Numerical analysis of unsteady flow of three-dimensional Williamson fluid-particle suspension with MHD and nonlinear the
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Numerical analysis of unsteady flow of three-dimensional Williamson fluid-particle suspension with MHD and nonlinear thermal radiations Madiha Bibi1,2,a , A. Zeeshan3 , M. Y. Malik4 1 Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan 2 Department of Mathematics, 150 N. University Street, Purdue University, West Lafayette, IN 47907, USA 3 Department of Mathematics and Statistics, FBAS, International Islamic University, Islamabad 44000,
Pakistan
4 Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia
Received: 12 August 2020 / Accepted: 10 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this study, the time-dependent incompressible 3D solid–liquid Williamson nonNewtonian fluid for momentum and heat flow is evaluated. Flow is generated through a sheet in both x and y directions and effected by MHD, dust particles and nonlinear thermal radiations. Problem controlling partial differential equations are converted into ordinary differential equations by using appropriate transformations. The finalized mathematical model is evaluated by ‘bvp4c method’. Numerical results are compared with already published data. Non-dimensional parameters such as first- and second-ordered radiation parameter, the interaction parameter between fluid and particles, the unsteadiness parameter, the stretching parameter, and others are discussed that how they affect the velocity and heat distribution. Physical quantities nusselt number and skin friction are also analyzed numerically as well as graphically.
1 Introduction In this article, the boundary layer flow for three dimensions is under consideration and the flow is because of stretching of a sheet. As boundary layer flows gained much importance and attention as per industrial requirement, for example metal and polymer extrusion, metal wires, plastic sheets and films production, etc. From literature review it is well known that front runner researchers of boundary layer flows due to stretching were Sakiadis [1], Crane [2], Grubka and Bobba [3]. They have started study for two dimensions but afterward as the research area growing vast then the study of flows shifted toward the three dimensions. Although its tough to handle 3-D but in reality flows exist in three dimensions. So taking in account the importance of three-dimensional flow Wang [4] took initiative to analyze the 3-D flow over a flat stretching surface. More recently, Arif et al. [5] evaluate the two-dimensional mass and heat flow over stretching surface for a viscoelastic fluid. In current article, threedimensional flow is discussed with time dependency. As the time dependency is a major factor which we cannot neglect in real-life problems. After time of two years Surma Devi et al. [6]
a e-mail: [email protected] (corresponding author)
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Eur. Phys. J. Plus
(2020) 135:850
studied the three-dimensional flows due to stretching for un
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