Numerical Study of Buoyancy-Driven Melting of a Williamson Non-Newtonian Phase Change Material in a Porous Enclosure Under Magnetic Field Effects

Abstract

This study numerically investigates buoyancy-driven melting of a Williamson non-Newtonian phase change material in a porous enclosure. The model incorporates an inclined magnetic field, thermal radiation, and viscous dissipation. The enthalpy–porosity method is employed to describe the melting process. The governing equations are solved using an implicit finite-difference scheme and successive over-relaxation. Grid and time-step refinement are performed to assess numerical sensitivity, while comparisons with limiting-case benchmarks are used to validate selected components of the numerical implementation. The effects of the governing parameters on fluid flow, heat transfer, and melting behavior are examined. Higher Grashof and Darcy numbers strengthen circulation and accelerate melting. In contrast, larger Williamson and magnetic parameters, together with greater mushy-zone resistance, suppress fluid motion and delay melting. Thermal radiation and viscous dissipation enhance heating and promote melting. Higher Stefan numbers also accelerate melting and increase the liquid fraction. Overall, the results demonstrate that buoyancy, permeability, thermal radiation, viscous heating, and Stefan number promote PCM melting, whereas magnetic damping, non-Newtonian effects, and mushy-zone resistance inhibit it, providing insight into the competing transport mechanisms governing PCM melting in porous enclosures.

1. Introduction

Phase change materials (PCMs) have attracted considerable attention as latent heat storage media because they can absorb and release large amounts of thermal energy within a narrow temperature range, making them attractive for thermal energy storage (TES) applications [1]. Integrating them into building envelopes has shown significant potential to improve indoor thermal regulation and reduce energy consumption [2], while coupling PCM systems with radiative cooling technologies can further enhance passive cooling performance and building energy efficiency [3]. Despite these advantages, the widespread application of PCMs remains limited by their inherently low thermal conductivity, which results in slow melting and solidification processes and consequently restricts charging and discharging rates [4].

To overcome the low thermal conductivity of PCMs, heat transfer enhancement strategies generally follow two complementary routes: material-level modification and structure- and medium-level regulation. The first includes incorporating nanoparticles [5] and constructing interconnected, highly conductive filler networks within the PCM. Solid–solid phase change composites containing aligned graphite networks for solar/electro-thermal energy storage demonstrate how directional conductive pathways can improve thermal transport [6]. Since their rheological properties strongly influence the melting behavior of these materials, particular attention has been devoted to pseudoplastic and viscoplastic fluids [7]. In permeable systems, non-Newtonian effects significantly influence fluid circulation and heat transfer [8], while magnetic fields, porous resistance, and internal heat generation further modify the transport characteristics of electrically conducting fluids [9]. These studies demonstrate that several physical mechanisms affect the thermal performance of PCM systems; however, their combined influence on latent heat thermal energy storage remains only partially understood [10].

Among non-Newtonian constitutive models, the Williamson model has received considerable attention because it accurately represents shear-thinning fluids encountered in numerous engineering applications [11,12]. Researchers have investigated the rheological behavior of Williamson fluids in porous and fractured media [13], as well as in heat transfer problems involving permeable surfaces [14]. Additional studies have demonstrated that magnetic fields can significantly alter the momentum and thermal transport characteristics of electrically conducting non-Newtonian fluids [15], while thermal radiation and Ohmic dissipation further influence velocity and temperature distributions [16]. Variable thermal conductivity [17], slip effects and nanoparticle transport [18], porous-medium resistance and internal heat generation [19], and recent artificial intelligence approaches for predicting radiative heat transfer [20] have all contributed to improving the understanding of Williamson-fluid transport. Nevertheless, these investigations have generally considered convective flow configurations rather than phase-change processes, and they have devoted only limited attention to the melting behavior of Williamson-type phase change materials.

The enhancement of heat transfer during PCM melting has also been extensively investigated from the perspective of thermal energy storage applications [21]. Thermal radiation can contribute significantly to heat transfer, particularly in high-temperature thermal systems [22], and its combination with PCM-based walls has been shown to improve overall thermal performance [23]. Numerical simulation of melting and solidification is commonly performed using the enthalpy–porosity method, one of the most widely accepted approaches for modelling convection, phase-interface evolution, and mushy-zone behavior [24,25]. Several studies have also shown that nanoparticles [26] and metallic fins [27] can substantially accelerate melting and solidification by enhancing heat transfer within PCM enclosures. More recently, investigations of non-Newtonian flows have shown that thermal radiation [28], magnetic fields and porous structures [29], and fractional constitutive models [30] considerably affect momentum and heat transport. The study in Ref. [31] investigated the melting behavior of a Carreau–Yasuda non-Newtonian PCM within a shell-and-tube heat exchanger containing an anisotropic porous medium. However, these studies have primarily focused on fluid-flow problems and have not addressed their combined influence on phase-change phenomena occurring in latent heat thermal energy storage systems.

Although previous studies have examined the melting behavior of non-Newtonian phase change materials in porous media, the combined effects of Williamson shear-dependent viscosity, an inclined magnetic field, thermal radiation, and porous resistance on coupled fluid flow and phase-change phenomena remain insufficiently understood. To address this research gap, the present study investigates how these interacting physical mechanisms influence fluid circulation, wall heat transfer, and melting progression in a two-dimensional enclosure. Phase change is represented using the established enthalpy–porosity method. The study focuses on parameter sensitivity, assessed through velocity fields, streamlines, isotherms, melting-front evolution, average liquid fraction, local skin-friction coefficient, and local Nusselt number. The formulation represents an idealized configuration rather than a specific storage device, and simpler cases can be recovered by omitting the relevant physical terms. The applicability and relative importance of these mechanisms depend on material properties and operating conditions. Accordingly, the findings describe trends within the investigated parameter ranges and provide a basis for further thermal energy storage research, subject to additional coupled validation and numerical uncertainty assessment.

2. Mathematical Formulation

2.1. Physical Model

Consider the transient melting of a Williamson phase change material inside a two-dimensional rectangular enclosure of length L and height H, as illustrated in Figure 1. The horizontal and vertical coordinates are denoted by x and y, respectively, with the origin at the lower-left corner. The enclosure contains a stationary, homogeneous, and isotropic porous medium saturated with PCM. Initially, the PCM is completely solid and has a uniform temperature T_i < T_s, where T_s is the solidus temperature. At t > 0, the lower wall is maintained at a constant temperature T_h > T_l, where T_l is the liquidus temperature.

Figure 1. Schematic of the physical model and boundary conditions.

The upper wall and both vertical walls are thermally insulated. All walls are stationary, impermeable, and satisfy the no-slip condition. Heating produces a liquid region adjacent to the lower wall. Buoyancy-driven motion develops as melting proceeds. Gravity acts vertically downward, while a uniform magnetic field of magnitude B₀ is applied at an angle γ to the horizontal direction. The two-dimensional approximation assumes negligible out-of-plane variations and provides a simplified framework for examining parameter effects on convection and melting. Front and rear end-wall effects and three-dimensional circulation are therefore excluded, and quantitative application to practical storage systems requires three-dimensional validation. The analysis adopts the following assumptions:

•

The flow is two-dimensional, transient, laminar, and incompressible.

•

The liquid PCM follows the Williamson shear-thinning model.

•

Density variations are neglected except in the buoyancy term.

•

Solid and liquid densities are taken as equal, so melting does not change the occupied volume.

•

The porous matrix and PCM are in local thermal equilibrium.

•

The porous region is represented by a simplified effective-continuum Brinkman–Darcy model. Porosity effects are incorporated into the effective coefficients; separate porosity corrections to inertia are neglected.

•

The magnetic Reynolds number is sufficiently small to neglect the induced magnetic field. No externally imposed electric field is considered.

•

Thermal radiation is represented by the linearized Rosseland diffusion approximation.

•

Viscous dissipation and Joule heating are included. Heating caused by Darcy drag and mushy-zone damping is neglected.

•

Temperature gradients across the mushy zone do not necessarily imply a local temperature difference between the porous matrix and PCM. However, finite interfacial heat exchange may produce thermal lag and slower melting than predicted by LTE when the matrix supplies heat to the PCM. A two-temperature LTNE model is required to quantify this deviation.

For a consistent effective-continuum description, ρ, c_p, k, L_h, and σ below denote the effective density, specific heat, thermal conductivity, latent heat per unit effective mass, and electrical conductivity used in the model. Thermophysical properties and Williamson constitutive parameters are assumed temperature-independent to isolate parameter effects, while apparent viscosity remains shear-rate dependent. Local thermal equilibrium assumes sufficiently rapid heat exchange between the porous matrix and PCM to maintain approximately equal local temperatures. These approximations require reassessment for strong temperature-dependent property variations or rapid melting, where a local thermal non-equilibrium model may be necessary.

2.2. Williamson Constitutive Model

The velocity vector and material derivative are defined as

$$\mathbf{u}=\left(u,v,0\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{D}{\mathit{Dt}}}={\displaystyle \frac{\partial }{\partial t}}+u{\displaystyle \frac{\partial }{\partial x}}+v{\displaystyle \frac{\partial }{\partial y}}.$$

(1)

The extra-stress tensor is

$$\mathrm{\tau }={\mathrm{\mu }}_{\mathrm{a}}{\mathrm{A}}_{\mathrm{1}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{A}}_{\mathrm{1}}=\nabla \mathrm{u}+{\left(\nabla \mathrm{u}\right)}^{\mathrm{T}},$$

(2)

where μ_a is the apparent viscosity. For the Williamson model,

$${\mu }_a={\mu }_{\infty }+{\displaystyle \frac{{\mu }_0-{\mu }_{\infty }}{1+\mathrm{\Gamma }\dot{\gamma }}},$$

(3)

in which μ₀ and μ_∞ are the zero- and infinite-shear-rate viscosities, respectively, and Γ is the Williamson time constant. The shear-rate magnitude is

$$\dot{\gamma }=\sqrt{{\displaystyle \frac{1}{2}}{\mathbf{A}}_1:{\mathbf{A}}_1}={\left[2{\left({\displaystyle \frac{\partial u}{\partial x}}\right)}^2+2{\left({\displaystyle \frac{\partial v}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}\right)}^2\right]}^{1/2}.$$

(4)

Here, subscripts denote partial derivatives. The stress components therefore become

$${\tau }_{\mathit{xx}}=2{\mu }_a{\displaystyle \frac{\partial u}{\partial x}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tau }_{\mathit{yy}}=2{\mu }_a{\displaystyle \frac{\partial v}{\partial y}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tau }_{\mathit{xy}}={\tau }_{\mathit{yx}}={\mu }_a\left({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}\right).$$

(5)

2.3. Magnetic Force

The applied magnetic field is

$$\mathbf{B}=B_0\left(\cos \gamma ,\sin \gamma ,0\right).$$

(6)

Under the adopted electrical approximation, Ohm’s law gives

$$\mathbf{J}=\sigma \left(\mathbf{u}\times \mathbf{B}\right)=\sigma B_0\left(u\sin \gamma -v\cos \gamma \right)e_z.$$

(7)

The corresponding Lorentz-force components are

$$\begin{array}{l}{F}_x=-\sigma B_0^2\left(u{\sin }^2\gamma -v\sin \gamma \cos \gamma \right),\\ F_y=-\sigma B_0^2\left(v{\cos }^2\gamma -u\sin \gamma \cos \gamma \right).\end{array}$$

(8)

The cross terms in Equation (8) are required for an inclined magnetic field.

2.4. Conservation of Mass and Momentum

The incompressible continuity equation is expressed as:

$${\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}=0.$$

(9)

The momentum equations in the horizontal and vertical directions are

$$\rho \left({\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}\right)=-{\displaystyle \frac{\partial p}{\partial x}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{\partial {\tau }_{\mathit{xx}}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{\mathit{xy}}}{\partial y}}\hspace{0.17em}-\hspace{0.17em}{\displaystyle \frac{{\mu }_a}{K}}u-S\left(f_l\right)u-\sigma B_0^2\left(u{\sin }^2\gamma -v\sin \gamma \cos \gamma \right),$$

(10)

$$\begin{array}{l}\rho \left({\displaystyle \frac{\partial v}{\partial t}}+u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}\right)=-{\displaystyle \frac{\partial p}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{\mathit{xy}}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{\mathit{yy}}}{\partial y}}-{\displaystyle \frac{{\mu }_a}{K}}v-S\left(f_l\right)v+\rho g\beta \left(T-T_m\right)-\sigma B_0^2\\ \left(v{\cos }^2\gamma -u\sin \gamma \cos \gamma \right).\end{array}$$

(11)

Here, K is the porous-medium permeability, β is the thermal expansion coefficient, and T_m is the reference melting temperature. The pressure p is measured relative to the hydrostatic pressure associated with the reference density. No external pressure gradient is imposed. The Darcy resistance in Equations (10) and (11) uses the local apparent viscosity. It remains distinct from the mushy-zone resistance: the former represents the permanent porous matrix, whereas the latter suppresses motion as the PCM solidifies.

The enthalpy–porosity method suppresses motion in the solid and mushy regions through the resistance coefficient

$$S\left(f_l\right)=C_m{\displaystyle \frac{{\left(1-f_l\right)}^2}{f_l^3+\epsilon }},$$

(12)

where C_m is the dimensional mushy-zone resistance constant with units of kg m⁻³ s⁻¹ and ε > 0 is a dimensionless regularization constant that prevents the denominator from vanishing as f_l → 0. The liquid fraction f_l varies from zero in the solid region to unity in the liquid region.

2.5. Enthalpy and Energy Equation

The specific total enthalpy is expressed as

$$h=c_p\left(T-T_m\right)+L_h{f}_l.$$

(13)

The liquid fraction is related to temperature by

$$f_l\left(T\right)=\left\{\begin{array}{ll}0,& T\le T_s,\\ {\displaystyle \frac{T-T_s}{T_l-T_s}},& T_s<T<T_l\\ 1,& T\ge T_l.\end{array}\right.,$$

(14)

Accordingly, f_l = 0 represents fully solid PCM; f_l = 1 corresponds to completely molten; and 0 < f_l < 1 defines the mushy region. The energy balance is written in conservative enthalpy form as

$$\rho \left[{\displaystyle \frac{\partial h}{\partial t}}+\nabla ·\left(\mathit{uh}\right)\right]=\nabla ·\left(k\nabla T-q_r\right)+\mathsf{\Phi }+Q.$$

(15)

2.6. Viscous Dissipation and Joule Heating

$$\begin{array}{l}\mathsf{\Phi }={\mu }_a\left[2{\left({\displaystyle \frac{\partial u}{\partial x}}\right)}^2\hspace{0.17em}+2{\left({\displaystyle \frac{\partial v}{\partial y}}\right)}^2+\hspace{0.17em}{\left({\displaystyle \frac{\partial u}{\partial y}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{\partial v}{\partial x}}\right)}^2\right],\\ Q=\sigma B_0^2{\left(u\sin \gamma -v\cos \gamma \right)}^2.\end{array}$$

(16)

Thus, Joule heating depends on the velocity component perpendicular to the magnetic field. The Rosseland radiative heat flux is

$$q_r=-{\displaystyle \frac{4{\sigma }^{\ast }}{3k^{\ast }}}\nabla \left(T^4\right).$$

(17)

where σ* is the Stefan–Boltzmann constant and k* is the mean absorption coefficient. When temperature differences are sufficiently small relative to the absolute reference temperature T_m

$$T^4\hspace{0.17em}\cong \hspace{0.17em}4T_m^{3}T\hspace{0.17em}-\hspace{0.17em}3T_m^4.$$

(18)

The Rosseland approximation requires an optical thickness k*H >> 1, while the linearization of T⁴ requires ΔT/T_m<<1, with temperatures expressed in kelvin. These conditions are assumed rather than demonstrated for a specific PCM–porous matrix system. The radiation results are therefore conditional on these assumptions, which represent a limitation of the present model.

Radiative diffusion is included in both coordinate directions, consistent with the two-dimensional temperature field. Substituting Equations (13) and (18) into Equation (15) gives

$$\rho c_p\left({\displaystyle \frac{\partial T}{\partial t}}+u{\displaystyle \frac{\partial T}{\partial x}}+v{\displaystyle \frac{\partial T}{\partial y}}\right)=\left(k+{\displaystyle \frac{16{\sigma }^{\ast }{T}_m^3}{3k^{\ast }}}\right)\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}\right)\hspace{0.17em}+\hspace{0.17em}\mathsf{\Phi }+Q-\rho L_h\left(f_{l,t},u{f}_{l,x},v{f}_{l,y}\right).$$

(19)

Equation (19) retains both storage and transport of latent enthalpy. An implementation that retains only f_l,t must explicitly justify neglecting latent-enthalpy advection.

2.7. Initial and Boundary Conditions

Initially, the PCM is stationary and completely solid:

$$u\left(x,y,0\right)=v\left(x,y,0\right)=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}T\hspace{0.17em}\left(x,y,0\right)=T_i<T_s,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_l\left(x,y,0\right)=0.$$

(20)

For t > 0, the boundary conditions are

$$\begin{array}{l}y=0:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u=v=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}T=T_h,\\ y=H:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u=v=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{T}_y=0,\\ x=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}L:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u=v=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{T}_x=0.\end{array}$$

(21)

The temperature jump at the lower wall represents sudden heating at t = 0⁺. No additional conditions u_t = v_t = 0 are imposed. Because the enclosure is closed, pressure is determined up to an additive constant. A convenient reference is

$${\displaystyle {\int }_{\mathsf{\Omega }}\mathit{pd}\mathsf{\Omega }=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathsf{\Omega }=\left(0,L\right)\times \left(0,H\right).}$$

(22)

Dimensionless quantities are specified by [18]:

$$\begin{array}{l}\overline{y}={\displaystyle \frac{y}{H}},\hspace{0.17em}\hspace{0.17em}\overline{x}={\displaystyle \frac{x}{H}},\hspace{0.17em}\hspace{0.17em}\overline{t}={\displaystyle \frac{\alpha t}{H^2}},\hspace{0.17em}\hspace{0.17em}\overline{u}={\displaystyle \frac{\mathit{uH}}{\alpha }},\hspace{0.17em}\hspace{0.17em}\overline{v}={\displaystyle \frac{\mathit{vH}}{\alpha }},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{p}={\displaystyle \frac{p{H}^2}{\rho {\alpha }^2}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\theta ={\displaystyle \frac{T-T_m}{T_h-T_m}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\alpha ={\displaystyle \frac{k}{\rho c_p}},\hspace{0.17em}G={\displaystyle \frac{H^2\dot{\gamma }}{\alpha }},\\ \mathit{Gr}={\displaystyle \frac{g\beta \left(T_h-T_m\right)H^3}{v_0^2}},\hspace{0.17em}\hspace{0.17em}\mathsf{\Upsilon }={\displaystyle \frac{L}{H}},v_0={\displaystyle \frac{{\mu }_0}{\rho }},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{Da}={\displaystyle \frac{K}{H^2}},\hspace{0.17em}\hspace{0.17em}M=B_{0}H\sqrt{{\displaystyle \frac{\sigma }{{\mu }_0}}},\hspace{0.17em}\hspace{0.17em}\mathit{We}={\displaystyle \frac{\Gamma \alpha }{H^2}},\mathrm{Pr}={\displaystyle \frac{v_0}{\alpha }},\\ \mathit{Ec}={\displaystyle \frac{{\alpha }^2}{H^2{c}_p\left(T_h-T_m\right)}},\mathit{Rd}={\displaystyle \frac{16{\sigma }^{\ast }{T}_m^3}{3k^{\ast }k}},\mathit{St}={\displaystyle \frac{c_p\left(T_h-T_m\right)}{L_h}},A={\displaystyle \frac{C_m{H}^2}{\rho \alpha }}.\end{array}$$

(23)

The dimensionless local resistance coefficient, apparent viscosity and stress tensor are

$$\begin{array}{l}{\displaystyle \frac{S\left(f_l\right)H^2}{\rho \alpha }}={\displaystyle \frac{A{\left(1-f_l\right)}^2}{f_l^3+\epsilon }},\hspace{0.17em}\eta ={\displaystyle \frac{{\mu }_a}{{\mu }_0}}={\gamma }_{\mu }+{\displaystyle \frac{1-{\gamma }_{\mu }}{1+\mathit{WeG}}},\hspace{0.17em}{\overline{\tau }}_{\mathit{xx}}=2\eta {\displaystyle \frac{\partial \overline{u}}{\partial \overline{x}}},\hspace{0.17em}{\overline{\tau }}_{\mathit{yy}}=2\eta {\displaystyle \frac{\partial \overline{v}}{\partial \overline{y}}},\hspace{0.17em}{\overline{\tau }}_{\mathit{xy}}=\eta \\ \left({\displaystyle \frac{\partial \overline{u}}{\partial \overline{y}}}+{\displaystyle \frac{\partial \overline{v}}{\partial \overline{x}}}\right).\end{array}$$

(24)

Here, (We) is the Williamson parameter, and (G) is the dimensionless shear-rate magnitude. The viscosity ratio is γ_μ, and η denotes the dimensionless apparent viscosity. Substituting these variables into the governing equations yields the dimensionless momentum and energy equations, presented in Equations (25)–(27). For simplicity, the overbars denoting dimensionless variables are omitted:

$$\begin{array}{l}{\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}=-{\displaystyle \frac{\partial p}{\partial x}}+\mathit{Pr}\left({\displaystyle \frac{\partial }{\partial x}}\left({\tau }_{\mathit{xx}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\tau }_{\mathit{xy}}\right)\right)-\mathit{Pr}{M}^2\left(u{\sin }^2\gamma -v\sin \gamma \cos \gamma \right)-\\ {\displaystyle \frac{\mathit{Pr}}{\mathit{Da}}}\eta u-A{\displaystyle \frac{{\left(1-f_l\right)}^2}{f_l^3+\epsilon }}u,\end{array}$$

(25)

$$\begin{array}{l}{\displaystyle \frac{\partial v}{\partial t}}+u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}=-{\displaystyle \frac{\partial p}{\partial y}}+\mathit{Pr}\left({\displaystyle \frac{\partial }{\partial x}}\left({\tau }_{\mathit{xy}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\tau }_{\mathit{yy}}\right)\right)-\mathit{Pr}{M}^2\left(v{\cos }^2\gamma -u\sin \gamma \cos \gamma \right)-\\ {\displaystyle \frac{\mathit{Pr}}{\mathit{Da}}}\eta v-A{\displaystyle \frac{{\left(1-f_l\right)}^2}{f_l^3+\epsilon }}v+\mathit{GrP}{r}^2\theta ,\end{array}$$

(26)

$$\begin{array}{l}{\displaystyle \frac{\partial \theta }{\partial t}}+u{\displaystyle \frac{\partial \theta }{\partial x}}+v{\displaystyle \frac{\partial \theta }{\partial y}}=\left(1+\mathit{Rd}\right)\left(\hspace{0.17em}{\displaystyle \frac{{\partial }^2\theta }{\partial x^2}}+{\displaystyle \frac{{\partial }^2\theta }{\partial y^2}}\right)+\mathit{PrEc}\eta G^2+\mathit{PrEc}{M}^2{\left(u\sin \gamma -v\cos \gamma \right)}^2-\\ {\displaystyle \frac{1}{\mathit{St}}}\left({\displaystyle \frac{\partial f_l}{\partial t}}+u{\displaystyle \frac{\partial f_l}{\partial x}}+v{\displaystyle \frac{\partial f_l}{\partial y}}\right).\end{array}$$

(27)

The resulting formulation is governed by several important dimensionless parameters, including the Prandtl number (Pr), Grashof number (Gr), Stefan number (St), Eckert number (Ec), magnetic parameter (M), and radiation parameter (Rd). The consistent dimensionless liquid-fraction relation is

$$f_l\left(\theta \right)=\left\{\begin{array}{l}0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\theta \le 0,\\ \theta ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0<\theta <1,\\ 1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\theta >1.\end{array}\right.$$

(28)

The initial and boundary conditions become

$$\begin{array}{l}t=0:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u=v=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\theta ={\theta }_i.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_l=0,\\ y=0:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u=v=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\theta =1,\\ y=1:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u=v=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_y=0,\\ x=0,\hspace{0.17em}\hspace{0.17em}\mathsf{\Upsilon }:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u=v=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_x=0.\end{array}$$

(29)

2.8. Engineering Performance

The engineering quantities of interest, the local skin-friction coefficient, and local Nusselt number are defined as follows:

$$C_f={\left.\eta \left({\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}\right)\right|}_{y=0},\hspace{0.17em}\mathit{Nu}=-\left(1+\mathit{Rd}\right){\left.{\displaystyle \frac{\partial \theta }{\partial y}}\right|}_{y=0}.$$

(30)

The average skin-friction coefficient, average Nusselt number, and average liquid fraction are

$${\overline{C}}_f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Upsilon }}}{\displaystyle {\displaystyle \underset{0}{\overset{\mathsf{\Upsilon }}{\int }}}{C}_f\left(x,t\right)\mathit{dx},}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{N}u\left(t\right)={\displaystyle \frac{1}{\mathsf{\Upsilon }}}{\displaystyle {\displaystyle \underset{0}{\overset{\mathsf{\Upsilon }}{\int }}}\mathit{Nu}\left(x,t\right)\mathit{dx}}.$$

(31)

$${\overline{f}}_l={\displaystyle \frac{1}{\mathsf{\Upsilon }}}{\displaystyle {\displaystyle \underset{0}{\overset{\mathsf{\Upsilon }}{\int }}}{\displaystyle {\displaystyle \underset{0}{\overset{1}{\int }}}{f}_l\mathit{dydx}.}}$$

(32)

The dimensionless stream function is defined by

$$u={\displaystyle \frac{\partial \psi }{\partial y}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}v=-{\displaystyle \frac{\partial \psi }{\partial x}}.$$

(33)

3. Numerical Methodology

3.1. Numerical Discretization

The coupled dimensionless equations are solved using a finite difference method on a uniform Cartesian grid. The computational domain is 0 ≤ x ≤ ϒ and 0 ≤ y ≤ 1, where ϒ = L/H. The spatial derivatives appearing in the governing equations were approximated using second-order central finite difference schemes. For a general variable ϕ, the first- and second-order derivatives are approximated by Equations (34) and (35), respectively:

$${\left({\displaystyle \frac{\partial \varphi }{\partial x}}\right)}_{i,j}={\displaystyle \frac{{\varphi }_{i+1,j}-{\varphi }_{i-1,j}}{2\Delta x}}, {\left({\displaystyle \frac{{\partial }^2\varphi }{\partial x^2}}\right)}_{i,j}={\displaystyle \frac{{\varphi }_{i+1,j}-2{\varphi }_{i,j}+{\varphi }_{i-1,j}}{\Delta x^2}},$$

(34)

$${\left({\displaystyle \frac{\partial \varphi }{\partial y}}\right)}_{i,j}={\displaystyle \frac{{\varphi }_{i,j+1}-{\varphi }_{i,j-1}}{2\Delta y}}, {\left({\displaystyle \frac{{\partial }^2\varphi }{\partial y^2}}\right)}_{i,j}={\displaystyle \frac{{\varphi }_{i,j+1}-2{\varphi }_{i,j}+{\varphi }_{i,j-1}}{\Delta y^2}}.$$

(35)

Time integration was performed using a first-order implicit scheme:

$${{\left({\displaystyle \frac{\partial \varphi }{\partial t}}\right)}_{i,j}}^{n+1}={\displaystyle \frac{{{\varphi }_{i,j}}^{n+1}-{{\varphi }_{i,j}}^n}{\Delta t}},$$

(36)

where n and n + 1 denote the previous and current time levels, respectively. Additional calculations with smaller time steps produced negligible changes in the velocity, temperature, and liquid-fraction fields, confirming temporal independence of the reported numerical results. For clarity, let m denote the inner iteration number at the current time level n + 1. This notation is distinct from n, which denotes the physical time level. Starting from the converged solution at time level n, provisional velocity components u* and v* are obtained from the discretized momentum equations using the currently available pressure field.

3.2. Iterative Solution Algorithm

Finite-difference discretization of the governing equations with initial and boundary conditions gives

$$\begin{array}{l}{u}^{k+1}=u^k-\Delta t{\delta }_x{p}^{k+1}+\Delta \mathit{tPr}\left({\delta }_x\left({\tau }_{\mathit{xx}}^{k+1}\right)+{\delta }_y\left({\tau }_{\mathit{xy}}^{k+1}\right)\right)-\mathit{Pr}{M}^2\left(u^{k+1}{\sin }^2\gamma -v^{k+1}\sin \gamma \cos \gamma \right)-\\ {\displaystyle \frac{\mathit{Pr}}{\mathit{Da}}}{\eta }^{k+1}{u}^{k+1}-A^{k+1}{\displaystyle \frac{{\left(1-f_l\right)}^2}{f_l^3+\epsilon }}{u}^{k+1},\end{array}$$

(37)

$$\begin{array}{l}{v}^{k+1}=v^k-\Delta t{\delta }_y{p}^{k+1}+\Delta \mathit{tPr}\left({\delta }_x\left({\tau }_{\mathit{xy}}^{k+1}\right)+{\delta }_y\left({\tau }_{\mathit{yy}}^{k+1}\right)\right)-\mathit{Pr}{M}^2\left(v^{k+1}{\cos }^2\gamma -u^{k+1}\sin \gamma \cos \gamma \right)-\\ {\displaystyle \frac{\mathit{Pr}}{\mathit{Da}}}{\eta }^{k+1}{v}^{k+1}-A^{k+1}{\displaystyle \frac{{\left(1-f_l\right)}^2}{f_l^3+\epsilon }}{v}^{k+1}+\mathit{GrPr}{\theta }^{k+1},\end{array}$$

(38)

$$\begin{array}{l}{E}_{i,j}^{k+1}=E_{i,j}^k+\Delta t\left(1+\mathit{Rd}\right)\left(\hspace{0.17em}{\displaystyle \frac{{\partial }^2{\theta }_{i,j}^{k+1}}{\partial x^2}}+{\displaystyle \frac{{\partial }^2{\theta }_{i,j}^{k+1}}{\partial y^2}}\right)+\mathit{PrEc}{\eta }_{i,j}^{k+1}{\left(G_{i,j}^{k+1}\right)}^2+\mathit{PrEc}{M}^2\\ {\left(u_{i,j}^{k+1}\sin \gamma -v_{i,j}^{k+1}\cos \gamma \right)}^2-{\displaystyle \frac{1}{\mathit{St}}}\left({\displaystyle \frac{\partial f_l}{\partial t}}+u_{i,j}^{k+1}{\displaystyle \frac{\partial f_l}{\partial x}}+v_{i,j}^{k+1}{\displaystyle \frac{\partial f_l}{\partial y}}\right).\end{array}$$

(39)

$$\begin{array}{l}{u}_{i,j}^0=v_{i,j}^0=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_{i,j}^0={\theta }_i,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_{l,i,j}^0=0,\\ u_{i,0}=v_{i,0}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_{i,0}=2-{\theta }_{i,1},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y=0,\\ u_{i,1}=v_{i,1}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_{i,N_y+1}={\theta }_{i,N_y},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}y=1,\\ u_{0,y}=v_{0,y}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_{0,j}={\theta }_{1,j},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x=0,\\ u_{1,y}=v_{1,y}=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\theta }_{N_x+1,j}={\theta }_{N_x,j},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x=\mathsf{\Upsilon }.\end{array}$$

(40)

Thus, the lower wall is maintained at θ = 1, while the remaining walls have zero normal heat flux. At each time step, the nonlinear equations are linearized through Picard iteration. The resulting scalar linear systems are solved using SOR. For the algebraic equation, the update is

$${\varphi }_p^{\left(m+1\right)}=\left(1-\omega \right){\varphi }_p^{\left(m\right)}+{\displaystyle \frac{\omega }{\mathit{ap}}}\left[\mathit{bp}-{\displaystyle \sum _{Q<p}\mathit{apQ}{\varphi }_p^{\left(m+1\right)}-}{\displaystyle \sum _{Q>p}\mathit{apQ}{\varphi }_p^{\left(m+1\right)}}\right]$$

(41)

where m is the linear iteration index and ω is the relaxation factor. The relative residual used to monitor convergence is defined by Equation (44). Figure 2 presents the convergence characteristics of the numerical procedure. Figure 2 (single-panel figure showing the velocity and temperature residual convergence).

$$R={\displaystyle \frac{{\displaystyle \sum _{i=1}^{N_x}{\displaystyle \sum _{j=1}^{N_y}\left|{\varphi }_{i,j}^{m+1}-{\varphi }_{i,j}^m\right|}}}{{\displaystyle \sum _{i=1}^{N_x}{\displaystyle \sum _{j=1}^{N_y}\left|{\varphi }_{i,j}^{m+1}\right|}}}},$$

(42)

where ϕ represents the dependent variable and (m) denotes the iteration number. The previous time-level solution provides the initial estimate. The viscosity and resistance coefficients are updated, followed by the momentum solution, pressure correction, and enthalpy solution. This sequence is repeated until the coupled system converges. Following the grid and time-step convergence studies, a uniform 100 × 100 computational grid and a dimensionless time step of Δt = 10⁻⁴ are adopted, and simulations were advanced to t = 1. The iterative process for solving the discrete equations is initialized using the prescribed initial and boundary conditions. At each time step (k + 1)th level, the initial values of velocities $u_{i,j}^{k+1},\hspace{0.17em}{v}_{i,j}^{k+1}$ and temperature ${\theta }_{i,j}^{k+1}$ from the previous time level t = t_k are utilized. MATLAB R20XX (The MathWorks, Inc., Natick, MA, USA) was selected for its efficient matrix operations and flexibility in implementing the coupled equations through a custom finite-difference scheme. A simulation required approximately 45 min; this runtime depends on the hardware, grid, time step, and convergence tolerance. No computational speed advantage over ANSYS Fluent or COMSOL is claimed.

Figure 2. Convergence of velocity and temperature residuals.

Figure 2. Convergence of velocity and temperature residuals.

3.3. Grid Independence and Convergence Analysis

The temporal sensitivity is assessed by comparing the average liquid fraction and local Nusselt number for three different time steps on a fixed computational grid at the same final simulation time. The corresponding results are presented in Table 1.

Table 1. Time-step convergence of the average liquid fraction and local Nusselt number.

Dimensionless Time Step	${\overline{\mathit{f}}}_{\mathit{l}}$	Nu
0.0002	0.7900	3.0500
0.0001	0.8100	3.1000
0.00005	0.8112	3.1049

Reducing the time step from 10⁻⁴ to 5 × 10⁻⁵ changes the average liquid fraction and local Nusselt number by approximately 0.15% and 0.16%, respectively, relative to the finer time-step results. Although the variation in the average liquid fraction between 2 × 10⁻⁴ and 10⁻⁴ is approximately 2.47%, the considerably smaller changes obtained with further refinement indicate improved temporal consistency. Therefore, a dimensionless time step of 10⁻⁴ is selected as a reasonable compromise between numerical accuracy and computational cost.

The grid-independence study is performed using four computational meshes (40 × 40, 60 × 60, 80 × 80, and 100 × 100) to investigate the influence of spatial resolution on the numerical results. As presented in Table 2, the local Nusselt number increases from 2.83 to 3.12, whereas the local skin-friction coefficient decreases from 0.820 to 0.735 with successive mesh refinement. Similarly, the average liquid fraction decreases from 0.86 to 0.80. Refining the mesh from 80 × 80 to 100 × 100 results in relative changes of approximately 0.64%, 1.50%, and 1.25% in the local Nusselt number, skin-friction coefficient, and average liquid fraction, respectively, using the finest-grid results as references. These relatively small variations indicate that the 100 × 100 mesh provides adequate spatial resolution for the present simulations. Consequently, the 100 × 100 grid is selected for subsequent numerical investigations.

Table 2. Grid independence study for the present numerical simulations.

Grid Size	Nu	Cf	${\overline{\mathit{f}}}_{\mathit{l}}$
40 × 40	2.83	0.820	0.86
60 × 60	3.03	0.778	0.83
80 × 80	3.10	0.746	0.81
100 × 100	3.12	0.735	0.80

The convergence history of the numerical algorithm is further examined by monitoring the velocity and temperature residuals over successive iterations. As illustrated in Figure 2, both residuals decrease continuously with increasing iteration number, indicating stable convergence behaviour of the numerical solution. After 100 iterations, the velocity and temperature residuals reach approximately 3.5 × 10⁻⁴ and 2.5 × 10⁻³, respectively. This decreasing residual behaviour demonstrates the progressive reduction in numerical errors during the iterative solution procedure.

3.4. Model Validation

The numerical implementation was assessed through limiting-case comparisons with Abbas et al. [18] and Brent et al. [25]. For the velocity comparison, phase change, porous-medium resistance, magnetic effects, thermal radiation, and Williamson non-Newtonian effects were neglected. The profiles in Figure 3a closely overlap over 0 ≤ y ≤ 1, with zero velocity at both boundaries and a maximum dimensionless velocity of approximately u = 1.0 at y = 0.5. For the melting comparison in Figure 3b, viscous dissipation was additionally neglected, with Gr = 10⁴, Pr = 6.2, and St = 0.2. At times 0.05, 0.10, 0.20, and 0.30, the present liquid fractions are 0.129, 0.242, 0.404, and 0.548, compared with the supplied reference values of 0.132, 0.247, 0.411, and 0.557, respectively. The corresponding relative differences are 2.27%, 2.02%, 1.70%, and 1.62%. The comparisons show good agreement, supporting the accuracy of the present numerical implementation for the tested limiting-case flow and melting problems.

Figure 3. Comparison of present results with literature data: (a) velocity profiles against Abbas et al. [18] and (b) liquid-fraction evolution against Brent et al. [25].

4. Results and Discussion

The present study provides a dimensionless parametric investigation of Williamson non-Newtonian PCM melting in a porous-enclosure model under buoyancy-driven natural convection, an inclined magnetic field, thermal radiation, and viscous dissipation. The momentum and energy equations are coupled with the enthalpy–porosity formulation to describe the solid–liquid phase transition. Results are examined through velocity contours, streamlines, temperature distributions, liquid-fraction evolution, and melting-front progression. The effects of the Williamson parameter (We), magnetic parameter (M), radiation parameter (Rd), Eckert number (Ec), Darcy number (Da), Stefan number (St), mushy-zone resistance parameter (A), and Grashof number (Gr) are assessed using the skin-friction coefficient (Cf), Nusselt number (Nu), and average liquid fraction ${\overline{f}}_l$. The baseline values are Gr = 10⁴, Pr = 6.2, We = 0.04, M = 1, Da = 10⁻², Ec = 0.1, Rd = 0.5, A = 0.8, and St = 0.2. The values A = 0.5, 0.8, 1.1, and 1.4 represent the dimensionless mushy-zone resistance parameter defined in Equation (23), with all other parameters held at their baseline values during this variation.

The impact of the mushy-zone resistance parameter A is illustrated in Figure 4a through velocity profiles. It is observed that increasing A significantly suppresses the velocity within the domain. the peak velocity decreases from approximately 0.69 at A = 0.5 to 0.44 at A = 1.4, indicating weaker circulation. This damping is strongest where the liquid fraction is low and vanishes in fully liquid regions where f_l = 1, although changes in the surrounding flow can still affect velocity there. The effect of the radiation parameter Rd on the temperature field is depicted in Figure 4b. As Rd increases from 0.2 to 1.5, the temperature distribution increases significantly due to enhanced radiative heat transfer. The influence of viscous dissipation, represented by the Eckert number Ec, is shown in Figure 4c. Increasing Ec leads to a rise in temperature throughout the domain due to the conversion of kinetic energy into thermal energy. This effect is particularly pronounced near the walls, where velocity gradients are high. The influence of the Darcy number on the computed flow and thermal fields is discussed later in this section, with the corresponding skin-friction coefficient, Nusselt number, and average liquid fraction reported in Table 3.

Figure 4. (a–c). Influence of (a) mushy-zone parameter A on velocity profile and (b) thermal radiation Rd, and (c) Eckert number Ec on temperature distribution.

Table 3. Variation in skin-friction coefficient, Nusselt number, and average liquid fraction for different physical parameters at t = 1.

Parameter	Values	(Cf)	(Nu)	(fl)
(We)	0.02	0.782	3.12	0.84
	0.04	0.746	3.10	0.81
	0.06	0.706	3.02	0.76
	0.08	0.685	2.98	0.73
(M)	0.5	0.812	3.18	0.86
	1.0	0.746	3.10	0.81
	1.5	0.689	3.05	0.74
	2.0	0.638	2.97	0.69
(Da)	10−5	0.600	2.35	0.51
	10−4	0.610	2.90	0.61
	10−3	0.730	3.05	0.78
	10−2	0.746	3.10	0.81
(A)	0.5	0.803	3.21	0.87
	0.8	0.746	3.10	0.81
	1.1	0.704	3.04	0.76
	1.4	0.662	2.94	0.70
(Rd)	0.2	0.702	2.92	0.72
	0.5	0.746	3.10	0.81
	1.0	0.781	3.32	0.90
	1.5	0.824	3.54	0.98
(Ec)	0.1	0.746	3.10	0.81
	0.2	0.756	3.15	0.85
	0.3	0.766	3.20	0.89
	0.4	0.776	3.25	0.94
(St)	0.10	0.684	2.89	0.68
	0.20	0.746	3.10	0.81
	0.50	0.803	3.45	0.91
	1.00	0.891	3.78	0.99
(Gr)	103	0.742	2.96	0.76
	104	0.746	3.10	0.81
	105	0.758	3.22	0.88

The influence of the Williamson parameter on the velocity contours, streamlines, and temperature distributions is presented in Figure 5. An increase in We corresponds to stronger non-Newtonian behavior and enhanced shear-dependent viscosity. Consequently, the effective flow resistance increases, which suppresses fluid motion and weakens the buoyancy-induced circulation within the enclosure. This reduction in convective transport leads to lower velocity magnitudes and weaker streamlines. Because convection is diminished, thermal energy accumulates within the PCM, resulting in thicker thermal boundary layers and higher temperatures throughout the cavity. The reduced convective heat transfer causes a decrease in the Nusselt number, while the melting process becomes slower due to weaker circulation and reduced heat transport from the heated wall into the bulk PCM.

Figure 5. Effect of We on velocity contours, streamlines, and isotherms in the porous PCM enclosure for We = 0.02, 0.04, 0.06, and 0.08.

Figure 6 presents the effect of the magnetic parameter M on the flow and thermal fields for M = 0.5, 1.0, 1.5, and 2.0. A clear suppression of the velocity field occurs with increasing M. The interaction between the electrically conducting liquid PCM and the externally applied magnetic field generates a Lorentz force that opposes fluid motion. Increasing the magnetic parameter therefore introduces stronger electromagnetic damping. The streamline patterns confirm the weakening of the recirculating flow as M increases. Since natural convection is responsible for transporting thermal energy away from the heated region, suppression of the velocity field reduces convective heat transfer. An inclined magnetic field could, in principle, be generated using an oriented permanent-magnet assembly or electromagnet, although installation size, cost, and electromagnet power and cooling requirements may constrain implementation. The present model prescribes an idealized field; practical feasibility requires separate assessment of the magnet system and PCM electrical conductivity.

Figure 6. Effect of M on velocity contours, streamlines, and isotherms in the porous PCM enclosure for M = 0.5, 1.0, 1.5, and 2.

The isotherms consequently become more conduction-dominated, with thermal energy concentrated more strongly near the heated region. This behavior is also reflected in the engineering quantities. Increasing M from 0.5 to 2.0 reduces C_f from 0.812 to 0.638 and Nu from 3.18 to 2.97. Over the same range, the average liquid fraction decreases from 0.86 to 0.69. Thus, stronger magnetic damping not only suppresses fluid circulation but also delays the propagation of the melting front.

The influence of porous-medium permeability is illustrated in Figure 7 through variations in the Darcy number Da. Increasing Da from 10⁻⁵ to 10⁻² corresponds to increasing permeability and therefore decreasing resistance imposed by the porous matrix. For the baseline case, Da = 10⁻² represents a highly permeable porous matrix with relatively low flow resistance, corresponding to a permeability scale of K ≈ 10⁻²H².

Figure 7. Effect of Da on velocity contours, streamlines, and isotherms for Da = 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻².

Figure 7. Effect of Da on velocity contours, streamlines, and isotherms for Da = 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻².

As a result, the liquid PCM is able to circulate more freely, producing higher velocity magnitudes and stronger recirculating motion. The enhanced circulation modifies the thermal field by transporting heat more effectively from the heated boundary into the molten region. The isotherms become more strongly compressed near the heated wall, indicating an increased wall-temperature gradient and therefore an enhanced heat-transfer rate. Improved fluid circulation also transfers energy more efficiently toward the solid–liquid interface and promotes melting. The quantitative data demonstrate this enhancement clearly. As Da increases from 10⁻⁵ to 10⁻², C_f increases from 0.600 to 0.746, Nu increases from 2.35 to 3.10, and the average liquid fraction rises from 0.51 to 0.81. Thus, permeability plays a beneficial role in both natural-convection strength and thermal-storage charging performance.

Figure 8 illustrates the influence of the Stefan number St for St = 0.1, 0.2, 0.5, and 1.0. The Stefan number represents the relative importance of sensible heating compared with the latent heat associated with phase transition. A larger value of St corresponds to a smaller latent-heat requirement relative to the available sensible thermal energy. The isotherms show a pronounced increase in temperature penetration as St increases. At relatively small St, a substantial fraction of the supplied thermal energy is consumed in overcoming the latent heat of fusion, which limits the temperature rise. At larger St, less energy is required for the phase transition relative to sensible heating, and thermal energy spreads farther into the PCM domain. Consequently, melting proceeds more rapidly. Table 3 confirms that the Stefan number has a particularly strong influence on thermal and phase-change performance. Increasing St from 0.1 to 1.0 increases C_f from 0.684 to 0.891, Nu from 2.89 to 3.78, and the average liquid fraction from 0.68 to 0.99. Among the parameter ranges examined in Table 3, St produces one of the strongest enhancements in the melting response. The observed sensitivity to St applies only to the ranges examined. In practical systems, St is constrained by the PCM properties and allowable operating temperatures; changing it may also affect other dimensionless parameters.

Figure 8. Effect of St on velocity contours, streamlines, and isotherms for St = 0.1, 0.2, 0.5, and 1.

Figure 9 illustrates the influence of the Grashof number on the velocity contours, streamlines, and isotherms. It is observed that increasing (Gr) significantly enhances the buoyancy force generated by thermal gradients, thereby strengthening the natural convection flow. Consequently, the velocity magnitude increases throughout the enclosure, as evidenced by the higher contour levels and intensified circulation patterns. The streamline plots reveal the formation of a stronger primary vortex that occupies a larger portion of the cavity as (Gr) increases, indicating enhanced fluid mixing and recirculation. Moreover, the isotherms become progressively distorted from their conductive configuration, demonstrating the transition from conduction-dominated to convection-dominated heat transfer.

Figure 9. Effect of Grashof number (Gr = 10³, 10⁴, and 10⁵) on velocity contours, streamlines, and isotherms.

At lower (Gr), the relatively parallel temperature contours are consistent with weaker convection; at higher (Gr), their deformation indicates greater redistribution of heat by fluid motion. These qualitative changes should not be interpreted as a proportional enhancement of wall heat transfer. Specifically, increasing (Gr) from 10³ to 10⁵ raises Nu from 2.96 to 3.22 (8.8%) and Cf from 0.742 to 0.758 (2.2%), whereas the average liquid fraction increases from 0.76 to 0.88. Thus, the reported results show stronger circulation and increased melting extent, accompanied by a modest improvement in wall heat transfer.

The temporal evolution of the liquid fraction is illustrated in Figure 10. At τ = 0, the PCM is entirely solid, with (f_l = 0) and a uniform temperature (T_i < T_s), consistent with Equations (20) and (29). After heating begins, a thin liquid layer develops adjacent to the heated wall, while most of the PCM remains solid. As time progresses, heat penetrates farther into the domain, the liquid region expands, and the solid–liquid interface moves away from the heated boundary. At τ = 1, the baseline average liquid fraction reaches 0.81, indicating substantial but incomplete melting, with approximately 19% of the PCM remaining solid. The evolution is consistent with an initially conduction-led melting process and an increasing contribution from convection as the liquid region develops. From a thermal energy storage perspective, the increasing liquid fraction represents progressive charging and greater latent-energy storage. However, this increase alone does not establish improved storage efficiency or demonstrate an enhancement attributable to the porous medium.

Figure 10. Temporal evolution of the liquid fraction on velocity contours, streamlines, and isotherms.

Table 3 summarizes the combined effects of the governing parameters on the skin-friction coefficient C_f, Nusselt number Nu, and average liquid fraction f_l. Increasing the Williamson parameter (We), magnetic parameter (M), and mushy-zone resistance parameter (A) decreases these quantities, indicating reduced wall shear, wall heat transfer, and melting extent. Conversely, increasing (Da), (Rd), (Ec), (St), and (Gr) increases the reported quantities. Specifically, increasing (Gr) from 10³ to 10⁵ raises (Nu) from 2.96 to 3.22 (8.8%) and (C_f) from 0.742 to 0.758 (2.2%), while (f_l) increases from 0.76 to 0.88. These results indicate modest changes in the wall quantities alongside a larger change in melting extent. Stronger circulation does not necessarily produce a proportional increase in wall heat transfer because energy transport also depends on the evolving temperature field and phase-change interface. The contribution of latent-heat absorption to this behavior requires assessment using transient heat-transfer results. Among the parameter ranges examined, (St) produces the largest increases in (Nu) and (f_l), from 2.89 to 3.78 and from 0.68 to 0.99, respectively.

5. Conclusions

This study examined the unsteady melting behavior of a Williamson non-Newtonian phase change material (PCM) embedded in a porous medium under the combined effects of an inclined magnetic field, thermal radiation, viscous dissipation, and porous resistance. The enthalpy–porosity method was employed to model the solid–liquid phase transition, while the governing momentum and energy equations were solved numerically using the finite difference method. The influence of key physical parameters on flow characteristics, heat transfer, and melting performance was analyzed through velocity contours, streamlines, isotherms, liquid fraction evolution, skin-friction coefficient, and Nusselt number. The major findings are summarized as follows:

•

Increasing the Williamson parameter suppresses fluid motion due to enhanced non-Newtonian resistance. Consequently, the skin-friction coefficient, Nusselt number, and average liquid fraction decrease, indicating weaker convective heat transfer and slower melting progression.

•

Increasing the magnetic parameter strengthens electromagnetic damping and suppresses circulation. The associated decreases in the Nusselt number and average liquid fraction indicate reduced wall heat transfer and melting extent.

•

Increasing the Darcy number reduces porous resistance and promotes fluid motion. Higher permeability is associated with increased skin-friction and Nusselt numbers and a larger molten fraction.

•

Increasing the mushy-zone resistance parameter suppresses motion in the partially melted region. The average liquid fraction decreases from 0.87 to 0.70 over the examined range, representing a relative reduction of approximately 19.5%.

•

Increasing the radiation parameter raises the interior temperature and promotes thermal penetration.

•

Increasing the Eckert number strengthens the dissipative heating contributions represented in the energy equation. The computed results show higher temperatures, Nusselt numbers, and liquid fractions.

•

The Stefan number produces the largest increases in the Nusselt number and average liquid fraction.

•

Increasing the Grashof number enhances buoyancy-driven flow, strengthens vortex formation and fluid mixing, and promotes the transition from conduction-dominated to convection-dominated heat transfer.

The present model has been assessed against simplified benchmark problems; however, quantitative validation against published non-Newtonian or nanoparticle-enhanced PCM melting results remains outstanding. Future work will evaluate appropriate limiting cases under matched conditions, using average liquid fraction at identical dimensionless times and total melting time as comparison measures. Further validation will also address the coupled magnetic and porous-medium effects. The current findings describe parameter-dependent trends within the adopted model, while material-specific design applications require additional numerical and experimental assessment. Potential extensions include temperature-dependent properties, nanoparticle-enhanced PCMs and three-dimensional configurations.