EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5616 ISSN 1307-5543 – ejpam.com Published by New York Business Global Impacts of Activation Energy and Modified Darcy Law on the Motion of Williamson Nanofluid Over a Stretching Riga Sheet Through a Porous Medium N. T. M. Eldabe1, A. M. Saeed2,∗, A.M. Ismael3, S. M. Ali4 1 Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt 2 Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Ain shams University, Abbasiya, Egypt 4 Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 34151 Saudi Arabia Abstract. The present analysis is motivated by the need to elucidate with more accuracy and sophistication the motion of Williamson nanofluid with activation energy and modified darcy law through porous medium over a stretching riga sheet. The problem is modulated mathematically by using the momentum,energy and concentration equations. The nonlinear partial differential equations describe the motion is transformed to nonlinear ordinary differential equations by using a suitable transformations. The obtained system of equations with boundary conditions inside the boundary layer are solved semi analytically by using homotopy perturbation method. The velocity, temperature and the concentration of the fluid as well skin-friction, Nusselt and Sherwood numbers are obtained as a functions of the physical parameters of the problem. The effects of these parameters on the solutions are discussed numerically and illustrated graphically through some figures. It is found that the parameters play a dramatic role to control the solutions. For example the velocity increases with increasing Williamson parameter, permiability parameter and modified Hermann number. On the other hand the fluid temperature increases with increasing both of Brownian parameter and Eckert number. In addition, increasing the activation energy and thermophoresis parameter increases the fluid concentration. 2020 Mathematics Subject Classifications: 76S05, 76D10, 80A32, 35Q79 Key Words and Phrases: Activation energy, riga sheet, nanofluid, magnetohydrodynamic flow ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5616 Email addresses: eng mohamed nabil125@hotmail.com ((N. T. M. Eldabe), abdulkafi.ahmed@qu.edu.sa (A. M. Saeed), ayaismael1990@gmail.com (A. M. Ismael), smsali@iau.edu.sa (S. M. Ali) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 2 of 22 1. Introduction The boundary layer refers to the layer of fluid in the immediate vicinity of a boundary surface where the effects of viscosity are significant. A range of velocities occurs across the boundary layer from maximum to zero, provided that the fluid is contact with surface. Generally, boundary layer flows affected by magnetohydrodynamic (MHD) play a critical role in manufacturing and technical processes, including the construction of MHD turbines, flow meters, and nuclear reactors. External magnetic fields are widely used to control high conductivity fluid flows, such as semiconductor melting or liquid metals, referred to as con- ventional MHD flow. This method is ineffective for fluids with low electrical conductivity, such as sea water. A Riga surface generates Lorentz force. Riga refers to a plate surface containing mutually placed magnets and electrodes. The plate is unique because it induces electromagnetic energy sufficient to generate Lorentz forces along the surface, thereby re- stricting the flow of slightly conducting fluid. The plate was originally constructed from an array of interspaced and obligatory magnets distributed in a span wise configuration. It can be utilized to prevent boundary layer tearing caused by radiation. In this regard, the Riga plate induced laminar flow has been examined in physical properties. Naseer et al. [18] studied heat transfer and the steady boundary layer flow of a hyperbolic tangent fluid flowing over a vertical potentially stretching cylinder in its axial direction. viscous dissipation and Joule heating influences on the boundary layer flow over a stretching ver- tical Riga plate of a micropolar nanofluid are discussed by Eldabe et al. [10]. Bilal et al. [5] investigated the thermal characteristics generated in a viscosity-dependent viscoelastic non-linear fluid as it flows over an isothermally inclined Riga surface. Shamshuddin and Narayana [22] discussed the effect of Joule heating and dissipation between two Riga plates with Cattaneo-Christov heat flux on the magnetohydrodynamic squeezing flow. Some re- cent attempts in this direction can be viewed via refs. [1, 19]. Non-Newtonian flows are more efficient due to their practical applications in physiological, technological processes and most industries. The properties of all non- Newtonian fluids are diverse, and the behaviors of all types of such fluids are depicted by a single relation. Williamson fluids elucidate such behaviors with more advantages than other fluids. Hameda et al. [12] in- vestigated the two-dimensional incompressible Williamson nanofluid hydro magnetic flow over a stretching sheet in a porous media. A new mathematical model in a micropolar Williamson nanofluid for the flow of an electro-osmotic boundary layer are studied by Eldabe et al. [11]. Over an exponentially stretched surface, Jangid et al.[15] studied the heat transference to expand the hydro-magnetic Williamson fluid flow with nano-particles. Many results of the non-Newtonian are discussed in these articles [4, 13]. A solid-liquid mixture of tiny size nanoparticles and base liquid is known as nanofluid. The colloids of base liquid and nanoparticles have important physical characteristics which enhance their potential role in the applications of ceramics, drug delivery, paintings, coatings etc. Nanofluids are declared as super coolants because their heat absorption capacity is much higher than traditional liquids Eldabe et al. [7] studied the peristaltic unsteady flow of nanofluid non-Newtonian with heat transfer in vertical non-uniform duct. The flow follows the Herschel Bulkley model in a non-Darcy porous medium, considering thermal diffusion M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 3 of 22 and mixed convection. In a symmetric channel with compliant wall characteristics, the mixed convective peristaltic flow of Carreau-Yasuda nanoliquid in the presence of slip conditions is discussed by Nisar et al [20]. Nowar [21] investigated the peristaltic flow of an incompressible conducting viscous electrically nanofluid through a porous medium in a vertical asymmetric channel with taking the Hall effects. Many results of the nanofluid are studied in these papers [3, 6, 8, 9, 14, 16]. The activation energy is a key concept in understanding the relationship between chemical reactions and the energy required for them to occur. The first mathematical connection between the rate of a chemical reac- tion and absolute temperature was given by Hood. Arrhenius extended Hood’s idea to obtain the relation between the rate of chemical reaction and temperature, and the equa- tion is usually called the Arrhenius equation. In the Arrhenius equation, the activation energy parameter refers to the minimum energy threshold that reactants must attain for a chemical reaction to occur and form new products. Muhammad et al. [17] investigated a mathematical analysis for three-dimensional Eyring-Powell thermal radiation nanofluid nonlinear with mass fluxes plus modified heat. The peristaltic waves of a non-Newtonian nanofluid along an asymmetric channel are investigated by taking activation energy and thermal radiation in consideration by Sara et al. [2]. This work attempts to fill the void of the movement of nano-non-Newtonian fluid obeying the Williamson model with heat and mass transfer through a porous medium over a stretching Riga sheet. The activation energy and modified Darcy law of the non-Newtonian fluid are considered. The non-linear partial differential equations that describe the motion were converted to ordinary differen- tial equations using suitable transformations. The homotopy perturbation technique was used to solve this system subject to appropriate boundary conditions. The influences of relevant parameters were discussed graphically. 2. Mathematical formulation The motion of non-Newtonian Williamson nanofluid with heat and mass transfer through porous medium with Williamson fluid over a stretching Riga plate is investi- gated. Choose Cartesian coordinates x and y, where x is in the direction of stretching sheet, and y is perpendicular to it as shown in figure (1). The activation energy, viscous dissipation and modified Darcy law for a porous medium are taken in our consideration. The constitutive equation of the Williamson model can be written as [12] τ = ( µ∞ + µ0 − µ∞ 1− Γν∗ ) A, (1) where A = ∇V + ∇TV ν∗ = √ 1 2 trace(A2), τ is the stress tensor , µ0 is the limiting viscosity , µ∞ is the viscosity at infinity, A is the first Rivlin-Ericksen tensor, Γ > 0 is a time constant and V (u, v) is velocity. In the case of µ∞ = 0 , Γν∗ < 1, we get τ = µ0(1 + Γν∗)A, (2) M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 4 of 22 Figure 1: Sketch of the problem. And have the following components τxx =µ0 1 + Γ √ 4 ( ∂u ∂x )2 + ( ∂u ∂y + ∂v ∂x )2  2 ∂u ∂x τyy =µ0 1 + Γ √ 4 ( ∂u ∂x )2 + ( ∂u ∂y + ∂v ∂x )2  2 ∂v ∂x τxy =µ0 1 + Γ √ 4 ( ∂u ∂x )2 + ( ∂u ∂y + ∂v ∂x )2  { ∂u ∂y + ∂v ∂x } (3) Since, the magnetic force is generated by the Riga plate, the Lorentz force f = J ∧ B is defined as magnetic force. According to the Grinberg hypothesis this magnetic force is defined as f = π 8 j0M0e −π l yi, (4) where j0 is the current density, M0 is the constant magnetic field and l is the width of the plate. The boundary layer equations for the velocity, temperature and concentration can be written as follows: ∂u ∂x + ∂v ∂y = 0 (5) u ∂u ∂x + v ∂u ∂y = µ0 ρf ∂ ∂y {1 + Γ ∂u ∂y }∂u ∂y + π j0M0 8ρf e− π l y − µ0 k0 ρf {1 + Γ ∂u ∂y }u (6) u ∂T ∂x + v ∂T ∂y = kc (ρc)f ∂2T ∂y2 +τ0 { DB ∂T ∂y ∂C ∂y + DT T∞ ( ∂T ∂y )2 } + (7) µ0 (ρc)f { 1 + Γ ∂u ∂y } ( ∂u ∂y ) 2 − Q0 (ρc)f (T − T∞) u ∂C ∂x + v ∂C ∂y = DB ∂2C ∂y2 + DT T∞ ∂2T ∂y2 − λ 0 ( T T∞ )m (C − C∞)e− Ea kaT (8) M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 5 of 22 where the last term in right hand side of equation (6) is due to the modified darcy law, where ρf is the fluid density, T is the fluid temperature, C is the fluid concentration, kc is the thermal conductivity, DB is the Brownion diffusion coefficient, DT is the ther- mophoretic diffusion coefficient, (ρc)p is the heat capacitance of the nanofluid, τ0 = (ρc)p (ρc)f , (ρc)f is the heat capacitance of the fluid, Q0 is the constant heat absorption, λ0 is chemical reaction coefficient, Ea is the activation energy, ka is the Boltzmann constant and m is the fitted rate constant. The appropriate boundary conditions are u = ax, v = 0, T = Tw, C = Cw, y = 0 u → 0, T → T∞, C → C∞, y → ∞ (9) Consider the following similarity solutions to transform the system of our equations from non-linear partial differential equations to non-linear ordinary differential equations, u = ax ∂f (η) ∂η , v = − √ aν f (η) , η = √ a ν y T = T∞ + θ (Tw − T∞) , C = C∞ + φ(Cw − C∞) (10) Where Tw is temperature at the Riga plate, Cw is the concentration at the Riga plate and T∞and C∞are the temperature and concentration at infinity respectively. Then equations (6-8) become f ′′′ + 2Wef ′′f ′′′ + ff ′′ − f ′2 − 1 k f ′ − We k f ′f ′′ +Mexp (−sη) = 0 (11) θ′′ + Prf θ′ +Nb Pr θ′φ′ +Nt Pr θ′ 2 + Pr N ( f ′′2 +We f ′′3 ) −QPrθ = 0(12) φ′′ + Sc f φ′ + Nt Nb θ′′ − Sc λ e−E (E +m) (w − 1) θ φ = 0 (13) With conditions f = 0, f ′ = 1, θ = 1, φ = 1, η = 0 f ′ = 0, θ = 0, φ = 0 η → ∞ (14) Where We =Γa 3 2 x√ ν is the Williamson parameter, k = k0a ν is the permeability parameter, M = π j0M0 8 ρf a2x is the modified Hermann number, s = √ π2ν l2a is the dimensionless parameter, Pr = µ0cf kc is the Prandtl number, Nb = τ0DB(Cw−C∞) ν is the Brownian motion, Nt = τ0DT (Tw−T∞) ν T∞ is the thermophoresis variable, Ec = a2x2 cf (Tw−T∞) is the dimensionless parameter, Sc = ν DB is the Schmidt number, λ = λ0 a is the chemical reaction parameter, E = Ea ka T∞ is the activation energy parameter and w = Tw T∞ is the dimensionless parameter, Q = Q0 (ρc)fa is the heat absorption parameter. M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 6 of 22 3. Method of solutions We shall use the homotopy perturbation method to obtain the semi-analytical solutions of equations (11-13) with boundary conditions (14). The initial solutions can be written as f0 (η) = 1− e−η, θ0 (η) = e−η and φ0 (η) = e−η. (15) Take the linear operator as (D3 − 1 kD) to obtain the first and second order solutions of the velocity, (D2 − Q Pr) for temperature and (D2 − s47) for concentration. Now by using homotopy technique the velocity, temperature and concentration can be written as follows f ′(η) = −s31e −η − s35e −2η + s36e −3η − s37e −sη − s38e −η √ 1/k + s39e −η(1+s) + s40 e −η ( 1+ √ 1/k ) + s41 η e−η √ 1/k , (16) θ (η) = s82e −η + s83 e−η √ Q Pr + s84e −2η + s840e −3η − s700e −4η − s72η e−η √ Q Pr + s74e −η(1+s) − s75 e−η(1+ √ Q Pr) + s76e −η ( 1+ √ 1/k ) + s77 e−η(1+ √ s47) − s770 e−η(2+ √ Q Pr) − s771e −η(2+s), (17) φ (η) = s111e −η + s112e −2η + s100e −3η + s1000e −4η − s113 e−η √ s47 + s101η e−η √ s47 + s102e −η ( 1+ √ 1/k ) + s103e −η(1+s) − s104 e−η √ Q Pr + s105 e−η(1+ √ s47) + s106 e−η(1+ √ Q Pr) , (18) 3.1. Skin friction coefficient The most important and interesting non- dimensional quantity that is the skin friction coefficient which is defined, evaluated and computed as τw = f ′′ +We f ′′2 (19) For Williamson fluid, and at the wall η = 0, the skin friction becomes τw = s138 (20) Which is constant and functions of the physical parameters of the fluid. M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 7 of 22 3.2. Nusselt number The Nusslt number in non-dimensional form can be written as Nu = −θ′ ∣∣ η=0 (21) at the wall η = 0, the Nusselt number becomes Nu = s143 (22) Which is constant and functions of the physical parameters of the fluid. 3.3. Sherwood number The Sherwood number in dimensionless can be written as Sh = −φ′∣∣ η=0 (23) at the wall η = 0, the Sherwood number becomes Sh = s148 (24) Which is constant and functions of the physical parameters of the fluid. Here, s0- s148 are defined in the appendix. 4. Results and discussion The aim of this section is to study the influences of activation energy with modified Darcy law for Williamson non-Newtonian fluid on the flow inside the boundary layer of a Riga plate through a porous medium. Heat and mass transfer are taken into account with heat absorption and chemical reaction. The system of equations is converted to nonlinear ordinary differential equations with boundary conditions using an appropriate transformation. The system is solved using the homotopy perturbation technique, and the velocity, temperature, and concentration distribution along with skin friction, Nusselt, and Sherwood numbers are obtained as a function of the physical parameters of the problem. Numerical calculations are used to investigate the influences of the physical parameters of the obtained solution. Additionally, the results are explained graphically through figures (2-21). Figures (2-5) are plotted to illustrate the effects of M, k, s and We on the velocity f ′(η) for various values of the parameters of interest. It is shown from the figures that the velocity improves prominently with increasing the parameters M, k and We, while the velocity decreases with increasing s. The physical interpretation of these observations is due to the fluid moves in the boundary layer of the magnetic plate in the presence of a positive Lorentz force, which attracts the fluid to the plate and works to increase its speed directly with M . On the other hand, in the porous medium, there exist voids between the molecules of the fluid which allows the fluid velocity to increase with increasing k. M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 8 of 22 It can also be noted that since the term of magnetism that increases the speed, as we mentioned above, is multiplied by an exponential term with a negative strength s, it is mathematically clear that the speed of the fluid will decrease with an increase of s. It is also clear that increasing the viscosity of the fluid, which occurs in the case of non-Newtonian fluids for the Williamson model, works to increase the velocity of the fluid in the boundary layer of the stretching sheet. Therefore, the velocity will increase with the increase of We. Variation of temperature is illustrated in Figures (6-12). Figure (6) shows the magnetic field effect on temperature; an increase in magnetic field leads to a decrease in temperature. Also, as the permeability k of the medium increases and the gaps between its molecules increase, the temperature decreases, as shown in Figure (7). Figure (8) clarifies that the temperature slightly increases with increasing Brownian parameters Nb, due to the addi- tional nanoparticles which enhance temperature. Variations of temperature components with s are shown in Figure (9). Temperature components increase with an increase in the constant s. Figure (11) illustrates that the temperature of the fluid rises as the value of Ec increases. This occurs because heat is generated within the fluid due to frictional heating as Ec increases. The effect of the Prandtl number on the temperature profile is shown in Figure (12). As the Prandtl number increases, the rate of thermal diffusion reduces, resulting in a constantly decreasing temperature profile. Physically, it follows that fluids with a high Prandtl number have high viscosity and low thermal conductivity, causing the fluid to thicken and decrease its velocity. Figures (10) and (17) show that as the Williamson parameter increases, the tempera- ture and concentration profiles, as well as the thickness of the boundary layer, also increase. This is due to reduced heat and mass transfer, as well as nanoparticle saturation within the boundary caused by drag against the non-Newtonian fluid stream. The presence of nanoparticles makes it ideal for use as a lubricant cooling agent, as they remain in the base fluid longer and enhance the flow characteristics of nanofluids. The concentration profile rises for large values of activation energy E, as shown in Fig- ure (13). An improvement in the concentration field results from the Arrhenius function deteriorating by snowballing the activation energy value, which promotes the generative chemical process. The effects of thermophoresis Nt on the concentration profile are de- picted in Figure (14). This figure shows that as the thermophoresis parameter increases, the concentration profile rapidly increases at all places in the flow domain. Figure (15) shows how the chemical reaction parameter affects the concentration profiles. This figure shows that increasing values of λ have a significant effect on the concentration distribution in the boundary layer. It is apparent that the increasing values of λ lower the concen- tration in the boundary layer. Physically, large values of λ decrease the solute boundary layer thickness and promote mass transfer. Sc relates the relative thickness of the mass transfer layer to the hydrodynamic boundary layer. As a result, as Sc increases, mass diffusivity reduces, leading to a lower concentration, as shown in Figure (16). As the permeability parameter is raised, the concentration profile decreases. When the porous medium’s permeability increases, it becomes more porous, which reduces the Darcian body force’s magnitude and increases the rate at which mass transfers into the porous medium, M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 9 of 22 as shown in Figure (18). Figure (19) shows that an increase in Eckert number leads to a higher concentration. As the concentration increases, more radiation should be absorbed, resulting in an increase in absorbance; thus, concentration and absorption are directly proportional, as shown in Figure (20). Figure (21) demonstrates that Brownian motion reduces both concentration and thickness of the solute boundary layer. It is owing to the fact that increases in Brownian motion enhance the random movement that spreads the nanoparticles and hence decreases concentration. Finally, the effects of the parameters of the problem on each of the skin friction τw, Nusselt number Nu, and Sherwood number Sh are discussed and illustrated in tables (1- 3). Skin friction coefficients τw are calculated using the formula in Eq. 20. In Table (1a), the Williamson parameter We variety, while other parameters are fixed.The current study shows that skin friction increases with larger values of the Williamson parameter. This is because increasing the Williamson parameter thickens and makes the fluid more viscous. The magnetic field parameter M variety, while other parameters are fixed as shown in Table (1b). As M increases, the skin friction coefficient τw also increases. This is because a working magnetic field tends to enhance flow motion and, consequently, surface friction force in nanofluid flow. It has been observed that as a result of changing the parameters, this leads to change in the values of skin friction τw, which indicates the importance of the presence of external influences in the case of flow of non-Newtonian nanofluid inside the boundary layer of Riga magnetized surface. This has important scientific applications in various fields, such as paints, reactors and some medical devices. The Nusselt number increases as the magnetic field parameter increases, as shown in table (2). The table (3) explains how different values of Schmidt number influence the Sherwood number. The Sherwood number increases with an increase in the Schmidt number. The Schmidt number has a greater impact on the Sherwood number compared with the Nusselt number. For your information, other influences on all parameters were. Figure 2: The variation of the velocity f ′(η) for various values of M. We = 0.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 3: The variation of the velocity f ′(η) for various values of k. We = 0.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1 M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 10 of 22 Figure 4: The variation of the velocity f ′(η) for various values of s. We = 0.5, M = 1.5, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 5: The variation of the velocity f ′(η) for various values of We. M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 6: The variation of the temperature θ(η) for various values of M. We = 0.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 7: The variation of the temperature θ(η) for various values of k. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5, P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1 M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 11 of 22 Figure 8: The variation of the temperature θ(η) for various values of Nb. We = 0.5, M = 1.5, s = 0.2, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 9: The variation of the temperature θ(η) for various values of s. We = 0.5, M = 1.5, Nb = 0.4, Nt = 0.3, Ec = 5, P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 10: The variation of the temperature θ(η) for various values of We. M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 11: The variation of the velocity θ(η) for various values of Ec. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 12 of 22 Figure 12: The variation of the velocity θ(η) for various values of Pr. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 13: The variation of the velocity ϕ(η) for various values of E. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5, P r = 0.5, Q = 0.3, λ = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 14: The variation of the velocity ϕ(η) for various values of Nt. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Ec = 5, P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 15: The variation of the concentra- tion φ(η) for various values of λ We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 13 of 22 Figure 16: The variation of the velocity ϕ(η) for various values of Sc. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, m = 0.5, w = 1, k = 0.3 Figure 17: The variation of the velocity ϕ(η) for various values of We. M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 18: The variation of the velocity ϕ(η) for various values of k. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1 Figure 19: The variation of the velocity ϕ(η) for various values of Ec. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 14 of 22 Figure 20: The variation of the velocity ϕ(η) for various values of Q. We = 0.5, M = 1.5, s = 0.2, Nb = 0.4, Nt = 0.3, Ec = 5 , P r = 0.5, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 Figure 21: The variation of the velocity ϕ(η) for various values of Nb. We = 0.5, M = 1.5, s = 0.2, Nt = 0.3, Ec = 5, P r = 0.5, Q = 0.3, λ = 0.7, E = 0.7, Sc = 0.1, m = 0.5, w = 1, k = 0.3 We k M Pr sc s Ec Nb Nt Q λ E τw 0.1 0.3 1.5 0.5 0.1 0.2 5 0.4 0.3 0.3 0.7 0.7 -0.566953 0.3 -0.395052 0.5 -0.238782 0.9 0.223161 Table (1a). The computation results for local skin friction coefficient for variation of We. We k M Pr sc s Ec Nb Nt Q λ E τw 0.5 0.3 0.5 0.5 0.1 0.2 5 0.4 0.3 0.3 0.7 0.7 -0.496554 1.0 -0.418833 1.5 -0.238782 2.2 0.185204 Table (1b). The computation results for local skin friction coefficient for variation of M . We k M Pr sc s Ec Nb Nt Q λ E Nu 0.5 0.3 0.5 0.5 0.1 0.2 5 0.4 0.3 0.3 0.7 0.7 0.522544 1.0 0.672974 1.5 0.823403 2.2 1.034 Table (2). The computation results for local Nusselt number for variation of M. M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 15 of 22 We k M Pr sc s Ec Nb Nt Q λ E Sh 0.5 0.3 1.5 0.5 0.08 0.2 5 0.4 0.3 0.3 0.7 0.7 0.894771 0.1 0.912382 0.3 1.03737 0.7 1.19726 Table (3). The computation results for local Sherwood number for variation of sc. 5. Conclusions This study provides a comprehensive analysis of the complex interaction governing the Williamson non-Newtonian nanofluid flow over a Riga plate while incorporating the effects of activation energy, modified Darcy law, and thermal absorption. Heat and mass transfer have been included. With the necessary similarity variables, the set of nonlin- ear partial differential equations describing the velocity, temperature, and concentration are transformed into ordinary differential equations. Semi-analytical solutions using the homotopy perturbation method are obtained. The study indicates that velocity increases with increasing Williamson parameter, permeability parameter, and modified Hermann number, while it decreases with an increase in s. Increasing both the Brownian parameter and Eckert number affects the fluid’s temperature, causing it to increase. Also, increasing the permeability parameter, Prandtl number, and modified Hermann number decreases the temperature. Furthermore, nanoparticles concentration increases as the activation energy parameter, Williamson parameter and thermophoresis parameter increase. It was also concluded that the enhancement in both the chemical reaction, Schmidt number and Brownian parameter leads to a decrease in nanoparticle concentration. The study of fluid flow in the boundary layer adjacent to different surface, which is an important topic in fluid mechanics, is of great importance for many applications in various scientific fields, whether industry, such as the manufacture of aircraft, space vehicles and reactors. Also, in the painting works for different surfaces, and the manufacture of medical devices, as well as medical treatments. For future work and since the study of fluids flow is of great importance, we will use different models of non-Newtonian fluids that link the stress-rate of strain relationship. Also, an external magnetic field can be applied to the fluid motion and the ohmic dissipation due to magnetic field will take into consideration. In addition, we will study the problem when the fluid flows through porous media obeys non-darcy law. Acknowledgements The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025). M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 16 of 22 References [1] N Abbas and W Shatanawi. Heat and mass transfer of micropolar-casson nanofluid over vertical variable stretching riga sheet. Energies, 15:4945, 2022. [2] SI Abdelsalam, A Magesh, P Tamizharasi, and AZ Zaher. Versatile response of a sutterby nanofluid under activation energy: hyperthermia therapy. International Journal of Numerical Methods for Heat & Fluid Flow, 34(2):408–428, 2024. [3] SI Abdelsalam, KS Mekheimer, and AZ Zaher. Alterations in blood stream by elec- troosmotic forces of hybrid nanofluid through diseased artery: aneurysmal/stenosed segment. Chinese Journal of Physics, 67:314–329, 2020. [4] KK Asogwa, BS Goud, NA Shah, and SJ Yook. 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Math, 18 (1) (2025), 5616 18 of 22 Appendix s0 = (2− 1/k)We,s1 = s0 2(4− 1/k) , , s2 = M s(s2 − 1/k) , s3 = k k − 1 , s4 = s1 + s3 − s2, s5 = √ k (s3 + 2s1 − s s2) , s6 = s4 − s5, s7 = 24s1 We− s1 − 6s1 We k , s8 = 4We s3 + 4s1 − 2s3 We k , s9 = s3+s6, s10 = s2−2We s2 s2 (s+ 1) , s11 = s s2 (s− 2)+ We s s2 k (s+ 1) , s12 = s10 + s11, s13 = −2 We s5 ( 1 + √ 1/k ) k + s5 We ( 1 + k √ 1/k ) k2 , s14 = s5 k2 ( k+ k22k2 √ 1/k ) , s15 = s13 + s14, s16 = s5 k , s17 = s2s2, s18 = s7 3 ( 9− 1 k ) , s19 = s8 2 ( 4− 1 k ) , s20 = s9 /(1− 1/k), s21 = s12 (1 + s) ( (1 + s)2 − 1 k ) , s22 = s15( 1 + √ 1/k )( 1 + 2 √ 1/k ) , s23 = k s16 2 , s24 = s17 s ( s2 − 1 k ) , s25 = s18+s19+s20+s21+s22−s24, s26 = √ k (3s18 + 2s19 + s20) , s27 = √ k ( s21 (1 + s)+ s22 ( 1 + √ 1/k ) − s23 − s s24 ) , s28 = s26+s27, s29 = s25−s28, s30 = 1+ s6 + s29, s31 = − (1 + s3 + s20) , s32 = − (s1 + s19) , s33 = s2 + s24, s34 = s5 + s28, s35 = 2s32, s36 = 3 s18 ,s37 = s s33, s38 = s34 √ 1/k+ s23, s39 = ( 1+ s)s21, s40 = ( 1 + √ 1/k ) s22, M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 19 of 22 s41 = s23 √ 1/k, s42 = Pr (1 +Nt+Nb+Ec) , s43 = Pr, s430 = PrEc W, s44 = s42 4−Q Pr , s45 = s43 1−Q Pr , s450 = s430 9−Q Pr , s46 = s44−s45−s450, s47 = sc λ e−E , s48 = s47 (E +m) (w− 1) , s49 = s48 − sc, s50 = sc− Nt Nb , s51 = s49 4− s47 , s52 = s50 1− s47 , 53 = s51 + s52, s54 = Pr (s6 + s45) , s55 = Pr (2s44 + s45 + s3 +Nb s52) , s56 = Pr (−Nb s45 − 2Nt s45 − 2Ec s3) , s57 = s56 − s55, s58 = Pr (2s44 − s1 − 2Nb s51) , s59 = Pr (2 Nbs44 + 4Nts44 − 8Ec s1) , s590 = 3Pr (s450 +Ec We s3) , s60 = s58 + s59 + s590 , s600 = 3Pr (s450 +Nb s450 + 2 Nt s450 − 4Ec We s1) , s62 = Pr s46 √ Q Pr , s64 = Pr ( s2 + 2Ec s2s2 ) , s65 = Pr √ Q Pr (s46 +Nb s46 + 2Nt s46 ) , s66 = Pr s5 ( 1 + 2Ec k ) , s67 = Pr Nb s53 √ s47, s670 = 3We Ec s5 Pr k , s671 = 3We Pr Ec s2 s2, M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 20 of 22 s68 = s54 1−Q Pr , s69 = s57 4−Q Pr , s70 = s60 9−Q Pr , s700 = s600 16−Q Pr , s72 = s62 2 √ Q Pr , s74 = s64 (1 + s)2 −Q Pr , s75 = s65 1 + 2 √ Q Pr , s76 = s66( 1 + √ 1/k )2 −Q Pr , s77 = s67( 1 + √ s47 )2 −Q Pr , s770 = s670( 2 + √ 1/k )2 −Q Pr , s771 = s671 (2 + s)2 −Q Pr , s78 = s68 + s69 + s70, s79 = s771 + s700 + s770 + s75 , s80 = s74 + s76 + s77, s81 = s78 − s79 + s80, s82 = 1+ s45 + s68, s83 = s46 − s81, s84 = s69 − s44, s840 = s70 + s450, s85 = sc (s52 + s6)− Nt Nb s45 s86 = sc (2s51 − s52 − s3) , s87 = ( 4 Nt)/Nb s44 + s48 s52 + s48 s45, s88 = s86 + s87, s89 = sc (s1 + 2s51)+ 9 Nt Nb s450, s90 = s48 (s51 − s44) , s91 = s90 − s89, s910 = s48 s450, s92 = sc s53 √ s47, s93 = sc s5, s94 = sc s2, s95 = Nt Nb s46 Q Pr, s96 = s53 (sc √ s47 − s48) , M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 21 of 22 s97 = s48 s46, s98 = s85 1− s47 , s99 = s8 4− s47 , s100 = s91 9− s47 , s1000 = s910 16− s47 , s101 = s91 2 √ s47 , s102 = s93( 1 + √ 1/k )2 − s47 , s103 = s94 (1 + s)2 − s47 , s104 = s95 Q Pr − s47 , s105 = s96 1 + 2 √ s47 , s106 = s97( 1 + √ Q Pr )2 − s47 , s107 = s98 + s99 + s100 + s1000, s108 = s102 + s103, s109 = s105 + s106, s110 = s107 + s108 + s109 − s104 , s111 = 1+ s98 + s52, s112 = s51 + s99 , s113 = s110 + s53, s114 = 2 s35, s115 = 3s36 , s116 = s s37, s117 = s38 √ 1/k+ s41, s118 = s39 (1 + s) , s119 = s40 ( 1 + √ 1/k ) , s120 = s41 √ 1/k , s121 = s31 + s114 − s115 + s116, s122 = s117 − s118 − s119, s123 = s121+s122, s124 = (s31) 2+(s114) 2+(s115) 2+(s116) 2, s125 = (s117) 2 + (s118) 2 + (s119) 2, s126 = 2s31 (s114 − s115 + s116) , s127 = 2s31 (s117 − s118 − s119) , s128 = 2s114 (s116 − s115 + s117) , s129 = 2s114 (s118 + s119) , s130 = 2s115 (s116 + s117 − s118 − s119) , M. El-dabe Nabil et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5616 22 of 22 s131 = 2s116 (s117 − s118 − s119) , s132 = 2s117 (s118 + s119) , s133 = 2s118 s119, s134 = s124 + s125 + s126 + s126 , s135 = s128 − s129 − s130, s136 = s131 − s132 + s133, s137 = s134 + s135 + s136, s138 = s123 +We s137, s139 = s82+s83 √ Q Pr +2s84, s140 = 4s700+s770 ( 2 + √ 1/k ) −3s840, s141 = s75 ( 1 + √ Q Pr ) − s74 (1 + s)+ s771 (2 + s) , s142 = s76 ( 1 + √ 1/k ) + s77 (1 + √ s47)+ s72, s143 = s139 − s140 − s141 + s142, s144 = s111 + 2s112 + 3s100, s145 = s102 ( 1 + √ 1/k ) − s113 √ s47, s146 = s103 (1 + s)− s101 − s104 √ Q Pr , s147 = 4s1000 + s106 ( 1 + √ Q Pr ) + s105 (1 + √ s47) , s148 = s144 + s145 + s146 + s147.