Modified Chebyshev wavelets approach for mixed convection flow due to oblique stagnation point along a vertically moving surface with zero mass flux of nanoparticles

Highlights

• Active and passive control of nanoparticles is determined when the fluid is strike nonorthogonal on a flat surface.

• Modified Chebyshev wavelets approach is applied to handle the nonlinear complex mathematical model.

• Nonorthogonality is measured at for various angles.

• Nusselt increases behavior is calculated in term of bar graphs.

• Streamlines are plotted for both orthogonal and nonorthogonal flows.

Abstract

In this article mixed convection flow of nanofluid near a vertical wall is studied for cases of active and passive control. Flow is governed by nonlinear Partial Differential Equations (PDEs), which are transformed into nonlinear ODEs by using similarity transformations. Nonlinear system of equations is solved via two different techniques which are: Modified Chebyshev wavelets scheme and Finite different method. The impact of emerging parameters including free parameter "A" on velocity, temperature and concentration profiles is plotted. Some values of the physical parameters against the active and passive controls for streamlines are calculated. Tables for values of A, components of skin friction, heat flux and mass flux against different parameters are also given. Streamlines are also plotted to predict the flow patterns at various locations inside flow. In this study we found that velocity of flow has same behavior in both active and passive controls, however, temperature and concentration profiles are more sensitive to the passive control.

Introduction

Stagnation point in a flow is a point where velocity of fluid is zero. It arises when an obstacle opposes flow and force the fluid to change its path. Stagnation point flow caught the attention of researchers due to its wide range practical applications in aerodynamics and industry like flight operations, emergency cooling of nuclear reactors and wind sensitive solar panels. Non orthogonal stagnation points are more feasible in practical situations as they provide us several possibilities for angle of inclination. Accurate solutions of such models can be found by analytical techniques. The accuracy of solution can be maximizing by using appropriate similarity transformations.

In the present era, developments are made in the field of heat transfer, and scientists are struggling for more efficient ways to enhance and control the heat transfer properties. Introduction of nanotechnology revolutionize the conventional ways of heat energy transfer. In this scenario, nanofluids (engineered fluids) attracted researchers to think about an efficient mode for heat transfer in the field of fluid dynamics. Choi [1] was the person who first ever introduced nanofluid in his article “enhancing thermal conductivity of fluids with nanoparticles”. After that Yang. et.al [2] explores the properties of heat transfer attained by nanofluids. Buongiorno [3] proposed a correlation for effects of nanofluids inside boundary layer. He postulated that the Brownian motion and thermophoresis affect the heat transfer process in nanofluids. His idea was further extended by many researchers in multi directions. W.A. Khan, I. Pop [4] studies boundary layer flow of nanofluids along a stretching sheet. Nadeem. S, and Haq. Ru [5] discussed the flow of nanofluid along a stretching sheet under influence of MHD and convective boundary conditions. Many other nanofluid models were introduced and explored by researchers, tackling different aspects of heat transfer properties. Tiwari, R. K., & Das, M. K. [6] introduced a nanofluid model in which nanoparticles volume fraction got primary importance. Corcione and Massimo. [7] discussed a novel approach to correlate the thermal conductivity of nanofluid. Muhammad et al. [8] analyzed squeezed flow of nanofluids with Cattaneo–Christov heat and mass flux. Idea of Corcione is implemented and explored by Ali et al. [9].

Nanoparticle’s distribution in a nanofluid can affect the heat transfer rate. Atlas et al. [10] discussed about active and zero mass flux of nanoparticles in a channel. They explored isotherms under both controls. Hayat et al. [11] developed series solutions for active and passive controls of Jeffery nanofluids. Abdul Halim et al. [12] talked about active and passive controls in his article about Williamson nanofluids. Halim et al. [13] discussed stagnation point flow of Maxwell nanofluid under both assumptions of active and passive controls. Rauf et al. [14] studied about active and passive interactions of nanoparticles on an oscillatory rotating disk. Active and passive control analysis of viscoelastic nanofluids is also done by Ramesh [15] with activation energy.

Oblique stagnation point flow was first ever discussed by Stuart [16] in which he discussed the viscous flow near a stagnation point. He explored that near a stagnation point the impinging fluid needs not to be vertical all times. Latterly, exact solutions for non-orthogonal stagnation point were discussed by Dorrepaal [17]. He explained many aspects of non-orthogonal flows. One of his remarkable findings was locating stagnation point. Amaouche and Boukari [18] investigated the non-orthogonal flow under influence of thermal convection. Lok [19] Investigated non orthogonal flow initiated by a stretching sheet. Lok and Pop [20] further analyzed micropolar fluid with non-orthogonal stagnation point along a vertical wall. Labropulu [21] Analyzed the unsteady model of non-orthogonal stagnation flow of viscoelastic fluids. D. Li et al. [22] investigated mixed convection flow of orthogonal stagnation point flow on a vertical wall. Javed, Tabish et al. [23] investigated unsteady oblique stagnation flow along oscillating plate. Jalilpour [24] extended flow of oblique stagnation point for nanofluids. Rashid M. et al. [25] discussed radiating Casson fluid for non-orthogonal stagnation point flow near stretching surface. Li et al. [26] discussed about existence of dual solution for the current topic along shrinking surface. Zainal, N.A et al [27] extended idea of oblique stagnation point flow for hybrid nanofluids. Abbasi, A et al. [28] extended this idea over lubricating surface. Mehmood et al. [29] studied MHD flow of said issue along convective stretching surface. Mixed convection flow gain attention due to its physical applicability in engineering process like metallurgy, drilling and rocket launching. Haq et al. [30] explored mixed convection flow along heated wall. Bharadwaj et al. [31] explained the said issue for uniform temperature of wall. Siavashi et al. [32] discussed the process of mixed convection for two-phase non-Newtonian fluid inside a square cavity. Many other researchers [32], [33], [34], [35], [36], [37], [38], [39] worked on related issues.

Inspired by the literature mentioned above, our vision in this paper is to extend the idea of non-orthogonal stagnation point flow of nanofluid along a vertical wall. The main purpose is to explore active and passive controls in nanofluid flowing near a vertical wall.