Effects of nonuniform heating and viscosity variation on transient natural convection around a vertical plate in a porous medium Open Access

C_p

specific heat at constant pressure

Da

Darcy number

Gr

Grashof number

g

gravitational acceleration

K′

permeability

k

thermal conductivity

L

characteristic length

Nu, $Nū$

local and average Nusselt numbers

Pr

Prandtl number

T

temperature of liquid (nondimensional)

T′

temperature (dimensional)

t

time (nondimensional)

t′

time

U, V

velocity components parallel and perpendicular to the plate (nondimensional)

u′, v′

velocity components parallel to and perpendicular to the plate

x, y

distances parallel and perpendicular to the plate (nondimensional)

x′, y′

distances parallel and perpendicular to the plate

β

coefficient of thermal expansion

ζ

viscosity parameter

μ, $μ̄$

dynamic and temperature-dependent viscosities

μ₀

characteristic viscosity

ν

kinematic viscosity

ρ

fluid density

τ, $τ̄$

local and average skin frictions

ϕ

porosity of porous medium

s

at the surface of the plate

∞

away from the plate

The study of the transient free convective flow of a viscous incompressible fluid over a vertical plate has attracted significant interest from researchers because of its great applicability throughout engineering and technology. This phenomenon is relevant in a variety of fields, including subsurface dynamics, cooling of nuclear reactors, heated wires, glass and polymer assembly, geophysics, geothermal energy, thermal insulation, petroleum and vapor zones, groundwater pollution, oil cistern engineering, underground distribution of chemical waste, and packed bed storage tanks. Siegel¹ examined unsteady free convective flow over a vertical plate using an integral method for two thermal boundary conditions. Later, Hellums and Churchill² explored transient effects in the case of a vertical plate by employing an explicit finite-difference technique. Takhar et al.³ presented a numerical solution for transient free convective flow over a vertical plate with variable surface temperature. A numerical analysis of transient viscous dissipative flows past an isothermal slanting plate was carried out by Ganesan and Palani.⁴ Kumar and Singh⁵ analyzed unsteady flow over a vertical plate in the presence of an induced magnetic field. Abu Zeid et al.⁶ investigated the effects of thermal radiation and mass transfer on unsteady natural convective flow over a moving vertical porous plate by means of numerical simulation using a cubic B-spline collocation technique.

From an application perspective, porous media are significant in areas such as the solidification of binary alloys, oil reservoir management, and geophysics. Detailed descriptions of the properties of porous media and their uses can be found in the volumes edited by Ingham and Pop^7,8 and by Vafai^9,10 and in the textbook by Nield and Bejan.¹¹ Takhar et al.¹² investigated magneto-convective flow over a non-isothermal inclined surface incorporating various suction and injection slots and embedded in a high-porosity medium. Chandran et al.¹³ explored convective flow within a partially porous cavity, examining both ramped and isothermal heating conditions on the cavity wall. Sarveshanand and Singh¹⁴ studied the effect of induced magnetism on magneto-convective flow between porous parallel plates. They found that the velocity of the fluid was reduced by the suction parameter. Several studies^15–18 have presented various mathematical models of flow in porous media under different physical conditions. Expanding on these approaches, Salahuddin et al.^19–21 focused on heat and mass transfer in channels with Carreau and Carreau–Yasuda nanofluids. Ali and co-workers^22–26 have explored MHD effects, heat transfer, and viscoelastic fluid behavior in roll coating processes, employing analytical, numerical, and AI-based approaches to enhance fluid flow analysis.

In most of the studies reviewed above, the viscosity was assumed to be constant. In practical situations, however, viscosity may be a function of temperature, pressure or distance.²⁷ The cohesive force between fluid molecules dominates the motion of a viscous fluid. Also, the cohesive force decreases with increasing temperature, consequently reducing the fluid viscosity. Therefore, from the application point of view, in describing fluid motion, we need to consider a viscosity that varies with temperature. Hossain et al.²⁸ investigated the effect of varying viscosity on natural convective flow through a consistently heated erect porous wall with thermal radiation and constant suction. Pantokratoras²⁹ investigated laminar natural convection along a vertical isothermal plate, accounting for the effects of varying viscosity. Palani and Kim³⁰ presented a numerical investigation of free convection over a vertical plate with variable viscosity and thermal conductivity. Seddeek³¹ investigated non-Darcian forced convective flow over a vertical flat plate in a porous medium with temperature-dependent viscosity. Astanina et al.³² studied transient natural convection in a porous cavity with a heat source. Florence³³ explored the behavior of a non-Newtonian fluid with variable viscosity and thermal conductivity in a porous medium. Recent studies by Ali and co-workers^34–36 have employed analytical and computational techniques to examine the effects of temperature-dependent viscosity in Eyring–Powell and viscoelastic fluids in the context of roll-rotating systems, web coating, and calendering.

Molla and Hossain³⁷ studied the influence of double-diffusive convection with temperature-dependent viscosity over an isothermal sphere. Alam et al.³⁸ and Makinde et al.³⁹ performed detailed studies of transient free convective magnetohydrodynamic flow on an inclined porous plate in the presence of thermophoresis and radiation effects. Chou et al.⁴⁰ studied the effects of temperature-dependent viscosity on free convective flow in a porous medium between two concentric spheres. Astanina et al.⁴¹ and Umavathi et al.⁴² investigated unsteady flows with temperature-dependent viscosity in a porous medium contained in a square cavity and in a vertical channel, respectively. Gupta et al.⁴³ studied the effects of temperature-dependent viscosity on magnetoconvective heat transfer in the presence of rotational speed modulation. They found that the viscosity factor increased the rate of heat transfer.

The investigation of convection in the presence of nonuniformly heated surfaces is important because of its relevance to many fields, such as the cooling of electronic components and solar collectors and various processes in the food industry. Roy and Basak⁴⁴ carried out a numerical analysis of free convective flows in square cavities with uniformly and nonuniformly heated walls. They found a sinusoidal variation in the rate of heat transfer in the case of nonuniform heating. Hernandez and Zamora⁴⁵ studied the effects of variable physical properties and nonuniform heating on natural convection in both symmetric and asymmetric vertical channels using numerical techniques. Natarajan et al.⁴⁶ performed numerical investigations of free convective flow within a trapezoidal enclosure, considering both nonuniform and uniform heating of the bottom wall. Several numerical studies^47–50 have examined nonuniform heating conditions at the boundaries in different geometries and fluid systems. Liu et al.⁵¹ investigated transient heat transfer in water heat pipes under nonuniform heating conditions.

An extensive review of the literature identifies a notable gap in existing research where previous studies have not explored the effects of nonuniform heating boundary conditions and temperature-dependent viscosity on transient free convection, although it is known, for example, that under nonuniform heating conditions, the rate of heat transfer will be different at different cross-sections of a channel containing a porous medium. The aim of the present study is to fill this gap and obtain valuable insights for practical applications by exploring transient free convective flow over a vertical wall in a porous medium, considering both nonuniform heating and temperature-dependent viscosity.

This is a novel investigation into the effects of temperature-dependent viscosity and nonuniform thermal boundary conditions on flow dynamics and heat transfer. Unlike those of previous research, the result presented here demonstrate that viscosity variations strongly influence velocity profiles and thermal distribution. Moreover, it is found that stronger thermal buoyancy accelerates fluid motion, whereas higher Prandtl numbers restrict thermal diffusion owing to reduced thermal conductivity, leading to a thinner thermal boundary layer and altered heat transfer characteristics. The governing equations are numerically solved using the Crank–Nicolson finite difference technique.⁵² The effects of key physical parameters, including the viscosity parameter and the Darcy, Prandtl, and Grashof numbers, and of time, on temperature, velocity, skin friction, and Nusselt number are illustrated through graphical representations. The main results reveal that a higher Grashof number intensifies flow velocity, while an increased Darcy number facilitates fluid motion within the porous medium. Enhanced thermal buoyancy leads to a rise in the Nusselt number, whereas greater viscosity variation reduces skin friction. Additionally, the numerical results for the average Nusselt number and average skin friction are summarized in tabular form for various parameter values. The insights obtained here have significant applications in thermal management systems, geothermal energy utilization, and the optimization of advanced heat exchanger designs.

We consider transient viscous incompressible free convective flow over a semi-infinite vertical plate embedded in a porous medium. A nonuniform heating condition is assumed at the plate, together with a temperature-dependent viscosity. The x axis is oriented vertically upward along the plate, while the y axis is perpendicular to the plate. The fluid and the plate are initially at the same temperature, denoted by $T∞′$⁠. When t′ > 0, the plate’s temperature rises in a nonuniform manner given by $T∞′+Ts′−T∞′1−cos2πx′/L$ and then remains at that level. Figure 1 depicts the physical configuration of the problem.

The expression for the local skin friction evaluates the shear stress exerted by the fluid on the plate’s surface, where μ is the dynamic viscosity and ∂U/∂y is the velocity gradient near the wall. This parameter is crucial for assessing drag forces and flow resistance in boundary layer analysis. Similarly, the local Nusselt number is a dimensionless measure of heat transfer at the surface, indicating the ratio of convective to conductive heat transfer. Higher values of Nu indicate improved convective heat transfer efficiency.

The average Nusselt number represents the overall heat transfer efficiency across the plate’s surface, providing insights into the system’s thermal performance. The average skin friction represents the mean shear stress exerted by the fluid, which is crucial for analyzing flow resistance and improving system design.

In this section, we present the numerical results obtained by solving the governing equations. The model examines transient free convection within a porous material, considering a temperature-dependent viscosity and nonuniform heating along a semi-infinite vertical plate. Various physical parameters have been identified during the modeling process, including Prandtl number, Grashof number, Darcy number, and viscosity parameter, as well as time. The computational results are illustrated through graphical plots of velocity, temperature, local shear stress τ, and local Nusselt number Nu at several cross-sections x = 0.25, 0.5, and 0.75. The numerical computations have been carried out over a range of physical parameters:^32,49,50 Grashof number $2≤Gr≤10$⁠, cross-section $0.25≤x≤0.75$⁠, time $0.5≤t≤15$⁠, Darcy number $0.001≤Da≤0.008$⁠, Prandtl number $0.733≤Pr≤7$⁠, and viscosity factor $0≤ζ≤4$⁠. Unless specified otherwise, the common parameter values used in the computations are Da = 0.005, Pr = 0.733, Gr = 5, t = 0.7, and ζ = 2.

Velocity and temperature plots for different values of the Darcy number (porous material parameter) Da are depicted in Figs. 2(a) and 2(b), respectively. As can be seen, increasing the small values of the Darcy number $Da=0.001,0.005,0.008$⁠) leads to a rise in fluid velocity and a decrease in temperature. This behavior can be attributed to the increase in permeability associated with higher Darcy numbers, allowing the fluid to flow more easily through the porous medium. Consequently, the velocity increases, while the temperature decreases. Additionally, both the thermal and momentum boundary layers expand with increasing cross-sectional area. A higher Darcy number signifies increased permeability, which lowers resistance to fluid flow and enhances velocity. With greater permeability, convective transport improves, resulting in a reduction in fluid temperature. The momentum boundary layer grows as fluid motion penetrates deeper, while the thermal boundary layer extends owing to enhanced heat transfer. An increase in cross-sectional area further expands both boundary layers, affecting overall heat transfer.

Figure 3 illustrates the velocity and temperature profiles for different values of the Grashof number Gr at several cross-sections $x=0.25,0.50,and0.75$⁠) along the plate. It can be seen that an increase in Gr results in higher velocity profiles, while the temperature profiles exhibit the opposite trend. As Gr increases, the thermal buoyancy force intensifies, leading to a noticeable rise in velocity. Notably, as Gr grows, both the momentum and thermal boundary layers rapidly become thinner, allowing the velocity to approach the free-stream value more quickly. Moreover, the free-stream velocity at x = 0.75 is higher than at other cross-sectional locations. Physically, a higher Grashof number indicates intensified buoyancy forces, which drive stronger fluid motion and increase velocity. As natural convection strengthens, heat is transferred more efficiently, causing a temperature drop near the surface. The dominance of buoyancy reduces the momentum boundary layer thickness, while the thermal boundary layer also contracts owing to enhanced heat dissipation. Variations in the cross-sectional position influence the flow behavior, with greater free-stream velocity at larger values of x owing to accumulated buoyancy effects.

The effects of the viscosity factor ζ on the velocity and temperature profiles are shown in Fig. 4. As can be seen from Fig. 4(a), the velocity profile rises with increasing viscosity parameter $ζ=0,2,4$ at all the considered cross-sections. In Fig. 4(b), the temperature exhibits a decreasing trend with increasing ζ at cross-sections x = 0.25 and 0.50, although at x = 0.75, it exhibits an increasing trend. This is because the fluid viscosity decreases as the temperature rises. Additionally, Fig. 4(a) reveals that the highest free-stream velocity is observed for ζ = 2.0 at the cross-section x = 0.75. From a physical perspective, ζ characterizes the dependence of viscosity on temperature, affecting both flow and heat transfer. A higher ζ lowers viscous resistance, leading to increased velocity and changes in boundary layer behavior. The temperature drop at smaller values of x is due to enhanced convective heat transfer, while at x = 0.75, reduced viscosity causes greater heat retention, influencing thermal distribution and energy transport.

Figures 5(a) and 5(b) illustrate the influence of the Prandtl number Pr on the velocity and temperature fields, respectively. As shown in Fig. 5(a), the velocity decreases as the Prandtl number increases. This occurs because a higher Pr leads to increased kinematic viscosity and reduced thermal diffusivity, thereby lowering the fluid velocity. It is clear from Fig. 5(b) that when Pr rises, the temperature falls. Physically speaking, with increasing Pr, the thermal conductivity decreases, reducing heat transfer. Furthermore, both the momentum and thermal boundary layers become thinner with increasing Pr. From a combined analysis of Figs. 2(b), 3(b), 4(b), and 5(b), it can be concluded that with nonuniform heating, the temperature is highest at the cross-section x = 0.5, lower at x = 0.75, and lowest at x = 0.25.

Figures 6(a) and 6(b) depict the variations in the velocity and temperature profiles, respectively, with y at different times t = 0.5, 0.7, 1, 2, 5, 7, 10, and 15 at a fixed cross-section x = 0.5. It can be observed that the maximum magnitudes of both velocity and temperature occur near the plate. As time increases, both the momentum and thermal boundary layers expand. Additionally, the values corresponding to y → ∞ are found farther from the plate. Beyond a certain point, no further changes in velocity and temperature occur with time, indicating that the system reaches a steady state.

Figures 7(a)–7(d) show the local skin-friction profiles vs x for different values of Da, Gr, ζ, and Pr, respectively. It can clearly be seen that the peak local skin-friction values are attained in the central zone of the plate. Moreover, the local skin friction increases with higher values of Da [Fig. 7(a)] and Gr [Fig. 7(b)], whereas it decreases with increasing values of ζ [Fig. 7(c)] and Pr [Fig. 7(d)]. Physically, the local skin friction represents the interaction between fluid motion and the plate surface. Higher Da enhances permeability, reducing resistance and increasing shear stress. Similarly, greater Gr strengthens buoyancy forces, accelerating fluid flow and increasing friction at the boundary. By contrast, increasing ζ lowers viscosity, weakening shear forces and reducing friction. A higher Pr limits thermal diffusion, stabilizing the boundary layer and further decreasing skin friction.

Figures 8(a)–8(d) illustrate the local Nusselt number profiles vs x for different values of Da, Gr, ζ, and Pr, respectively. It can be seen that that increases in all four parameters Da, Gr, ζ, and Pr lead to an increase in the local Nusselt number. From a physical perspective, the Nusselt number represents the effectiveness of convective heat transfer relative to conduction. A higher Da increases permeability, reducing resistance to fluid flow and enhancing heat transfer. Greater Gr strengthens buoyancy-driven convection, facilitating the movement of thermal energy. An increase in ζ lowers viscosity, accelerating fluid motion and increasing heat dissipation. Likewise, a higher Pr reduces thermal diffusivity, steepening the temperature gradient and intensifying convective heat exchange. These variations highlight the intricate relationship between flow dynamics and heat transfer efficiency.

Table I presents the numerical values of the average Nusselt number and average skin friction for different parameter values and at different times. It is evident that for given values of the parameters, with increasing time, the average skin friction increases, but the average Nusselt number decreases. At a fixed time t = 0.7, increases in both the Darcy and Grashof numbers increase the average skin friction, while increases in the Prandtl number and viscosity parameter decrease it. Furthermore, at t = 0.7, increases in all the considered parameters lead to an increase in the average Nusselt number. From a physical standpoint, the average skin friction represents surface shear stress accumulation, while the average Nusselt number represents overall heat transfer efficiency. A longer time duration intensifies flow interactions, leading to a higher shear stress but a reduced thermal gradient, thereby lowering the Nusselt number. Increasing Da enhances permeability, allowing easier fluid movement and raising shear stress. Similarly, higher Gr amplifies buoyancy-driven convection, accelerating flow and increasing friction. Conversely, a rise in Pr limits thermal diffusion, stabilizing the boundary layer and reducing shear stress, while an increase in ζ lowers viscosity, weakening momentum transfer and further decreasing friction. The overall rise in the average Nusselt number for increases in all the parameters highlights the influence of permeability, buoyancy, and viscosity on convective heat transfer.

This article has investigated transient free convection along a semi-infinite vertical plate in a porous medium, considering nonuniform heating and temperature-dependent viscosity. A numerical solution for the flow model has been obtained by applying the Crank–Nicolson technique to solve the nondimensionalized momentum and energy equations. The results have been presented graphically, highlighting the effects of significant physical parameters, namely, the viscosity parameter and the Darcy, Grashof, and Prandtl numbers, and of time on the flow characteristics. The key findings of this study can be summarized as follows:

1. An increase in the Prandtl number reduces both the velocity and temperature profiles.

2. Increases in the Grashof and Darcy numbers decrease the temperature field, while they have the opposite effect on the velocity profile.

3. The temperature decreases with an increasing viscosity parameter at different cross-sections, whereas the velocity shows an increasing trend.

4. Both velocity and temperature increase with time, eventually reaching a steady state.

5. The local convective heat transfer coefficient and local skin friction increase with increasing Darcy and Grashof numbers.

6. The thicknesses of both the momentum and thermal boundary layers decrease with increasing values of all the considered parameters.

7. The average skin friction increases with increasing Da and Gr, but decreases with increasing Pr  and ζ.

8. The maximum free-stream velocity is found at the fixed cross-section x = 0.75 for ζ = 2.0.

This study has some limitations that could be addressed in future work. The analysis here is based on a two-dimensional flow model, which does not account for three-dimensional effects. It assumes laminar flow, excluding turbulence, and only considers a temperature-dependent viscosity, neglecting other fluid property variations. The analysis does not include factors such as thermal radiation, chemical reactions, or variable thermal conductivity. Additionally, ideal boundary conditions are assumed, overlooking surface roughness and permeability variations that can affect flow dynamics.

Future work could extend the model to three-dimensional flow, incorporate turbulence effects, and explore hybrid nanofluids and non-Newtonian fluids. Including thermal radiation, chemical reactions, and variable thermal conductivity would enhance the analysis. Machine learning techniques could optimize heat transfer in porous media, and scaling up to industrial applications such as geothermal systems and heat exchangers would broaden the study’s practical impact.

The analysis presented here enhances understanding of transient free convection in porous media with temperature-dependent viscosity and nonuniform heating, which is crucial for various engineering and industrial applications. The results help in optimizing heat transfer efficiency in systems such as those for cooling of electronic devices, in solar energy devices, and in geothermal processes. Additionally, it provides a framework for improving insulation materials, designing efficient heat exchangers, and improving fluid flow control in biomedical and environmental applications. The study also contributes to the theoretical modeling of convection phenomena, aiding researchers in developing more accurate and efficient thermal management solutions.

The authors have no conflicts to disclose.

Anurag: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Anand Kumar: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Richa Rajora: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Vijay Kumar Sukariya: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Ashok Kumar Singh: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal).

The authors confirm that the data supporting the findings of this study are available within the article.