Effect of asymmetric slip and Newtonian cooling on MHD, radiation, convective and heat and mass transfer flow of ethylene glycol based copper (CU) nanofluid in vertical channel with variable properties

Introduction

The transfer of warmth and mass difficulties are predicted to have increasingly notable usages in the late decades due to their widespread use and value in technological areas. Over the decades, many scholars have conducted significant investigations in MHD, natural and induced convection, heat and mass streams, taking into account the effects of chemical processes as well as Soret and Dufour impacts. Despite claiming that the journals on the issue are exhaustive, an effort is made below to highlight the key studies, while confining the references to the context and parameters of this study. Anghel et al. [8] , and Muralidhar [36] and Raju et al. [43] were investigated the Dufour and Soret effects on a free convection boundary layer with heat and mass transfer flow across a vertical surface buried in a porous material using a regular perturbation technique. Nanoparticles with high thermal conductivity can be elements the oxides, the carbides, or nitrides, while the base fluid can be water, kerosene, motor oil, toluene, ethylene, or triethylene glycol, depending on the procedure utilized. Several authors have been studied and experimental examined by Choi and Eastman [15] , Choi [16] , Choi et al. [17] , Das et al. [18] were probably the first to employ a mixture of nanoparticles and base fluid that such fluids were designated as nano-fluids. Thermal radiation is the means whereby a warm surface emits energy in all instructions in the outline of electromagnetic waves that arrangements in alluring and exciting waves. Mondal et al. [35] , Prakash et al. [42] , Umamaheswar et al. [48] , and Vedavathi et al. [50] , Alam and Ahammad [5] and Kumar et al. [30] have investigated the influence of radiation on heat transport difficulties with exothermic chemical processes and heat generation. A research investigation of heat radiation on free convection flow has a variety of use cases, including high-speed flights, space vehicle reentry, and thermal power plants. The natural conduction of air within conduits is the focus of extensive investigation in the past few decades due to its technical uses, including electronic cooling systems, nuclear reactors, and exchanges of heat.

The vertical channel is a commonly used set up in heating tools, such as solar power collectors and cooling devices for digital and micro-digital equipment by Gill and Casal [23] , Ostrach [40] , Cebeci et al. [13] , Datta and Jana [19] , Barletta [10] , and Barletta et al. [11] . Non-Darcy effects on transpiration in porous surfaces have received a lot of focus lately due to testing done with different blends of particles and water wrapping an array of dealing with variables, that reveal that data collected for platforms outside of a glass liquid at tiny Rayleigh parameters conflict with theorems depending within the Darcy flow model. This discrepancy in radiation transfer findings was quite thoroughly explored in Cheng et al. [14] , many a few more. The condition with Newton's law of thermodynamics occurs in what are often known as conjugate convective flows, in which heat is given to the flowing fluid via a bounding surface with a limited heat capacity. Several researchers, including Merkin [34] , Lesnic et al. [32] , Rao and Reddy [45] , have examined a different but still applicable guiding system for convective flow across boundaries close to a straight exterior. In these studies, it was thought that flow was started by Newtonian warming from the boundary, meaning that the heat transfer from the surface was identical at the personal surface temperature. The no-slip boundary condition (the assumption that a liquid clings to a solid border) is commonly used in flow issues involving viscous fluids. Slip may occur on the boundary when the fluid is particulate, as in emulsions, suspensions, foams, and polymer solutions. Many scholars have examined slip flows in different flow configurations. Abbasbandy and Shirzadi [1] , Rashidi et al. [44] , Hayat et al. [24] , Hayat et al. [26][27][28] , Ellahi et al. [21] , Awais et al. [5] , Adetayo et al. [4] , Anghel and Takhar [8] , Falade [22] use the homotopy analysis approach to create asymmetric Navier slips. In chemistry and physics, activation energy is the smallest quantity of energy necessary for a molecule to undergo a chemical reaction. A reaction's activation energy (Ea) can be represented as joules per mole (J/mol), kilojoules per mole (kJ/mol), or kilocalories per mole. Arrhenius [7] , a Swedish scientist, coined the word "activation energy" in 1889. Several writers have studied the energy of activation, including Abu-Nada et al. [2] and Abu-Nada and Eiyad [3] , Tiwari [47] , Kathyani and Subramanyam [28] , Narayana and Ramakrishna [39] , Nagasasikala [39] , and Devasena [20] , Lalramngaihzuali and Rao [31] , and Kumar et al. [29] , who addressed the effects of energy of activation on the hydromagnetic convective In most of the studies the viscosity of the fluid was assumed to be constant, When the effects of variable viscosity is taken into account, the flow characteristics are significantly changed compared to the constant property case. Ali (2006) was briefly discussed the effect of variable viscosity on mixed convection heat transfer along a vertical moving surface are studied by Abu-Nada et al. [2] and Abu-Nada et al. [3] , Vajravelu et al. [49] , Makinde [34] , Satya Narayana and Ramakrishna [45] , Nagasasikala [39] , Devasena [20] . Asymmetric slip and congested boundaries that characterize the MHD non-darcy turbulent mass and thermal flow of a Cu nanofluid based on ethylene in a vertical channel with asymmetrical warmth sources. Using Galerkine Finite element method with quadratic interpolation functions the non-linear, coupled equations governing the flow, heat and mass transfer have been executed. The velocity, temperature & nanoconcentration, Stress, Nusselt and Sherwood numbers have exhibited through graphs and tables.

Formulation of the Problem

The fluid consists of an ethylene glycol base fluid and tiny copper nanoparticles in a vertical porous tube with heat radiation. A homogeneous magnetic field with strength Ho is applied normal to the plate. Because of the fully matured assumption, the flow variables are just y-dependent. Fig - 2.1 depicts both the problem under examination and a coordinate system.

Under Rosseland approximation can be written in dimensional form as :

The boundary conditions [42,43] are as follows

α -Navier slip, μ f -viscosity fluid, u-axial velocity, P-fluid pressure, ρ f is the density of the nanoparticle, v o =channel porosity, Cp-constant pressure,(T,C)are the nanofluid temperature, concentration and kf is the thermal conductivity of the material respectively,(T o ,T 1 )-temperature,(C o ,C 1 )Concentration and γ o,1 -Newtonian cooling. D B is the molecular diffusivity, are the effective thermal expansion of the nanofluid which are given by.

2.5

where the addendums nf, f and s address the thermo actual properties of the nanofluid, base liquid and the nanosolid particles individually and  is the strong volume part of the nanoparticles. The thermo actual properties of the nanofluid are given in Table 2.1.

The thermo physical properties of the nanofluids are given in Table 2.1 (See Ozotop and Abu-Nada [41] ).

This is done by expanding 4 T ′ in a Taylor series about top wall temperature Ti as follows:

Neglecting higher order terms in the above equation beyond the first order in ( )

In view of the equations (2.6) and (2.8),equation(2.2) becomes

and solution of eq (1) becomes v=-v 0 (10) Introducing the following dimensionless variables:

appropriate boundary conditions of ordinary differential equations are

The transformed boundary conditions (3) reduce to 0 1 1 5 5

how good u defines the undefined flow rate, α 0,1 are the undefined slip factors at the barrier, S is the fluid suction/injection factor due to duct pores, and Bi0,1 are the Biot coefficients.

( )

is the Activation energy parameter 2.5 1 2

The limiting case α 0,1 →∞ parallels to the seamless lubricated plate surface.

Finite Element Analysis

Using quadratic polynomial estimation functions, a study of finite elements is performed along the axial distance across the vertical channel. For various changes in the controlling parameters, the behavior of the pressure, temperature, and velocity profiles is being computationally studied. Throughout the finite element analysis, the Galerkin method was used in the variational formulation of each element to get the global linked matrices for the degrees of freedom, quantity, and velocity. Select any element ek and assign the values of u k , and C to it (u k , θ k , and C k ).

The error residuals are defined as

[( ) ] Pr

The linear combinations in terms of respective local nodal values are

The partial differential equations (3.1)-(3.2) are transformed into matrix form of solutions using Galarkin's approach, yielding 3x3 local stiffness matrices. Through the use of inter element continuity, a state of equilibrium, and a substitution of nationwide nodal values, all of these local matrices are integrated into a global matrix. The global matrices that were produced were then solved iteratively until integration or |u i+1 -u i |<10 -6 , was achieved.

Comparison

In the absence of convection (G=0), heat sources (A11=0=B11), variable viscosity (B=0), activation energy (E1=0), δ=0 the results are in good agreement with Falade John [22] Para-meter Falade John [22] Present results τ(-

Results and Discussion

An investigation of the effects of changing viscosity, thermal radiation, asymmetric slip, and non-uniform warmth source on the mass transfer and convective heat flow of an EG-based nanofluid in an upward channel under Newtonian cooling is being attempted in this work. Different parametric modifications have been considered with regard to Temperature (θ), Velocity (u), and Nano concentration(C). in γ<0. This is owing to the fact that thickness of the thermal and solutal boundary layers become thinner with values of γ>0 and opposite effect is noticed in γ<0.

11. Fig. 12b-12c are activation energy parameter (E1) and temperature difference ratio (δ). From the profiles we find that the thickness of thermal and solutal boundary layers become thicker with rising values of E1 and δ. This results in an upsurges in θ and C with higher values of E1 and δ in the flow region. 12. Figs.13b-13c are Radiation absorption (Q1) and index number (n). The effect of Q1 is to upsurge θ and C. Grows in n leads to a growth in θ and C in the entire flow region. This amount to the fact that the thermal and solutal boundary layers becomes thicker with rising values of Q1 and n.

13. The Shear stress(τ) is exhibited in table.3 at the walls η=±1 for different parameters G,M,K,S,Rd,Ec,B,A11,B11,B,φ, ∆ ,α0,α1,Bio,B11 and Pr. An upsurge in G/φ/Rd/A11 grows stress at η=±1 in Eg based Cu nanofluid. Higher the Lorentz force (M)/Suction parameter (S) decays stress at η=+1 & grows at η=-1 in Eg based Cu nanofluid. Increase in B/R/∆/Ec diminishes the stress at η±1. K reduces stress at η=-1 and enhances at η=+1 in Eg Cu nanofluid. An increase in αo, α1 decays the stress at left wall while at η=1, it upsurge with αo and decays with α1. Upsurge in Biot number (Bi0) leads to a fall in stress at η±=1 while stress grows at η=-1 and diminishes at η=+1 with higher values of Biot number (Bi1). τ with Pr, the thermal diffusivity smaller the stress at the walls. 14. Table 3 are

Conclusions

The influence of activity energy, variable viscosity, solar radiation, and dissipation on the convective warmth and mass flow of an Eg-based Cu with asymmetric slips & convective limits at lack of irregular heating elements. The nonlinear, mixed problems were solve by the Finite-element approach using quadratic interpolation function. The results of the above review are:

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