Forced convective tangent hyperbolic nanofluid flow subject to heat source/sink and Lorentz force over a permeable wedge: Numerical exploration

Research Article Aisha M. Alqahtani, Muhammad Bilal*, Muhammad Bilal Riaz, Wathek Chammam*, Jana Shafi, Mati ur Rahman, and Adnan Forced convective tangent hyperbolic nanofluid flow subject to heat source/sink and Lorentz force over a permeable wedge: Numerical exploration https://doi.org/10.1515/ntrev-2024-0014 received September 11, 2023; accepted March 25, 2024 Abstract: The magnetohydrodynamics tangent hyperbolic nanofluid (THNF) flow with the mutual impact of melting heat transfer and wedge angle over a permeable wedge is investigated numerically in the present study. Electronic devices generate excessive heat during operations, so THNF is often employed to regulate them. THNF has the ability to neutralize heat with greater efficacy, thereby reducing the probability of overheating. The influence of thermal radiation, Soret and Dufour, and heat source/sink is also observed on the fluid flow. The modeled equations are simplified to the lowest order through the similarity conversion. The obtained set of dimensionless equations is further calculated numerically by employing the parametric continuation method. The computational findings of the present study are compared to the published results for accuracy purposes. It has been detected that the results are precise and reputable. Moreover, from the graphical results, it has been perceived that the effectofpermeability factor (K p ) reduces the fluid flow. The rising effect of wedge angle factor enhances the energy dissemination rate and shearing stress; however the augmentation of Weissenberg number drops skin friction and energy transference rate. Keywords: exponential heat source/sink, thermal radiation, tangent hyperbolic nanofluid, numerical approach, Lorentz force, permeable wedge Nomenclature A second order tensor B 0 constant magnetic field ( )B xvariable magnetic field C concentration C f skin friction C s heat capacity of solid surface Cwwall concentration ∞ C ambient concentration D B Brownian diffusion DT thermophoresis diffusion D 4 Dufour number ′f non-dimensional velocity ′ K constant permeability K p medium permeability kT fluid thermal diffusion ratio () K xvariable permeability k thermal conductivity κ ⁎ absorption constant L e Lewis number M magnetic field mwedge angle parameter N u Nusselt number P rPrandtl number Q e heat source/sink factor Aisha M. Alqahtani: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh, 11671, Saudi Arabia  * Corresponding author: Muhammad Bilal, Sheikh Taimur Academic Block-II, Department of Mathematics, University of Peshawar, 25120, Khyber Pakhtunkhwa, Pakistan, e-mail: [email protected] Muhammad Bilal Riaz: IT4Innovations, VSB–Technical University of Ostrava, Ostrava, Czech Republic; Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon * Corresponding author: Wathek Chammam, Department of Mathematics, College of Science, Majmaah University, Al-Majmaah, 11952, Saudi Arabia, e-mail: [email protected] Jana Shafi:Department of Computer Engineering and Information, College of Engineering in Wadi Alddawasir, Prince Sattam Bin Abdulaziz University, Wadi Ad-Dawasir, 11991, Saudi Arabia Mati ur Rahman: Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon; School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, China Adnan: Department of Mathematics, Mohi-ud-Din Islamic University, Nerian Sharif, AJ and K, 12080, Pakistan Nanotechnology Reviews 2024; 13: 20240014 Open Access. © 2024 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. Qe⁎exponential heat source/sink R d thermal radiation Re x Reynolds number S hxSherwood number S rSoret number T fluid temperature T 0 surface temperature T m surface temperature due to melting ∞ T ambient temperature ( ) Ux velocity of boundary layer Wi Weissenberg number Ωwedge angle λ nanofluid latent heat υ fluid viscosity ∗ σ Stefan–Boltzmann constant η similarity variable β 1 Hartree pressure gradient n power law index α thermal diffusivity Λ heat capacity σ electrical conductivity θ dimensionless temperature ( ) ψxy,stream function ϕ dimensionless concentration 1 Introduction In the context of chemical engineering principles, there is an intriguing concept called the tangent hyperbolic (TH) fluid model which falls under the category of non-Newtonian (NN) theories. This approach offers practical benefits being both straightforward to use and robust in its outcomes. It is grounded in a realistic understanding of fluid behavior rather than mere speculation. Numerous scientificpublicationshave explored this fluid concept by illuminating its importance [1–3]. Although a universal explanation for the convolutions of non-Newtonian fluids cannot be applieduniformly,theTH concept introduced by Pop and Ingham [4] holds notable importance. Reddy et al. [5] evaluated the fluid flow of magnetohydrodynamic Carreau nanofluid across an extended sheet by incorporating nonlinear thermal radiation, threedimensional effects, and convection conditions. Al-Khaled et al. [6] explored the consequence of TH nanofluid (THNF) motion along a progressively moving surface driven by periodic motion. Zaib et al. [7] reported the consequences of magnetization and thermal radiation on entropy production of a TH magnetite iron oxide nanofluid along a vertical porous panel. Their examination factored in viscous dissipation and Joule heating. Prakash et al. [8] scrutinized the bioconvection flow of a THNF exhibiting non-Newtonian characteristics above a two-way elongating surface. This study involved steady and incompressible states and integrated mutually orthogonal electrical and magnetic domains. Abdal et al. [9] investigated heat and mass transmission throughout a temporally fluctuating TH nanoliquid flow over a flexible Riga wedge. Stagnation point, heat sourcing, and activation energy were all taken into consideration. This inquiry also computed flow under altered Hartmann number, especially relating to an unstable TH liquid current. The central focus aimed to enhance heat conduction within the main control volume operating as the medium for heat and mass transfer. Tharapatla et al. [10] described the transmission of heat and mass in the progression of hydromagnetic THNFs through an inclined permeable plate exhibiting thermal stratification. Sindhu and Gireesha [11] executed an assessment to ascertain the attributes of thermal conduct in the context of stagnation point flow of a TH fluid across a vertical surface using a rheological model. Fatunmbi et al. [12] introduced an examination of a dual stratification mechanism relevant to mixed nonlinear convection fluid motion involving magneto-hyperbolic tangent responsive movement across a 2D elastic apparatus within a soaked porous medium. Choudhary et al. [13] deliberated the configurations of hydromagnetic dual convection stagnation flow within a TH fluid. Naseer et al. [14] explored the sustained boundary layer motion and heat exchange of a TH fluid coursing across a vertically and exponentially elongating cylinder in its axial path. Some studies related to NN are recently delivered in various research works [15–21]. Mass and heat transfer play an essential functionality within several industrial uses for instance fusion, magma solidification, casting processes, defrosting frozen ground, and permafrost melting [22]. The study of fluid flow through wedge-shaped geometries holds significant importance due to its wide-ranging applications, including polymer processing, groundwater pollution, nuclear power plants, geothermal industry, ship design and simulation, packed bed reactors, and more [23]. Dadhich et al. [24] considered the effects of viscous dissipation, nanoparticle’svolumefraction,thermal radiation, wedge angle parameter, and heat absorption/generation on Sisko fluid flow containing dispersed nanomaterials over a wedge plate. Their findings revealed that thermal transfer decreases with rising magnetization and Eckert number while the mass transfer rate increases with higher pressure gradient parameter. Roy and Pop [25], Zainal et al. [26], and Alharbi et al. [27] investigated the flow of nanoliquids over shrinking/stretching wedges with energy and mass transfer considerations. Usman et al. [28] and Jyothi et al. [29] extended the study to incorporate nonlinear thermal effects in the nanoliquid flow over a wedge, revealing that 2Aisha M. Alqahtani et al. increasing the temperature ratio parameter leads to higher temperature profiles and reduced concentration boundary layer thickness. The dissipative effects on Jeffrey fluid flow over a wedge were explored by Dharmaiah et al. [30]. Butt et al. [31] employed an innovative model involving inversely multiquadric radial-based artificial neural networks to study magnetized nanoliquid flow over a wedge. Rana et al. [32] scrutinized the 3D hybrid nanoliquid flow consisting of MWCNTs and MgO over a wedge. Some valuable results are presented in the previous literature [33–35]. In fluid mechanics, a magnetic field refers to a region of space where a force is exerted on a moving electrically charged fluid, typically a conducting fluid such as a liquid or plasma. This force is referred to as the magnetic force. The interconnection between the magnetization and the moving charged particles within the fluid leads to various phenomena, including the generation of electric currents, the induction of electric fields, and the alteration of fluid flow patterns. Magnetic fields are pivotal in the exploration of hydromagnetics, a study focused on examining the behavior of fluid flow in the presence of magnetic fields [36–38]. Gul et al. [39] studied the vertical magnetic effects on a delicate aqueous solution layer incorporating iron trioxide ( ) Fe O 34 and CNT nanoparticles mixed with water (H 2 O) where the magnetic influence was oriented vertically within the flow regime. Liu et al. [40] investigated how magnetism effects of Sobolev boundary layer equations for a two dimensional hydromagnetic system lack resistivity while finding solutions using Prandtl-type equations obtained from a resistivity-excluded magnetohydrodynamics (MHD) system with no-slip velocity boundary, and determined that awell-defined local time solution exists in Sobolev spaces even without requiring velocity monotonicity. Arain et al. [41] studied the movement of fluid inside circular disks that spin. These disks are placed within a specific range and are filled with a mixture called Reiner-Rivlin suspension. The Reiner-Rivlin suspension can conduct electricity and also contains tiny organisms that move in response to disk spinning subjected to activation energy and heat radiation. Hakeem et al. [42] observed the convection-driven motion of ternary hybrid nanoparticles dispersed in distinct fundamental fluids around a vertically oriented downward-pointing spinning cone in the existence of a magnetic effect. Furthermore, the study analyzed the effects of non-Newtonian characteristics using the Casson model. Javid et al. [43] investigated the impact of a magnetic field on the phenomenon of doublediffusive convection occurring in the intricate peristaltic movement of a fluid through an expanding channel with 2D geometrical representation. Furthermore, the study considered the effects of porosity and the rheological properties of the fluid within the analysis. Varun Kumar et al. [44] presented the impact of magnetization on the flow of a Casson nanoliquid over a curved surface and also examined the mass and heat transmission rate with the effects of temperature differences, and heat generation. Sreedevi and Reddy [45] performed a numeric exploration of convective heat transfer utilizing the Tiwari-Das model nanoparticle liquid inside a cubic enclosure considering thermal emission and a magnetic force. Arshad et al. [46] surveyed the motion of a viscous fluid with energy transference over a porous plate in the existence of constant magnetization. Patil and Goudar [47] conducted an investigation into the importance of unsteady hydromagnetic flow through an infinite yawed cylinder by considering the entropy generation. Waqas et al. [48] and Reddy et al. [49] discussed the effects of hydromagnetic heat transfer characteristics of an incompressible fluid which were resolved numerically for a progressively elongating horizontal cylinder submerged in a permeable medium, considering the presence of internal heat generation/sink. Moradi et al. [50] examined the vertical bioconvective flow having gyrotactic microbes. Their findings indicated that a rise in the thermophoresis coefficient induces a reaction exerted on adjacent particles resulting in their movement from hotter to cooler areas. Additionally, the concentration of microorganisms within an area increases as the thermophoresis coefficient increases. Some remarkable results are further presented in previous literature [51–55]. The literature reveals a notable gap in comprehensive research that addresses the flow of thermally responsive TH MHD nanoliquids over a permeable wedge under the influence of magnetic field, heat source, and thermal radiation. Consequently, our study seeks to numerically analyze how the combination of wedge angle and energy transfer during melting influences the TH MHD nanoliquid flow across a permeable wedge. This investigation also takes into account the impacts of Soret and Dufour effects. The governing equations governing the flow of the THNF are formulated as partial differential equations (PDEs) which are subsequently solved using the parametric continuation method (PCM) numerical method. The computed results obtained through PCM are compared with the existing published research to validate the accuracy of this approach. In Section 2, the problem is formulated and its physical description is provided (Figure 1). 2 Mathematical formulation The unsteady 2D forced convective electrically conducting incompressible THNF flow with melting heat transfer and Forced convective THNF subject to heat source/sink and Lorentz force 3 thermal diffusion across a porous wedge is studied. T m signifies the surface temperature due to melting process and Cwindicates the wall concentration. ()=Ux ax m( →a positive constant) is the nanofluid velocity above the boundary layer. To the surface of the wedge, the variable magnetic field () ) ( = − B xBx 0m1 2is applied perpendicularly. The permeability of wedge surface is specified as () () =′ −− K xKx . m1 The wedge angle factor is expressed as = − m ββ2 1 1 ( →β 1Hartree pressure gradient). ∞ T is the temperature of nanofluid away from the surface. Here < ∞ TT m as well as <TT . m0The basic equations of tangential hyperbolic fluid is defined as [( ) ( ) ]=+ + ∞∞ τ μμ ΓΩ μΩtanh   , n 0(1) where τ and Γ is the extra stress tensor and material constant, where Ω  can be defined as =∑∑ =ΩΩΩ A   22 , kmmk km (2) where A is the invariant of the second order strain rate tensor and can be defined as () =+ Atr L L 2 , T2(3) where ()= L gradV , Vis the fluid velocity. Consider that = ∞ μ0 , and <ΓΩ  1 , then τ becomes, [( )] [( )] [()] ==−++ =−+ + τμΩΓΩ μΩ ΓΩ μΩ nΓΩ    11  11. nn 00 0 (4) The THNF flow equations based on the above assumptions are modeled as [56–58] ∂ ∂+∂ ∂= u xv y0 , (5) () () () ()(() ) ⎟ ⎜ ⎜⎟ ⎜⎟ ⎜⎟ ⎛ ⎝∂ ∂⎞ ⎠+⎛ ⎝ ∂ ∂⎞ ⎠=⎛ ⎝∂∂⎞ ⎠ +⎛ ⎝+⎞ ⎠− +⎡ ⎣ ⎢⎛ ⎝ ⎛ ⎝ ∂ ∂⎞ ⎠−⎞ ⎠+⎤ ⎦ ⎥∂ ∂ uu xvu yUx Ux x σB x ρυ Kx Ux u υΓ u ynu y 211, 2 f 2 2 (6) () () ⎟ ⎜ ⎜⎟ ⎜⎟ ⎛ ⎝∂ ∂⎞ ⎠+⎛ ⎝ ∂ ∂⎞ ⎠=∂ ∂+∂ ∂ +⎧ ⎨ ⎩ ∂ ∂∂ ∂+⎛ ⎝ ∂ ∂⎞ ⎠ ⎫ ⎬ ⎭−∂ ∂ ++− ⎛ ⎝−⎞ ⎠ ∞ ∞ uT xvT yαT yκD CC C y ΛD C yT yD TT yρC q y Q ρC TT a υny 1 exp , TB sp Br 2 2 2 2 T2 p e ⁎ phnf f (7) ⎜⎟ ⎛ ⎝∂ ∂⎞ ⎠+⎛ ⎝ ∂ ∂⎞ ⎠=∂ ∂+∂ ∂+∂ ∂ ∞ u C xvC yκD TT yD TT yDC y . BT m 2 2T2 2B 2 2(8) The boundary conditions (BCs) are ()(( ) ) () === ∂ ∂=− += = →=→→→∞ ⎫ ⎬ ⎪ ⎭ ⎪ ∞ TTu v k T yρv x C T T λCC y uUxv CCTT y ,0,0, ,0 ,at0, ,0, , as . msm0 w(9) The term q r is stated as =− ∂ ∂ q σ κT y 4, r ⁎ ⁎ 4(10) and ≅− + ∞∞ TTTT34, 443 (11) where κ ⁎ is the mean absorption constant. By incorporating equations (10) and (11), equation (7) becomes () () ⎟ ⎜ ⎟ ⎜ ⎜⎟ ⎜⎟ ⎛ ⎝∂ ∂⎞ ⎠+⎛ ⎝ ∂ ∂⎞ ⎠=⎛ ⎝+⎞ ⎠ ∂ ∂+∂ ∂ +⎧ ⎨ ⎩ ∂ ∂∂ ∂+⎛ ⎝ ∂ ∂⎞ ⎠ ⎫ ⎬ ⎭ ++− ⎛ ⎝−⎞ ⎠ ∞ ∞ ∞ uT xvT yααT ρC κ T yκD CC C y ΛD C yT yD TT y Q ρC TT a υny 16 3 exp . ⁎2 p⁎ 2 2TB SP 2 2 BT2 e ⁎ phnf f (12) The stream function ( ) ψxy,is defined as ∂ ∂=∂ ∂=− ψ yuψ xv, . (13) The similarity variables are Figure 1: Fluid flow over a permeable wedge. 4Aisha M. Alqahtani et al. ()() ()( ) () () () () =⎛ ⎝+⎞ ⎠=−+ =⎛ ⎝+⎞ ⎠ − −= ∞∞ ∞ ∞ ηmUx xv yT θηT T T ψvxU x mfη CC CC ϕη 1 2,, 21, . 1 2m 1 2 w (14) By substituting equation (14) in equations (6, 8, 9, 12), we get (( )) ()() () −−″‴+ −′ ++−′+″ +−′= nWiff mf mf Kff Mf 11 21 11 10, 2 p(15) {} () ⎛ ⎝+⎞ ⎠″+ ″+ ′′+ ′ + ′ +−= θDϕϕθθfθ Qnη 14Rd 3Pr Nb Nt exp 0, 42 e (16) ″+ ′ + ′′ ⎛ ⎝+⎞ ⎠= ϕ ϕf θPr Le Pr Le Sr Nt Nb 0 . (17) The transformed BCs are () () () () () () () () ′= = ′+ = == ′→ → → →∞ ⎫ ⎬ ⎪ ⎭ ⎪ fη θη Bθη Pfη ϕη η fη θη ϕη η 0, 1, 0, 1at 0, 1, 0, 0as . r (18) The non-dimensional parameters are () () () ()( ) () () () () () ()() () ==− +− =′+ =+=+ =−= =− =− −= == − − ⎫ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ∞ ∞ ∞ ∞ ∞∞ ∞ v αBCT T λCT T KaK m v Uxm Γ xMσB ρam DT TΛ vT QQ aρC DC CΛ v DκD C C CC T T α D σT κκ κD T T TvC C Pr , , 1 2, Wi 1,21, Nt , , Nb , ,Le , Rd 4,Sr . fm 0 sm 0 p 32 0 2 f Tm ee ⁎ pf Bw 4TB w sp m B ⁎3 ⁎TBm mw (19) where θ , ′f , and ϕ are the non-dimensional form of temperature, velocity, and concentration, respectively. L e and Pr is the Lewis and Prandtl number, Wi and N b is the Weissenberg number and Brownian diffusion, S rand Nt is the Soret number and thermophoresis factor, and D 4 and K p is the Dufour number and surface permeability, M , Q e , and R d is the magnetic field, heat source/sink factor, and thermal radiation, respectively. Melting heat expression is expressed as () () = − +− ∞ B. CfT T λCT T pm sm 0 ()−CT T λ sm 0 indicates the Stefan number for solid states and ()−CfT T λ pm0 is the Stefan number for liquid. The engineering interest quantities are stated as (())( )() () () =″+−″ =− ′ =− ⎛ ⎝+⎞ ⎠′ Cnfnf ϕ θ Re Wi 201 0, Sh Re 0, Nu Re 1Rd 4 30. x x x f2 (20) In Section 3, the modeled equations are numerically solved. 3 Numerical solution The essential steps of PCM approach is expressed as [59,60] Step 1: JJ J JJ JJ () () () () () () () () ==′=″ ==′ ==′ fη fη f η θη θη ϕη η ϕη ,,, ,, ,. 12 3 45 67 (21) By substituting equation (21) in equations (15)–(17) and (18), we get JJ J JJJ J (( )) () () () −− + − + +−++−= nWi mm KM 11 21 1 110 , 33 22 2 p13 2 (22) JJJJJJJ{} () ⎛ ⎝+⎞ ⎠++++ +−= D Qnη 14Rd 3Pr Nb Nt exp 0, 547755 215 e (23) JJJJ J ++ ⎛ ⎝+⎞ ⎠ −= Pr Le Pr Le Sr Nt Nb Kr 0. 7715 6 (24) Table 1: Validation of published work with the present outcomes B m Le Endalew and Sarkar [61] Present work 1.0 0.5 1.0 1.74864 1.7486453 0.6 1.77403 1.7740325 2.0 1.75051 1.7505174 3.0 1.75123 1.7512336 2.0 1.66138 1.6613848 3.0 1.60525 1.6052524 Forced convective THNF subject to heat source/sink and Lorentz force 5 The transformed boundary conditions are as follows JJ JJ J JJ J () () () () () () () () == += == →→→→∞ ⎫ ⎬ ⎪ ⎭ ⎪ ηηBηPη ηη ηηηη 0, 1, 0, 1at 0, 1, 0, 0as . 24 5 r1 6 246 (25) Step 2: Introducing constraint p JJ JJ JJ J (( )) ()() ()() −− + − ++− +−+−= nmmK pM 11Wi 21 11 11 0, 33 222 p 13 2 (26) JJJJ J JJ {() }() ⎛ ⎝+⎞ ⎠++ −+ ++−= Dp Qnη 14Rd 3Pr Nb 1 Nt exp 0, 54775 5 2 15 e (27) JJJJ()+−+ ⎛ ⎝+⎞ ⎠=pPr Le 1 Pr Le Sr Nt Nb 0 . 7175 (28) Step 3: Applying numerical implicit scheme By using the implicit numerical scheme as given below () () −=−= −=−= ⎫ ⎬ ⎪ ⎭ ⎪ +++ +++ UU ηAU I ηA U U WW ηAW I ηA W W Δ,or Δ , Δ,or Δ . iiiii iiiii 111 111 (29) Figure 6: Wi vs () f η′ . Figure 2: K p vs () f η′ . Figure 4: Mvs () f η′ . Figure 3: nvs () f η′ . Figure 5: mvs () f η′ . 6Aisha M. Alqahtani et al. Finally, we get the iterative form as () ()( ) =− =− + ⎫ ⎬ ⎭ +− +− UIηAU WIηAWηR Δ, ΔΔ . ii ii 11 11 (30) Table 1 reveals the validity of the present results (equations (26)–(28)) with published work. 4 Results and discussion The TH MHD nanoliquid flow over a permeable wedge under the influence of magnetic field, heat source, and thermal radiation are studied. The governing equations of the THNF are formulated as PDEs, which are subsequently solved using the numerical approach PCM. The computed results obtained through PCM are demonstrated through figures. The core observations are discussed as follows: Figures 2–6 show the significances of permeability parameter K p , power law index n, magnetic term M, wedge angle parameter m, and Wi on the fluid velocity ()′fη . Figure 2 shows that the influence of the K p of surface reduces the fluid flow. Physically, K p is a characteristic of porous surfaces, which enables liquid particles through them. But this consequence resists the flow, resulting in a reduction in the speed of fluid. Figures 3 and 4 illustrate that the fluid flow ()′fη increases with the increasing values of power law index and drops with the impact of magnetic parameter. Physically, the resistive effect known as Lorentz force generates, due to magnetic field, which declines the fluid flow ()′fη . Figures 5 and 6 demonstrate that the flow velocity increases with the effect of wedge angle m, whereas decreases with the impact of Wi. Physically, the ascending wedge angle’s impact on the flow can increase fluid velocity. As the wedge angle rises, the flow area shrinks, leading the fluid to speed up as it moves across the smaller space. The accelerated rate causes a boost in fluid velocity (Figure 5). The Weissenberg number Figure 7: D 4 vs ()θη . Figure 9: Nt against ()θη . Figure 8: Nb vs ()θη . Figure 10: Pr vs ()θη . Forced convective THNF subject to heat source/sink and Lorentz force 7 is a non-dimensional number employed in the analysis of viscoelastic substances to quantify the relative significance of elastic vs viscous impacts. It measures the amount of polymer extending and relaxation in the fluid. The Wi affects the flow behavior and distortion of viscoelastic fluids. Figures 7–12 display the effect of D 4 , Nt, Nb, Rd, Wi, and Pr on the temperature outline ()θη . Figures 7–9 elucidate that the temperature field improves with the intensifying effect of Nt, D 4 , and Nb respectively. Actually, the Dufour number is employed in mass and heat distribution, particularly for convective transportation. It determines the significance of thermal diffusion vs mass dispersion in a fluid. It expresses a ratio of mass diffusion factor to the mass thermal diffusion factor. In the present case, the effect of D 4 augments the fluid temperature ( ) θη.Figures8 and 9 reveal that the energy field enhances with the intensifying effectofNtandNb.Physically,Brownianmotionisthe arbitrary motion of fluidatomsembeddedinafluid caused by interactions with encompassing molecules, whereas the thermophoresis, refers to the movements of atoms in a fluid caused by variations in temperature. Figures 10 and 11 depict that the energy curve of THNF drops with the impact of Pr; but increases with the effect of Rd. Physically, the high Prandtl fluid has less thermal diffusivity, hence, the fluid temperature decreases with the Figure 11: Rd vs ()θη . Figure 13: Nb vs () ϕ η . Figure 14: Le vs () ϕ η . Figure 15: Nt vs () ϕ η . Figure 12: Wi vs ()θη . 8Aisha M. Alqahtani et al. increasing effect of Pr as revealed in Figure 10. The heat radiation factor acts as a heating mediator for the fluid, i.e., the impact of radiation on the movement of fluid across a wedge leads to the improvement of energy field ( ) θη as presented in Figure 11. Figure 12 exposes the result of Wi on the energy curve ()θη . It can be seen that the rising values of Wi enhances the fluid temperature. Figure 13 exemplifies the influence of Nb on the mass curve. The influence of Nb depicts that the concentration field () ϕ η falls with the impact of Nb. Figures 14 and 15 show the influence of Le and Nt vs the mass field. The increasing effect of Le decreases the mass profile; however, the upshot of Nt amplifies the mass curve () ϕ η . Table 2 shows the numerical outcomes for skin friction ()′′f0 , Sherwood ( )− ′ϕ0, and Nusselt number () − ′θ0 . It can be noticed that the increasing effect of wedge angle factor increases the energy dissemination rate and shearing stress; however, the augmentation of Wi drops skin friction and energy transference rate. 5 Conclusion The MHD tangent hyperbolic nanoliquid flow with the mutual impact of melting heat transfer and wedge angle over a permeable wedge is investigated numerically. The influence of thermal radiation, Soret and Dufour, and heat source/sink is also observed on the fluid flow. The modeled equations are simplified to the lowest order through the similarity conversion. The obtained set of dimensionless equations is further calculated numerically by employing the PCM. The computational findings of the present study are compared to the published results for accuracy purposes. Moreover, the results are presented through tables and figures. The key deductions are as follows: •The influence of K p reduces the fluid flow. •The fluid flow ()′fη increases with the increasing values of power law index and drops with the impact of magnetic parameter. •The flow velocity increases with the influence of wedge angle m, whereas decreases with the impact of Wi. •The temperature field improves with the intensifying effect of Nt, D 4 , and Nb. •The energy field enhances with the intensifying effect of Nt and Nb. •The temperature curve of THNF decreases with the impact of Pr, but increases with the effect of Rd and Wi. •The influenceofLeandNtdecreasesthemassprofile () ϕ η . •The increasing effect of wedge angle factor enhances the energy dissemination rate and shearing stress, but the augmentation of Wi drops skin friction and energy transference rate. Table 2: Numerical outcomes for skin friction () f ′′ 0 , Sherwood ( ) ϕ′0 , and Nusselt number ()θ′0 . m W i n Pr N b Nt D 4 Sr M Le Rd (()) f ′′ 0 (())θ′0 (())ϕ′0 0.1 0.3 0.1 2.0 0.1 0.1 0.4 0.3 0.5 0.1 1.0 1.37280 4 0.326823 0.275324 0.3 15.38542 0.327932 0.275733 0.5 0.3 1.337263 0.326412 0.275282 0.6 1.339636 0.322983 0.273196 0.9 0.3 1.372824 0.326263 0.275028 0.5 1.360023 0.328364 0.276786 0.7 3.0 1.372736 0.326825 0.275244 4.0 1.345468 0.313012 0.426132 5.0 0.1 1.372298 0.326383 0.427587 0.2 1.361522 0.246484 0.411287 0.3 0.1 1.372852 0.326272 0.247528 0.2 1.363294 0.263596 0.058810 0.3 0.4 1.372886 0.226923 0.275366 0.8 1.320621 0.060430 0.522274 1.2 0.3 1.372812 0.326212 0.275181 0.5 1.380527 0.379971 0.018282 0.7 0.5 1.472907 0.326363 0.275914 1.0 1.306823 0.327895 0.276185 1.5 0.1 1.372133 0.326907 0.275897 0.2 1.362410 0.254135 0.732238 0.3 1.372361 0.326373 0.275359 1.0 1.372605 0.326173 0.275234 2.0 2.452132 0.384444 0.345891 Forced convective THNF subject to heat source/sink and Lorentz force 9