The energy PDE in Eq. (6) and PST boundary conditions are converted into a non-dimensional ODE and non-dimensional boundary condition using Eqs. (11) and (12) as follows: (23)Θηη+PrFΘη−2PrFηΘ=−PrEc1QFη2, (24)Θ(0)=1andΘ(∞)=0.

Upon introducing the new variable ξ=−e−ηαPrα2 into Eqs. (23) and (24), we get (25)2dΘ+(−ξd−a#0+1)Θξ+ξΘξξ=−Ec1α2d2QξPr, (26)Θ(−Prα2)=1andΘ(0)=0.where a#0=Prdα2+PrSα.

The analytical solution of Eq. (25), subject to the boundary condition (26), is expressed in terms of the confluent hypergeometric function as a function of ξ as (27)Θ(ξ)=(1+Ec1Prd2α2Q2(2−a#0)M(a#0−2,1+a#0,−Prα2d)(−Prα2)a#0)ξa#0M(a#0−2,1+a#0,ξd)+(−Ec1α2d2PrQ2(2−a#0)ξ2).

Upon replacing the variable ξ with η, we get (28)Θ(η)=(−C1+1)e−a#0αη×M(a#0−2,1+a#0,−Prdα2e−αη)M(a#0−2,1+a#0,−Prdα2)+C1e−2αη.

The dimensionless gradient of the wall temperature for the PST case is given by Θη(0) is (29)Θη(0)=(1−C1)(Prdαa#0−21+a#0×M(a#0−1,2+a#0,−Prdα2)M(a#0−2,1+a#0,−Prdα2)−a#0α)−2C1α.where C1=−Ec1Prd2Q2α2(2−a#0), and M is the Confluent Hypergeometric function [40] defined by M(a~,b~,x)=1+∑n=1∞(a~)nxn(b~)nn!,(a~)n=a~0(1+a~0)(2+a~0),⋯,(n−1+a~0),(b~)n=b~0(1+b~0)(2+b~0),⋯,(n−1+b~0).

The local Nusselt number, as defined in [31,40], is Nu1 for PST case (30)Nu1=xqwκ(T~w−T~∞),Whereqw=−κ(∂T~∂y)y=0.

Using the similarity variables in Eqs. (11), (30) yields the following equation: (31)Nu1Rex−(12)=−Θη(0).

The energy PDE in Eq. (6) and PHF boundary conditions are converted into a non-dimensional ODE and non-dimensional boundary condition using Eqs. (11) and (12) as follows: (32)Gηη+PrFGη−2PrFηG=−PrEc2QFη2, (33)Gη(0)=−1andG(∞)=0.

Substituting ξ into Eqs. (32) and (33) yields (34)2dG+(−ξd−a#0+1)Gξ+ξGξξ=−Ec2α2d2QξPr, (35)Gξ(−Prα2)=−αPrandG(0)=0.

The analytical solutions of Eqs. (34) with (35) are expressed in terms of ξ and η as (36)G(ξ)=(−αPr−Ec2d2Q(2−a#0)((a#0−21+a#0)dM(a#0−1,2+a#0,−Prα2d)+a#0M(a#0−2,1+a#0,−Prα2d)(−α2Pr))(−α2Pr)−a#0)×ξa#0M(a#0−2,1+a#0,dξ)−Ec2α2d2PrQ2(2−a#0)ξ2, (37)G(η)=C2e−a#0αηM(a#0−2,1+a#0,−Prdα2e−αη)+C1e−η2α.

The dimensionless gradient of the wall temperature for the PHF case is given by (38)G(0)=C2M(a#0−2,1+a#0,−Prdα2)+C1.where C2=(−1+2C1α)(−αa#0M(a#0−2,1+a#0,−Prα2d)+(a#0−21+a#0)dPrαM(a#0−1,2+a#0,−Prdα2))

The local Nusselt number is defined (see [31,40]) as Nu2 for PHF case (39)Nu2=x(νb)qwκ(T~w−T~∞),Whereqw=E0(1l2)y=0.

Using the similarity variable and the PHF boundary condition in Eq. (39) yields the following equation: (40)Nu2Rex−(12)=1G(0).

The boundary constraints stated in Eq. (14) are used to determine the precise solution to the dimensionless momentum equation. This solution contains an unknown parameter, denoted as α, which needs to be identified to resolve Eq. (13). By substituting Eq. (15) into expression (13), the momentum expression transforms into a cubic polynomial equation (see Eq. (16)) with respect to α. According to the fundamental theorem of algebra, solving for α yields three solutions for Eq. (13), denoted as α1,α2 and α3. It is vital to emphasize that only positive values of α offer physically relevant solutions to this problem [5,33,40].

Fig. 2 depicts the physically viable solution α corresponding to the momentum equation in relation to the suction parameter (S). It is observed that only one of the values α1,α2 and α3, considered α1 provides a meaningful solution. Therefore, there is only one solution to the momentum equation for the current problem. The solution α1 was employed to obtain exact answers to both the momentum and heat energy equations. Furthermore, Fig. 2a shows that the α1 value increases as the λ1 increases. From Fig. 2b,c, it is also noted that the α1 value drops as the β and δ∗ increase. The impact of the Q is shown in Fig. 2d; as the Q increases, so does the α1 value.

The velocity profile of the Jeffrey fluid for various values, considering the combined effects of λ1, β, and Q with δ∗, is graphically illustrated in Fig. 3a–c, respectively. Fig. 3a presents a comparative analysis of the λ1 and the δ∗. It is observed that the fluid’s velocity decreases with an increase in λ1, which is attributed to the thinning of the momentum boundary layer. Conversely, the fluid’s velocity rises with an increase in the δ∗, highlighting the magnetic field’s influence on the fluid’s viscosity. The physical reason for the thinning effect due to increased λ1 can be linked to the structural changes in the fluid’s molecular level interaction. As the λ1 increases, the fluid molecules experience a delayed viscous response to applied stress, allowing them more time to align and stretch along the flow direction, which reduces the overall resistance, thins the momentum boundary layer, and decreases the velocity.

Fig. 3b displays the effects of the β and the δ∗ on the velocity profile. An increase in the β enhances both the fluid velocity and the momentum boundary layer thickness. Since the β is related to the stretching parameter b, it indicates that a higher β corresponds to a greater rate of stretching of the sheet, which in turn reduces the fluid’s resistance to motion. This results in higher fluid mobility within the boundary layer near the surface, leading to increased velocity and boundary layer thickness. Furthermore, the increase in magnetic viscosity parameter also boosts the fluid’s velocity, compounding with the effects of the β.

The combined impact of the Q and the δ∗ on the velocity profile is shown in Fig. 3c. An increase in the Q results in a reduction of the velocity distribution. This decrease in the velocity profile is due to the Lorentz magnetohydrodynamic drag force, which acts perpendicular to the magnetic field and inhibits the flow, whereas an increase in the δ∗ enhances the fluid velocity.

The local skin friction coefficient is illustrated in Fig. 4 for various values of λ1, β, and δ∗. The x-axis represents the Q, and the y-axis represents the local skin friction coefficient. The skin friction coefficient decreases as Q rise, while it increases with the increase of λ1 (Fig. 4a). This occurs because the fluid takes longer to reach its equilibrium state; thus, the increased movement within the fluid generates more friction, leading to a higher skin friction coefficient. Fig. 4b illustrates the influence of the β on the local skin friction coefficient. It is observed that a rise in the β results in a diminish in the local skin friction coefficient. This is because the β is directly proportional to the rate of stretching of the sheet, and a higher β decreases the resistance to fluid motion, thereby reducing the local skin friction coefficient. The impact of the δ∗ on the local skin friction coefficient is shown in Fig. 4c. An increase in the δ∗ leads to a decrease in the local skin friction coefficient. The relationship between local skin friction coefficient and various parameters, including λ1, β, δ∗, and Q, is tabulated in Table 2. The computed numerical values of these physical quantities are systematically presented in a tabular format, providing a quantitative perspective that will be useful for future reference.

The temperature profile is analyzed under two separate boundary conditions: the PST and PHF cases. In this context, PST and PHF are represented by different symbols Θ(η) and G(η). The temperature profile incorporates the effects of ohmic dissipation, introducing the δ∗ and Q into the non-dimensional energy equation. Fig. 5 illustrates the combined effects of λ1 and the δ∗ on the temperature profile for the PST case in Fig. 5a and the PHF case in Fig. 5b. As the λ1 increases, the temperature profile decreases, which occurs because higher λ1 thins the thermal boundary layer, reducing the temperature. However, an increase in the δ∗ raises the temperature due to the influence of the magnetic field in the fluid, affecting both the PST and PHF cases.

Fig. 6 illustrates the combined effects of the β and the δ∗ on the temperature profile for both boundary conditions. A rise in the β and δ∗ leads to higher temperature profiles (Fig. 6a,b). This effect is due to the intensified motion of fluid particles, which thickens the thermal boundary layer. The enhanced fluid motion slightly raises the temperature at the sheet surface, particularly as the β increases. Additionally, the rise in the δ∗ promotes smoother fluid flow, further increasing the thermal boundary layer thickness and temperature.

Fig. 7 shows the temperature profile for the PST and PHF cases, considering the effects of the Q and δ∗. In both Fig. 7a,b, the profiles show an increase in temperature for the PST and PHF cases, due to the effect of the magnetic field. An increase in Q causes the expansion of the flow region in both scenarios. This expansion results from the resistive Lorentz force induced when a transverse magnetic field interacts with an electrically conducting fluid, reducing fluid velocity in the flow region and enhancing the temperature distribution.

Fig. 8a,b presents the temperature distribution for the PST and PHF cases, respectively, along with the effects of the Eckert number and δ∗. A higher Eckert number results in an expanded thermal boundary layer, increasing the significance of dissipative energy and broadening the temperature profile. Additionally, energy dissipation causes a sharper temperature gradient in the PST case and a rise in wall temperature in the PHF case. Energy dissipation from electrical resistance generates a larger temperature field and an increase in the Eckert number. Notably, there is a considerable temperature overshoot as the Eckert number increases, reflecting the relationship between the flow’s kinetic energy and the enthalpy difference in the boundary layer. Internal friction heating between fluid molecules, combined with ohmic heating, which converts mechanical and electrical energy into thermal energy, heats the fluid in the stretching sheet flow. A higher δ∗ and Eckert number produces more thermal energy, resulting in a higher temperature of the stretched sheet and a thicker thermal boundary layer.

Fig. 9 shows the variation of the local Nusselt number for different parameters, including λ1, β, and δ∗. Fig. 9a shows that as the λ1 grows, so does the local Nusselt number. As retardation time grows, the fluid has more time to absorb heat, resulting in a larger temperature differential at the surface and consequently a higher local Nusselt number, which decreases with Q.

Fig. 9b demonstrates the behavior of the local Nusselt number in relation to the Q for different values of the β. It has been noted that an increase in the β results in a decrease in the local Nusselt number. Fig. 9c illustrates that as δ∗ increases, the local Nusselt number decreases, thereby enhancing heat transfer within the boundary layer. Fig. 10 depicts the relationship between 1G(0) which is a local Nusselt number for the PHF case and the Q. Fig. 10a demonstrates that when λ1 increases, the 1G(0) value rises significantly since it is the reciprocal of the temperature profile in the PHF case (η=0). Fig. 10b,c shows the 1G(0) values drop when the β and δ∗ rise, respectively. This small drop is caused by the magnetic field. An increase in the Q increases the Lorentz force, which opposes fluid motion and reduces the temperature gradient, hence decreasing the 1G(0) value due to the Lorents force. Table 3 shows the non-dimensional temperature profiles for the PST and PHF cases, along with the local Nusselt numbers for both cases, under different parameters such as λ1, β, δ∗, Q, Eckert number, and Pr.