Mathematical modelling of Darcy–Forchheimer MHD Williamson nanofluid flow above a stretching/shrinking surface with slip conditions

Nomenclature

U w

stretching velocity ( m s − 1 )

u and v

x and y components of the velocity ( m s − 1 )

T w

surface temperature

Ec

Eckert number

T ∞

ambient temperature

ψ

stream function

T w

wall temperature

k

thermal conductivity

Re x

Reynold number

ρ nf

nanofluid density

λ = 0

static state of the surface

B 0

strength of the magnetic field

( ρ c p ) nf

capacity of heat in nanofluids

λ > 0

stretching state of the surface

λ < 0

shrinking state of the surface

η

similarity variable

ϕ

volume fraction of the nanoparticles

f

base fluid

PDEs

partial differential equations

HAM

homotopy analysis method

S

suction parameter

x , y

plane coordinate axis

C f x

coefficient of skin friction

ξ

velocity slip parameter

F r

Darcy–Forchheimer number

κ

vertex viscosity

Nu x

Nusselt number

ξ T

thermal slip parameter

α

thermal diffusivity

δ

Williamson parameter

f ′

velocity without dimension

K

couple stress parameter

J

current’s density

θ

temperature without dimension

Pr

Prandtl number

τ w

shear stress of the wall surface

J × B

Lorentz force

MWCNT

multi-wall carbon nanotube

ODEs

ordinary differential equations

SWCNT

single-wall carbon nanotube

1 Introduction

One type of fluid dynamics study that uses nanofluids on porous surfaces is the Darcy–Forchheimer nanofluid flow. It characterizes the flow properties by using both Darcy’s equation and Forchheimer extension. When Darcy–Forchheimer nanofluid flow is taken into account, the mutual impact of thermal properties improved by the nanoparticles and the flow dynamics via the porous surface with both Darcy and Forchheimer contributions is analysed. This work has significant effects for engineering fields such as improved oil recovery, geothermal energy extraction, and sophisticated cooling system design. In order to study these flows, scientists typically employ complex mathematical models and numerical simulations that account for factors like heat transmission, interactions between fluid particles, and the influences of many physical constants. Ganesh et al. [1] studied the Darcy–Forchheimer flow of a hydromagnetic nanofluid over a stretching/shrinking sheet in a thermally stratified porous medium for second-order slip, viscosity, and ohmic dissipation effects. Seddeek [2] examined how viscous dissipation and thermophoresis affected Darcy–Forchheimer mixed convection in a fluid-saturated porous medium. Hayat et al. [3] examined the Darcy–Forchheimer function and Cattaneo–Christov heat flow with different thermal conductivities. Pal and Mondal [4] examined how different viscosities and non-uniform heat sources and sinks cause species to hydromagnetically disperse over a porous Darcy–Forchheimer medium. Ahmed et al. [5] examined the use of the Darcy–Forchheimer relation in numerical computation for gyrotactic bacteria in magnetohydrodynamics (MHD) radiative Eyring–Powell nanomaterial flow via a static/moving wedge. Rasool et al. [6] studied the role of the Rosenland radiative process in the presence of Lorentz and Darcy–Forchheimer forces. Khan and Alzahrani [7] examined silicon dioxide and molybdenum disulphide-containing Darcy–Forchheimer porous media, and the impacts of radiation and free convection, as well as second-order velocity slip and entropy production highlighted. Alotaibi and Eid [8] examined the thermal analysis of 3D electromagnetic radiative nanofluid flow using blowing and suction according to the Darcy–Forchheimer methodology. A single layer of carbon atoms is rolled into a cylinder to form single-walled carbon nanotubes (SWCNTs), whereas multiple layers of carbon atoms are rolled into concentric cylinders to form multi-walled carbon nanotubes (MWCNTs). These are the two basic forms of carbon nanotubes (CNTs). Because of their extremely high thermal conductivity, CNTs can enhance the nanofluid’s total capacity for heat transfer. Convective heat transfer can be improved by CNTs’ special qualities, which make these nanofluids appropriate for use in heat exchangers and cooling systems. Electronics cooling, automotive cooling systems, and other industries, where effective heat removal is essential, use CNTs. The enhanced heat transfer properties of CNT nanofluids make them suitable for use in a range of heat exchanger types used in industrial operations. Potential uses for higher efficiency in geothermal, solar, and other renewable energy sources. Because of their unique properties, CNTs may be used in medical imaging and drug delivery. Islam et al. [9] examined how MHD Darcy–Forchheimer fluid is produced by gyrotactic Casson nanoparticle bacteria on a stretched surface with convective boundary conditions. Hayat et al. [10] examined how CNTs affected a nonlinear stretching sheet in the stagnation point forward with different thicknesses. Khan et al. [11] studied the Darcy–Forchheimer hybrid nanofluid flow (MoS₂, SiO₂) that produces entropy. Huda et al. [12] examined the effects of the Cattaneo–Christov model on the Darcy–Forchheimer flow of the ethylene glycol base fluid across a moving needle. Hussain et al. [13] examined the Darcy–Forchheimer elements of the Keller box approach for a CNT nanofluid passing via a stretched cylinder. Sreedevi et al. [14] examined mass and heat transmission CNTs, both single and multiwall, around a vertical cone utilizing convective boundary conditions and MHD. Cho et al. [15] examined the electrical and electrochemical sensors based on CNTs for monitoring water quality. Bruzaca et al. [16] investigated the thermal and electrical characteristics of composites made of CNT polymers in different aspect ratios. Jiang et al. [17] investigated the nanofluid based on CNTs’ effective heat conductivity. Zhang et al. [18] presented an overview of the advantages and disadvantages of CNTs, a class of nanomaterials that is becoming more and more popular. Afsarimanesha et al. [19] provided a review of the most current advancements in nanogenerators based on CNTs. Rehman et al. [20] examined the effects of viscous dissipation on the time-dependent MHD Casson nanofluid on stretched surfaces. Rehman et al. [21], considering the effects of heat radiation and viscous dissipation, investigated the MHD stagnation point flow analytically across a stretching surface. Rehman et al. [22] examined the blood-based nanofluids in mixed convection boundary layer: heat transfer study under the influence of viscous dissipation. Rehman et al. [23] examined the impact of Marangoni convection on blood-based hybrid nanofluid, including CNTs that couple stress and visibly dissipate heat transfer. Shah et al. [24] studied micropolar hybrid-nanofluid flow with melting heat transfer over a radially stretchable porous rotating disc with Darcy–Forchheimer MHD rotational symmetry. Jameel et al. [25] studied radiative hybrid nanofluid flow with Hall effect over exponential stretching/shrinking plate: statistical and entropy optimization modelling. Shah et al. [26] studied fourth-grade fluid model over a Riga plate, gyrotactic microorganisms, and heat transfer study of water conveying MHD SWCNT nanoparticles. Shah et al. [27] performed numerical analysis of radiative hybrid nanofluid flow across a power law stretching/shrinking sheet with a suction effect using sodium alginate, alumina, and copper. Khan et al. [28] studied heat generation from natural convection heat transfer of a hybrid nanofluid in a permeability quadrantal enclosure. Rooman et al. [29] studied Ree–Eyring nanofluid flow in a conical gap between porous rotating surfaces using statistical modelling with Hall effect and entropy production. Baleanu et al. [30] studied a novel fractional model and non-singular derivative operator-based optimum control of tumor-immune surveillance. Ebrahimzadeh et al. [31] studied enhancing water pollution management through a comprehensive fractional modelling framework and optimal control techniques. Baleanu et al. [32] studied fractional investigation of time-dependent mass pendulum. Rehman et al. [33,34] studied the boundary layer flow of both nanofluid and hybrid nanofluid semi-numerically. Heat transfer, material processing, and biological systems are just a few of the scientific and engineering domains where the study of nanofluid flow across a stretching or contracting surface is important. Here are the main ideas emphasizing their significance. Glass fibre drawing, metal casting, and polymer extrusion all involve stretching surfaces.

The primary goal of this article is to study the mathematical model that incorporates the Darcy–Forchheimer model, MHD, slip conditions, and viscous dissipation effects to describe the flow of Williamson CNT nanofluid over a stretching/shrinking surface. We have considered the combined effects of thMWCNT and SWCNT nanoparticles with water as a base fluid in the porous medium. The current study, according to the authors, is novel and inventive. By using similarity transformations, the system of PDEs is transformed into a system of ODEs. With the help of Mathematica software, the transformed equations are semi-numerically solved using the homotopy analysis method. The impacts of various applied parameters are shown graphically as well as in table form. This study is intended to assist novice researchers studying nanofluids, especially those examining several solutions that result from the nonlinearity of equations at various values of the applied parameters.

2 Mathematical formulation

Let us consider 2D MHD Darcy–Forchheimer Williamson (MWCNT–SWCNT/water) micropolar nanofluids flowing across a stretching/shrinking surface. The location of the coordinate flow model, considering the heat transfer properties of the mechanism, is depicted in Figure 1. The energy equation makes use of the viscous dissipation effect. J × B is the Lorentz force, where J is the current density. The Ohm’s law is J = σ ( E + V × B ) , where B is the magnetic field and E is the electric field, and we assume that E = 0 .

Figure 1

Geometry of the problem flow.

The equations of the flow modelled problem, constructed on the above-mentioned hypothesis, are expressed as follows:

The flow problem boundary conditions are as follows:

where the velocity components in the y and x axes, respectively, are represented by the variables u and v . Also, γ nf = μ nf μ f + κ 2 is the vertex viscosity, and F = C b x κ is the coefficient of nonuniformed porous media. J is the density of micro-inertia, and B 0 is the strength of the magnetic field. λ = 0 denotes a static state of the surface, λ > 0 denotes a stretching state of the surface, and λ < 0 denotes a shrinking state of the surface, there is consideration for both stretching and shrinking situations of the surface.

The thermophysical properties of the nanofluids are as follows:

The following similarity transformation is used to convert the PDEs (1)–(4) into ODEs with boundary conditions (5):

Using similarity transformations from their dimension system to a non-dimensional system and the thermophysical properties of the nanofluid, the fundamental flow equations for continuity, velocity, and temperature are changed. When Equation (7) is substituted in Equations (2)–(5), a balanced continuity equation is obtained:

with modified BCs given by

where δ = 2 a ν a Γ is the Williamson number, Pr = μ f k f ( c p ) f is the Prandtl number, Ec = u w 2 ( c p ) f ( T w − T ∞ ) is the Eckert number, M = σ B 0 2 c ρ f is the magnetic parameter, κ is the vertex viscosity, F r = c b κ is the Darcy–Forchheimer number, ξ = A c ν f is the velocity slip parameter, ξ T = B c ν f is the temperature slip parameter, and S = − v 0 c ν is the suction parameter.

The engineering quantities of this study are as follows:

where

Applying Equation (7) results in the following non-dimensional form:

where Re = u w υ is the local Reynolds number.

3 Solution methodology

One useful method for handling nonlinear problems is the series solution. Most non-linear issues in the scientific and technical domains are difficult to solve using other perturbation methods. The recently released HAM programs, BVPh. 1.0 and BVPh. 2.0, enhance the convergence of the presented issues. It is difficult to utilize the BVPh. 2.0 package up to the 100th iteration, but these packages are helpful for rapid convergence. The first solution is handled using the subsequent method:

The linear operators in the above-mentioned situation are defined as

They possess the following subsequent characteristics:

N F and N θ are the nonlinear operators and are given as follows:

The fundamental idea of HAM described in the studies [25,26,27,28] for Equations (19)–(20) is as follows:

such that r ∈ [ 0, 1 ] , when r = 1 and r = 0 we have

The Taylor’s series expansions about g = 0 of F ( η , r ) and θ ( η , r ) are

The secondary limitations h F and h θ are chosen such that the series (26) converges at r = 1 , substituting r = 1 , we obtain

4 Results and discussion

This section elaborates on the graphical description for SWCNT and MWCNT nanofluids. MWCNTs are shown as solid lines, and SWCNTs are shown as dashed lines. The Forchheimer parameter, Williamson parameter, stretching parameter, magnetic field parameter, velocity slip parameter, volume friction of the nanoparticles, temperature slip parameter, and Eckert number are the key results that are obtained from temperature and velocity equations and presented in Figures 2–11. Using a magnetic field, it is possible to conduct the heat transfer analysis on an electrically conducting fluid. The governing Equations (8)–(10) are considered, and their approximate analytical solutions are examined. The dimensionless NODEs are simulated using the Mathematica bvp2 package. The MWCNT and SWCNT nanoparticles were mixed with water, the base fluid, to form two different nanofluids. Figure 1 depicts the geometry of the flow problem. The influence of the magnetic field parameter via the nanofluid velocity field is shown in Figure 2. As the magnetic field increases, the velocity of the field decreases. The velocity field of a nanofluid can be greatly affected by the magnetic field parameter, especially in applications where MHD is involved. The electrically conductive nanofluid is subject to a Lorentz force created by using a magnetic field. Depending on the direction of the magnetic field and the fluid’s velocity, this force can either accelerate or decrease the fluid. The nanofluid’s velocity normally decreases when a magnetic field is added. This is due to the Lorentz force’s retarding effect, which raises the flow’s overall resistance. The nanofluid’s properties related to convective heat transport may be affected by the magnetic field. Convective heat transfer rates usually decrease with decreasing velocity. Generally, as the magnetic field parameter increases, the boundary layer’s thickness increases as well. This causes the velocity gradient to become more noticeable close to the surface. Electronic cooling systems can benefit from the use of magnetic fields to regulate the flow of nanofluids. Magnetic fields are employed in solar collectors and nuclear reactors to maximize the flow of nanofluids and improve heat transfer. The effect of Williamson factor using the nanofluid velocity field is shown in Figure 3. The velocity field decreases as the Williamson parameter increases. The Williamson fluid model, which explains the behaviour of non-Newtonian fluids with shear-thinning characteristics, is linked to the Williamson parameter. The yield stress and shear-thinning viscosity of this model are its distinguishing features. Shear thinning, in particular, a non-Newtonian behaviour of nanofluids, is explained by the Williamson fluid model. The velocity profile of such fluids is impacted by the viscosity’s reduction with increasing shear rate. When the Williamson parameter is present, the nanofluid’s velocity usually decreases. This is because the fluid’s effective viscosity increases with lower shear rates, increasing the low resistance. On-Newtonian factors may cause the boundary layer thickness to increase, resulting in a more gradual velocity gradient close to the surface. The impact of volume friction caused by nanoparticles via the nanofluid velocity field is depicted in Figure 4. As the volume friction of the nanoparticles increases, the velocity field decreases. The volume friction of the nanoparticles has a major effect on the nanofluid’s velocity. The concentration of these nanoparticles affects the fluid’s density, viscosity, and thermal conductivity. The volume percentage of friction caused by the nanoparticles in the nanofluid determines its effective viscosity. As the viscosity increases, the nanofluid’s flow resistance increases and its velocity decreases. Figure 5 shows the effect of the velocity slip factor on the velocity field of the nanofluid. As the velocity slip parameter increases, the velocity field also increases. In contrast to the no-slip condition, the slip condition permits a higher fluid velocity closer to the wall. A thinner boundary layer and a sharper velocity gradient close to the surface may arise from this. Because the velocity difference between the fluid and the boundary is lower when slip is present, the shear stress at the boundary is reduced. Because there is less friction at the borders and more room for the fluid to flow, the slip condition in channels or pipes can increase the overall flow rate. There may be an impact on the possibility of flow separation. Slip can occasionally stop or postpone flow separation, resulting in a more stable flow. The effect of stretching parameter on the nanofluid velocity field is shown in Figure 6. The velocity field of the nanofluid increases with increasing stretching parameter. In many fluid dynamics problems – particularly those requiring boundary layer flows across stretching surfaces, such as those in polymer sheet production or extrusion processes – the stretching parameter plays a crucial role. The stretching parameter is an important factor in determining the velocity field when considering nanofluids. There is a greater velocity gradient close to the surface as a result of surface stretching. This causes the fluid velocity to quickly transition from zero at the surface (no-slip condition) to the free stream velocity, creating a boundary layer. The fluid close to the border is accelerated by the stretching surface, which raises the boundary layer’s velocity. This leads to a velocity profile that is steeper than that of non-stretching surfaces. The stretching parameter may have an impact on the flow’s stability. Increased stretching rates have the ability to stabilize the flow, which lowers the chance of turbulence and flow separation. Figure 7 illustrates how the Forchheimer parameter affects the nanofluid velocity profile; with increasing Forchheimer parameter, the velocity of the nanofluid decreases. By adding a dimensionless component to Darcy’s law to account for non-linear drag forces, the Forchheimer parameter describes the inertial effects in a porous media flow. When dealing with higher velocities or more intricate porous materials, this parameter becomes important. The inertial resistance to flow is represented by a non-linear factor that is added to Darcy’s equation by the Forchheimer parameter. This lowers the nanofluid’s velocity by increasing the total resistance it must overcome to pass through the porous material. The flow velocity of the nanofluid within the porous medium reduces as a result of the increasing flow resistance. At greater velocities where inertial effects are strong, this impact is more evident. The effect of temperature slip parameter via the nanofluid temperature field is displayed in Figure 8. As the temperature slip parameter increases, the fluid’s temperature decreases. In a nanofluid flow, the temperature slip parameter is a crucial factor to take into account, particularly at the micro and nanoscales where the traditional no-slip boundary conditions for temperature may not always hold. A temperature gradient that is not zero at the border is implied by the existence of a temperature slip parameter. This has an impact on the nanofluid’s temperature distribution. In comparison to the no-slip scenario, the temperature gradient close to the boundary is decreased as a result of temperature slip. A distinct temperature profile in the boundary layer may result from this. In general, the temperature slip condition causes the rate of heat transfer from the fluid’s boundary to decrease. The total heat flux is decreased by the thermal resistance that the slip introduces. Nonetheless, under some conditions, the slip can improve heat transfer by limiting the amount of heat buildup close to the border. Figure 9 shows the effect of the magnetic field parameter using the nanofluid temperature field. The temperature of the fluid increases as the magnetic field parameter increases. Applying a magnetic field to a nanofluid can have a significant effect on its temperature field, especially if the nanofluid contains magnetic nanoparticles. Magnetic fields have the ability to increase the convective heat transport in nanofluids. The capacity of the magnetic field to apply forces to magnetic nanoparticles may improve the fluid’s mixing and circulation. This could enhance convectional heat dispersion and increase temperature consistency. The presence of magnetic nanoparticles in the nanofluid can significantly alter its effective viscosity due to the influence of the magnetic field. A decrease in fluid flow velocity caused by an increase in viscosity due to the alignment of nanoparticles in response to the magnetic field may have an effect on convective heat transfer and the temperature field. The effective thermal conductivity of a nanofluid can be altered by the positioning and distribution of magnetic nanoparticles in a magnetic field. The temperature field may be affected by reduced temperature gradients due to more efficient heat conduction from higher thermal conductivity. The magnetic field may have an impact on the nanofluid’s flow patterns. Changes in the flow patterns, including the formation of vortices or fluctuations in the boundary layer’s thickness, can have a significant effect on the temperature distribution. The influence of the volume friction of nanoparticles on the nanofluid temperature field is shown in Figure 10. As the volume friction of the nanoparticles increases, the nanofluid’s temperature field increases as well. One important factor that affects a nanofluid’s temperature field and overall thermal performance is the volume fraction of nanoparticles in the fluid. The effective thermal conductivity of a base fluid is usually increased by adding nanoparticles to the nanofluid. The thermal conductivity increases with higher volume fractions of the nanoparticles, which can enhance the fluid’s capacity for heat transmission. Improved heat transfer rates in applications like heating and cooling systems are possible because of the nanofluid’s enhanced thermal conductivity. The effective viscosity of the nanofluid typically increases with higher nanoparticle volume fractions. This may have an impact on the flow characteristics, perhaps resulting in greater pressure drops and a greater need for pumping power. Figure 11 illustrates how the Eckert number affects the nanofluid’s temperature field. It is evident that when the Eckert number increases, the temperature profile also increases. A dimensionless metric called the Eckert number is used to measure how much heat conduction and kinetic energy dissipation matter occur in a fluid flow. Increased thermal energy produced by viscosity dissipation causes the temperature gradients in the nanofluid to increase. In close proximity to interface and boundary zones that experience significant shear rates, this phenomenon may be more noticeable. Viscous dissipation takes over as the primary source of thermal energy at large Eckert numbers and has a major impact on the temperature field. The fluid’s kinetic energy is mostly transformed into thermal energy. Insignificant localized heating is not seen in the temperature field, particularly in high-velocity areas. The thermophysical characteristics of the nanoparticles and water are displayed in Table 1. Table 2 shows the effects of the skin friction values on the vertex viscosity parameter, magnetic field parameter, and Williamson parameter. It was found that when the values of the Williamson, magnetic field, and vertex viscosity parameters increased, the skin friction value also increased. The fluid particle moment encountered higher resistance as the values of the Williamson, magnetic field, and vertex viscosity parameters increased. The influence of the temperature slip parameter, the Eckert number, and the vertex viscosity parameter is shown in Table 3 via the Nusselt number. The Nusselt number increases in tandem with the increasing magnitude of the temperature slip parameter, the Eckert number, and the vertex viscosity parameter. Table 4 shows a plot of the residual error based on the number of iterations (Figure 12).

Figure 2

Effect of the magnetic field parameter on the nanofluid velocity field.

Figure 3

Effect of the Williamson parameter on the nanofluid velocity field.

Figure 4

Impact of the nanoparticle volume friction on the nanofluid velocity field.

Figure 5

Effect of the velocity slip parameter on the nanofluid velocity field.

Figure 6

Effect of the stretching parameter on the nanofluid velocity field.

Figure 7

Effect of the Forchheimer parameter on the nanofluid velocity field.

Figure 8

Effect of the temperature slip parameter on thetemperature field of the nanofluid.

Figure 9

Effect of the magnetic field parameter on the temperature field of the nanofluid.

Figure 10

Effect of the nanoparticle volume friction on the temperature field of the nanofluid.

Figure 11

Effect of the Eckert number on the temperature field of the nanofluid.

Figure 12

Comparison of the current work with that of Seethamahalakshmi et al. [35].

5 Conclusion

This work aims to explore the approximate analytical analysis of a two-dimensional, time-independent, incompressible water-based Darcy–Forchheimer Williamson CNT nanofluid containing MWCNT and SWCNT nanoparticles over a stretching/shrinking surface. In this study, water serves as the base fluid, and MWCNT and SWCNT are employed as nanoparticles. The behaviour of the important parameters is shown through graphs, and the governing equations for temperature and velocity are approximated analytically using HAM. The primary findings of the present investigation are as follows. The relationship between the nanofluid’s velocity field and the magnetic field, nanoparticle volume friction, Williamson, Forchheimer, and magnetic field parameters is inverse. The velocity field of the nanofluid is directly related to the stretching parameter and velocity slip parameter. Temperature field increases with the Eckert number, magnetic field parameter, and volume friction of the nanoparticles. Temperature field decreases with increasing temperature slip parameter. As the vertex viscosity parameter, magnetic field parameter, and Williamson parameter increase, the skin friction also increases. As the vertex viscosity parameter, Eckert number, and temperature slip parameter increase, the Nusselt number also increases. As the number of iterations increases, the residual error decreases. Future research directions for this work involve incorporating both Newtonian and non-Newtonian hybrid nanofluids, as well as considering both uniform and non-uniform conditions within the discussed mathematical model. Additionally, the problem’s structure can be modified. Various combinations of base fluids and nanofluids could be explored, and different numerical methods may be employed to analyse this model. In many practical applications, thermal radiation and chemical reactions significantly influence the nanofluid behaviour, but they may not be included in the model. The analysis may assume constant slip conditions and porous media properties, whereas in real-world applications, these parameters can vary spatially.