This study examines the unsteady convective squeezing flow of a viscous, tri-hybrid nanomaterial fluid through two perforated Riga plates. This flow regime occurs between the compressed infinitely parallel Riga plates and is influenced by thermal radiation, varying thermal conductivity, magnetic fields, and viscous dissipation. The top plate moves with velocity toward the lower plate, generating a squeezing motion.

The symbol h¯(t) represents the height of the channel or the separation between the horizontal plates. The range of variation in the separation variable is 0 to h¯, where h¯ is much higher than the thickness of the wall’s border layer. While the upper surface can make a movement vertically toward the bottom plate for a suitable compression, the upper plate remains immovable. The vertical displacement of the proposed channel may be expressed using the following criterion, per the findings of Khaled and Vafai [34]. The following equations, where h0 is a constant, which expresses the function h¯(t) in two distinct contexts. (1)h¯(t)={h0(s+at)−1a,ifa>0,h0exp⁡(−st),ifa=0.}

The proposed model is structured utilizing the x−y plane, with the x−axis show to be the actual direction of flow. The physical depiction of the problem at hand is exposed in Fig. 1.

The following are the governing continuity equation (principle of conservation of mass), Navier-Stokes equation (principle of conservation of momentum), and temperature equation (principle of conservation of energy). Based on the literature [34–36].

Continuity Equation: (2)ux+vy=0,

Momentum Equation: (3)ut+uux+vuy=u¯t+u¯u¯x+μThnfρThnfuyy−1KμThnfρThnf(u¯−u)−Bx¯2σThnfρThnf(u¯−u)+1ρThnfπJ0M08e−πly,

Energy Equation: (4)Tt+uTx+vTy=kThnf(T)(ρCp)ThnfTyy−16σ∗∗T¯∞33k∗∗1(ρCp)ThnfTyy−μThnf(ρCp)Thnf(uy)2+Q0(ρcp)Thnf(T−T∞),where the Rosseland approximation yields a radiative unidirectional heat flux (qr∗), as follows by [36], this means that heat transport is largely in one direction. This approximation is commonly utilized when the medium is thought to be in radiative equilibrium. (5)qr∗=−4σ∗∗3k∗∗(T¯4)y

It is anticipated that the fluid temperature corresponds to T closely be similar to T∞ by expanding the Taylor series expansion T¯4 near to T∞. By leaving out all the higher-order terms starting from the second order, T¯4 is stated as: (6)T¯4≅4T¯∞3−3T¯∞4.

The current model considers the following boundary conditions (BCs): (7)Atlowerplate(y=0):u(y)=0;v(y)=vw(t);−κThnf(Ty(y))=hf(Tw−T∞),ForfreestreamCondition(y→∞):u(y)=u¯;T(y)=T∞.}

In Eq. (7), when y=0, the Riga plates are supposed to be immobile. This means that the flow in x−axis is zero. Along y direction, the velocity of the fluid equals the vw(t), i.e., wall permeable velocity and the heat flux on the wall qw(x) is the same as the temperature gradient. As y approaches to infinity, the velocity and temperature distributions in the x−direction approach that of the unconstrained stream.

Role and Modelling of Porosity and Permeability

In the current study, the porous medium is modelled using Darcy’s law, which introduces a resistance force proportional to the fluid velocity. This is captured in the momentum equation through the term: −1KμThnfρThnf(u−u¯)

Here, K denotes the permeability of the porous medium, which quantifies how easily fluid can flow through it. A lower K implies higher flow resistance, simulating denser or more compact porous materials.

While permeability directly enters the governing equations, porosity is inherently embedded in the effective thermophysical properties of the ternary hybrid nanofluid. It influences how heat and momentum are transported through the medium by altering the apparent density, heat capacity, and viscosity. These effects are considered in the definitions of ρThnf,μThnf, and (ρcp)Thnf which vary based on the solid–fluid volume fractions and porosity-dependent mixing rules.

The combined effect of porosity and permeability controls the extent of momentum suppression and heat dispersion within the porous region, which is crucial for realistic modelling of nanofluid behaviour in engineered materials and biological tissues.

Similarity Variables:

Suitable similarity variables are proposed to solve the gained nonlinear coupled boundary value problem numerically. The stated equations are converted into a nondimensional form employing the dimensionless similarity variables. They were added in order to make the equations simpler and facilitate analysis. Eq. (8), which represents these dimensionless variables, is based on a related work by Muhammad et al. [36]. Using these scaling modifications, a nonlinear ordinary differential model is created using the 2D, unstable TNF flow Eqs. (2)–(4) and the boundary conditions given by Eq. (7). (8)u¯=bx;vw(t)=v0b;Bx¯(t)=Bo¯b;qw(x)=qo¯x;kThnf(T)=[1+ε{(T−T∞)kfqo¯xbνf}],θ(ζ)=(T−T∞)kfqo¯xbνf;b=(s+at)−1;u=bxf′(ζ);v=−bνff(ζ).}when Eq. (8) is applied to Eqs. (2)–(4) and (7), they are modified as follows. (9)γ1⁢f′′′+f⁡f′′+a⁢ζ2⁢f′′−(f′)2+a⁢(f′−1)+γ1⁢λ⁢(1−f′)+γ2⁢M⁢(1−f′)+γ3⁢Z⁢exp⁡(−S∗⁢ζ)+1=0, (10)1Pr{γ4(1+εθ)+4γ53Rd}θ′′+(f+aζ2)θ′−(f′+a2)θ+γ6Ec(f′′)2−Qθ=0.

Furthermore, Eq. (11) incorporates the boundary constraints from Eq. (7): (11)Atζ=0,f′(ζ)=0,f(ζ)=−S,γ4θ′(0)=−Bi(1−θ(ζ)),Asζ→∞,f′(ζ)=1,θ′(ζ)=0.}where, (12)γ1=(μThnfμf)γ3;γ2=(σThnfσf)γ3;γ3=ρfρThnf;γ4=(kThnfkf)γ5;γ5=(ρCρ)f(ρCρ)Thnf;γ6=μThnfμfγ5,λ=νfxu¯K;M=σfBo¯ρf;Rd=4σ∗∗T¯∞33k∗∗kf;Pr=(ρCρ)fkf;Z=πJ0M8ρfu¯b;Ec=(u¯)2kf(Cρ)fqo¯xbνf;S=ν0νf.]

Based on the similarity transformation, the parameters shown in Eq. (12) are taken into consideration. Using TNF correlation, Table 2 lists the values for γi where i=1–6. Here, M the magnetic, λ represents a porous medium, Rd the thermal radiation, Z the Hartmann number, Ec the Eckert number, Pr the Prandtl number, and f0 the surface permeability velocity.

Engineering Parameters

Engineering quantities such as skin friction coefficient (Cf) and Nusselt number (Nu) are employed in the research of heat/mass transport and liquid mechanics. These numbers are frequently utilized in a variety of engineering applications and are pertinent to diverse flow types. These can be assessed independently for the upper and lower plates in the manner described below. The skin friction coefficient (Cf) and local Nusselt number (Nu) are provided by: (13)Cf=τwρf(u¯)2;Nu=xqwkf(Tw−T∞)when τw and qw correspond to the shear stress and heat flux parameters, respectively, (14)τw=μThnf(uy)|y=0;qw=−kThnf(Ty)|y=0+(qr∗)|y=0

So, Nu and Cf are modified using a similarity transformation as: (15)R⁢ex12⁢Cf=(μT⁢h⁡n⁢fμf)⁢f′′⁡(0);R⁢ex−12⁢N⁢u=(1+4⁢R⁢d⁢kfkT⁢h⁡n⁢f)⁢(−1θ′⁡(0))when examining the behavior of fluid flow and heat transfer at boundaries, Eq. (14) is important. It provides the information required to assess and predict the thermal and momentum characteristics of a system. Where the local Reynolds number is defined as Rex=ux¯νf.

Thermophysical Characteristics:

As thermophysical characteristics are extremely important since they allow us to more accurately calculate efficiency. According to the literature, heat transfer characteristics are dependent on the formulation, whereas viscosity is temperature-dependent and thermal conductivity is concentration-dependent.

Density ρThnf=(1−ϕ1){(1−ϕ2)[(1−ϕ3)ρf+ϕ3ρ3]+ϕ2ρ2}+ϕ1ρ1

Viscosity μThnf=μf(1−ϕ1)2.5(1−ϕ2)2.5(1−ϕ3)2.5

Thermal conductivity kThnfkhnf=k1+khnf(n−1)−(n−1)(khnf−k1)ϕ1k1+(n−1)khnf+ϕ1(khnf−k1)where,

khnfknf=k2+(n−1)knf−ϕ2(n−1)(knf−k2)k2+knf(n−1)+(knf−k2)ϕ2, knfkf=k3+(n−1)kf−(kf−k3)(n−1)ϕ3k3+kf(n−1)+(kf−k3)ϕ3

Thermal expansion coefficient (ρβ)Thnf=(1−ϕ1){(1−ϕ2)[(1−ϕ3)(ρβ)f+ϕ3(ρβ)2]+ϕ2(ρβ)2}+ϕ1(ρβ)1

The impact of (0.2≤Z≤0.6) on the velocity distribution f′(ζ) is depicted in Fig. 3. The magnetic field between the plates is strengthened as Z is increased. The stronger magnetic effect between the plates drives the fluid flow. The performance of Riga plates will resemble that of a regular channel when Z goes back to zero. The Hartmann number greatly influences the velocity profile in the presence of a magnetic field. The interaction between electromagnetic and viscous forces, whose relative intensity is measured by the Hartmann number, is where the physical relevance is found. Therefore, it may be possible to effectively use the increased magnetism between the plates to speed up the flow in the squeezing phase. As a result, if Z disappears, velocities are reduced, and the Riga plates become normal plates.

The effect of the suction parameter (0.2≤S≤0.6) can be seen in Fig. 4, one can notice that as the suction rises, the overall fluid velocity upsurges. The act of sucking a fluid in, or suction, has a large influence on flow behaviour and velocity profiles. Faster fluid flow close to the surface is usually the result of decreasing the thickness of the boundary layer (BL) and increasing the velocity gradient at the surface of the plate. The boundary layer is thinned by suction, which removes fluid close to the surface. Because of the suction, the velocity profile steepens at the surface, indicating that the fluid is travelling more quickly toward the wall than it would in a suction-free situation. Even if the velocity increases close to the wall, if the suction is high, the total velocity profile may be lower farther from the wall.

The impact of a porous parameter (0.1≤λ≤1.7) on a velocity profile is seen in Fig. 5. The porosity characteristic reduces fluid velocity in the velocity profile. Generally speaking, a higher porosity parameter (which signifies greater permeability of the porous material) results in higher drag force, which lowers velocity.

The volume fraction (0.00≤ϕ1,ϕ2,ϕ3≤0.12) of SiO2, ZnO, and MWCNT nanoparticles dominate the velocity field as seen in Fig. 6. As volume fractions grow, disturbances like eddies and flow separations seen to be more noticeable. The volume percentage of nanoparticles in a fluid usually reduces the velocity profile, owing to the higher viscosity and friction induced by the nanoparticles. Essentially, the nanoparticles thicken the fluid and increase its resistance to flow, slowing it down.

The effect of the heat source parameter on the thermal field θ(ζ) is portrayed in Fig. 7. The thermal boundary layer expands as the fluid temperature increases as a result of a heat source. Temperature intensifies the thermal energy added due to heat sources. Thermal energy is introduced into a system via a heat source, such as a resistor that dissipates electrical power. This energy raises the molecules’ kinetic energy, which raises the thermal effect and causes the thermal boundary layer to grow.

The effect of the thermal radiation parameter, (0.3≤Rd≤2.3), on the thermal distribution, θ(ζ), across the gap is shown in Fig. 8. The dimensionless heat equation Eq. (10) in the increased thermal diffusion term includes the parameter Rd. It specifies how much heat transport from thermal radiation and thermal conduction contributes to each other. The contribution of heat radiation disappears with Rd=0. Thermal radiation becomes more and more dominant over thermal conduction when Rd>1. In traditional boundary layer flows, this would result in increased flow energy and temperature, but in the existing model, raising the radiation parameter has the reverse effect. The decreasing temperatures linked to a larger radiation heat flow impact are caused by the squeezing effect, which suppresses thermal diffusion. The decline of temperature between the two Riga plates is almost linear when unidirectional radiative flux is not present. However, the connection gets increasingly nonlinear as the radiative influence increases, especially toward the downside Riga plate.

The impact of thermal Biot number, (0.1≤Bi≤0.5), on temperature profiles, θ(ζ), is seen in Fig. 9. This quantity, which is a variation of the traditional thermal Biot number, appears in the boundary conditions at the bottom Riga plate. Physically speaking, when this value is less than 0.2, it indicates that temperature gradients are minimal within the lower Riga plate and that heat conduction there is substantially quicker than heat convection outside of it. This is in line with the “thermally thin” situation. The “thermally thick” situation occurs when Bi>0.2, meaning that thermal conduction within the downside Riga plate proceeds at a significantly slower rate than thermal convection far from its plate into the pierced flow regime. The temperature across the gap significantly rises as Boit number increases. Increased thermal convection warms the flow, resulting in higher convection from the downside Riga plate to the squeezing regime.

Fig. 10 demonstrates the impact of the Eckert number (0.1≤Ec≤0.5), i.e., energy dissipation parameter effects on thermal profiles θ(ζ). The relative contribution of kinetic energy dissipated to the boundary layer (BL) enthalpy difference is represented by the Eckert number. Viscous dissipation is assumed to be zero for Ec≈0. A higher temperature is the outcome of a gradually larger conversion of kinetic energy to heat as Ec rises. This conversion mostly occurs through viscous dissipation, in which the flow’s mechanical energy is converted to heat, raising the temperature due to frictional forces between the walls and the fluid.

The transport of heat to the fluid is influenced by varied thermal conductivity (0.2≤ε≤0.6), as seen in Fig. 11. Higher thermal conductivity areas of the fluid will heat faster and subsequently radiate heat to the fluid’s cooler areas. It also lessens the fluid’s internal temperature gradients. This is because the fluid’s parts with higher thermal conductivity act as temperature sinks, taking up heat from the hotter segments and transferring it to the colder ones.

The inspiration of the Prandtl number (7.0≤Pr≤15.4) on θ(ζ) is demonstrated in Fig. 12. The fluid’s momentum diffusivity and thermal diffusivity are related by the Prandtl number Pr. Thus, the fluid flow with a greater Pr has low thermal diffusivity, which causes the temperature gradient to steepen and decrease. Designing thermal systems i.e., heat exchangers, requires particular nanofluids with variable Pr values to ensure optimal thermal performance.

The inspection of suction parameter, (0.2≤S≤0.6), on the thermal distribution θ(ζ) can be noticed in Fig. 13. The Temperature often fall when a fluid is drawn into a porous media by suction. In particular, raising the suction parameter lowers the temperature and may also lower the local skin friction coefficient and heat flow. In a physical sense, suction cools the system by removing heat.

The present analysis is benchmarked against the study by Rammoorthi and Mohanavel [27] by employing parameter values consistent with their work. The reliability of the results is validated through the graphical comparisons presented in Figs. 14 and 15. As shown in Table 2, the skin friction coefficient and Nusselt number exhibit sensitivity to various dimensionless parameters. Specifically, both (Rex)1/2Cf and (Rex)−1/2Nux increase with higher values of Z,λ,ϕ1,ϕ2,ϕ3, and M while they decline with increasing b. Conversely, parameters such as Ec,Rd, and ε exert a negligible effect on skin friction. The Nusselt number increases with Rd and decreases with both Ec and ε. These observed patterns offer critical insights into the influence of physical parameters on thermal transport behaviour.

In this section, 3D surface plots against various flow factors are illustrated in Figs. 16 and 17. The following 3D surface plots show how in a magnetohydrodynamic tri-hybrid nanofluid flow system, the local Nusselt number (Nu) as well as the local Skin Friction Coefficient (Cf) are affected by M and ϕ that are magnetic parameter and nanoparticle volume fraction, respectively. Nu grows linearly with the growth of M and the nanoparticle volume fraction ϕ. Nu increases from around 2.7 to 5.3 at M=0.5 when ϕ is increased from 0 to 0.1. Nu rises from approximately 2.7 to 5.2 when M is increased from 0.5 to 2.5 at ϕ=0. Thermal conductivity is improved by a higher nanoparticle concentration. Increasing M improves flow stability, which raises the effectiveness of heat transfer. Cf grows with magnetic parameter M and volume fraction ϕ, although at a smaller gradient than Nu. At M=0.5, Cf rises from approximately 1.5 to 2.3 as ϕ goes from 0 to 0.1. For constant ϕ=0, Cf rises from approximately 1.5 to 2.2 as M rises from 0.5 to 2.5. The generation of Lorentz force due to the magnetic field increases flow hinderance, resulting in increased skin friction. Nanoparticles thicken the boundary layer, resulting in more viscous effects.