Influence of non-uniform heat source/sink and variable viscosity on mixed convection flow of third grade nanofluid over an inclined stretched Riga plate

Abstract

The present study focuses on the impact of non-uniform heat source/sink and temperature dependent viscosity modeled by Reynolds on Cattaneo-Christov heat flow of third grade nanofluid subject to an inclined stretched Riga plate. Fourth order R-K and shooting methods have been implemented to obtain the numerical solution of the transformed boundary layer equations. The achievability of the present study is that the material constants associated with third grade fluid augment the fluid motion and boils down the fluid temperature leading to ascending velocity boundary layer and descending thermal boundary layer. And viscosity parameter enhances the heat transfer rate from the plate. Furthermore, augmented space and temperature dependent heat source upsurges the fluid temperature and the related thermal boundary layer thickness.

1. Introduction

It was the time when the World was looking forward to greater research contributions from across the Globe; Choi et al. [1] played an important role in discovering experimentally the concept of nanofluids. Nanofluids are the novel type fluids that contain nanoparticles, such as metals, oxides, carbides, and nitrides, with sizes less than 100 nm have higher thermal conductivity compared to that of the conventional base fluid, such as water, engine oil, and ethylene. Such extraordinary characteristics of nanofluids (fluids with more heat transfer capability) make them potentially useful to serve as better coolant in nuclear reactor, micro manufacturing, refrigeration, automotives, heat exchangers, aircrafts and space applications and other high energy devices, antibacterial treatment, wound treatment, asthma treatment, targeted drug release/drug delivery [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22].

Looking at the Riga-plate arrangement it produces a Lorentz force of exponentially decaying nature which induces the flow over the plate. Pantokratoras and Magyari [23] were the pioneers to study electro-magnetohydrodynamic flow over a horizontal Riga plate. Later, Ahmad et al. [24] studied the flow of nanofluid past a Riga plate considering the influence of nanoparticles size on skin friction. Hakeem et al. [25] investigated the impact of exponentially variable viscosity and permeability on Blasius flow of Carreu nanofluid over an electromagnetic plate.

In fact, third grade fluid imparts features such as normal stresses and shear thinning/shear thickening phenomena. Examples of such fluids are molten plastics, slurry flows, highly viscous silicon oils and dilute polymer solutions and many more. Hayat et al. [26] explored in their investigation that velocity field disparages with enhancing of melting parameter and reducing wall thickness parameter. The deviations of flow variables for third-grade fluid are significant compared to the Newtonian fluid flow (Ready et al. [27]).

The interesting aspect of CCHF model is that thermal relaxation was added to Fourier’s law by Cattaneo [28]. The Cattaneo’s model was further improved by Christov [29] after which this model is known as CCHF model. Augmentation in thermal relaxation time contributes to low temperature (Imtiaz et al. [30]). Hayat and Nadeem [31] declared in their study that the heat source/sink that exerts strong influence on HTR associated with the TBL. Heat source/sink alters the HTR and thereby changes the structure of TBL. The fluid temperature and wall shear stress increase due to increase in heat source/sink parameter in stagnation point flow past stretching sheet (Sharma et al. [32]).

In brief, the present study is clearly meant for exploring the effects of non-uniform heat source/sink and temperature dependent variable viscosity on the flow of third grade nanofluid flow over an inclined Riga plate. Further, CCHF model is invoked to obtain the behavior of relaxation time. An appropriate numerical solution of the developed similarity transferred coupled non-linear differential equations has been devised by using fourth order R-K method through shooting technique. The influence of various significant emerging parameters concerned presented through appropriate graphs and discussion.

2. Formulation of the problem

In the current study, we consider the steady incompressible flow of third grade nanofluid over an inclined stretched Riga-plate. A Cartesian coordinate system is chosen such that x-axis is along the plate and y-axis is normal to it (Figure 1). The plate is stretched by equal and opposite forces with velocity U_w (x) = ax . Electromagnetic field of the Riga-plate induces an exponentially descending Lorentz force parallel to the plate surface.

The equations of continuity, momentum and energy governing the flow of third grade nanofluid in the presence of non-uniform heat source/sink are [8,14,26,31]:

Figure 1. Flow configuration and coordinate system.

Figure 1. Flow configuration and coordinate system.

According to CCHF theory, we reveal [13]:

where λ is the relaxation time of heat flux/thermal relaxation of the fluid, V is the velocity vector and q is the heat flux. It is the time taken for the shear stress in a viscoelastic material to reduce 1/e of its initial value.

Setting λ = 0 in eq (5), classical Fourier’s law can be obtained. Considering the flow as steady and incompressible (∇.V = 0), eq (5) takes the form

q + λ (V.∇q − q.∇V) = −k∇T

(6)

The energy equation then becomes

According to Grinberg [33], the force density averaged over the span wise coordinate z takes the form

Hence, is the Grinberg term of the momentum equation (2) which does not depend upon the flow velocity.

The requisite boundary conditions are:

The non-uniform heat sourc/sink is expressed as

Where Q₀ and Q₁ are the coefficients of space and temperature dependent heat source/sink respectively. The case Q₀ > 0 and Q₁ > 0 contributing internal heat generation while Q₀ < 0 and Q₁ < 0 contribute internal heat absorption.

The suitable transformations employed for the purpose are:

Where U = Uw +U∞ is the composite velocity. The incompressibility condition is satisfied. According to Reynolds model, the expression for temperature dependent viscosity is

which can be expressed as

μ_f(θ) = μ₀ [1− (mθ) + o (m² )]

(12)

With the help of eqs (9), (10) and (12) and neglecting higher order terms in eq. (12) and using the resulting expression, eqs (2), (4), (7) and (8) take the form:

The local skin friction coefficient

Where denotes the wall shear stress.

The non-dimensional local skin friction coefficient

with denotes the wall mass flux.

The non-dimensional local Sherwood number

3. Numerical Methodology

The system of equations (13) – (15) together with the equation (16) are solved by using 4^th order RK method along with shooting technique. A relevant short logic diagram depicting the algorithm of the solution is mentioned below.

4. Results and Discussion

The present study primarily focuses on the effect of non-uniform heat source/sink and variable viscosity on mixed convection CCHF of TGNF over an inclined stretched Riga plate. An appropriate numerical solution of the transformed governing equations is obtained. The values of the parameters considered for the present simulation are: Ω₁ = Ω₂ = Ω₃ = 0.1, λ_c = λ_T = 0.3, ϒ = 0.1 , Λ = 45°, m = 0.5, δ = 0.2, Γ= 0.1, ε = 0.6 Pr = 1, Nb = Nt = 0.5, Q₀ = Q₁ = 0.1 and Le = 0.5 . The present analysis imparts well exploration regarding the effects of different pertinent parameters on velocity, temperature, concentration, skin friction and Sherwood number profiles through suitable graphical representation and pretty discussion.

Figure 2 and Figure 3 bring to the focus the behavior of λ_T and λ_C on fluid velocity. It is noticed that an accelerated fluid motion takes place due to enhancement in λ_T and λ_C which in turn establishes thicker MBLs in the respective flow domains. Figure 4 reveals that an augmented m contributes to the accelerated fluid motion giving rise to the same environment that due to the presence of thermal and solutal buoyancies. The fundamental reason behind this ascending trend is that viscous force dominates over the inertial force. Increasing values of Γ yields the rising flow contributing to high wall velocity gradient (Figure 5). For the more and more the angle of inclination of the plate, VBL thickness gets reduced (Figure 6). As ε rises, the flow diverges from the plate towards the ambient fluid with gradually more descending trend (Figure 7). Figure 8, Figure 9 and Figure 10 portray the influence of Ω₁, Ω₂ and Ω₃ on velocity profiles respectively. Fluid velocity gets diminution near the plate within the range (0 ≤ η ≤ 0.5) and it shows a reverse trend followed by a transition at η = 1.5 under the impact of increasing Ω₁ . Indeed, material parameters are inverse relation to viscosity. That is why increase in Ω₂ and Ω₃ belittles the fluid viscosity thereby favors the fluid motion.

It is obvious from Figure 11, Figure 12 and Figure 13 that increase in Ω₁ and Ω₂ undermine the fluid temperature θ(η) while that of Ω₃ upsurges it. A unique feature of these θ(η) profiles is the appearance of non-linearity which is due to the presence of non-linear terms in governing energy equation. Figure 14 demonstrates the non-linear decaying nature of the θ(η) profiles due to increase in m leading to shrinking of TBL in the entire flow regime. A non-linear θ(η) profile due to distinguished behavior of Pr is visualized in Figure 15. It is noticed here that increasing Pr makes the θ(η) less (due to low thermal diffusivity compared to viscosity) which in turn reduces TBL thickness. However, high Prandtl fluid (Pr = 6) exhibits slow response to such diminishing trend. Further, symmetrical diminishing trend of non-linear θ(η) profiles due to increase in ϒ is envisioned in Figure 16. This is because of low temperature profile under the influence of CCHF model. Figure 17 and Figure 18 offer the non-linear, however, increasing behavior of non-dimensional θ(η) due to enhancement in N_b and N_t respectively. The rationally behind this is that higher Brownian diffusion and thermophoresis force contribute to stronger fluid temperature. Behavior of non-dimensional θ(η) due to the influence of non-uniform heat source/sink (Q₀ and Q₁) is reflected from Figure 19 and Figure 20 respectively. In other words, increase in Q₀ and Q₁ upsurge the θ(η) in non-linear fashion throughout the growing TBL associated with enhanced wall temperature gradient.

Figure 21, Figure 22 and Figure 23 illustrate that hiked Ω₁ and Ω₂ upsurge the fluid concentration ϕ (η) while that of Ω₃ undermine it. Further, Figure 24 indicates an increasing non-linear behavior of ϕ (η) due to rise in m generating ascending CBL. Figure 25 displays that increase in Le belittles the ϕ (η) developing descending CBL. This is only due to lower mass diffusivity. Fluid concentration ϕ (η) profiles enhance in response to rise in N_b and N_t , however, enhancement is more in case of Nt (Figure 26 and Figure 27). This is due to the effective motion of nanoparticles from the plate to the fluid.

Variation of skin friction coefficient against m for different Ω₁, Ω₁ and Ω₃ is shown in Figure 28. It is conveyed from this figure that gets augmented due to increase in Ω₁ and Ω₂ while the wall shear stress shows diminution due to enhancement in Ω₃ . Meanwhile, Figure 29 provides the variation of the wall shear stress against m for different Γ and Λ . What is found here is that increase in Γ enhances the wall shear stress. As far as the behavior of Sherwood number is concerned, influence of Ω and Ω on it are diametrically opposite (Figure 30).

Augmented Ω₁ enhances the mass transfer rate from the plate while that of Ω₂ significantly reduces the same. Ascending trend of mass flux is prominent for comparatively higher Ω₁ (Ω₁ = 0.3).

Figure 2. f ′(η)Vs λ_T.

Figure 2. f ′(η)Vs λ_T.

Figure 3. f ′(η)Vs λ_c.

Figure 3. f ′(η)Vs λ_c.

Figure 4. f ′(η)Vs m.

Figure 4. f ′(η)Vs m.

Figure 5. f ′(η)Vs Γ.

Figure 5. f ′(η)Vs Γ.

Figure 6. f ′(η)Vs Λ..

Figure 6. f ′(η)Vs Λ..

Figure 7. f ′(η)Vs ε.

Figure 7. f ′(η)Vs ε.

Figure 8. f ′(η)Vs Ω_1.

Figure 8. f ′(η)Vs Ω_1.

Figure 9. f ′(η)Vs Ω_2.

Figure 9. f ′(η)Vs Ω_2.

Figure 10. f ′(η)Vs Ω_3.

Figure 10. f ′(η)Vs Ω_3.

Figure 11. θ(η)Vs Ω1.

Figure 11. θ(η)Vs Ω1.

Figure 12. θ(η)Vs Q_2.

Figure 12. θ(η)Vs Q_2.

Figure 13. (η)Vs Q_3.

Figure 13. (η)Vs Q_3.

Figure 14. θ(η)Vs m.

Figure 14. θ(η)Vs m.

Figure 15. (η)Vs Pr.

Figure 15. (η)Vs Pr.

Figure 16. θ(η)Vs ϒ.

Figure 16. θ(η)Vs ϒ.

Figure 17. θ(η)Vs Nb.

Figure 17. θ(η)Vs Nb.

Figure 18. θ(η)Vs N_t.

Figure 18. θ(η)Vs N_t.

Figure 19. θ(η)Vs Q_0.

Figure 19. θ(η)Vs Q_0.

Figure 20. θ(η)Vs Q_1.

Figure 20. θ(η)Vs Q_1.

Figure 21. ϕ (η)Vs Q_1.

Figure 21. ϕ (η)Vs Q_1.

Figure 22. ϕ (η)Vs Ω_2.

Figure 22. ϕ (η)Vs Ω_2.

Figure 23. (η)Vs Ω_3.

Figure 23. (η)Vs Ω_3.

Figure 24. (η)Vs m.

Figure 24. (η)Vs m.

Figure 25. ϕ (η)Vs Le.

Figure 25. ϕ (η)Vs Le.

Figure 26. ϕ (η)Vs N_b.

Figure 26. ϕ (η)Vs N_b.

Figure 27. ϕ (η)Vs N_t.

Figure 27. ϕ (η)Vs N_t.

Figure 28. Effects of Ω1, Ω2 and Ω3 on Re 2 C_f against m.

Figure 28. Effects of Ω1, Ω2 and Ω3 on Re 2 C_f against m.

Figure 29. Effect of Λ and Γ on Re 2 Cf against m.

Figure 29. Effect of Λ and Γ on Re 2 Cf against m.

Figure 30. Effects of Ω₁ and Ω₂ on the Sherwood number against Le.

Figure 30. Effects of Ω₁ and Ω₂ on the Sherwood number against Le.

5. Outcomes at a glance

The present study deals with the influence of non-uniform heat source/sink and temperature dependent viscosity on Cattaneo-Christov heat flow of third grade nanofluid past an inclined stretched Riga plate. The present investigation comes out with some important outcomes.

In the current analysis, accelerated fluid motion and thicker momentum boundary layers are due to increment in λ_T , λ_C , m , Γ. It has identified that increase in Ω1 ascends the fluid motion while that of Ω₂ and Ω₃ exhibits reverse effect. It acknowledges that augmented Ω₁ and Ω₂ accounts for shrinkage of thermal boundary layer while that of Ω₃ establishes ascending thermal boundary layer. There are reports that enhanced Nb and Nt upsurges the fluid temperature while hike in Le belittles the fluid concentration leading to a descending concentration boundary layer. Further, increment in Q₀ and Q₁ contributes to increment in wall temperature gradient and yields an augmented wall shear stress. Finally, rise in Ω₁, Ω₂ and m has caused to upsurge the concentration while that of Ω₃ boils down it.