Sage Journals: Discover world-class research

Abstract

Darcy-Forchheimer transport of nanofluid comprising Ethylene glycol in Cattaneo-Christov heat flux across an extended cylinder with various slips is investigated in this paper. The influence of autocatalytic chemical reactions in the governing equations enrich the novelty of the proposed mathematical formula. The current challenge also includes an entropy minimization study of the governing equations. For the conversion of the nonlinear system to ODEs, a relevant transformations technique is used. For the calculation of a nonlinear set of linear equations, the bvp4c (shooting) method is coupled with the MATLAB program. Graphical drawings are used to examine the effects of the leading factors on occupied fields. The results demonstrate that a high magnetic parameter boosts the thermal profile while lowering the velocity. In addition, the velocity is reduced when the slip parameter is estimated. Furthermore, it was determined that as the thermal relaxation parameter was raised as the entropy number increases. The tabular data is included to support the current mathematical model.

Keywords

Cattaneo-Christov heat flux Darcy-Forchheimer flow nanofluid multiple slips entropy optimization autocatalytic chemical reaction

Introduction

The investigation of the dehydration process is very important in the semiconductor sector because excessive heat production can damage or destroy equipment or devices. An increase in heat transfer between such a coolant and a heated surface is comparable to improving convective transfer in cooling systems. All thermophysical properties enhancement measures will affect these two metrics, convective heat transfer efficiency, and fluid/wall exchange surfaces. The fins primarily enhance the thermal transfer surface, while the heat transfer fluid’s conductance affects the heating and cooling coefficient. To improve the efficiency of a cooling system, many strategies have been used. Large surfaces, including fins, are a dependable, cost-effective, and commonly utilized technique of heat dissipation. To maintain a cool, large, the high-powered system needs more heat dissipation. The second approach for eliminating heat is integrating liquid aerodynamic performance with high conductivity of increased resistance. Metal-based water nanofluid has greater thermal conductivity and convective heat transfer than the base liquid. This strategy is undeniably the most appealing for increasing heat transport. In 1993, Masuda et al.¹ were the first to use ultrafine solid particles for distribution in a base fluid. Choi² coined the term “nanofluid” in 1995, which traditionally refers to a fluid with nanomaterials suspended in it. Many studies have found that substituting a coolant with a nanofluid enhances a base fluid’s thermal efficiency.^3,4 The thermal conductivity of most solids is larger than the fluid’s heat transfer coefficients. The dispersion of nanoparticles to the conventional fluids resulted in greater thermophysical properties and improved thermal performance when compared to the base fluid. Muhammad et al.⁵ studied the impact of compressing flow on the melting of nanofluids. Goudarzi et al.⁶ investigated the effect of nanomaterials with convection and radiation on nanofluid. Sadaf and Abdelsalam⁷ studied the effects of nanofluid flow in a curved non-uniform disc with flow separation features. Hussein et al.⁸ looked at using a nanofluid to boost the thermal output of photovoltaic panels. Gul et al.⁹ studied the impact of MHD dipoles on heat transfer using a nanofluid. Abbas et al.¹⁰ explored the Yamada–Ota and Xue theories of nanofluid on spinning needles. Waini et al.¹¹ investigated with a forward flow of nanofluid toward such a squeezing cylinder. Non-Newtonian fluids are widely employed in industry and business, which has prompted academics to do research in this area. The chemical industry, which includes paint manufacture, palm increased oil, and conditioner production, and even the food industry, which includes mayonnaise preparation, are all examples of key applications for these fluids. Eyring–Powell fluid and Viscoelastic fluid are the most important non-Newtonian fluids. The Eyring–Powell liquid was investigated in this article. Exercising non-Newtonian fluids is the subject of the following scientific projects. Broad-spectrum of products methodologies, including satellite communications vehicles, missile social reintegration, various propulsion equipment for airplanes, an atom bomb power plant, solar energy assimilation, in combustion application fields such as Engine design, furnaces, fires, solar constructions, etc., all require radiative heat transfer inside the flow. The mass and heat transfer scenario happened when the concentration variation of species in a mixture transfers them from a high-concentration area to a low-concentration area. There are various operations in this amazing period, such as absorption, heat resistance, and food preparation, as well as alcohol distillation and water content dispersion over grooves fields, all of which need a mass transfer. Abbas et al.¹² developed an entropy optimized MHD micropolar nanofluid with slip phenomena and osmotic pressure. Kotha et al.¹³ investigated the heat generation on MHD flow of nanofluid generated by microorganisms. Shahid et al.¹⁴ used heat radiation to study the MHD flow of nanofluid through a porous layer. Awan et al.¹⁵ investigated nonlinear heat and mass transfer effectiveness in the magnetohydrodynamics flow of nanofluid using the impacts of a solar cell. Mishra and Kumar¹⁶ investigated the effects of velocity and slip flow using a straining cylinder with entropy production and Joule heating. Li et al.¹⁷ explored microbe investigation and Wu’s slip-over flow of nanofluid across a surface. Morteza Mousavi et al.¹⁸ studied the impacts of Joule heating on Magnetohydrodynamics flow nanofluid with heat transmission in microfluidic systems. The MHD generator creates converts electrical energy to kinetic energy directly. The main difference between an MHD generator and a traditional electric alternator is that a magnetohydrodynamics generator seems to utilize ionized fluids as conducting polymers. MHD is the investigation of the magnetic properties of conducting fluids. The connection between fluid metals or magnetized particles inside the present and the electrostatic force is considered via MHD interaction. The electrodynamics Maxwell formulas are coupled with fluid equations that incorporate the Lorentz force in the MHD model. In principle, the Lorentz force and inductive electric current have opposing production mechanisms. An increment in the Joule heating variable causes the temperature to rise, concentration to fall, and velocity to rise. Below are some studies in the fields listed in this paragraph. In a Magnetohydrodynamic generator, an ionized fluid is anticipated to migrate at a specific velocity through such a strong magnetic field, creating an electromagnet that may be used to harvest electric energy by inserting two conductors across the fluid stream. Alshber and Nabwey¹⁹ examined heat and mass transfer phenomena caused by the MHD flow of nanofluid around a rotating frame. The features of MHD flow of nanofluid with heat production were investigated by Oyelakin et al.²⁰ The effects of dissipative heat energy on MHD flow across a nonlinear sheet were investigated by Baag et al.²¹ More work on nanofluid and MHD is carried out.^22–30 Figure 1 shows a schematic representation of the microbiological, antineoplastic, wound healing, and angiogenic capabilities of zinc oxide nanoparticles in veterinary sciences. ZnO nanoparticles have also been employed in tissue regeneration, as a food additive, and as a feed ingredient.

Figure 1.

Application of ZnO NPs.

This communication analyses the Darcy–Forchheimer permeable medium ﬂo $ZnO$ w of nanofluid with the effects of homogenous and heterogeneous reactions, MHD, and multiple slips over a cylinder. Here zinc $Zn$ , zinc oxide $ZnO$ , and base fluid Ethylene glycol $C_{2} H_{6} O_{2}$ are used. The similarity transformations are utilized to solve the collection of nonlinear differential equations arising from the controlling system in curvilinear coordinates. MATLAB’s built-in methodology Bvp4c is used to mathematically and visually assess the flow problem’s outcomes. No investigation on the issue has been performed to our knowledge. As a consequence, this technique will be used to collect essential intake from such flows, which will aid in the resolution of several technical difficulties.

Physically and mathematically flow modeling

Flow description

Here we considered the incompressible flow of nanofluid including zinc $Zn$ , zinc oxide $ZnO$ , and base fluid Ethylene glycol $C_{2} H_{6} O_{2}$ with the effects of multiple slip conditions through a stretched cylinder. The significance of homogenous and heterogeneous reactions with Darcy–Forchheimer permeable medium is investigated here. The importance of entropy generation and Cattaneo-Christov (C-C) theory of heat profile and with a magnetic field is also examined. The cylinder’s coordinates are chosen so that r and z in the horizontal and vertical directions, correspondingly, correspond to the cylinder radius see Figure 2.

Figure 2.

Flow geometry.

Chaudhary and Merkin³¹ proposed the isothermal homogeneous reaction with cubic autocatalysis in nanofluid flow, which is expressed as follows:

C + 2 D \to 3 D,

(1)

Hence the heterogeneous reaction can be calculated as follows:

C \to D .

(2)

The homogeneous reaction rate is $(k_{1} H b^{2})$ k1Hb2, while the heterogeneous reaction rate is $(k_{s} H)$ . The H and b are chemical species with concentrations of C and D.

Dimensional non-linear equations

The assumed model for the governing flow is given as^32–35:

{(rw)}_{z} + {(ru)}_{r} = 0,

(3)

ρ_{nf} (w w_{z} + u w_{r}) = μ_{nf} (w_{rr} + \frac{1}{r} w_{r}) - σ_{F} B_{0}^{2} w - \frac{μ_{nf}}{k} w - F w^{2},

(4)

ρ_{nf} (w u_{z} + u u_{r}) = μ_{nf} (u_{rr} + \frac{1}{r} u_{r} - \frac{u}{r^{2}}),

(5)

\begin{matrix} (w T_{z} + u T_{r}) = \frac{k_{nf}}{{(ρ c_{p})}_{nf}} (T_{rr} + \frac{1}{r} T_{r}) \\ - λ_{2} (\begin{matrix} w^{2} T_{zz} + u^{2} T_{rr} + 2 uw T_{rz} \\ + w w_{z} T_{r} + w u_{z} T_{r} + u w_{r} T_{z} \\ + u u_{r} T_{z} + u u_{r} T_{r} \end{matrix}) - \frac{q^{*}}{{(ρ c_{p})}_{nf}}, \end{matrix}

(6)

u H_{r} + w H_{Z} = D_{A} (H_{rr} + \frac{1}{r} H_{r}) - k_{1} H b^{2},

(7)

u b_{r} + w b_{z} = D_{B} (b_{rr} + \frac{1}{r} b_{r}) + k_{1} H b^{2},

(8)

The relevant boundary is

\begin{matrix} u = 0, w = W_{w} + L w_{r}, \\ T = T_{w} + I T_{r}, \\ D_{A} H_{r} = k_{s} H, \\ D_{B} b_{r} = - k_{s} H at r = a \\ u \to 0, T \to T_{\infty}, H \to a_{0}, \\ b \to 0 as r \to \infty \end{matrix}} .

(9)

Transformation variables

\begin{matrix} u = - \frac{ca}{\sqrt{ζ}} f (ζ), w = 2 zcf' (ζ), θ (ζ) = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, \\ Δ T = T_{w} - T_{\infty}, ζ = {(\frac{r}{a})}^{2}, g = \frac{H}{a_{0}}, h = \frac{b}{a_{0}} \end{matrix}} .

(10)

Dimension-less equations

With similarity transformation (10) the equations (4)–(9) obtained the following form

\begin{matrix} B_{1} (ζ f ″' + f ″) + \frac{B_{2} Re}{2} (ff ″ - {f'}^{2}) \\ - Re F_{r} f'^{2} - \frac{γ}{4} \frac{Re B_{1}}{B_{4}} f' - \frac{M}{4} f' = 0 \end{matrix}

(11)

\begin{matrix} B_{3} (θ' + 2 ζ θ ″) + A * f' + β θ - 2 α Re \Pr (\frac{f^{2}}{ζ} θ ″ + ff' θ') \\ - Re B_{4} \Pr f θ' = 0, \end{matrix}

(12)

2 ζ g ″ + 2 g' + 2 Sc Re fg' - Sc Re Kg h^{2} = 0,

(13)

2 ζ h ″ + 2 h' + 2 \frac{Sc}{δ} Re fg' - \frac{Sc}{δ} Re Kg h^{2} = 0,

(14)

The dimensionless boundary conditions are

\begin{matrix} f' (1) = 1 + λ f ″, f (1) = 0, \\ θ (1) = 1 + Ω θ' (1), \\ g' (1) = - K_{s} g (1), \\ δ h' (1) = - K_{s} g (1), \\ f' (\infty) = 0, θ (\infty) \to 0, \\ g (\infty) \to 1, h (\infty) \to 0 \end{matrix}} .

(15)

Reduced parameters

In the above equations, the Prandtl number is $\Pr (= \frac{ν_{f}}{α_{f}}),$ the Reynolds number is denoted by $Re (= \frac{c a^{2}}{2 υ_{f}}),$ the porosity parameter is $γ (= \frac{a^{2}}{4 k}),$ $δ (= \frac{D_{B}}{D_{A}})$ heat absorption parameter, the Schmidt number is $Sc (= \frac{υ}{D_{A}}),$ the velocity slip parameter is $λ (= \frac{2 L}{a}),$ $Ω (= \frac{2 I}{a}),$ the thermal relaxation parameter is $α (= λ_{2} a)$ , the Strength of homogenous reaction $K (= \frac{k_{1} a_{0}^{2}}{c})$ , and the strength of the heterogeneous reaction $K_{s} (= \frac{k_{s} a_{0}}{D_{B}})$ .

The dispersion if $(δ = 1)$ the (D_A & D_B) coefficients are assumed to be the same, calculated based on the following relationship:

g (ζ) + h (ζ) = 1 .

(16)

From equation (13)–(14)

2 ζ g ″ + 2 g' + 2 Sc Re g' f - Sc Re Kg {(1 - g)}^{2} = 0,

(17)

With

g' (1) = - K_{s} g (1), g (\infty) \to 1 .

(18)

Skin friction coefficient is categorized as

C_{f} = \frac{τ_{w}}{\frac{1}{2} ρ_{f} W_{w}^{2}},

(19)

τ_{w} = μ_{f} w_{r} |_{r = a} .

(20)

Here, appended the dimensionless form of the drag force

C_{f} = f ″ (1),

(21)

Numerical scheme

The nonlinear non-dimensional converted problem equations and boundary conditions were addressed using the nonlinear shooting technique and the MATLAB built-in function bvp4c.^36–38 For $ζ_{max} = 7$ asymptotic convergence is reported to be reached. bvp4c is a boundary value problem solver in the MATLAB package. This problem’s numerical results are all applied to a 10⁻⁶ error tolerance. The following variables are used to transform a system of partial differential equations into a system of first-order ordinary differential equations:

\begin{matrix} f = s_{1}, f' = s_{2}, f ″ = s_{3}, f ″' = {s_{3}}^{'}, \\ θ = s_{4}, θ' = s_{5}, θ ″ = {s_{5}}^{'}, \\ ϕ = s_{6}, ϕ' = s_{7}, ϕ ″ = {s_{7}}^{'} \end{matrix}} .

(22)

\begin{matrix} s s_{1} = (\frac{1}{ζ B_{1}}) \\ [- B_{1} s_{3} - B_{2} Re (s_{1} s_{3} - s_{2}^{2}) + Re F_{r} s_{2}^{2} + \frac{λ Re}{2 B_{4}} B_{1} s_{2} + M s_{2}], \end{matrix}

(23)

\begin{matrix} s s_{2} = \\ (\frac{- B_{3} s_{5} - A^{*} s_{2} - β s_{4} + 2 α Re B_{4} \Pr s_{1} s_{2} s_{5} + Re B_{4} \Pr s_{1} s_{5}}{2 ζ B_{3} - 2 α Re B_{4} \Pr \frac{s_{1}^{2}}{ζ}}), \end{matrix}

(24)

s s_{3} = \frac{1}{2 ζ} [- 2 s_{7} - 2 Sc Re s_{1} s_{7} + Sc Re K s_{6} {(1 - s_{6})}^{2}],

(25)

\begin{matrix} s_{2} (1) - 1 - λ s_{3} (1), s (1), s_{4} (1) - 1 - Ω s_{5} (1) \\ s_{7} (1) - \frac{\partial K}{2 \sqrt{ζ}} s_{6} (1), s_{2} (\infty), s_{4} (\infty), s_{6} (\infty) - 1 \end{matrix}},

(26)

Entropy profile

The entropy generation is stated as.^39,40

\begin{matrix} S_{G}''' = \underset{Heat transfer irreversibility}{\underset{︸}{\frac{k_{nf}}{T_{\infty}^{2}} T_{r}^{2}}} + \underset{Viscous dissipation irreversibility}{\underset{︸}{\frac{μ_{nf}}{T_{\infty}} w_{r}^{2}}} - \underset{porous medium irreversibility}{\underset{︸}{\frac{μ_{nf}}{T_{\infty} k} w}} + \underset{Joule heating irreversibility}{\underset{︸}{\frac{σ B_{0}^{2}}{T_{\infty}} w^{2}}} \\ + \underset{mass transfer irreversibility}{\underset{︸}{\frac{R D_{A}}{T_{\infty}} (H_{r} T_{r}) + \frac{R D_{A}}{a_{0}} {(H_{r})}^{2} + \frac{R D_{B}}{T_{\infty}} (b_{r} T_{r}) + \frac{R D_{B}}{a_{0}} {(b_{r})}^{2}}} \end{matrix}

(27)

The rate of entropy generation is $N_{S} = \frac{{S^{‴}}_{G}}{{S^{‴}}_{0}}$

S ″'_{0} = \frac{4 k_{f} {(T_{w} - T_{\infty})}^{2}}{L^{2} T_{\infty}^{2}} .

(28)

By using similarity transformation (10), the entropy generation is transformed into the following dimensionless form

\begin{matrix} N_{S} = \frac{{S ″'}_{G}}{{S ″'}_{0}} \\ = (\begin{matrix} B_{3} \frac{ζ}{m} {θ'}^{2} (ζ) - B_{1} \frac{λ}{4} f' + \frac{B_{1}}{m} \frac{Br}{Γ} ζ {f ″}^{2} + \frac{Br}{m Γ} M^{2} {f'}^{2} (ζ) \\ + L_{1} θ' g' + \frac{L_{1} g'}{Γ} + L_{2} θ' h' + \frac{L_{2} {h'}^{2}}{Γ} \end{matrix}) . \end{matrix}

(29)

\begin{matrix} N_{S} = \frac{{S ″'}_{G}}{{S ″'}_{0}} \\ = (\begin{matrix} B_{3} \frac{ζ}{m} {θ'}^{2} (ζ) - B_{1} \frac{λ}{4} F' + \frac{B_{1}}{m} \frac{Br}{Γ} ζ {f ″}^{2} + \frac{Br}{m Γ} M^{2} {f'}^{2} (ζ) \\ + L_{1} θ' g' + \frac{L_{1} {g'}^{2}}{Γ} + L_{2} θ' g' + \frac{L_{2} {g'}^{2}}{Γ} \end{matrix}), \end{matrix}

(30)

\begin{matrix} m & = \frac{a^{2}}{L^{2}}, Br = \frac{μ_{f} W_{w}^{2}}{k_{f} (Tw - T_{\infty})}, Γ = \frac{T_{w} - T_{\infty}}{T_{\infty}} \\ L_{1} & = \frac{R D_{A} a_{0}}{k}, L_{2} = \frac{R D_{B} a_{0}}{k} . \end{matrix}

(31)

The Brinkman number is supposed as

Br = \frac{μ_{f} W_{w}^{2} (z)}{k_{f} (T_{w} - T_{\infty})} = \frac{μ z^{2} c^{2}}{k_{f} (T_{w} - T_{\infty})} .

(32)

Br = \frac{μ c^{2}}{k_{f} (T_{w} - T_{\infty})} .

(33)

The Bejan number is defined as

Be = (\begin{matrix} \frac{\underset{Heat transfer irreversibility}{\underset{︸}{\frac{k_{nf}}{T_{\infty}^{2}} T_{r}^{2}}} + \underset{mass transfer irreversibility}{\underset{︸}{\frac{R D_{A}}{T_{\infty}} (H_{r} T_{r}) + \frac{R D_{A}}{a_{0}} {(H_{r})}^{2} + \frac{R D_{B}}{T_{\infty}} (b_{r} T_{r}) + \frac{R D_{B}}{a_{0}} {(b_{r})}^{2}}}}{\underset{Heat transfer irreversibility}{\underset{︸}{\frac{k_{nf}}{T_{\infty}^{2}} T_{r}^{2}}} + \underset{Viscous dissipation irreversibility}{\underset{︸}{\frac{μ_{nf}}{T_{\infty}} w_{r}^{2}}} - \underset{porous medium irreversibility}{\underset{︸}{\frac{μ_{nf}}{T_{\infty} k} w}} + \underset{Joule heating irreversibility}{\underset{︸}{\frac{σ B_{0}^{2}}{T_{\infty}} w^{2}}}} \\ + \underset{mass transfer irreversibility}{\underset{︸}{\frac{R D_{A}}{T_{\infty}} (H_{r} T_{r}) + \frac{R D_{A}}{a_{0}} {(H_{r})}^{2} + \frac{R D_{B}}{T_{\infty}} (b_{r} T_{r}) + \frac{R D_{B}}{a_{0}} {(b_{r})}^{2}}} \end{matrix})

(34)

The non-dimensional form of the Bejan Number

Be = (\begin{matrix} \frac{B_{3} \frac{ζ}{m} {θ'}^{2} (ζ) + L_{1} θ' g' + \frac{L_{1} {g'}^{2}}{Γ} + L_{2} θ' g' + \frac{L_{2} {g'}^{2}}{Γ}}{B_{3} \frac{ζ}{m} {θ'}^{2} (ζ) + \frac{B_{1}}{m} \frac{Br}{Γ} ζ {f ″}^{2} - B_{1} \frac{λ}{4} f' + \frac{Br}{m Γ} M^{2} {f'}^{2} (ζ)} \\ + L_{1} θ' g' + \frac{L_{1} {g'}^{2}}{Γ} + L_{2} θ' g' + \frac{L_{2} {g'}^{2}}{Γ} \end{matrix}) .

(35)

Results and discussion

The governing prominent parameters such as magnetic parameter, thermal relaxation parameter, Brinkman number, the velocity slip parameter, Reynolds number, the nanoparticles volume fraction, the thermal slip parameter, and temperature difference parameter against the temperature, concentration distribution, velocity, and entropy generation are deliberated through graphs by MATLAB software (see Table 1 and Table 2). The results of $f'$ the positive values $M$ are deliberated in Figure 3. The velocity concentration $f'$ has negative nature for the positive magnitude of the $M$ . The outcomes of a velocity profile $f'$ for the distinct values of the velocity slip parameter are plotted in Figure 4. From the curves, it is observed that the larger valuation of the velocity slip parameter $λ$ reduces the velocity distribution $f'$ . Figure 5 displays the consequences of the volume fraction of nanoparticles $ϕ$ on the temperature concentration $θ$ . It is noted that the thermal profile $θ$ upsurge for the larger values of volume fraction of nanoparticles $ϕ$ . The thickness of the associated boundary layer is enhanced with the augmentation of the volume fraction of nanoparticles $ϕ$ . Noticeably, the basic element is that the increment in nanoparticles expands the heat conveyance and cohesive forces among the fluid particles; as a result, the fluid temperature is improved. Figure 6 is illustrated to reveal the effects of magnetic parameters $M$ on the thermal profile $θ$ . The mounting valuation $M$ enhanced the thermal profile $θ$ . Figure 7 depicts the characteristics of the temperature slip $Ω$ on the $θ$ . From the figure, results are concluded that the swelling values of the temperature slip parameter $Ω$ fall the temperature concentration $θ$ . The salient features of the thermal relaxation parameter $α$ on temperature concentration $θ$ are premeditated in Figure 8. It is apprehended that the escalating variations of the thermal relaxation parameter $α$ reduce the $θ$ . Figure 9 designates the features of $β$ on $θ$ . It is regarded that the temperature distribution $θ$ declined for fluctuating values of the heat source-sink parameter $β$ . Figure 10 is grabbed to picture the influence of the $Sc$ on the $ϕ$ . From the arcs, it is reflected that the higher variations of the Schmidt number have negative trends for concentration. The upshots of the strength of heterogeneous reaction $K_{s}$ on the concentration distribution are pondered in Figure 11. The increasing value strength of heterogeneous reaction $K_{s}$ diminishes the in the fluid. The influence of growing values of Reynolds number $Re$ on the entropy generation is delighted in Figure 12. The entropy generation was boosted with the evaluated variation of the Reynolds number. Figure 13 exposed the impacts of the Brinkman number $Br$ on the $N_{G}$ . The amended variation in the Brinkman number $Br$ augmented the entropy generation of the nanofluid. Figures 14 and 15 present the results of streamlining for nanofluid when the velocity slip parameter is $λ = 0.0$ and $λ = 0.8$ . Figures 16 and 17 show the significance of streamlining for nanofluid when the magnetic parameter is $M = 0.0$ and $M = 1.0$ . Figures 18 and 19 analyze the effects of the Contour line for Prandtl number, thermal radiations parameter, velocity slip parameter, and heat source-sink parameter via Nusselt number. Figures 20 and 21 analyze the effects of 3D plotting for Prandtl number, thermal radiations parameter, velocity slip parameter, and heat source-sink parameter via Nusselt number.

Table 1.

The properties of nanofluid.

Properties	Notations	Nanofluid
Dynamic viscosity	$B_{1} = (\frac{μ_{nf}}{μ_{f}}) = μ_{nf}$	$μ_{nf} = \frac{μ_{f}}{{(1 - ϕ)}^{2.5}},$
Heat capacity	$B_{4} = (\frac{{(ρ c)}_{nf}}{{(ρ c)}_{f}}) = {(ρ c)}_{nf}$	${(ρ c)}_{nf} = (1 - ϕ) {(ρ)}_{f} + {(ρ)}_{s} ϕ,$
Density	$B_{2} = (\frac{ρ_{nf}}{ρ_{f}}) = ρ_{nf}$	$ρ_{nf} = {(ρ c)}_{f} (1 - ϕ) + {(ρ c)}_{s} ϕ,$
Thermal conductivity	$B_{3} = (\frac{k_{nf}}{k_{f}}) = k_{nf}$	$\frac{k_{n f}}{k_{f}} = \frac{(1 - ϕ) + 2 ϕ \frac{k_{s}}{k_{s} - k_{f}} I n (\frac{k_{s} + k_{f}}{2 k_{f}})}{(1 - ϕ) + 2 ϕ \frac{k_{f}}{k_{s} - k_{f}} I n (\frac{k_{s} + k_{f}}{2 k_{f}})},$

Table 2.

Thermophysical properties of nanoparticles $Z n O & Z n$ and base fluid $C_{2} H_{6} O_{2}$ .^41,42

Physical properties	$C_{2} H_{6} O_{2}$	$ZnO$	$Zn$
$ρ (\frac{kg}{m^{3}})$	1115	5600	7140
$C_{P} {(\frac{kgK}{J})}^{- 1}$	2430	495.2	390
$K {(\frac{mK}{W})}^{- 1}$	0.253	13	116
$σ (\frac{s}{m})$	$1.10 \times 10^{- 4}$	0.01	$1.69 \times 10^{7}$

Figure 3.

Aspects of velocity profile for $M$ .

Figure 4.

Aspects of velocity profile for $λ$ .

Figure 5.

Aspects of thermal profile for $ϕ$ .

Figure 6.

Aspects of thermal profile for $M$ .

Figure 7.

Aspects of thermal profile for $Ω$ .

Figure 8.

Aspects of thermal profile for $α$ .

Figure 9.

Aspects of thermal profile for $β$ .

Figure 10.

Aspects of concentration profile for $Sc$ .

Figure 11.

Aspects of concentration profile for $K_{s}$ .

Figure 12.

Aspects of entropy profile for $Re$ .

Figure 13.

Aspects of entropy profile for $Br$ .

Figure 14.

Aspects of streamlining for $λ = 0.0$ .

Figure 15.

Aspects of streamlining for $λ = 0.8$ .

Figure 16.

Aspects of streamlining for $M = 0.0$ .

Figure 17.

Aspects of streamlining for $M = 1.0$ .

Figure 18.

Aspects of Contour line for $\Pr & R d$ .

Figure 19.

Aspects of Contour line for $λ & β$ .

Figure 20.

Aspects of the 3D plot for $λ & β$ .

Figure 21.

Aspects of the 3D plot for $\Pr & R d$ .

Conclusions

Here the Darcy–Forchheimer flow of $Z n O / C_{2} H_{6} O_{2} & Z n / C_{2} H_{6} O_{2}$ based nanofluid with influences of multiple slips effects, and HOM–HET chemical reaction across a cylinder was studied. The proposed mathematical formulation is also subjected to an entropy generation and Cattaneo-Christov heat flux assessment. The following are the model’s significant assumptions:

• The velocity distributions profile decreased for the rising values of porosity parameter and magnetic parameter $Z n O / C_{2} H_{6} O_{2} & Z n / C_{2} H_{6} O_{2}$ based nanofluid

• The temperature distributions increased for the growing variations of magnetic parameter and volume fraction of nanoparticles

• The heat profile diminished for the higher values of thermal relaxation parameter and slip parameter for $Z n O / C_{2} H_{6} O_{2} & Z n / C_{2} H_{6} O_{2}$ based nanofluid

• The concentration distributions profile declined for the rising estimation of Schmidt number and heterogeneous reaction

• The entropy generations profile is enhanced for the higher values of Reynolds number and Brinkman number for $Z n O / C_{2} H_{6} O_{2} & Z n / C_{2} H_{6} O_{2}$ based nanofluid

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia for funding this work through Large Groups Project under grant number RGP.2/206/43.

ORCID iD

Umar Farooq

References

Masuda

Ebata

Teramae

, et al. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Dispersion of Al₂O₃, SiO₂ and TiO₂ ultra-fine particles. Netsu Bussei 1993; 7: 227–233.

Choi

SUS

. Enhancing thermal conductivity of fluids with the nanoparticle. ASME Fluids Eng. Div 1995; 231: 99–105.

Lee

Choi

SUS

, et al. Measuring thermal conductivity of fluids containing oxide nanoparticles. ASME J Heat Transfer 1999; 121: 280–289.

Wang

Zhou

Peng

. A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles. Int J Heat Mass Transf 2003; 46: 2665–2672.

Muhammad

Hayat

Alsaedi

, et al. Melting heat transfer in squeezing flow of base fluid (water), nanofluid (CNTs+ water), and nanofluid (CNTs+ CuO+ water). J Therm Anal Calorim 2021; 143: 1157–1174.

Goudarzi

Shekaramiz

Omidvar

, et al. Nanoparticles migration due to thermophoresis and Brownian motion and its impact on Ag-MgO/water hybrid nanofluid natural convection. Powder Technol 2020; 375: 493–503.

Sadaf

Abdelsalam

. Adverse effects of a hybrid nanofluid in a wavy non-uniform annulus with convective boundary conditions. RSC Adv 2020; 10: 15035–15043.

Hussein

Habib

Muhsan

, et al. Thermal performance enhancement of a flat plate solar collector using hybrid nanofluid. Sol Energy 2020; 204: 208–222.

Gul

Khan

Bilal

, et al. Magnetic dipole impact on the hybrid nanofluid flow over an extending surface. Sci Rep 2020; 10: 8474.

10.

Abbas

Malik

Nadeem

, et al. On extended version of Yamada–Ota and Xue models of hybrid nanofluid on moving needle. Eur Phys J Plus 2020; 135: 1–16.

11.

Waini

Ishak

Pop

. Hybrid nanofluid flow towards a stagnation point on a stretching/shrinking cylinder. Sci Rep 2020; 10: 9296.

12.

Abbas

Khan

Kadry

, et al. Fully developed entropy optimized second order velocity slip MHD nanofluid flow with activation energy.. Comput Methods Programs Biomed 2020; 190: 105362.

13.

Kotha

Kolipaula

Venkata Subba Rao

, et al. Internal heat generation on bioconvection of an MHD nanofluid flow due to gyrotactic microorganisms. Eur Phys J Plus 2020; 135: 1–19.

14.

Shahid

Huang

Khalique

, et al. Numerical analysis of activation energy on MHD nanofluid flow with exponential temperature-dependent viscosity past a porous plate. J Therm Anal Calorim 2021; 143: 2585–2596.

15.

Awan

Raja

MAZ

Mehmood

, et al. Numerical treatments to analyze the nonlinear radiative heat transfer in MHD nanofluid flow with solar energy. Arab J Sci Eng 2020; 45: 4975–4994.

16.

Mishra

Kumar

. Velocity and thermal slip effects on MHD nanofluid flow past a stretching cylinder with viscous dissipation and Joule heating. SN Appl Sci 2020; 2: 1–13.

17.

Waqas

Imran

, et al. A numerical exploration of modified second-grade nanofluid with motile microorganisms, thermal radiation, and Wu’s slip. Symmetry 2020; 12: 393.

18.

Morteza Mousavi

Ehteshami

Ali Rabienataj Darzi

. Two-and-three-dimensional analysis of Joule and viscous heating effects on MHD nanofluid forced convection in microchannels. Therm Sci Eng Prog 2021; 25: 100983.

19.

Alshber

Nabwey

. Rough set approach for identifying the combined effects of heat and mass transfer due to MHD nanofluid flow over a vertical rotating frame. Mathematics 2021; 9: 1798.

20.

Oyelakin

Adeyeye

Sibanda

, et al. On a new block method for an MHD nanofluid flow with an exponentially decaying internal heat generation. Int J Numer Methods Fluids 2021; 93: 1816–1824.

21.

Baag

Nayak

Mishra

. 2021). Effects of dissipative heat energy and chemical reaction on MHD nanofluid flow over a nonlinearly stretching sheet. In: Recent trends in applied mathematics: select proceedings of AMSE 2019, pp.415–426. Singapore: Springer .

22.

Waqas

Wakif

Al-Mdallal

, et al. Significance of magnetic field and activation energy on the features of stratified mixed radiative-convective couple-stress nanofluid flows with motile microorganisms. Alex Eng J 2022; 61: 1425–1436.

23.

Sheikholeslami

Zareei

Jafaryar

, et al. Heat transfer simulation during charging of nanoparticle enhanced PCM within a channel. Phys A Stat Mech Appl 2019; 525: 557–565.

24.

Farooq

Waqas

Khan

, et al. Thermally radioactive bioconvection flow of Carreau nanofluid with modified Cattaneo-Christov expressions and exponential space-based heat source. Alex Eng J 2021; 60: 3073–3086.

25.

Eid

Mahny

Dar

, et al. Numerical study for Carreau nanofluid flow over a convectively heated nonlinear stretching surface with chemically reactive species. Phys A Stat Mech Appl 2020; 540: 123063.

26.

Song

Waqas

Al-Khaled

, et al. Bioconvection analysis for Sutterby nanofluid over an axially stretched cylinder with melting heat transfer and variable thermal features: A Marangoni and solutal model. Alex Eng J 2021; 60: 4663–4675.

27.

Amanulla

Wakif

Saleem

. Numerical study of a Williamson fluid past a semi-infinite vertical plate with convective heating and radiation effects. Diffus Found 2020; 28: 1–15.

28.

Waqas

Muhammad

Noreen

, et al. Cattaneo-Christov heat flux and entropy generation on hybrid nanofluid flow in a nozzle of rocket engine with melting heat transfer. Case Stud Therm Eng 2021; 28: 101504.

29.

Shaheen

Ramzan

Alshehri

, et al. Soret–Dufour impact on a three-dimensional Casson nanofluid flow with dust particles and variable characteristics in a permeable media. Sci Rep 2021; 11: 1–21.

30.

Waqas

Farooq

Ibrahim

, et al. Numerical simulation for bioconvectional flow of burger nanofluid with effects of activation energy and exponential heat source/sink over an inclined wall under the swimming microorganisms. Sci Rep 2021; 11: 1–15.

31.

Chaudhary

Merkin

. A simple isothermal model for homogeneous-heterogeneous reactions in boundary-layer flow. I equal diffusivities. Fluid Dyn Res 1995; 16: 311–333.

32.

Hayat

Hussain

Alsaedi

, et al. Magnetohydrodynamic flow by a stretching cylinder with Newtonian heating and homogeneous-heterogeneous reactions. PLoS One 2016; 11: e0156955.

33.

Malik

Salahuddin

Hussain

, et al. Homogeneous-heterogeneous reactions in Williamson fluid model over a stretching cylinder by using Keller box method. AIP Adv 2015; 5: 107227.

34.

Hayat

Qayyum

Imtiaz

, et al. Impact of Cattaneo-Christov heat flux in Jeffrey fluid flow with homogeneous-heterogeneous reactions. PLoS One 2016; 11: e0148662.

35.

Hayat

Ahmed

Muhammad

, et al. Computational modeling for homogeneous-heterogeneous reactions in three-dimensional flow of carbon nanotubes. Results Phys 2017; 7: 2651–2657.

36.

Higham

. MATLAB Guide. New Delhi, India: SIAM, 2005.

37.

Khan

Ali

. Recent developments in modeling and simulation of entropy generation for dissipative cross material with quartic autocatalysis. Appl Phys A 2019; 125: 397.

38.

Wang

Khan

Asghar

, et al. Entropy optimized stretching flow based on non-Newtonian radiative nanoliquid under binary chemical reaction. Comput Methods Programs Biomed 2020; 188: 105274.

39.

Butt

Ali

. Entropy analysis of magnetohydrodynamic flow and heat transfer due to a stretching cylinder. J Taiwan Inst Chem Eng 2014; 45: 780–786.

40.

Hayat

Nawaz

Alsaedi

. Entropy analysis for the peristalsis flow with homogeneous–heterogeneous reaction. Eur Phys J Plus 2020; 135: 1–16.

41.

Ahmad

Faisal

Zan-Ul-Abadin

, et al. Unsteady 3D heat transport in hybrid nanofluid containing brick shaped ceria and zinc-oxide nanocomposites with heat source/sink. Nanocomposites 2022; 8: 1–12.

42.

Hafeez

Hayat

Alsaedi

, et al. Numerical simulation for electrical conducting rotating flow of Au (Gold)-Zn (zinc)/EG (ethylene glycol) hybrid nanofluid. Int Commun Heat Mass Transf 2021; 124: 105234.

Significance of Cattaneo-Christov heat flux in Darcy-Forchheimer transport of nanofluid with entropy optimization

Abstract

Keywords

Introduction

Physically and mathematically flow modeling

Flow description

Dimensional non-linear equations

Transformation variables

Dimension-less equations

Reduced parameters

Numerical scheme

Entropy profile

Results and discussion

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References