Coupled heat and mass transfer mathematical study for lubricated non-Newtonian nanomaterial conveying oblique stagnation point flow: A comparison of viscous and viscoelastic nanofluid model

Nomenclature

( a' , b' )

dimensionless constants

(C ₁,C ₂)

constant of integration

C _f

dimensionless skin friction coefficient

C _w

surface concentration

C ∞

free surface concentration

D B

Brownian diffusion coefficient

D T

thermophoresis diffusion coefficient

j _w

mass flux

k

thermal conductivity

k ₀

second grade fluid parameter

k ^*

mean absorption constant

k ₁

dynamic coefficient of viscosity

Le

Lewis number

N_b

Brownian motion parameter

Nt

thermophoresis parameter

Nu _X

Nusselt number

p

pressure

Pr

Prandtl number

Q

flow rate

q _w

radiative heat flux

Re

Reynolds number

R _d

radiation parameter

Sh

Sherwood number

T _w

surface temperature

(u,v)

components of velocity

(U,V)

axial and tangential velocities of lubricant

u _e

axial velocity component at free stream

v _e

tangential velocity component at free stream

T w

surface temperature

T ∞

free surface temperature

δ ( x )

lubricated surface width

ρ

density

μ

dynamic viscosity

α

thermal diffusivity

τ

effective heating nanoparticles capacitance to nanofluid heat capacity

σ *

Stefan–Boltzmann constant

α 1

viscoelastic parameter

β c

chemical reaction parameter

λ

slip parameter

γ

shear at the free stream

τ w

wall shear stress component

1 Introduction

In fluid mechanics, stagnation point flow is defined as the movement of fluid near the solid surface. Fluid splits up into two surfaces or deviates its route when it approaches the surface. Physical importance of stagnation point flows is noteworthy since they utilize to commute velocity gradients as well as estimate mass or heat transfer and skin friction in the stagnation zone. A comprehensive literature survey regarding stagnation points flow owing to its eminent applications has been presented by numerous investigators. Hiemenz [1] scrutinized the two dimensional motion of viscous fluid in the zone of stagnation point on a flat plate. Homann [2] discussed the axisymmetric flow in the zone of stagnation point on flat surface. Hannah [3] analyzed the Homan flow of stagnation point over disk which was the broadening work of Homan’s genuine inquiry involving the plate flow following the stagnation phenomenon. The moving surface according to the stagnation point situation, a two-dimensional, viscous and incompressible flow was examined by Rott [4]. On a flat plate, Weidman and Mahalingam [5] did their research relating to the axisymmetric flow of stagnation point problem. Mahapatra and Gupta [6] examined stagnation point flow approaching the stretched surface. The flow analysis for a viscous fluid impinging obliquely on stretched sheet was carried out by Lok et al. [7]. Nadeem et al. [8] investigated the dilemma of stagnation point’s flow for the Jeffrey fluid on shrinking sheet. Weidman and Sprague [9] investigated flow caused by a plate moving in a normal to stagnation point flow. Irrotational stagnation point flows by Hiemenz and Homan were contemplated. Two Hiemenz planar stagnation point flows that were normal to a uniform plate were explored by Weidman [10] for their interaction.

The approaching impinging flow on a surface creates a point of stagnation where the fluid velocity drops to zero. The fluid flow surrounding this stagnation point is known as stagnation point flow. In fact, stagnation point phenomenon preserved the extension of Hiemenz phenomenon. Hiemenz flow, also known as the oblique stagnation point flow, involves the flow of a liquid that adjusts to the surface at a right angle. The prime investigation on this subject was carried out by Stuart [11]. Tamada [12] studied the stagnation point flow impinging obliquely over a oscillating flat surface. The outcomes for two-dimensional oblique stagnation point flow are presented through calculating exact solutions by Dorrepaal [13]. Reza and Gupta [14] analyzed the non-orthogonal situation for the stagnation point where pressure gradient plays an important role. Weidman and Putkaradze [15] investigated the stagnation point impact for circular cylinder without considering the role of displacement thickness and pressure variation. The micropolar material flow with oblique stagnation point observations was determined by Lok et al. [16].

Heat is an energy form which distributes energy to the substance’s molecules in an accelerative way, and gathers this kinetic energy on a massive level. Consequently, as this energy approaches its maximum or minimum point, the molecules or atoms are liberated from interatomic forces of attraction and alter their state. Heat transfer is a mechanism that happens when the temperatures of two substances change i.e., from warm to cold or from cold to warm substance. Heat transport and flow using the second grade fluid across a stretched sheet were studied by researchers Dandapat and Gupta [17]. The transmission of heat in a flow of stagnation point across a stretched sheet was studied by Mahapatra and Gupta [18]. Analysis of analytical solutions for the exchange of thermal energy and axisymmetric flow in a viscoelastic fluid across a stretched sheet was done Hayat and Sajid [19]. Attia [20] determined the progress of heat transfer near the stagnation point region in a porous medium for second grade fluid. In a boundary layer stagnation point flow toward an extending or contracting sheet, melting heat transfer was examined by Bachok et al. [21]. The heating determination deduced for Jeffrey fluid via flat surface with non-orthogonal stagnation framework was examined by Arshad et al. [22]. In another attempt, Bano et al. [23] described a mathematical model to discuss the heat transfer characteristics over a stretching cylinder. Labropulu et al. [24] reported the effects of elasticity involving the heat transfer pattern. Abbasi et al. [25] studied about radiated phenomenon for obliquely moving surface.

The nanofluids attract the researchers due to many applications in industry and biomedical science like cancer therapy, cooling and heating processes, etc. Many organic and inorganic fluids like grease, oil and ethyl glycol have poor thermal and electric conductivity. Many techniques are proposed in the literature to enhance the thermal and electric properties of these working fluids. By adding nanoparticles up to size 1–100 nm having large thermal and electric conductivity in the base fluid is commonly used by many researchers and scientists. These fluids were characterized as nanofluids for the first time by Choi and Eastman [26]. Later on, Das et al. [27] carried out a study to prove that by addition of 1% of nanoparticles in a base fluid can enhance its thermal properties up to 100% more than the base fluid. Buongiorno [28] deduced the nanofluid transport applications via theoretical study. Later on, many theoretical studies used the Buongiorno model to investigate the stagnation point flows of nanofluids. Hamid et al. [29] identified the pattern of stagnation point for nanoparticles flow via stretched surface. Roşca et al. [30] investigated the response of the same phenomenon for boundary effected flow with flat surface due to nanofluid. Abbasi et al. [31] studied about slip impact with circular moving cylinder. Abbasi et al. [32] reported the stagnation outcomes of Maxwell nanomaterial over a stretching cylinder with different heat transfer features. Mahmood et al. [33] examined the effects of lubrication while elaborating the oblique stagnation onset of flat surface conveying the viscous fluid flow. It is noted that the temperature of nanofluid is large over lubricated surface as compared to rough surface. Abbasi et al. [34] imposed the microorganisms suspension for nanofluid via obliquely lubricated space. The representation of dual solution for lubricated nanofluid via oscillating surface was reported by Nadeem and Khan [35]. Bao et al. [36] predicted the hybrid nanofluid heat transfer performances for solar collectors. Selim et al. [37] disclosed the magnetic force association for fluctuating the thermal prospective of nanomaterials. Hanafi et al. [38] pronounced the mathematical model for expressing the nanofluid with suspension of aluminum and copper nanoparticles. Ghafouri and Toghraie [39] performed the experimental analysis for nanofluid in view of variable thermal conductivity. Sundar [40] presented a comprehensive review of hybrid nanofluid with heat exchanger applications. The physical significance of micro-channels are depicted in previous literature [41–43] and the impact of hydro-dynamics in flow with different geometries are reported in previous research works [44–46].

After presenting an inspired analysis on current topic, it is mentioned that different research studies on nanofluids are presented in various flow configurations. However, the thermal applications of nanofluids due to obliquely lubricated surface have not been focused yet. The applications of nanoparticles are interesting in lubricated phenomenon. With these motivations in mind, the aim of the current work is to analyze the two-dimensional stagnation point flow of second grade fluid on a flat rigid obliquely lubricated surface. This phenomenon is commonly observed when a jet of fluids strikes obliquely on a lubricated surface. The main aim of our study is to examine the oblique stagnation point flow of second grade nanofluid over a lubricated surface in the presence of linear thermal radiation. The reactive species with first order relations have been utilized to analyze the phenomenon. The motivations behind choice of second grade fluid is satisfied with novel rheology and interesting dynamics. The problem is illustrated and modeled with partial differential system. The implicit finite difference scheme along with quasi linearization is used to simulate the different flow and heat transfer features. The physical impact of problem with variation in parameters is observed.

2 Problem formulation

The current mathematical model is used to simulate 2D stagnation point assessment of viscoelastic fluid characterized by second grade fluid over a flat lubricated surface. For mathematical formulation, Cartesian coordinate system is used such that the fluid flowing along the x-axis and y-axis is perpendicular to the lubricated surface with variable width δ ( x ) . Furthermore, the temperature T w and concentration of the nanoparticles C w are same at the surface of the sheet and at y = 0 . The free surface thermal and concentration constraints are identified by T ∞ and C ∞ , respectively. The axial and tangential velocity components at free stream are u e and v e (Figure 1).

The investigation of nanofluid in the present problem is predicated by using Buongiorno two phase fluid model along with Rosseland approximation for linear thermal radiation. In order to describe the diffusion rates the chemical reaction of first order is taken in the concentration of nanoparticles.

After using the aforementioned restrictions and conditions, the formulated equations are [33,35] as follows:

where ρ stands for the density, u and v are the components of velocity, k 0 is the second grade fluid parameter, μ represents the dynamic viscosity, p is the pressure, α is the thermal diffusivity, τ = ( ρ c ) p / ( ρ c ) f expressed the effective heating nanoparticles capacitance to nanofluid heat capacity of nanoparticles, D B is the Brownian diffusion coefficient, D T is the thermophoresis diffusion coefficient, σ * is Stefan–Boltzmann constant and k * is the mean absorption constant.

The no-slip boundary condition at the surface implies

where U and V are the axial and tangential velocities of lubricant.

In order to obtain the boundary condition at the lubricant and non-Newtonian fluid interface, the continuity of shear stresses and velocities is used.

in which

where k 1 is the dynamic coefficient of viscosity. For linear lubricated velocity U ( x , y ) , the relation is

with lubricant thickness δ ( x ) .

Now the relation between the thickness of the lubricant and flow rate Q is

using Eqs. (9) and (10), Eq. (7) takes the form

The continuity of horizontal component of velocity components implies

Substituting Eq. (12) in Eq. (11), the continuity of shear stresses takes the form

If a' and b' are dimensionless constants, the velocities at free stream take the form [1]

New variables are:

After equating the coefficient of x 1 0 and x 1 and integrating once with respect to y, the modeled equation can be written as follows:

where C 1 and C 2 are the constants of integration, α 1 = k 0 a / ρ ν represents viscoelastic parameter, Pr = ν / α is the Prandtl number, Le = ν / D B is the Lewis number, Nt = τ D T ( T w − T ∞ ) T ∞ ν is the thermophoresis parameter, Nb = τ D B ( C w − C ∞ ) C ∞ ν is the Brownian motion parameter, R d = 4 σ * T ∞ 3 k * is the radiation parameter and β c = K 1 ( C w − C ∞ ) ν is the chemical reaction parameter.

The associated boundary conditions in dimensionless form are

where λ = k 1 ν a 2 n x 2 n − 1 / μ a 3 2 ( 2 Q ) n is the slip parameter. Also, γ = b / a is the shear at the free stream. After implementing the boundary conditions at infinity, the constants of integration are C 1 = 1 and C 2 = γ ( β − ε ) , where ε = η ∞ − f ( ∞ ) and β is the free parameter. Now expressing the wall shear force

with dimensionless form

3 Solution methodology

The simulations are predicted with Keller Box method for solving the Eqs. (15)–(18). The Keller Box technique is supported with finite difference scheme attaining the second-order accuracy. The Keller Box scheme is interesting and various multidisciplinary problems are computed via this approach. The numerical scheme is implemented in four steps.

Step 1: Reducing the governing Eqs. (15)–(18) subject to associated boundary conditions in Eqs. (20)–(22) in the system of first-order equations. For this, we consider F ′ = U 1 , U 1 ′ = V 1 , V 1 ′ = W , G ′ = P 1 , P 1 ′ = Q 1 , Q 1 ′ = S 1 , θ ′ = Y 1 and ϕ ′ = Z 1 , the governing equations takes the form

Step 2: Replacing the derivatives by central finite difference approximation and all dependent and independent variables by taking average of them, we get

Step 3: As Eqs. (34)–(36) are nonlinear, Newton linearization scheme is followed for converting the problem to linearized form.

The associated boundary conditions take the form

where