Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 175 https://internationalpubls.com Darcy-Forchheimer Flow of Casson Nanofluids Towards a Spinning Disk with Non-Uniform Heat Source Chepiala Pushpalata1, M. Monica2, K. Satyanarayana3, Ch. Kishore Kumar4, B. Shankar5, Mattipelli Ramachandru6* 1Department of Mathematics, TSWRDCW, Siddipet, Telangana, India-502103, ch.pushpalatha14@yahoo.com 2Department of Mathematics, TSWRDCW, Vikarabad, Telangana, India-501504, monica.medikare@gmail.com 3Department of Mathematics, UCS, Osmania University, Telangana, India-500007, satyamaths123@gmail.com 4Department of Mathematics, Nizam College (A), Hyderabad, Telangana, India-500001, kishoresai09@gmail.com 5Department of Mathematics, CVR College, Hyderabad, Telangana, India-501510, bandarishanker@yahoo.co.in 6*Department of Humanities &science, UCET, Mahatma Gandhi University, Telangana, India-508254, ramanmaths7@gmail.com *Corresponding Author: Email: ramanmaths7@gmail.com Article History: Received: 28-07-2024 Revised: 08-09-2024 Accepted: 17-09-2024 Abstract: This study investigates the Darcy-Forchheimer flow of a Casson nanofluid over a spinning disk, incorporating the effects of a non-uniform heat source. Such flows are significant in various industrial and engineering applications, including cooling systems, lubrication technologies, and chemical processing involving complex fluids. The governing boundary layer equations, initially in partial differential form, were transformed into a set of ordinary differential equations (ODEs) using similarity transformations. These ODEs were then solved numerically using the bvp4c MATLAB solver. The influence of various physical parameters on the velocity, temperature, and concentration profiles was thoroughly analyzed and visualized through graphs. Additionally, the Nusselt and Sherwood numbers were computed and presented in tabular form to quantify heat and mass transfer rates. Keywords: Darcy Forchhiemer, Casson fluid, nanoparticles, spinning disk, non- uniform Heat source. 1. Introduction Non-Newtonian fluids have special flow characteristics that make them useful in many industrial applications, such as coating sheets, polymers, and optical fibers. Additionally, because of their capacity to withstand flow under stress, these fluids are essential to brake and damper systems. A particular kind of non-Newtonian fluid called Casson fluid has drawn a lot of interest recently because of its unique behavior, which makes it useful in situations where elastic and viscous qualities are equally relevant. Copley [1] and Blair [2] illustrated the fundamental shear characteristics of blood in arteries by employing the Casson fluid model. This fluid model stated to fit rheological data. According to the study of Nadeem et al. [3] and Kandasamy and Pai [4] Casson fluid exhibits the yield stress. The analytical study of the non-co-axial effects of transportation of mass subjected to first-order chemical reaction was examined by Jabbar et al. [5] Recently the numerical simulation of non-coaxial rotation of a Casson fluid towards a circular disc was examined by Alqarni et al. [6]. mailto:ch.pushpalatha14@yahoo.com mailto:monica.medikare@gmail.com mailto:kishoresai09@gmail.com mailto:ramanmaths7@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 176 https://internationalpubls.com The study of the flow over rotating disk involves numerous industrial applications like Turbine disks, rotary type machine systems and application of rotating disk boundary-layer flow finds direct use in Chemical Vapour Deposition (CVD) reactors. The immense applications of this area attract many researchers. Von Karman [7] was the first one to give a prominent result on boundary layer flow over a rotating disk. He gave a differential equation model for the flow problem by using the appropriate integral procedure. Inspired by his study, Cochran [8] has found a series solutions for the same problem. Later, Benton [9] has improved Cochran's problem by considering the time-dependent case. Freidoonimehra et al. [10] discussed the behavior of MHD flow over a porous rotating disk using a semi-numerical/analytical method called HAM. Whereas a numerical solution for the Buongiorno model for the flow over a rotating disk along with velocity, thermal, and solutal slips was found by Mustafa [11]. Shehzad et al. [12] numerically explored the flow of a rotating disk in both upward and downward motion. Recently, the irreversibility analysis of flow of a hybrid nanofluid through a rotating disk by considering thermal radiation and magnetic field was done by Kumar & Mondal [13]. Different from the regular, Ali et al. [14] studied the unsteady hydro magnetic flow towards an inclined rotating disc by using the neural network approach. At high velocities, fluid flow through porous medium becomes nonlinear, a phenomenon known as the Forchheimer effect. It incorporates inertial effects and goes beyond Darcy's law. By predicting fluid flow in densely packed, low-porosity reservoirs, it aids in oil recovery and maximises extraction. It helps to improve the quality of the final product in ceramic processing by helping to understand flow resistance in complicated, low-porosity materials. In non-Newtonian situations, when normal flow models are inadequate, it is useful overall. It occurs when there is a tightly packed medium with a lower porosity. A study conducted by Shenoy [15] examined the mixed, natural, and forced convection phenomena that can occur in non-isothermal structures. They were immersed in a type of porous medium that was saturated with a power-law fluid. Later, the two-dimensional Darcy- Forchheimer flow of Maxwell fluid towards a convectively heated sheet was studied by Sadiq & Hayat [16]. A study conducted by Vishnu Ganesh et al. [17] analyzed the effects of ohmic dissipations, second-order slip, and viscosity on the hydro magnetic nanofluid's flow in a porous medium. They found that the flow was directed toward a shrinking or stretching surface. Sadiq et al. [18] performed a similar study by using the Darcy-Forchheimer model on a convectively heated sheet. In a study by Khan et al. [19], they determined the optimal flow rate for the Carreau-Yasuda fluid on a flat surface with first-order velocity slip and Darcy-Forchheimer flow. A study conducted by Rasool et al. [20] revealed the influence of copper and alumina on the flow behavior of an electromechanical nanofluid made up of motor oil through a porous media known as Darcy- Forchheimer. After the thorough examination, practically the high porosity was involved in the case of non- Newtonian fluids, which are hard to find out, and therefore, the Brinkman effects are not important while dealing with non-Newtonian fluids. Hence, in the present study we have considered the Darcy- Forchheimer model for the Casson fluid over a rotating disk to get better results. 2. Mathematical Formulation Consider a 3D steady magnetohydrodynamic (MHD) Fluid dynamics is involved in the study of the motion of Casson Nanofluid, which is a semiconducting suspension of metallic particles at the Nano Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 177 https://internationalpubls.com scale. Particular issues pertaining to the slip dynamics have been identified. The angular velocity Ξ© and constants of the disk determine its rotation at z = 0. The electric field's influence can also be observed due to the Hall current. Rheological of casson fluid is as follows Nayak et al., [21] and Waqas et al. [22]. πœπ‘–π‘— = { 2 (πœ‡π΅ + 𝑃𝑦 √2πœ‹ ) 𝑒𝑖𝑗 ; πœ‹ > πœ‹π‘ 2 (πœ‡π΅ + 𝑃𝑦 √2πœ‹π‘ ) 𝑒𝑖𝑗; πœ‹ < πœ‹π‘ (1) Here 𝑒𝑖𝑗 denotes the rate component of (𝑖, 𝑗)π‘‘β„Ž, πœ‹ denotes the product component of the deformation rate itself, πœ‹π‘ is based upon non- Newtonian relation, 𝑃𝑦 denotes fluid yield- stress and πœ‡π΅denotes non-Newtonian relation fluid plastic dynamic-Viscosity. If πœ‹ < πœ‹π‘ expression (1) can be modified into πœπ‘–π‘— = πœ‡π΅((1 + 1 𝛾 ) 2𝑒𝑖𝑗. The upper half of plate is filed with nano fluid. The surface temperature 𝑇𝑀 is higher than the ambient fluid temperature π‘‡βˆž. The volumetric concentration 𝐢𝑀 for ambient fluid is𝐢∞ .The radiation is also embedded in the energy equation profile in the existence of Thermophoretic and Brownian diffusion effects. Boundary conditions were considered. The heat flow activation energy from C-C in a porous material was also studied. The governing equations are Waqas et al. [22] and Lv et al. [23] πœ•π‘’ πœ•π‘Ÿ + 𝑒 π‘Ÿ + πœ•π‘€ πœ•π‘§ = 0 (2) πœŒπ‘“ (𝑒 πœ•π‘’ πœ•π‘Ÿ βˆ’ 𝑣2 π‘Ÿ + 𝑀 πœ•π‘’ πœ•π‘§ ) = βˆ’ πœ•π‘ƒ πœ•π‘Ÿ + πœ‡π‘“ (1 + 1 𝛽 ) ( πœ•2𝑒 πœ•π‘Ÿ2 + 1 π‘Ÿ πœ•π‘’ πœ•π‘Ÿ βˆ’ 𝑒 π‘Ÿ2 + πœ•2𝑒 πœ•π‘§2) βˆ’ πœŽπ‘“π΅0 2𝑒 βˆ’ 𝜈 π‘˜βˆ— 𝑒 βˆ’ πΉπ‘Ÿπ‘’2 (3) πœŒπ‘“ (𝑒 πœ•π‘£ πœ•π‘Ÿ + 𝑒𝑣 π‘Ÿ + 𝑀 πœ•π‘£ πœ•π‘§ ) = πœ‡π‘“ (1 + 1 𝛽 ) ( πœ•2𝑣 πœ•π‘Ÿ2 + 1 π‘Ÿ πœ•π‘£ πœ•π‘Ÿ βˆ’ 𝜈 π‘Ÿ2 + πœ•2𝑣 πœ•π‘§2) βˆ’ πœŽπ‘“π΅0 2𝑣 βˆ’ 𝜈 π‘˜βˆ— 𝑣 βˆ’ πΉπ‘Ÿπ‘£2 (4) πœŒπ‘“ (𝑒 πœ•π‘€ πœ•π‘Ÿ + 𝑀 πœ•π‘€ πœ•π‘§ ) = βˆ’ πœ•π‘ƒ πœ•π‘§ + πœ‡π‘“ (1 + 1 𝛽 ) ( πœ•2𝑀 πœ•π‘Ÿ2 + 1 π‘Ÿ πœ•π‘€ πœ•π‘Ÿ + πœ•2𝑀 πœ•π‘§2 ) πœŽπ‘“π΅0 2𝑀 βˆ’ 𝜈 π‘˜βˆ— 𝑀 βˆ’ πΉπ‘Ÿπ‘€2 (5) 𝑒 πœ•π‘‡ πœ•π‘Ÿ + 𝑀 πœ•π‘‡ πœ•π‘§ = π›Όπ‘š ( πœ•2𝑇 πœ•π‘Ÿ2 + πœ•2𝑇 πœ•π‘§2 + 1 π‘Ÿ πœ•π‘‡ πœ•π‘Ÿ ) + (πœŒπ‘)𝑝 (πœŒπ‘)𝑓 {𝐷𝐡 ( πœ•π‘‡ πœ•π‘Ÿ πœ•πΆ πœ•π‘Ÿ + πœ•π‘‡ πœ•π‘§ πœ•πΆ πœ•π‘§ ) + 𝐷𝑇 π‘‡βˆž (( πœ•π‘‡ πœ•π‘Ÿ ) 2 + ( πœ•π‘‡ πœ•π‘§ ) 2 )} βˆ’ πœ†2 (𝑒2 πœ•2𝑇 πœ•π‘Ÿ2 + 𝑀2 πœ•2𝑇 πœ•π‘§2 + 2𝑒𝑀 πœ•2𝑇 πœ•π‘Ÿπœ•π‘§ + (𝑒 πœ•π‘’ πœ•π‘Ÿ + 𝑀 πœ•π‘’ πœ•π‘§ ) πœ•π‘‡ πœ•π‘Ÿ + (𝑒 πœ•π‘€ πœ•π‘Ÿ + 𝑀 πœ•π‘€ πœ•π‘§ ) πœ•π‘‡ πœ•π‘§ ) + π‘žβˆ— (6) Where π‘žβˆ— = π‘˜βˆžπ‘ˆπ‘€ π‘§πœˆ(πœŒπ‘)𝑓 {π΄βˆ—(𝑇𝑀 βˆ’ π‘‡βˆž)𝑓 + π΅βˆ—(𝑇 βˆ’ π‘‡βˆž)} (7) 𝑒 πœ•πΆ πœ•π‘Ÿ + 𝑀 πœ•πΆ πœ•π‘§ = 𝐷𝐡 ( πœ•2𝐢 πœ•π‘Ÿ2 + πœ•2𝐢 πœ•π‘§2 + 1 π‘Ÿ πœ•πΆ πœ•π‘Ÿ ) + 𝐷𝑇 π‘‡βˆž ( πœ•2𝑇 πœ•π‘Ÿ2 + πœ•2𝑇 πœ•π‘§2 + 1 π‘Ÿ πœ•π‘‡ πœ•π‘Ÿ ) (8) Where the kinetic energy is equal to πœ‡π‘“ πœŒπ‘“ . Here πœ‡π‘“ stands for the dynamic viscosity and the base liquid density πœŒπ‘“, π›Όπ‘š = π‘˜1 (πœŒπ‘)𝑓 stands for the thermal diffusivity, here thermal conductivity represented by π‘˜1, the liquid heat capacity and nanoparticle heat capacity are represented by (πœŒπ‘)𝑓 and (πœŒπ‘)𝑝 respectively, Casson parameter 𝛽, the temperature of the fluid is represented by T, the concentration of nano-size metallic particles is by C, πœ†2 is Thermal relaxation factor, π‘žβˆ— is the non Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 178 https://internationalpubls.com uniform heat source/sink, π΄βˆ—, π΅βˆ— are source and sink coefficients respectively.𝐷𝑇 , 𝐷𝐡 are Brownian diffusion co-efficient, and Thermophoresis diffusion co-efficient respectively. The boundary conditions are expressed as the energy levels of the fluid. At z=0 : 𝑒 = 𝐿1 πœ•π‘’ πœ•π‘§ v = L1 βˆ‚v βˆ‚z + π‘ŸΞ© , w = 0, T = Tw + L2 βˆ‚T βˆ‚z , 𝐷𝐡 πœ•πΆ πœ•π‘§ + 𝐷𝑇 π‘‡βˆž πœ•π‘‡ πœ•π‘§ = 0 As zβ†’ ∞: 𝑒 β†’ 0, 𝑇 β†’ π‘‡βˆž,𝐢 β†’ 𝐢∞ , 𝑃 β†’ π‘ƒβˆž (9) Here, 𝐿1, 𝐿2represent the velocity slip, thermal slip respectively, and we introduce the similarity transformations 𝑒(πœ‚) = π‘ŸΞ©π‘“β€²(πœ‚) 𝑣(πœ‚) = π‘ŸΞ©π‘”(πœ‚) 𝑀(πœ‚) = βˆ’2√Ων 𝑓(πœ‚) πœ‚ = √ Ξ© 𝜈 𝑧 πœƒ(πœ‚) = π‘‡βˆ’π‘‡βˆž π‘‡π‘€βˆ’π‘‡βˆž πœ‘(πœ‚) = πΆβˆ’πΆβˆž πΆπ‘€βˆ’πΆβˆž 𝑃(πœ‚) = π‘ƒβˆžβˆ’π‘ƒ Ωμf (10) Where πœ‚ is the similarity variable𝑓(πœ‚), 𝑔(πœ‚) are represented as non-dimensional velocities and, πœƒ(πœ‚), πœ™(πœ‚) are the dimensional temperature function, and dimensionless concentration function respectively. Equation (2) is already satisfied by equation (10), now eqns (3), (4), (5), (6), (7) and (8) becomes (1 + 1 𝛽 ) 𝑓′′′ βˆ’ 𝑓′2 + 2𝑓𝑓′′ + 𝑔2 βˆ’ 𝑀𝑓′ βˆ’ 𝐾𝑓′ βˆ’ πΉπ‘Ÿπ‘“β€²2 = 0 (11) (1 + 1 𝛽 ) 𝑔′′ βˆ’ 𝑓′𝑔 + 𝑓𝑔′ βˆ’ 𝑀𝑔 βˆ’ 𝐾𝑔 βˆ’ πΉπ‘Ÿπ‘”2 = 0 (12) πœƒβ€²β€² + 2π‘ƒπ‘Ÿπ‘“πœƒβ€² + π‘ƒπ‘Ÿ(π‘π‘πœƒβ€²πœ™β€² + π‘π‘‘πœƒβ€²2 + 𝛾1(𝑓2πœƒβ€²β€² + π‘“π‘“β€²πœƒβ€²) + 𝐴𝑓 + π΅πœƒ = 0 (13) πœ™β€²β€² + 2π‘ƒπ‘ŸπΏπ‘’π‘“πœ™β€² + 𝑁𝑑 𝑁𝑏 πœƒβ€²β€² = 0 (14) Similarly, boundary conditions are transferred into πœ‚ = 0 β‡’ 𝑓 = 0, 𝑓′ = 𝛾𝑓′′, 𝑔 = 1 + 𝛾𝑔′, πœƒ = 1 + π›Όπœƒβ€², πœ™β€² + 𝑁𝑑 𝑁𝑏 πœƒβ€² = 0 πœ‚ β†’ ∞ β‡’ 𝑓′ β†’ 0, 𝑔 β†’ 0, πœƒ β†’ 0, πœ™ β†’ 0 (15) Here 𝑀 = πœŽπ‘“π΅0 2 πœŒπ‘“Ξ© for the magnetic parameter, 𝐾 = 𝜈 π‘˜βˆ—πœŒπ‘“Ξ© for the porosity parameter, πΉπ‘Ÿ = πΉπ‘Ÿ πœŒπ‘“ for the Forchheimer parameter, π‘ƒπ‘Ÿ = 𝜈 π›Όπ‘š prandlt number, 𝑁𝑏 = ( (πœŒπ‘)𝑝 (πœŒπ‘)𝑓 ) ( 𝐷𝐡 𝜈 ) (𝐢𝑀 βˆ’ 𝐢∞) for Brownian parameter 𝑁𝑑 = ( 𝐷𝑇 𝜈 ) (𝑇𝑀 βˆ’ π‘‡βˆž) for Thermophoresis parameter, 𝛾1 = 4πœ†2Ξ© for the Thermal relaxation time, 𝐴 = π΄βˆ— ( π‘˜βˆžπ‘ˆπ‘€ π‘§πœˆ(πœŒπ‘)𝑓 ) (𝑇𝑀 βˆ’ π‘‡βˆž) , 𝐡 = π΅βˆ— ( π‘˜βˆžπ‘ˆπ‘€ π‘§πœˆ(πœŒπ‘)𝑓 ) are represented as space, temperature-dependent heat generation and absorption parameters respectively, 𝐿𝑒 = π›Όπ‘š 𝐷𝐡 for lewis number, 𝛾 = 𝐿1√ 2Ξ© 𝜈 for velocity slip parameter, 𝛼 = 𝐿2√ 2Ξ© 𝜈 for thermal slip parameter. Mass transfer, heat transfer, and non-dimensional skin friction rates can be expressed. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 179 https://internationalpubls.com π‘…π‘’π‘Ÿ 1 2𝐢𝑓 = 𝑓′(0), 𝑅𝑒𝑑 1 2𝐢𝑔 = 𝑔′(0), π‘…π‘’π‘Ÿ βˆ’1 2 𝑁𝑒 = βˆ’πœƒβ€²(0), π‘…π‘’π‘Ÿ βˆ’1 2 π‘†β„Ž = πœ™β€²(0) (16) Where the Reynolds number is calculated by taking into account the π‘…π‘’π‘Ÿ = 2(π‘Ÿ Ξ©) 𝜈 . 3. Numerical Procedure A convenient shooting method is used to treat the boundary conditions [15] of the fluid. For the initial value problems [11] to [14], we can substitute the numbers 𝑦1 = 𝑓 , 𝑦2 = 𝑓′, 𝑦3 = 𝑓′′, 𝑦4 = 𝑔, 𝑦5 = 𝑔′, 𝑦6 = πœƒ, 𝑦7 = πœƒβ€², 𝑦8 = πœ™, 𝑦9 = πœ™β€². we then obtained the following initial value problems into first-order differential equations. 𝑦1 β€² = 𝑦2; 𝑦2 β€² = 𝑦3; 𝑦3 β€² = 1 (1+ 1 𝛽 ) (𝑦2 2 βˆ’ 2𝑦1𝑦3 βˆ’ 𝑦4 2 + 𝑀𝑦2 + 𝐾𝑦2 + πΉπ‘Ÿπ‘¦2 2); 𝑦4 β€² = 𝑦5 ; 𝑦5 β€² = 1 (1+ 1 𝛽 ) (𝑦2𝑦4 βˆ’ 𝑦1𝑦5 + 𝑀𝑦4 + 𝐾𝑦4 + πΉπ‘Ÿπ‘¦4 2); 𝑦6 β€² = 𝑦7; 𝑦7 β€² = βˆ’ π‘ƒπ‘Ÿ 1+π‘ƒπ‘Ÿπ›Ύ1𝑦1 2 (2𝑦1𝑦7 + 𝑁𝑏𝑦7𝑦9 + 𝑁𝑑𝑦7 2 + 𝛾1𝑦1𝑦2𝑦7 + 𝐴𝑦1 + 𝐡𝑦1); 𝑦8 β€² = 𝑦9; 𝑦9 β€² = βˆ’2π‘ƒπ‘ŸπΏπ‘’π‘¦1𝑦9 βˆ’ 𝑁𝑑 𝑁𝑏 πœƒβ€²β€² Boundary conditions can be transformed as follows 𝑦1(0) = 0, 𝑦2(0) = 𝛾𝑠1 , 𝑦3(0) = 𝑠1, 𝑦4(0) = 1 + 𝛾𝑠2, 𝑦5(0) = 𝑠2, 𝑦6(0) = 1 + 𝛼𝑠3, 𝑦7(0) = 𝑠3, 𝑦8(0) = 𝑠4, 𝑦9(0) = βˆ’ ( 𝑁𝑑 𝑁𝑏 ) 𝑦7(0), 𝑦2(∞) β†’ 0, 𝑦4(∞) β†’ 0, 𝑦6(∞) β†’ 0, 𝑦8(∞) β†’ 0 The fifth-order RK-technique has been utilized to integrate the given equations.𝑠1, 𝑠2, 𝑠3 π‘Žπ‘›π‘‘ 𝑠4 are the initial values of the slope while integrating the above system. The slopes 𝑠1, 𝑠2, 𝑠3 π‘Žπ‘›π‘‘ 𝑠4 are iteratively by using Newton method. Numerical results are evaluated at πœ‚π‘šπ‘Žπ‘₯ = 20 For the scope of slip parameters, this method fulfills the far-field conditions. We were able to obtain numerical results from bvp4c of MATLAB. 4. Numerical outcomes and conversation This paper visualizes the scope of slip parameters' features can be represented in equations (11) - (14) namely, porosity parameter, Forchhiemer flow parameter, magnetic parameter, Casson fluid parameter, Prandtl number, Brownian motion parameter, Thermophoresis parameter, Lewis number against radial velocity profile, tangential velocity, temperature distribution profile and volumetric concentration profile. Controlling of flow parameters has a few study ranges such as 0 < 𝑀 < 1, 0 < πΉπ‘Ÿ < 1, 0 < 𝐾 < 1 etc. Here combined porosity and magnetic parameters. The study of the various aspects of the magnetic properties of a medium that’s used in the flow of nanofluids is carried out. Fig.1 The exact nature of M can also be determined by comparing its properties with that of the radial velocity field𝑓′(πœ‚). The velocity distribution of a revolving disk can be affected by the upper limit of the magnetic parameter influence. As its resistivity and Lorentz force increase, the effects of this variable's rise become more apparent. Fig. 2 illustrates the effect of the magnetic parameter M on the distribution of the tangential velocity𝑔(πœ‚). It shows that increasing its value causes the velocity distribution to decrease. The performance of the Casson parameter 𝛽 is evaluated Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 180 https://internationalpubls.com in terms of its relation to the radial velocity 𝑓′(πœ‚) in Fig.3. Magnitude variations in the parameter's value can cause a drop in the velocity profile of 𝑓′(πœ‚). The plot presented in Fig. 4 shows the Casson parameter 𝛽 value relative to the distribution of the tangential velocity 𝑔(πœ‚). The value decrease shown in 𝑔(πœ‚) is an interesting aspect of the plot. The Darcy-Forchheimer parameter πΉπ‘Ÿ influence on the Radial velocity 𝑓′(πœ‚) speed profile is shown in Fig. 5. The reduction in the profile is the result of higher estimates for this parameter. The Fig.6 shows πΉπ‘Ÿ variable's identifiable speed circulation range in the tangential velocity 𝑔(πœ‚). The distribution and other layers speed profile also decreased due to the expanding value. Fig.7 shows the various K permeability factor estimates made using a rotating disc’s distribution of Nanofluid. Since K increases 𝑓′(πœ‚) decreases for the nano fluid we understood here that the higher values of permeability constraint indicate lower permeability of the porous medium. The consequence of raising the porosity parameter K is shown in Fig. 8, whereby the nanofluid’s flow g(Ξ·) decreases. Fig.9 The thermal relaxation coefficient 𝛾1 influences the temperature profile πœƒ(πœ‚) of a fluid. High values of this parameter can cause the fluid’s temperature to decrease. Its insulating properties can help in reducing the temperature. The distribution of temperatures is also studied by analyzing how A and B, as well as the sink and heat source, affect this phenomenon. In Fig.10 &11The increasing values of A and B can cause the temperature profileπœƒ(πœ‚) to rise. The heat source parameter can then cause the nanofluid to generate more heat. The Brownian factor known as Nb influences the temperature profile πœƒ(πœ‚) by determining its distribution. Another well- known thermal and spatial parameter, the Prandtl number Pr, shows the transfer rate of heat from a solid to a flowing liquid. Temperature profiles begin to decrease as the Prandtl numbers are raised in Fig.12. The distribution of concentration profile πœ™(πœ‚) across Fig.13 shows a decreasing trend as the Prandtl number Pr increases. The relationship between conductivity and thermal diffusivity can be represented by the Prandtl number. Because of this, the increased Prandtl values can cause thermal diffusivity to decrease. Fig.14 depicts the random motion of microscopic particles increases as the Brownian moment factor rises. The heightened random Brownian motion Nb leads to increased collisions among nano particles, causing the conversion of kinetic energy into heat energy. Consequently; an increase the Brownian motion factor Nb results in an elevated temperature distribution with in fluid. Fig.15 The concentration profile πœ™(πœ‚) can be described as a distinct characteristic by increasing Nb values. The relationship between the concentration profile πœ™(πœ‚) and temperature distribution πœƒ(πœ‚) can be shown in Fig.16. It's revealed that the thermophoresis factor Nt causes an increase in this parameter's profile. A plot against Fig.17 shows the Nt features and how they affect the distribution. In Fig.18 the concentration distribution πœ™(πœ‚) and the Lewis parameter Le can be moved together to show the relationship between their values. It's important to note that the rising Le value causes a decrease in the distribution. This illustrates the link between the various features of Pr and the concentration field. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 181 https://internationalpubls.com Fig.1 Impact of M on radial velocity𝑓′(πœ‚). Fig. 2 Impact of M on tangential velocity 𝑔(πœ‚). Fig.3 Impact of Casson parameter 𝛽 on 𝑓′(πœ‚). Fig.4 Impact of Casson parameter 𝛽 on 𝑔(πœ‚) Fig.5 Various of Forchhiemer parameter πΉπ‘Ÿ on 𝑓′(πœ‚). Fig.6 Various of Forchhiemer parameter πΉπ‘Ÿ on 𝑔(πœ‚) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 182 https://internationalpubls.com Fig.7 Impact of Porosity parameter K on radial velocity 𝑓′(πœ‚). Fig.8 Impact of Porosity parameter K on 𝑔(πœ‚) Fig.9 Influence of Thermal relaxation parameter 𝛾1 on πœƒ(πœ‚). Fig.10 Influence of heat source parameter A on πœƒ(πœ‚) Fig.11 Influence of heat sink parameter B on πœƒ(πœ‚). Fig.12 Influence of Prandtl number Pr on πœƒ(πœ‚) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 183 https://internationalpubls.com Fig.13 Influence of Prandtl number Pr on πœ™(πœ‚). Fig.14 Influence of Brownian factor Nb on πœƒ(πœ‚) Fig. 15 Influence of Brownian factor Nb on Ο• (πœ‚). Fig.16 Influence of Thermophoresis parameter Nt on ΞΈ(πœ‚) Fig.17 Influence of Thermophoresis parameter Nt on Ο•(πœ‚). Fig.18 Influence Lewis number Le on Ο•(πœ‚). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 184 https://internationalpubls.com Table 1. Computational values of radial velocity 𝑓’(πœ‚), tangential velocity 𝑔(πœ‚) computed against various estimates M, K, Fr, 𝛼, 𝛽, 𝛾 M K Fr 𝛽 𝛼 𝛾 𝑓′(πœ‚) 𝑔(πœ‚) 0.2 0.2 0.2 - 0.2 0.2 0.49356 0.88414 0.4 0.44601 0.98320 0.6 0.40793 1.07683 0.8 0.37703 1.16536 0.4 0.44601 0.98320 0.6 0.40793 1.07683 0.8 0.37703 1.16536 0.4 0.47187 0.96897 0.6 0.45304 1.04822 0.8 0.40295 0.72191 2 0.50974 0.91313 4 0.53311 0.95498 7 0.49356 0.88414 0.3 0.48231 0.92732 0.5 0.46214 1.00923 0.7 0.44452 1.08605 0.2 0.44601 0.98320 0.4 0.40793 1.07683 0.6 0.37703 1.16536 Table 2. Numerical values of βˆ’πœƒβ€²(0) computed against various estimates Pr, Nb, Nt, Le, A, B, 𝛾1, 𝛼, 𝛽 Pr Nb Nt A B 𝛾1 𝛼 𝛽 Le βˆ’πœƒβ€²(0) 0.3 0.3 0.1 0.1 0.1 0.1 0.2 0.2 0.1 0.08928 0.5 0.08587 0.9 0.07984 0.1 0.06065 0.2 0.07032 0.4 0.09500 0.2 0.01321 0.3 0.02861 0.4 0.06584 0.15 0.02451 0.2 0.03524 0.25 0.09572 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 185 https://internationalpubls.com 0.15 0.01347 0.2 0.14270 0.25 0.33113 3 0.08814 5 0.09439 7 0.10154 1 0.08410 4 0.09109 7 0.10154 0.2 0.02705 0.3 0.14069 0.4 0.1 0.25694 5. Conclusions The researchers studied the swirling disk with a Casson nanofluid in the presence of a Cattaneo- Christov thermal flux. They found that the disk's flow and the associated boundary conditions have convective effects. The findings of the study support the use of bvp4c for solving the boundary value problem numerically. (i) The temperature profile decreases with higher estimates of the thermal relaxation parameter. 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