrm{A}}_{17}+{\mathrm{A}}_{18}+{\mathrm{A}}_{19}+{\mathrm{A}}_{20})};$$
$${\mathrm{C}}_{2}=\frac{{-(\mathrm{A}}_{10}+{\mathrm{A}}_{11}+{\mathrm{A}}_{12}+{\mathrm{A}}_{13}+{\mathrm{A}}_{14}+{\mathrm{A}}_{15})}{{\mathrm{A}}_{16}({\mathrm{A}}_{17}+{\mathrm{A}}_{18}+{\mathrm{A}}_{19}+{\mathrm{A}}_{20})};$$

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Jha, B.K., Gambo, D. & Adam, U.M. Fractional Analysis of Unsteady Slip Flow of Viscous Fluid Confined to the Boundaries of an Annulus Driven by Exponentially Decaying/Growing Time-Dependent Pressure Gradient. Int. J. Appl. Comput. Math 9, 16 (2023). https://doi.org/10.1007/s40819-022-01486-z

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