Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet

Nomenclature

AAE

Arrhenius activation energy

Al₂O₃

aluminum oxide

c

electrode width

C

concentration

CeO₂

cerium oxide

C p

heat capacitance (J kg⁻¹ K⁻¹)

CRs

chemical reactions

Ea

activation energy

HNF

hybrid nanofluids

j 0

current density

K

surface permeability

k

thermal conductivity (kg ms⁻³ K⁻¹)

k r

rate of chemical reaction

M 0

magnetization of the Riga plate

NN

non-Newtonian

NPs

nanoparticles

Q _h

Hartmann number

q r

heat radiation

SG

second-grade

SA

sodium alginate

RP

Riga plate

T

temperature

u , v

velocity component

V T

thermophoretic velocity

WB

Walter’s B

α 1 > 0

SG fluid

α 1 < 0

WB fluid

β 1

magnitude and width of electrodes

ρ

density (kg m⁻³)

ρ C p

thermal capacity

λ < 0

dwindling sheet

λ > 0

stretching sheet

σ

electrical conductivity

1 Introduction

The Riga plate (RP), a device or plate made up of a flat surface with specified electrodes and patterns, is commonly used in fluid dynamics to investigate the behavior of electrically conductive fluids including electrolytes and liquid metals. By supplying voltage, the Riga plate creates an electric field that generates electrohydrodynamic (EHD) flows, enabling researchers to alter fluid motion, improve mixing, and manage instabilities in a variety of applications. Stretching Riga plates, for example, can improve techniques for mass transfer and allow for accurate droplet manipulation in microfluidics, making them particularly useful in biomedical applications and chemical reactors. Furthermore, their use in cooling systems improves thermal management by facilitating effective heat transfer via fluid flow regulation [1,2,3]. Cui et al. [4] studied the nanofluid (NF) flow in a porous media over a Riga surface, with the impact of convective boundary conditions, and activation energy. It was discovered that the Hartman and Eckert numbers increase with both temperature and velocity profiles. Algehyne et al. [5] assessed the magnetohydrodynamics (MHD) hybrid nanofluids (HNF) flow consisting of MgO–Ag/water with motile microbes across a Riga sheet with the impact of slip conditions and activation energy. It was noticed that the velocity slip conditions drop the thermal curve and velocity rate of NF. Mohanty et al. [6] investigated the influence of interfacial nanolayer width on multi-walled carbon nanotubes and nano-diamond incorporated in water-based HNF fluid flow along an inclined stretching cylinder. Farooq et al. [7,8] studied the MHD NF flow consisting of SiO₂ and MoS₂ nanoparticles (NPs) with temperature-dependent viscosity across a stretching sheet. It was observed that increasing the magnetic factor lowers fluid velocity while increasing the temperature curve. Bilal et al. [9] numerically analyzed the mass distribution and energy transfer through NF flow over an expanding Riga plate. It was noticed that the fluid flow rate lowers with the impact of the power law index and improves with the outcome of the modified Hartmann number. Xin et al. [10] investigated how the discharge of pollutants concentration influences the non-Newtonian (NN)-NF flow across a permeable RP and discovered that the energy transmission rate of the second-grade (SG) fluid enhances up to 6.25%, while Walter’s fluid B rises to 7.85%. Gangadhar et al. [11] calculated that the improved slip and convective heat conditions influence the ternary NF flow across a RP and concluded that the variation in Hartmann number drops the fluid velocity. Bilal et al. [12] numerically simulated the pollutant discharge concentration impact on NN HNF flow for the treatment of wastewater for industrial uses across a Riga surface.

NN fluids, which feature complex flow behavior that differs from Newton’s viscosity law, are used in a variety of sectors. In medicine, they are utilized in the manufacture of pharmaceuticals and biological compounds, where their distinct features enhance the delivery of drugs and results for patients. NN fluids are essential in food processing for items such as sauces and dressings, where their viscosity can be adjusted for the best consistency and texture. Furthermore, in the construction sector, materials such as cement and various adhesives depend upon NN behavior to improve workability and efficiency. NN fluids are also used in the creation of body armor and smart fluids for automobiles, where their viscosity may alter under stress to provide better protection and functionality [13]. Mohanty et al. [14,15] investigated the irreversibility analysis of 3D Casson HNF flow induced by a turning disc employing a Runge–Kutta–Fehlberg-based shooting method. Farooq et al. [16] evaluated the stagnation point mixed convective ternary HNF flow over a vertical Riga sheet employing Al₂O₃, SiO₂, and TiO₂ NPs in the base fluid. When the mixed convection factor is employed, the thermal profile of HNF decreases, while the distribution of velocity improves. Cham and Mustafa [17] provided a thorough assessment of the effect of viscoelastic factor and partial slip conditions on the axial fluid flow caused by a plate or cylinder deforming with linear acceleration. Farooq et al. [18] addressed the thermal transportation of NN fluids containing carbon nanotubes NPs across a stretched surface, taking into account viscosity dissipation, porous medium, heat source, and magnetic field parameters. It was discovered that increasing the SG fluid factor increases the velocity dissemination, whereas increasing the magnetic and porosity parameters reduces the velocity dispersion. Ahmad et al. [19] discussed the importance of irreversible processes in the regular Homann flow of viscoelastic liquids. The heat and mass balancing characteristics were further characterized by the addition of Brownian motion and thermophoretic variables. The average entropy rate decreases as the magnetic factor and Brinkman number improves. Bilal et al. [20] addressed the micropolar NF flow over a vertically orientated, nonlinearly stretchy Riga surface, considering varying thermal conductivity and Brownian diffusion with the Levenberg–Marquardt back propagation technique. The flow rate was observed to increase when the Hartmann number, velocity slip, and buoyancy force parameters. Mohanty et al. [21] evaluated the thermal and flow properties of a cross-ternary HNF flow consisting of single-walled carbon nanotubes, GO, and MWCNT NPs through a stretching cylinder.

Chemical reactions (CRs) and Arrhenius activation energy (AAE) play critical roles in fluid mathematical models, especially in processes including fluids that react during flow [22]. Studying the Arrhenius equation, which depicts the rate of CRs as exponentially dependent upon temperature, assists engineers to precisely model the behavior of products and reactants under various thermal scenarios. This is critical for improving reaction kinetics and the performance of reactors and transmission lines. In technologies such as catalytic converters, fluid dynamics combined with reaction kinetics aid in the design of more effective catalysts by disclosing how temperature affects reaction rates, hence assisting in pollutant removal [23]. Moreover, in biological processes, fluid models that incorporate CRs allow scholars to investigate complex phenomena such as the transfer of nutrients in fluids, resulting in advances in pharmaceuticals and biotechnology. Ultimately, incorporating CR kinetics and AAE into fluid models improves our knowledge and optimization of chemical processes in a variety of domains [24]. Ullah et al. [25] investigated the effect of melting heat exchange and AAE on an unstable Prandtl-Eyring model. It was concluded that the impact of AAE accelerates the fluid concentration field. Sudarmozhi et al. [26] evaluated the Maxwell double-diffusive convective fluid flow in the presence of CRs, heat radiation, and AAE on a perforated vibrating flat plate. It was discovered that as the heat formation rate increases, the temperature profile decreases, whereas the concentration profile diminishes with the variation in CRs factor. Some further related results are calculated in previous studies [27,28,29,30].

An investigation of the physical characteristics of NN-HNF comprising Al₂O₃ and CeO₂ NPs found that Al₂O₃ and CeO₂ NPs boosts the capability of designing structures as well as processes for an extensive variety of industry applications, thereby enhancing the product’s durability and performance. Alumina NPs are widely recognized for their physical and thermal features. When preparing the HNF, empirical research to quantify thermal conductivity, viscosity, stability, and reliability is essential to govern the best combination for practical applications. Combining Al₂O₃ NPs with CeO₂ NPs, commonly referred to as cerium dioxide or cerium(IV) oxide, enhances viscosity and transference of heat, making it ideal for cooling and thermal systems. Therefore, in the current analysis, the NN-HNF flow including Al₂O₃ and CeO₂ NPs over a porous stretching or shrinking Riga sheet is studied. The flow phenomena are modeled in the form of nonlinear partial differential equations (PDEs), which are reduced into the dimension-free form through the similarity conversion. The solution is obtained by using the numerical approach parametric continuation method (PCM). The results are compared to the numerical outcomes of the published studies. In the coming sections, the flow is mathematically formulated and solved.

2 Mathematical formulation

An incompressible and steady 2D NN-HNF flow across a porous stretching or shrinking Riga sheet is studied. The HNF is synthesized by the scattering of CeO₂ and Al₂O₃ NPs. The Riga plate is stretching with velocity u w = a x , toward the x direction, where a is considered to be constant as shown in Figure 1. Here T & C are the temperature and NN fluid concentration of NN HNF. The electromagnetic effect defined as F m = ( F m , 0,0 ) : F m = exp π M 0 j 0 8 ρ nf − π c y [31] is employed on the extending and dwindling surface. Based on the above assumptions, the flow is formulated as [32,33]

Figure 1

Graphical sketch of NN-HNF flow (a) and stretching sheet (Riga sheet) (b).

Here V T is the thermophoretic velocity and q r is the heat radiation, which are defined as

The boundary conditions (BCs) for Eqs. (2)–(4) are introduced as

The similarity variables are introduced as

where M 0 and k r _are the magnetization and the rate of CR, E _a and K are the activation energy and surface permeability, λ presents the stretching and dwindling velocity of the sheet, ( λ < 0 is the shrinking case and λ > 0 is the stretching case), α 1 > 0 indicates the SG fluid and α 1 < 0 shows the Walter’s B fluid, c and j 0 are the electrode width and current density of electrode current, respectively.

By substituting Eq. (7) in Eqs. (1)–(4), (6), we obtain

The BCs are

where K 1 < 0 elaborates the Walter’s B fluid, while K 1 > 0 is the SG fluid.

In Tables 1 and 2, ρ is the density, C p is the heat capacity, ρ C p is the thermal capacity and σ is the electrical conductivity, where ϑ 1 , ϑ 2 , and ϑ 3 are defined as

In above equations, K 1 = α 1 a ρ f ν f is the viscoelastic factor, τ = − K ( T w − T ∞ ) T r is the thermophoretic parameter, Q h = π j 0 M 0 8 ρ f a U w is the modified Hartmann number, Rd = 4 σ T ∞ 3 k k f is the thermal radiation term, E a = E e κ T ∞ is the factor of activation energy, δ ⁎ = D B D A is the diffusion coefficient of mass, K r = k r 2 c shows the rate of CR, β 1 = π 2 ν f c 2 a is the magnitude and width of electrodes, S c = υ f D f is the Schmidt number, λ 1 = ν f K a is the porosity factor, Q e = Q e ⁎ l a ( ρ C p ) f is the heat source/sink factor, Re = x 2 a ν f is the Reynolds number, and Pr = ν f α f is the Prandtl number.

The physical interest quantities, such as Nusselt number and skin friction are stated as [31]

By inserting Eq. (6) in Eq. (13), we obtain

3 Stability analysis

Considering that there are multiple solutions to a physical model vs one parameter makes it necessary to assess the physical validity of each solution. The stability evaluations could be useful in applications related to engineering. In this way, we investigate the stability of dual solutions. As a result, an additional variable relating to the beginning value problem is appropriate for determining which solution will be obtained in actuality.

Therefore, we recall Eqs. (2), (3), (6), and (7) as (Weidman et al. [34] and Merkin [35])

The similarity variables are also modified as

By employing Eq. (17) in Eqs. (15) and (16)

Similarly, the BCs are also modified as

Consider, f ( η ) = f 0 ( η ) & θ ( η ) = θ 0 ( η ) in order to check the stability of Eqs. (8) and (9). The succeeding terms satisfying the main model as (Merrill et al. [36] and Weidman and Sprague [37])

Here H ( η ) and F ( η ) are very small as compared to θ 0 ( η ) and f 0 ( η ) , where γ is the small eigenvalue that we shall calculate.

By inserting Eq. (21) in Eqs. (18) and (19), we obtain

Consider, τ = 0 , for the stability of the steady-state flow solutions.

To derive the eigenvalues, we keep F ′ 0 ( η ) → 0 , as η → ∞ as (Harris et al. [38]). Moreover, the current work demonstrated the presence of non-uniqueness solutions within a confined range of the shrinkage factor, but a unique solution was established in that stretching case. The stability study found that the first solution has an extremely small positive eigenvalue, indicating that it was both stable and practicable.

4 Numerical solution

The PCM is a powerful tool used in mathematical computation along with various applied sciences to study how equation solutions modify when parameter varies. Here are a few notable applications of PCM, for example, it is frequently employed for investigating bifurcations in dynamical systems, fluid mechanics, engineering mechanical systems, control optimization problems, mathematical geometry, machine learning, computation in the field of neuroscience, and both dynamic and static system modelling. These applications highlight the adaptability and utility of PCM in a range of study subjects and technical domains, making it an invaluable tool for researchers analyzing systems with complex behaviors [39,40]. To employ the PCM, the following phases are used:

Step 1: Reduction of order

By inserting Eq. (28) in Eqs. (8)–(11) and (12), we obtain

The BCs are

Step 2: Introducing parameter “ p ”

The BCs remained the same as

Step 3: By using the implicit scheme as

Finally, we obtain the iterative form:

5 Results and discussion

In the current analysis, the NN-HNF flow including Al₂O₃ and CeO₂ NPs over a porous stretching or shrinking Riga sheet is studied. The flow phenomena are modeled in the form of nonlinear PDEs, which are reduced into the dimension-free form through the similarity conversion. The solution is obtained by using the numerical approach PCM. The results are compared to the numerical outcomes of the published studies.

Figure 2(a)–(d) reveal the variation in NN-HNF velocity profile f ′ ( η ) vs viscoelastic factor K 1 , modified Hartmann number Q 1 , stretching/shrinking surface factor, and porous surface parameter λ 1 respectively. Figure 2(a) shows the impact of a viscoelastic factor K 1 on the velocity field of HNF (Al₂O₃ and CeO₂/SA) for the stretching case ( λ = 1.0 ) . It can be seen that the flow rate boosted in both cases (SG fluid K 1 > 0 and Walter’s B fluid K 1 < 0 ). Physically, the increased fluid velocity in SG fluids and WB fluids is mostly owing to their viscoelastic character, which allows for flow adaptation, reduced effective viscosity by shear-thinning behaviors, and their capability to accumulate and utilize elastic energy. This combination produces much better flow properties than Newtonian fluids, which eventually contribute to higher fluid velocity under varied situations. Figure 2(b) demonstrates the impact of Q 1 on the velocity profile for SG fluid over a strehing sheet ( λ = 1.0 ) . It can be seen that the fluid velocity declines with the variation in Q 1 . Physically, as the Q 1 rises, the magnetic effect on the fluid flow gets stronger, resulting in a phenomenon known as MHD damping. When a magnetic field is placed orthogonal to the flow direction, it creates Lorentz forces that act against the velocity of the fluid. These forces increase with the strength of the magnetic field, which efficiently resists the fluid’s movement. As a result, larger Q 1 indicates increasing magnetic resistance, resulting in a considerable decline in fluid velocity. Figure 2(c) shows that the velocity curve is enhanced with the impact of λ . Fluid velocity increases as the sheet stretches and shrinks due to a direct effect of surface mobility at the boundary. Stretching a sheet improves the surface area that comes in contact with the fluid, causing a suction effect to entrain and accelerate the fluid around it. Stretching diminishes the thickness of the boundary layer, resulting in larger velocity gradients and enhanced shear stress, propelling the fluid further away from the surface. In contrast, a shrinking sheet can draw fluid toward itself, raising local velocity as fluid is drawn in. Thus, stretching and shrinking movements alter momentum transfer and flow properties, resulting in higher fluid velocity. Figure 2(d) illustrates the effect of the permeability parameter on the velocity field. Physically, increasing values of λ 1 reduces the fluid velocity due to the resistance applied to the flow field. The pores increase the surface area for the HNF flow, thus when the fluid travels through pores with a larger surface area, it generates friction and so reduces fluid velocity.

Figure 2

NN-HNF velocity profile f ′ ( η ) vs (a) viscoelastic factor K 1 , (b) modified Hartmann number Q 1 , (c) stretching/shrinking surface factor, and (d) porous surface parameter λ 1 .

Figure 3(a)–(c) displays the variation in thermal profile θ ( η ) vs the variation in CeO₂ and Al₂O₃ NPs, heat radiation Rd , and exponential heat source/sink Q e . Figure 3(a) shows that the energy curve of NF and HNF both diminishes with the addition of CeO₂ and Al₂O₃ NPs. The addition of CeO₂ and Al₂O₃ NPs to a fluid reduces the fluid temperature due to the increased thermal conductivity and efficient heat transfer capabilities of these NPs. As the NPs spread in the fluid, they increase surface area for transferring heat as well as enhance thermal interactions between fluid molecules. CeO₂ and Al₂O₃ NPs have higher thermal conductivity than the base fluid, allowing them to absorb and distribute heat more effectively. Furthermore, the inclusion of NPs can destabilize the fluid’s thermal boundary layer, allowing greater circulation and conduction of heat, which results in lower temperature relative to the unmodified fluid. Figure 3(b) shows that the energy field augments with the effect of Rd . Physically, Rd has the effect of increasing fluid temperature, mostly due to the inhalation of radiant energy from the source. In situations with high thermal radiation, particularly near a hot surface or in direct sunshine, the fluid absorbs infrared radiation, increasing its thermal energy. This absorption boosts the fluid’s temperature as it accumulates energy, particularly when the fluid’s emissivity is suitable for the effective absorption of heat. Similarly, the thermal field also enhances with the impact of Q e as shown in Figure 3(c).

Figure 3

Thermal profile θ ( η ) vs the variation in (a) CeO₂ and Al₂O₃ NPs, (b) heat radiation Rd , and (c) exponential heat source/sink Q e .

Figure 4(a)–(d) displays the variation in fluid concentration distribution ϕ ( η ) vs activation energy E a , CR K r , Schmidt number S _c, and thermophoretic velocity factor τ respectively. Figure 4(a) and (b) reveal that the effect of E a enhances the ϕ ( η ) , while the variation in K r drops the fluid concentration. Physically, the impact of the E a improves the fluid ϕ ( η ) because more activation energy is frequently associated with faster reaction rates and enhanced mass transmission in a fluid. When the activation energy of a CR increases, it usually takes more energy for the reactants to collide efficiently and transition into products. As the system temperature or energy inputs increase, so do the concentrations of reactive species, resulting in a greater concentration profile within the fluid. Furthermore, increased energy promotes molecular interactions and diffusion, which aids in the transportation of reactants and products across the fluid. As a result, the distribution of concentration increases steeply, which improves the ϕ ( η ) . Figure 4(c) illustrates that the ϕ ( η ) drops with the impact of S _c. Physically, the kinetic viscosity of fluid increases with the variation in S _c, that is why the fluid concentration drops with the impact of S _c. Figure 4(d) also presents that the ϕ ( η ) declines with the effect of r. Physically, the ϕ ( η ) decreases with the effect of τ , because thermophoresis leads particles to move from high-temperature regions onto low-temperature regions, resulting in a reduction in the quantity of suspended particles in heated areas of the fluid. When the thermophoretic velocity is high, particles move in response to temperature gradients, effectively dispersing them from hotter, more concentrated zones. This migration causes a reduction in ϕ ( η ) .