Verification of Dual Solutions for Water and Kerosene-Based Carbon Nanotubes over a Moving Slender Needle

Abstract

This article focuses on the boundary layer for an axisymmetric flow and heat transfer of a nanofluid past a moving slender needle with single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). In this study, the streamlines of the flow are symmetrically located along the needle’s surface. Water and kerosene are two types of base fluids that are considered in this study. This analysis is presented with needle thickness, the ratio of velocity, nanoparticle volume fraction, and Prandtl number. The partial differential equations (PDEs) are transformed into dimensionless ordinary differential equations (ODEs) by adopting relevant similarity transformations. The bvp4c package is implemented in MATLAB R2018a to solve the governing dimensionless problems numerically. The behaviors of various sundry variables on the flow and heat transfer are observed and elaborated further. The magnitude of the skin friction, heat transfer rate, as well as velocity and temperature distributions are demonstrated in graphical form and discussed. It is worth mentioning that kerosene-based CNTs have the largest skin friction coefficient and heat transfer rate compared to water-based CNTs. The thin wall of the needle and the single-walled carbon nanotubes also contributes to high drag force and heat transfer rate on the surface. It is revealed from the stability analysis that the first solution exhibits a stable flow. Obtained results are also matched with the present data in the restricting situation, and excellent agreement is noticed.

MHD flow of nanofluid over moving slender needle with nanoparticles aggregation and viscous dissipation effects

“…In Table 3, the results of this study are compared to the results that Soid et al 39 and Salleh and colleagues 38 talked about. When we worked with shooting in Mathematica, Salleh 38 used the bvp4c function in MATLAB. As can be observed, the values of .1emfnormal″false(efalse) are completely consistent with the investigations that are currently being conducted.…”

“…The transverse magnetic field Bfalse(xfalse)=B0x, which is replicated normal to the stretching/shrinking needle, is calculated using the applied magnetic field intensity B0. Based on the assumptions, the governing equation for system can be written as: (see 12,38 ) The following are the boundary conditions: 38 …”

“…For the purpose of simplifying equations (1) to (4), dimensionless quantities are introduced (see 38 ): In this case, ψ is a representation of an axisymmetric stream function, and u and v are denoted as: 38 And vf is the kinematic viscosity of the fluid. Further, U=Un+U∞ combined velocity of free stream and the needle, η represents variable that measures similarity, and θfalse(ηfalse) represents a temperature that is dimensionless.…”

“…Other important factors to consider include Prandtl number Pr, needle thickness e , as well as the flow-to-needle velocity ratio parameter ε,0.25emEc for Eckert number parameters and M is magnetic parameter which may be written as follows: where ε>0.25em0 implies that needle travels parallel to the direction of free flow, and ε<0 implies that the needle is moving in the opposite direction of the free flow. The physical amounts at stake, the coefficient of skin friction and transfer of heat rate (local Nusselt number) as follows: 38 where τn is the wall shear stress, and qn is the wall heat flux, which may be expressed as below:Equation plugging (5) and (6) into Equation (12) and using Equation (11), the following can be obtained:where Rex=uxvf is the local Reynolds number.…”

“…For the purpose of simplifying equations ( 1) to ( 4), dimensionless quantities are introduced (see 38 ):…”