Based on the idea of peristalsis, peristaltic pumping is a technique for transferring fluid through a flexible tube by means of rhythmic compression and relaxation. Latham [1] explained the properties of peristalsis, clarified the directions for its use in contemporary equipment such as heart-lung technology, roller machine, circular tubes, and evaluated the characteristics of fluid movement in the peristaltic pump. Barton and Raynor [2] examined the peristaltic wave in a tube and discussed two different cases of wall disturbance wavelength. Riaz et al. [3] considered how thermodynamic dissipation rate affected the Williamson viscosity model’s peristaltic transport within an asymmetrically deformable conduit. After solving the flow equations using perturbation approaches, they found that the heat pattern is getting better for both the Biot and Brinkman numbers. Srinivas et al. [4] discussed the fluid-particulate suspension through the Peristaltic motion of fluid subjected to magnetic field effects. Jagadesh et al. [5] described the peristaltic flow using Arhenius activation energy and included magnetohydrodynamics across a porous non-uniform channel. Other researchers have observed the phenomenon using different fluids under different contexts, covering a wide range of physiological and mechanical characteristics [6–9].

Non-Newtonian fluids are substances whose viscosity varies due to certain physical quantities and whose flow behavior deviates from that of a viscous fluid. The analysis of non-Newtonian liquid models has drawn increased attention from scientists and engineers currently owing to its comprehensive approach to multiple basic problems from a variety of industries, including petroleum, petrochemical, geology, and biological sciences. Akhtar et al. [10] studied how convective and conductive transport phenomena affect the wave-induced flow of a Casson rheological elliptical conduit fluid and found precise solutions using an exact method. According to their findings, the temperature profile boundary conditions take into account the fact that peak convection happens in the middle of the conduit but decreases near the boundaries, and finally reaches zero. Khan et al. [11] investigated theoretical examinations of pulsatile flow of Williamson solutions in a microchannel with bended configurations, along with the properties of electro-osmosis and entropy production. They noted that there are a few fascinating features of the curved microchannel that apply to bio-microfluidic devices. Hussain et al. [12] presented a dynamical description to investigate the Sisko fluid flow under magnetic influence across an extending container through the collective influences of viscous energy loss and Joule heating in nanoparticle-based Sisko fluid. This investigation found that the friction increases significantly when curvature and magnetic field are included.

Nanofluid (fluid with nanosized particles) in peristalsis has applications in industrial and biomedical domains, including nuclear systems, cancer treatment, radiation therapy, targeted drug administration, magnetic resonance imaging (MRI), biological sensors, solar water heating, automobile cooling, home refrigerators, etc. Choi and Eastman [13] was the first study that used nanofluid to improve the heat conduction ability of the base liquid. Akbar [14] perceived the Sisko peristaltic flow of nanoliquid in an asymmetrical channel and examined the numerical results for the problem. A key observation was an increase in peristaltic pressure rise due to the large impact of the Hartmann and Grashof numbers. Trisaksri and Wongwises [15] conducted an in-depth evaluation of the thermal transmission properties of nanoparticles and found that base fluids had a significantly lower thermal conductivity than nanofluids with various concentrations of nanoparticles. Ullah et al. [16] investigated the influence of thermal radiation on ternary hybrid nanofluid flow with activation energy through a computational numerical method. Their study emphasized the significance of radiation and energy activation in improving the thermal transport properties of nanofluids. Akram et al. [17] explored an analysis of the magnetic field on pseudoplastic nanoliquid with peristalsis in a tapered channel. Recently, a large number of researchers have studied magneto-nanofluids [18–22].

Double-diffusive convection is characterised by two distinct asymmetric buoyancy variations, referred to as density gradients and solutal gradients. It is utilised in engineering for the design of heat exchangers, chemical reactors, and materials processing, where managing the coupled interfacial quantity and temperature transfer is essential. Double diffusion in a deformable channel, Ali et al. [23] demonstrated the peristaltic flow of nanofluids and perceived that the deformation raises the temperature of the fluid and axial velocity. Alolaiyan et al. [24] showed how double diffusion convection affected a third-grade fluid flow in a wavy, curved shape with nanoparticles. This study reveals that the buoyancy parameter displays a decrease in temperature; the fluid becomes hotter when the values of the modified Dufour parameter and conventional buoyancy ratio are elevated. Shivappa and Giddaiah [25] examined a dual distribution regarding the vermiculation of nanofluids under the effects of a magnetic field, a filtering medium, and radiant heat. They concluded that the characteristics of the relaxation-retardation time scale affect velocity. Recent studies have explored double-diffusive nanofluid flows using advanced computational methods and artificial neural networks [26–30].

An electric field applied across a fluid containing ions and in touch with a charged solid surface, like the wall of a microchannel or a capillary tube, causes electroosmotic flow (EOF). It is a fundamental electrokinetics phenomenon that is particularly pertinent to bioengineering, lab-on-a-chip systems, and microfluidics. Later, the first methodical study of this problem was published by Chakraborty [31]. Chakraborty’s research focuses on using an electrokinetic body force to accurately control peristaltic transport. Tripathi et al. [32] extended the Chakraborty model further to account for the peristaltic electrokinetic transport of water-based nanosuspensions flowing in a complex constricted (micro) channel with thermal and buoyant effects generated by Joule heating. Using penetrable walls, Misra et al. [33] looked into how a micropolar fluid flows electroosmotically in a microscopic channel under periodic oscillations. Their results showed that the size of small oscillations is quite sensitive to changes in the width of the microchannel. The pumping features of electrokinetically regulated power-law fluid transport via peristaltic propelling in small, flexible channels were studied by Goswami et al. [34], who showed that electro-osmosis mostly affected peristaltic flow with minimal wave amplitude and that ensnarement could be controlled by a magnetic field. McKnight et al. [35] looked into how electrodes affected peristaltic electrokinetic pumping. Other researchers have experimented with the use of electro-optic techniques for fluid movement in channels or tubes [36–40].

It has been observed from the literature that there has not been extensive research conducted on the theory of nanofluids for the fluid models exhibiting both Newtonian and non-Newtonian properties, like Sisko fluid. In addition to this, electroosmotic and thermal radiation effects are not significantly discussed in such flow patterns. It is also significantly observed that most of the above studies produced analytical and numerical techniques that can be useful only in the basic flow problems. In the current study, the authors intend to work on the complex problem of Sisko nanoluid by assuming double diffusion convection, which is very rare for such high-profile fluids. Moreover, this research develops a bridge to this crucial research gap through the presentation of a machine learning method known as Morlet-Wavelet Neural Networks (MWNNs) to tackle the problem. The innovation in this research arises through the integration of the theory of the Sisko nanofluid flow with the MWNNs to augment the mathematical modeling and the accuracy in the solution. To prepare a unique scenario, an induced magnetohydrodynamic field is also assumed in the problem. The primary objectives are to derive an efficient MWNNs model, to conduct an extensive error analysis to ascertain the reliability of the solution, and to execute an extensive graphical analysis of the crucial parameters of the flow to gain deeper physical insight.

The Sisko fluid model provides a mathematical description of viscoelastic liquids, which manifests together viscous and elastic effects. The stress tensor for this model is provided by [41]. (1)τ=−PI+S (2)S=(a+m(Π)b−1)A1 (3)A1=L+LT,L=gradV,Π=12trac(A12)where trace(A12) is the second invariant strain tensor, and A1 is the Rivlin-Erickson tensor.

Let’s look at an asymmetrical channel of dimensions d1+d2 containing an elastic Sisko fluid with nanoparticles (see Fig. 1). The flow is induced by an annular wave that propagates along the channel walls at the uniform speed c. On the wall to the right (at y=H1), thermal data points, solvent concentration and nanoparticle fraction are determined as T=T0, C=C0, Θ=Θ0, while on the left wall (at y=H2) are T=T1, C=C1, Θ=Θ1. Furthermore, there is a uniform magnetic field with an intensity of A0 acting perpendicular to the flow, which results in the generation pertaining to magnetically induced field A(aX(X,Y,t),aY(X,Y,t),0). The combined magnetic flux is stated as A+(aX(X,Y,t),A0+aY(X,Y,t),0). The subsequent formulas are used to determine the wall surfaces. (4)Y=H1=d1+a1cos⁡(2πλ(x−ct)) (5)Y=H2=−d2−b1cos⁡(2πλ(x−ct)+ϕ)

In the equations displayed earlier a1 and b1 signify the oscillation amplitudes, λ is defined as the wavelength and t indicates the time period.

When double diffusion, viscous dissipation and radioactive heat flux are considered, the heat energy, nanoparticle proportion, and solute concentration of solute are described as [41–43]. (6)∇⋅V=0 (7)ρf(dVdt)=∇⋅τ+μe(A+⋅∇)A+−∇(12μe(A+)2)+ρeE+g(ρf0(1−Θ0)(βT(T−T0)+(C−C0)βC)−(ρp−ρf0)(Θ−Θ0)) (8)(ρc)f(dTdt)=k∇2T+(ρc)p(DB(∇Θ⋅∇T)+(DTT0)(∇T)2)+S⋅L−∇⋅qr+DTC∇2C+σ(E⋅E) (9)dCdt=Ds∇2C+DCT∇2T (10)dΘdt=DB∇2Θ+(DTT0)∇2T

The term qr defined in Eq. (8) is the flow of thermal radiations (solar energy) and is determined by implementing the Rosseland mean diffusion approach as [41]. (11)qr=−4σ∗3k∗∂T4∂Y

By eliminating the higher powers of T in the taylor series of T4, we get differntial form of Eq. (11) as: (12)∂qr∂Y=−16σ∗T033k∗∂2T∂Y2where k∗ symbolizes for Rosseland mean absorptivity and σ∗ refers to Stefan’s constant.

As we can see that the obtained mathematical relations are highly nonlinear and mutually dependent which cannot be perfectly handled by typical analytical or numerical methods. In this regard, the Morlet Wavelet Neural Networks (MWNNs) play an essential role in the attainment of greater accuracy and efficiency in neural network-based solutions to complex mathematical models. In comparison with traditional activation functions, the Morlet wavelet possesses the advantage of improved localization in time and frequency domains, which enables MWNNs to capture intricate patterns and dynamic behavior effectively. This renders them extremely well-suited for the solution of differential equations, signal processing, and engineering applications. MWNNs also improve convergence rates and reduce computational errors when paired with optimization methods like Particle Swarm Optimization (PSO). Their flexibility and accuracy also render them an effective tool in scientific computing and artificial intelligence applications. Fig. 2 demonstrates the framework of the considered Artificial Neural Netwroks (ANN) model.

Artificial neural networks with a Morlet wavelet nonlinear activation function are employed to address the problem and to achieve higher accuracy than typical numerical methods. Morlet-Wavelet Neural Networks (MWNNs) are important to capture intricate nonlinear structures because they are capable to integrate wavelet transforms with learning through neural networks. The fact that they are localized oscillatory improves the approximation to functions, making them very efficient to use in solving differential equations and scientific computation. The design methodology step by step is represented in Fig. 3, the fitness function in Fig. 4a indicates the convergence behavior during different independent runs, establishing the stability and efficiency of the optimization process. The Fig. 4b,c of the TIC and MSE indicates the behavior of the Theil’s Inequality Coefficient and Mean Squared Error, showing the predictive accuracy of the model during across multiple runs. The boxplot of the MSE values during independent runs in Fig. 4d compares the values, showing the robustness and variance of the model. The boxplot of the TIC during independent runs in Fig. 4e further indicates the reliability of the solution. The Fig. 4f of the normal distribution of the MSE indicates the statistical distribution of the errors, establishing the stability of the performance and the normality assumption. In Fig. 4g of the normal distribution of the TIC provides further analysis of the behavior of the predictive error. The graphical analysis collectively indicates the reliability of the model during different independent runs. The final results represent that the optimization process has stability while providing minimum variance to the measures of the error. This extensive statistical evidence proves the efficacy of the proposed neural network technique to the given problem. The proposed framework advances existing methods by providing higher accuracy and faster convergence chatcompared to classical numerical techniques. Its efficiency makes it suitable for solving complex nonlinear fluid flow models that are otherwise challenging with traditional methods.

The integrated MWNNs-PSO-NNA program is used in MATLAB to solve the equations, and the MWNNs-PSO-NNA command is utilized for nonlinear expressions. MWNNs-PSO-NNA, which is essentially A function in Wolfram Language, can solve differential equations in numerical form and a variety of partial derivatives as well as initial value problems. Within this segment, many variables that influence the velocity curve, isotherm distribution, concentration gradient, nanoparticle volume fraction, Lorentz force function, gradient of fluid pressure, and pressure escalation rate profile are displayed in Figs. 5–16.

Using different values of the Helmholtz-Smoluchowski velocity (Uhs), Hartmann number (M), Sisko parameter (m∗) and dimensionless nanoparticle Grashof number (GrF), Fig. 5a–d examines the phenomenon of flow velocity. As presented in Fig. 5a, fluid velocity has a tendency to increase when embedded in zone y∈[−0.45,0.1] that represents a surface velocity or driving force that actively accelerates the fluid and decreases when it is in the sector y∈[0.1,0.63] because of the growing influence of Uhs. According to Fig. 5b, due to enhanced resistance, the velocity profile dropped in the interval y∈[−0.45,−0.1] and y∈[0.3,0.63], while it grew in the region y∈[−0.1,0.3] where the predominant shear-thinning effects lower flow resistance as a result of the increasing m∗. The velocity profile in Fig. 5c improves within the regions y∈[−0.45,−0.1] and y∈[0.3,0.63], while decreasing in the region y∈[−0.1,0.3] as a result of the increasing M because motion is opposed by powerful Lorentz forces. However, it is important to note that the magnetic field does not always act as a retarding effect; in certain flow configurations it may assist the motion, as stated in [53]. The velocity falls in the region y∈[−0.9,−0.1] due to opposing effects and rises in the region y∈[−0.1,0.9]as mobility is improved in that area when buoyancy matches the direction of flow when the GrF value increases, as shown graphically in Fig. 5d. It can also be observed that this velocity is much greater than that of measured in [41]. The reason for this increase is that in the current study, electroosmosis (which was not in [43]) is creating an acceleration to the flow velocity, as in the presence of this effect, the electric body force becomes stronger enhancing the fluid motion. Fig. 6a–d shows 3D velocity profiles that behave similarly to the 2D velocity profiles. A more thorough picture is offered by the 3D graphs, which draw attention to the spatial differences and the consistency of the trends shown in the 2D graphs. While highlighting the complexity of 3D modeling for comprehending velocity distributions in intricate flow systems, this resemblance highlights the dependability of the 2D analysis and provides improved clarity for interpreting physical occurrences.

Fig. 7a–d shows how temperature distribution affects the non-Newtonian parameter (m∗), Dufour parameter (NTC), the parameter for Joule heating (β) and also Prandtl number (Pr). As the values of β increase, it is clear from Fig. 7a indicates that the pattern of temperature declines due to the fact that more heat is effectively transferred or conveyed. This finding aids in electroosmotic pumps, microelectromechanical systems (MEMS), and microchip cooling, where effective heat removal reduces overheating and boosts device dependability. The concentration distribution in Fig. 7b is increased by a greater Pr, showing that the diffusion of species is slowed by lower molecular diffusivity in comparison to momentum diffusivity. This observation is pertinent to the design of heat exchangers, lubrication systems, and nanofluid thermal management. The effects of m∗ and NTC on concentration distribution are represented in Fig. 7c,d. As the value of m∗ grows, non-Newtonian effects increase resistance, reducing fluid volume in the channel as shown in Fig. 7c and as NTC increases, which improves thermal-diffusion coupling, which intensifies thermophoretic effects and promotes larger concentration gradients and concentration distribution in Fig. 7d. This result for the Sisko fluid factor will be beneficial for regulating rheological behavior, which regulates flow homogeneity and cooling rates in polymer manufacturing, biofluids, and slurry transport. Impact of NTC remains crucial for heat exchangers based on nanofluids, fuel cells, and chemical reactors, where reaction efficiency is impacted by thermal–mass coupling. Furthermore, temperature distribution is enhanced due to additional radiative heat flux, which contributes to the overall energy transport; however, excessively strong radiation effects may reduce the accuracy of linearization assumptions, so the chosen range balances practicality with numerical stability. Fig. 8a–d processes 3D thermal distribution profiles that perform equivalent to the 2D profiles. By more accurately depicting spatial changes, validating the designs seen in the 2D graphs, and offering a more thorough comprehension of the thermal behavior, the 3D representation improves visualization.

The features of solutal concentration are shown in Fig. 9a–d on Soret constraint (NCT), Thermal radiation (Rd), Brownian motion (Nb) and Brinkman number (Br). Fig. 9a presents the findings on the Soret parameter NCT. The concentration profile falls when the NCT increases because a larger NCT upsurges the thermal diffusion effect, which causes particles to move from hotter to colder locations. Because of this redistribution, the concentration profile generally declines since there is less particle accumulation in the primary concentration zone. As discussed in the Fig. 9b–d, the concentration profile boosts consequent to the rise in Rd,Nb and Br. Physically, a higher Rd increases the system’s thermal radiation, which raises the temperature and encourages more robust particle diffusion, both of which increase the concentration. A greater Nb denotes improved dispersion due to intensified Brownian motion, in which nanoparticles collide randomly. In a similar vein, a higher Br value denotes stronger thermophoretic forces, which push particles from hotter to cooler areas and enhance accumulation. When combined, these actions cause the concentration profile to rise. Consistent with the 2D plots, the 3D solutal concentration profiles shown in Fig. 10a–d show that the solutal concentration decreases in response to increasing distance from the surface. In addition to verifying the trends seen in the 2D profiles, the 3D representation offers a more thorough picture of spatial variations and a more profound understanding of concentration distribution patterns.

Fig. 11a–d illustrates how the volume fraction of nanoparticles influences a number of fluid properties, including thermophoresis parameter (Nt), Thermal radiation (Rd), Dufour parameter (NTC) and Soret parameter (NCT). The movement of nanoparticles caused by temperature gradients intensifies with increasing Nt leading to the accumulation of nanoparticles in particular areas. Because increased thermophoretic processes drive particle motion, the fluid’s nanoparticle volume fraction rises in the way shown in Fig. 11a. Fig. 11b shows the effect of Rd on the nanoparticle profile. Improving thermal radiation raises the system’s energy content, which encourages thermally driven diffusion and particle motion. This intensifies thermal effects, which elevates the nanoparticle profile. On the other hand, NTC and NCT produce adverse consequences because they promote diffusion mechanisms that hinder the formation process, which lowers the concentration of nanoparticles in certain areas, as shown in Fig. 11c,d. The nanostructure 3D volume fragment profiles in Fig. 12a–d show patterns resembling those in the 2D graphs, with the volume fraction falling with increasing distance from the surface. The 3D display successfully draws attention to spatial changes, enhancing the coherence of the 2D analysis and offering a better understanding of nanoparticle configuration.

One of the fundamental cosmic powers, the magnetic force is produced by the dynamic movement of charges and is induced by the electromagnetic force. Moreover, when two charged items get close to one another, a magnetic attraction force is produced. The magnetic Reynolds number affects the correlative influence of the magnetic force (Rm) and electrical field (E) is depicted in Fig. 13a,b. As the magnetic Reynolds number increases, Fig. 13a below illustrates that the function of magnetic force often rises as well, suggesting that the influence of electromagnetic induction on fluid behavior is growing in significance. In Fig. 13b, increases in electric field intensity cause the charged fluid particles to accelerate, negating the constraining effect of the magnetic force and lowering the magnetic force function profile.

Fig. 14a–d shows how the pressure differential dpdx alongside the x direction is affected by the non-Newtonian parameter (m∗), Hartmann dimensionless parameter (M), Grashof number for thermal convection (Grt) and Joule heating parameter (β). The pressure gradient rises as the m∗ increases as evident from Fig. 14a. This implies that there is more resistance or slower fluid movement across the medium when m∗ is larger. Density or flow path alterations could be the cause of this. Whereas Fig. 14d shows the opposite effects, ultimately as a result of more direct or smoother channels, suggesting a faster flow and less resistance. The pressure gradient profile in Fig. 14b improves in the domains x∈[0.0,0.2] and x∈[0.85,1.0] as a result of the increasing M due to improved flow conditions or lower resistance, while it falls in the region x∈[0.2,0.85]. As shown in Fig. 14c, the pressure gradient rises across the regions x∈[0.0,0.3] and x∈[0.75,1.0] and falls in the region x∈[0.3,0.75] by increasing Grt which indicates less resistance or a more effective flow, perhaps as a result of smoother channels or less friction.

As seen in Fig. 15a–d, the impact of numerous factors on the pressure rise per cycle Δp plotted vs. flow rate Q. The pumped phases are divided into consecutive sections in order to highlight almost all the aspects of pressure rise. a) The region undergoes peristalsis when Q>0 and Δp>0: Pressure in this area is frequently modulated by peristalsis waves, which move fluid in the direction of the flow process. b) The augmented region persists when Q>0 and Δp<0: The flow is improved in this region by the pressure created by the peristaltic force. c) The retrograde area resides when Q<0 and Δp>0: Due to peristalsis compression impedes the flow in this region, pressure rises in the opposite direction of the flow. d) the free pumping zone occurs when Δp=0 and Q=0: In the given region, the only flow channels are peristalsis surfaces. The tensile force increases in the reverted, augmented, and peristaltic transfer portions as the Brinkmann number Br rises indicate higher heat production and viscous dissipation in the fluid, as demonstrated in Fig. 15a. In Fig. 15d, the effect of the Brinkmann number Br, per wavelength pressure increase in the reverse of the Prandtl number Pr, which represents thermal diffusivity; this can lessen pressure changes and thermal gradients. Raising the non-Newtonian parameter m∗ reduces resistance and pressure in the increased pumping region by altering the fluid’s shear-thinning or shear-thickening behavior. Meanwhile, with non-Newtonian effects, the retrograde zone faces higher pressure because of more energy dissipation, according to Fig. 15b. As observed in Fig. 15c, raising the Hartmann number heightens the magnetic field’s effect on the fluid that conducts electricity. This outcome in a pressure increase in the retrograde and free pumping zones because of improved electromagnetic resistance to flow and a pressure decrease in the augmented pumping region because the magnetic field suppresses flow velocity gradients, lowering the pressure needed to drive the fluid.

In peristaltic movement, trapping is one of the most crucial mechanisms. When streamlines split under certain conditions, a circulating bolus is created, a process known as trapping. Because the trapped boluses are enclosed by the peristaltic waves, they travel at the same fixed pace as the wave. This mechanism facilitates the way food moves through the digestive tract and the formation of blood thrombi. Fig. 16a–l shows several values of the resistive heating parameter (β), Helmholtz velocity (Uhs), non-Newtonian constant (m∗), and Hartmann number (M). As seen in Fig. 16a–c, the captive bolus shrinks in size as β grows, indicating a physical mechanism where the extent of bolus creation is limited by stronger restoring forces or limitations. As Uhs, m∗ and M increase, Fig. 16d–l shows the effect of these variables, giving the impression that the boluses grow in number and volume simply due to improved flow dynamics or energy input that encourages more fluid mixing and instability.

To validate the current results and to ensure the accuracy of MWNNs, Table 1 is prepared by neglecting the effects of electroosmosis Uhs and the Sisko fluid factor m∗. From the table, we can verify that the current results are showing a great matching with the results of Afsar et al. [38] which clarify that MWNNs produced accurate results.

The objective of this study was to design a framework for simulating the effects of thermally radiative Sisko nanofluid and viscous dissipation on peristaltic waves in an asymmetric channel, considering double diffusion convection, electroosmosis, and induced magnetohydrodynamics. The governing equations for a Sisko nanofluid incorporating simultaneous diffusion, convection, and nanoparticle dynamics were formulated and simplified under the assumption of long wavelength. This yielded a set of partial differential equations that are nonlinear. To acquire numerical solutions, an integrated unsupervised Morlet wavelet neural network approach was employed. Convergence was evaluated using statistical metrics such as the mean squared error and the Theil inequality coefficient. Across fifty independent runs, the convergence values ranged between 10⁻⁵ to 10⁻⁷ for the Theil inequality coefficient and 10⁻⁷ to 10⁻¹⁰ for the mean squared error, demonstrating that the proposed method is both reliable and rapidly convergent. The graphical results highlight the influence of numerous physical parameters on velocity, temperature, concentration, and pressure fields, thereby providing deeper insight into the transport mechanisms of Sisko nanofluid flows. Key findings from that research include the following facts: ⮚

The velocity displays an upsurge as Helmholtz–Smoluchowski velocity escalations, whereas for the nanoparticle Grashof number, the velocity undergoes an initial slowdown, then a speed up. These results demonstrate that in the presence of electroosmotic factors, velocity significantly upgraded as conpared with the non-electroosmotic scenario [43].

It should be noted that the present study is restricted to a two-dimensional channel geometry under limitations of lubrication and long wavelength approximations. Extending the model to three-dimensional domains and validating it with experimental data remain important directions for future research.