Stability analysis of heat transfer in nanomaterial flow of boundary layer towards a shrinking surface: Hybrid nanofluid versus nanofluid

Submitting author affiliation:
The Islamia University of Bahawalpur, Bahawalpur, Pakistan

Beilstein Arch. 2021, 202188.

Published 15 Dec 2021

This preprint has not been peer-reviewed. When a peer-reviewed version is available, this information will be updated.


Many boundary value problems (BVPs) have dual solutions in some cases containing one stable solution (upper branch) while other unstable (lower branch). In this paper, MHD flow and heat transfer past a shrinking sheet is studied for three distinct fluids: kerosene hybrid nanofluid, kerosene nanofluid, and kerosene nanofluid. The partial differential equations (PDEs) are turned into ordinary differential equations (ODEs) using an appropriate transformation and then dual solutions are obtained analytically by employing the Least Square method (LSM). Moreover, stability analysis is implemented on the time-dependent case by calculating the least eigenvalues using Matlab routine bvp4c. It is noticed that negative eigenvalue is related to unstable solution i.e., it provides initial progress of disturbance and positive eigenvalue is related to stable solution i.e., the disturbance in solution decline initially. The impacts of various parameters, skin friction coefficient, and local Nusselt number for dual solutions are presented graphically. It is also noted that the results obtained for hybrid nanofluids are better than ordinary nanofluids.

Keywords: Stability analysis; hybrid nanofluid; shrinking sheet; dual solutions; analytical procedure

How to Cite

When a peer-reviewed version of this preprint is available, this information will be updated in the information box above. If no peer-reviewed version is available, please cite this preprint using the following information:

Rehman, A. U.; Abbas, Z. Beilstein Arch. 2021, 202188. doi:10.3762/bxiv.2021.88.v1

Download citation
Download RIS (Reference Manager) Download BIB (BIBTEX)

© 2021 Rehman and Abbas; licensee Beilstein-Institut.
This is an open access work licensed under the terms of the Beilstein-Institut Open Access License Agreement (, which is identical to the Creative Commons Attribution 4.0 International License ( The reuse of material under this license requires that the author(s), source and license are credited. Third-party material in this work could be subject to other licenses (typically indicated in the credit line), and in this case, users are required to obtain permission from the license holder to reuse the material.