Cilia and electroosmosis induced double diffusive transport of hybrid nanofluids through microchannel and entropy analysis

Sufian Munawar; Najma Saleem; Dharmendra Tripathi

doi:10.1515/nleng-2022-0287

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Cilia and electroosmosis induced double diffusive transport of hybrid nanofluids through microchannel and entropy analysis

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Published/Copyright: April 12, 2023

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From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

A mathematical model is presented to analyze the double diffusive transport of hybrid nanofluids in microchannel. The hybrid nanofluids flow is driven by the cilia beating and electroosmosis in the presence of radiation effects and activation energy. Cu–CuO/blood hybrid nanofluids are considered for this analysis. Phase difference in the beatings of mimetic cilia arrays emerge symmetry breaking pump walls to control the fluid stream. Analytical solutions for the governing equations are derived under the assumptions of Debye–Hückel linearization, lubrication, and Rosseland approximation. Dimensional analysis has also been considered for applying the suitable approximations. Entropy analysis is also performed to examine the heat transfer irreversibility and Bejan number. Moreover, trapping phenomena are discussed based on the contour plots of the stream function. From the results, it is noted that an escalation in fluid velocity occurs with the rise in slippage effects near the wall surface. Entropy inside the pump can be eased with the provision of activation energy input or by the consideration of the radiated fluid in the presence of electroosmosis. The results of the present study can be applicable to develop the emerging thermofluidic systems which can further be utilized for the heat and mass transfer at micro level.

Keywords: thermal analysis; electric double layer; metachronal waves; hybrid nanofluid; activation energy; magnetic field; symmetry breaking walls

Nomenclature

Latin symbols

a: mean width of channel (m)
B ₀: magnetic field (A/m)
Be: Bejan number
c: speed of wave transmission (m/s)
C _P: specific heat capacity (J/(kg K))
e: electric charge (C)
Ec: Eckert number
E _x: applied electric field (V/m)
F: normalized flow rate (m³/s)
g _c: mass Grashof number
g _r: thermal Grashof number
Ha: Hartmann number
k: thermal conductivity (W/m K)
K: electroosmotic parameter
K*: absorption coefficient (1/m)
k _b: Boltzmann constant (J/K)
L: Knudsen number
L*: slip length (m)
n ₀: average concentration of ions (1/kg)
N _b: Brownian motion parameter
N _S: entropy generation number
N _t: thermophoresis parameter
p: normalized pressure field (N/m²)
P ¯: pressure (fixed frame) (N/m²)
Pr: Prandtl number
Q: mean flow rate (m³/s)
R _n: radiation parameter
S gen ‴: entropy generation rate (J/K)
S _G0: characteristic entropy (J/K)
S _p: Joule heating parameter
t ̅: time variable (s)
T ¯: fluid temperature (fixed frame) (K)
T ₀: reference temperature (K)
T _ave: electrolytic-solution temperature (K)
(u, v): dimensionless velocity vector
U _HS: Helmholtz–Smoluchowski velocity
( U ̅ , V ̅ ): velocity vector V (fixed frame) (m/s)
( u ̅ , v ̅ ): velocity vector (moving frame) (m/s)
We: Weissenberg number
X ₀: particle locale (m)
( X ̅ , Y ̅ ): space coordinates (fixed frame) (m)
( x ̅ , y ̅ ): space coordinates moving frame (m)
(x, y): dimensionless spatial coordinates
z: valence of ions

Greek symbols

α: eccentricity parameter
β: wave number
β _t: thermal expansion coefficient (1/K)
ε: cilia length (m)
ϵ: medium permittivity (C²/N m²)
ϵ ₀: permittivity of free space (C²/N m²)
ρ: density (kg/m³)
Δp _λ: pressure rise per wavelength
∆T: temperature difference (K)
λ: wavelength (m)
χ: directional angle of magnetic field
θ: normalized temperature profile
σ: electrical conductivity (S/m)
σ*: Stefan-Boltzmann constant
μ: dynamic viscosity (kg/m s)
ϕ: nanoparticles concentration (mole/m³)
Φ ̅: electroosmotic potential function
τ: normalized temperature difference
ζ: wall electric potential (kg m²/s³ A)
Ψ: stream function

Subscripts

hnf: hybrid nanofluid
e: electric charge
f: base fluid
1: copper (Cu)
2: copper-oxide (CuO)

1 Introduction

For a long time, numerous scientists have been using electrokinetic actuation for liquid flowing through BioMEMS and Lab-on-a-Chip microdevices, which is significant in biotechnology application and hydrodynamics [1]. Electrokinetic phenomenon takes place when the motion of bulk fluids or selected particles are embedded in fluids with an electric field application. Microfabrication that relies on electrokinetic phenomena provides effective manipulation techniques, which match the characteristic dimension of the micro and nano ranges. In this regard, various theoretical studies were proposed in the past to describe the electrokinetic modulate for the flow control for composition modulation in a microchannel [2], micro mixing [3], concentration [4], and control of the net flow rate in the microchannel [5]. Besides, interfacial phenomena such as the electrical double layer (EDL) influence the flow behavior to a large extent because the specific dimensions of these channels are limited to the microscopic range [6]. Electroosmotic transport occurs when EDL is forced to move by an external electric field, which causes solvent molecules to move due to viscous drag. Most Electroosmotic flow (EOF) research treats electroosmotic pressure-driven media as Newtonian fluids, which makes sense given that most electrolyte or buffer solutes used in microfluidic devices are Newtonian fluids. However, we need the production of EOF from non-Newtonian fluids to regulate various types of physiological fluid such as saliva, blood, and polymer solution. Some mathematical models have been communicated to see the rheological effects on EOF in the microchannel and capillaries [7,8,9,10,11,12,13,14].

Investigation of non-Newtonian fluid flow is a prime area of research employed in chemical material (polymer fluid), biofluid (blood plasma), food processing (magma) etc. To simulate some actual fluid flow situations, various non-Newtonian fluid models have been studied, such as Maxwell fluid, Viscoelastic fluid, Couple stress fluid, and Carreau fluid in the physiological conduit [15,16,17,18]. Shibeshi and Collins addressed that Power-law, Casson, and Carreau-Yasuda models represent the viscous property of blood [19]. These models are not universally accepted to simulate blood behavior but are considered for blood flow. In the present article, Carreau fluid is considered in view of the significance of such model as a combination of Newtonian fluid and power-law fluid. There are some relevant models of the Carreau fluid flow as follows: (i) the steady EOF of Carreau fluid in a sinusoidal wavy microchannel is presented by Noreen et al. [20], (ii) thermal analysis of Carreau fluid was performed for the propulsion of cilia under a uniform magnetic field by Munawar and Saleem [21], and (iii) some important contributions of nanomaterial on the Carreau model are presented in previous research [22–26].

In recent years, a lot of scientific work have been done to improve the heat transport aspect through nanomaterials in the industrial and technological fields. Numerous scientists are trying to understand nanomaterials’ unpredictable thermal properties, which is a mixture of nanoparticles and a common base liquid. However, hybrid nanofluids are being developed from mixing nanomaterial in which metal nanoparticles of various sizes and materials are embedded in the base fluid to achieve better thermal performance and synergy. The influence and thermophysical properties of conventional nanofluids and hybrid nanofluids were reviewed by Das [27]. This study reported that the hybrid nanofluids are highly stable and have better thermal conductivity than nanofluids. Sajid and Ali [28] investigated the impact of hybrid nanofluid and measured the stability and enhancement experimentally and using ANN. This study showed that the selection of hybrid nanofluid plays a significant role in attaining stability. Shah and Hafiz [29] discussed the application of hybrid nanofluids in the solar energy system and their limitations and challenges.

The EOF of hybrid nanofluids (alumina (Al₂O₃)-nanofluid, titania (TiO₂)–water nanofluid, Cu–water nanofluid) in the asymmetric channel was reported by Prakash et al. [30]. Their study illustrates that the Cu–water nanofluid has better thermal conductivity as compared to others. Some more literature covers the range and aspects of hybrid nanofluids [31,32,33,34]. Several researchers have performed by considering the activation energy on hybrid nanofluid to accomplish a chemical reaction between the nanomaterials. For illustration, Suganya et al. [35] applied the activation energy on the Cu–TiO₂/water hybrid nanofluid flow. This study reported that the effect of activation energy modulates the heat transfer rate. Ahmad and Nadeem [36] considered the activation energy for chemical reaction with entropy generation in the hybrid nanofluid. Besides, the impacts of the hall and ion slip current on the fluid properties and temperature distribution are discussed. Raju et al. [37] investigated the effect of activation energy on the transformed oil-based hybrid nanofluid. In same directions, most recently, some of the important studies [38,39,40,41,42,43,44,45,46] on nanofluids/hybrid nanofluids have been reported in literature and discussed for various industrial applications.

In recent technology, the combination of the electric field, magnetic field, and cilia beating propagation is highly demanded in designing and manufacturing of the artificial cilia to generate substantial fluid flow [47,48]. Motivated by the applications in recent technology, some of the interesting models [25,49–52] have been developed and discussed for various physical significance. The entropy generation was also discussed. The ability to manipulate the flow motion in the transverse direction due to the inclined surface in the y-direction is referred to as symmetric break [53]. Understanding the underlying properties and limitations of the symmetric break becomes critical for successful symmetric break slippery walls applications. It is clear from the above literature that there is no study which investigated the range of effects of electroosmosis and cilia actuated double-diffusive transport on hybrid nanomaterial with symmetry-breaking slippery walls involving activation energy. To fulfil this gap, a more generalized mathematical model which involves most of the above physical phenomena is presented and their parametric effects are discussed in detail. Such models could be utilized in development of new technology in the direction of artificial cilia and cilia embedded electroosmotic pumps.

2 Mathematical model

2.1 Geometrical model

Let us consider two-dimensional double diffusive pumping flow through a microfluidic pump in the presence of electric field E = (E _x, 0, 0) applied in axial direction and Lorentz force ( J × B ). It is also assumed that the constant magnetic field B = (0, B ₀, 0) and the electric current density J = σ _hnf(V × B) for hybrid nanomaterial induced Lorentz force. We also suppose that the channel is loaded with electrically conducting radiated hybrid nanomaterial obtained by dispersing 4% volume fraction of copper (Cu) and copper oxide (CuO) solid nano granules in Carreau fluid (taken as blood). The channel inner coating is lined with hair like elastic threads named as cilia having slip surface with the effective slip length L*. The upper and lower walls are kept at respective temperatures T ₁ and T ₂ and the concentration levels C ₁ and C ₂. The current in the medium is induced due to cilia stimulating asymmetric metachronal waves, propagating with constant velocity c along the pump wall (Figure 1). These periodic waves are driven by power and recovery knocks of cilia. We configured the fluid transport pattern in a Cartesian coordinate system contemplating X ̅ -axis in the direction of the wave transmission and the Y ̅ -axis in the orthogonal direction of fluid motion. The geometry of the symmetry-breaking flexible walls of microchannel are depicted in Figure 1 and described as follows [53,54]:

(1) Y ̅ = F i ( X ̅ , t ̅ ) = d i + a i ε cos 2 π λ ( X ̅ − c t ̅ ) + ϕ i = H ̅ i .

Figure 1

A physical configuration of the cilia-induced EOF in microchannel.

According to an experimental report contributed by Sleigh [55], the cilia heads attain elliptical motion pattern, and are horizontally placed at

(2) G ̅ i ( X ̅ , t ̅ ) = X 0 + d i ε α sin 2 π λ ( X ̅ − c t ̅ ) + ϕ i .

The horizontal and vertical velocity components of cilia at the lower and upper channel walls can be attained from Eqs. (1) and (2) as follows:

(3) U ¯ 0 , i = ∂ G ̅ i ∂ t ̅ X 0 = − 2 π λ a i c ε α cos 2 π λ ( X ¯ − c t ̅ ) + ϕ i 1 − 2 π λ a i ε α cos 2 π λ ( X ¯ − c t ̅ ) + ϕ i ,

(4) V ¯ 0 , i = ∂ F i ∂ t ̅ X 0 = − 2 π λ a i c ε α sin 2 π λ ( X ¯ − c t ̅ ) + ϕ i 1 − 2 π λ a i ε α cos 2 π λ ( X ¯ − c t ̅ ) + ϕ i ,

where subscript i = 1, 2 corresponds to the upper and lower walls of the channel, respectively, a _i indicates the wave amplitude, d _i indicates the channel width (total channel width is d ₁ + d ₂), λ represents the wavelength of metachronal wave, α stands for the eccentricity measurement for cilia’s elliptical course of motion, t is the time, ε is the cilia length, X ₀ is the position of the liquid particle, and ϕ _i represents the phase difference. For the upper wall, let wave to be out of phase (ϕ ₁ = 0) and for lower wave, we consider ϕ ₂ = π/3.

2.2 Governing problem in laboratory frame of reference

The governing equations of the intended hybrid nanofluid (Cu–CuO/Carreau fluid) transport model in the laboratory frame of reference are listed as follows [56]:

(5) ∂ U ¯ ∂ X ¯ + ∂ V ¯ ∂ Y ¯ = 0 ,

(6) ρ hnf ∂ U ¯ ∂ t ̅ + U ¯ ∂ U ¯ ∂ X ¯ + V ¯ ∂ U ¯ ∂ Y ¯ = − ∂ P ¯ ∂ X ¯ + ∂ S ¯ XX ∂ X ¯ + ∂ S ¯ XY ∂ Y ¯ − σ hnf B 0 2 U ¯ + ρ e E x + g ( ρ β t ) hnf ( T ¯ − T 0 ) + g β c ρ hnf ( C ¯ − C 0 ) ,

(7) ρ hnf ∂ V ¯ ∂ t ̅ + U ¯ ∂ V ¯ ∂ X ¯ + V ¯ ∂ V ¯ ∂ Y ¯ = − ∂ P ¯ ∂ Y ¯ + ∂ S ¯ XY ∂ X ¯ + ∂ S ¯ YY ∂ Y ¯ ,

(8) ( ρ C P ) hnf ∂ T ¯ ∂ t ̅ + U ¯ ∂ T ¯ ∂ X ¯ + V ¯ ∂ T ¯ ∂ Y ¯ = k hnf ∂ 2 T ¯ ∂ X ¯ 2 + ∂ 2 T ¯ ∂ Y ¯ 2 + S ¯ XX ∂ U ¯ ∂ X ¯ + S ¯ XY ∂ U ¯ ∂ Y ¯ + ∂ V ¯ ∂ X ¯ + S ¯ YY ∂ V ¯ ∂ Y ¯ − ∂ q ̅ r ∂ X ¯ + ∂ q ̅ r ∂ Y ¯ + σ hnf ( E x 2 + B 0 2 U ¯ 2 ) + ( ρ C P ) hnf D T T 0 ∂ T ¯ ∂ X ¯ 2 + ∂ T ¯ ∂ Y ¯ 2 + D B ∂ T ¯ ∂ X ¯ ∂ C ¯ ∂ X ¯ + ∂ T ¯ ∂ Y ¯ ∂ C ¯ ∂ Y ¯ ,

(9) ∂ C ¯ ∂ t ̅ + U ¯ ∂ C ¯ ∂ X ¯ + V ¯ ∂ C ¯ ∂ Y ¯ = D B ∂ 2 C ¯ ∂ X ¯ 2 + ∂ 2 C ¯ ∂ Y ¯ 2 + D T T 0 ∂ T ¯ ∂ X ¯ 2 + ∂ T ¯ ∂ Y ¯ 2 − ω 2 ( C ¯ − C 0 ) T ¯ T 0 n e − E r k r T ,

where P ¯ signifies the fluid pressure, V = ( U ̅ , V ̅ ) is the velocity vector, T ¯ stands for the fluid temperature, and C ̅ is the mass concentration.

For the Carreau nanofluid model, the extra stress tensor is described as follows [57]:

(10) S ¯ = − μ hnf ( 1 + ( Γ γ ̇ ) 2 ) n − 1 2 γ ̇ ¯ ,

with γ ̇ ¯ = 1 2 ∑ i ∑ j γ ̇ ¯ ij γ ̇ ¯ ji = 1 2 Π .

From Eq. (5), the stress components can be obtained as follows:

(11) S ¯ XX = − 2 μ hnf 1 + n − 1 2 Γ 2 γ ̇ 2 ∂ U ¯ ∂ X ¯ ,

(12) S ¯ XY = − μ hnf 1 + n − 1 2 Γ 2 γ ̇ 2 ∂ U ¯ ∂ Y ¯ + ∂ V ¯ ∂ X ¯ ,

(13) S ¯ YY = − μ hnf 1 + n − 1 2 Γ 2 γ ̇ 2 ∂ V ¯ ∂ Y ¯ ,

where μ _hnf signifies the dynamic viscosity of hybrid nanofluid, n indicates the power-law index, Γ is the time relaxation parameter, and Π corresponds to the second invariant strain tensor.

2.3 Thermophysical characteristics of Cu–CuO/blood

The mathematical equations describing the thermophysical features of hybrid nanomaterial (Cu–CuO/blood) are given as follows [31]:

(14) ρ hnf = ρ f 1 − φ 1 + φ 1 ρ 1 ρ f ( 1 − φ 2 ) + φ 2 ρ 2 ,

(15) ( ρ β t ) hnf = ( ρ β ) f ( 1 − φ 2 ) 1 − φ 1 + φ 1 ( ρ β ) 1 ( ρ β ) f + φ 2 ( ρ β ) 2 ,

(16) ( ρ C p ) hnf = ( ρ C p ) f ( 1 − φ 2 ) 1 − φ 1 + φ 1 ( ρ C p ) 1 ( ρ C p ) f + φ ( ρ C p ) 2 ,

(17) σ hnf σ f = 1 + 3 φ ( φ 1 σ 1 + φ 2 σ 2 − φ σ f ) ( φ 1 σ 1 + φ 2 σ 2 + 2 φ σ f ) − φ σ f ( φ 1 σ 1 + φ 2 σ 2 − φ σ f ) ,

(18) k hnf k bf = k 2 + ( S − 1 ) k bf − ( S − 1 ) φ 2 ( k bf − k 2 ) k 2 + ( S − 1 ) k bf + φ 2 ( k bf − k 2 ) ,

with

(19) k bf k f = k 1 + k f ( S − 1 ) − φ 1 ( S − 1 ) ( k f − k 1 ) k 1 + k f ( S − 1 ) + φ 1 ( k f − k 1 ) .

The dynamic viscosity is

(20) μ hnf = μ f ( 1 − φ 1 ) 2 . 5 ( 1 − φ 2 ) 2 . 5 ,

where ρ, μ, σ, ρC _P, ρβ, and k stand for liquid density, liquid viscosity, electrical conductivity, heat capacity, coefficient of thermal expansion, and thermal conductivity. The subscripts 1 and 2 signify the copper and copper oxide solid nano-scaled particles, hnf signifies the hybrid nanofluid, whereas f is designated to the conventional liquid. The parameter S suggests the structure of solid nanogranules, such that, its value equal to 5.7 represents lens-shaped nanogranules and 4.7 ties to rod-shaped nanogranules. To calculate the numerical values of these properties, we use Table 1. Moreover, φ conveys the total volume fraction of nanogranules in blood and is defined as the sum of φ ₁ (volume fraction of Cu nanogranules mixed in base liquid) and φ ₂ (volume fraction of CuO nanogranules mixed in base liquid).

Table 1

Thermophysical features of hybrid nanofluid [50]

Physical quantities	Base fluid (f)	Solid nanosized particle properties
	Blood	Cu	CuO
	F	s ₁	s ₂
σ (1/Ωm)	0.8	59.6 × 10⁶	2.7 × 10⁻⁸
ρ (kg/m³)	1,063	8,933	6,320
C _p (J/kg K)	3,594	385	531.8
K (W/m K)	0.492	400	76.5

2.4 Thermal radiation

Radiation is accepted as an important mode of heat transfer used in thermal therapeutic processes and biofluid transports. The radiative heat flux in the fluid flow direction is insignificant than the radiative heat flux in the orthogonal direction of fluid stream. The radiative heat flux vector q ̅ r will adopt the following form after applying the Rosseland approximation [25]:

(21) q ̅ r = − 4 σ * 3 K * ∂ T ¯ 4 ∂ Y ¯ ,

where σ* stands for Stefan-Boltzmann constant and K* is the absorption coefficient of the considered fluid. The temperature differences inside the stream are appropriately slight. Thus, the expansion of the Taylors’s series of T ⁴ about temperature difference and neglecting second order and higher order terms lead to T ¯ ⁴ ≅ 4(T ₁ − T ₀)³ T ¯ − 3(T ₁ − T ₀)⁴. Consequently, the radiative heat flux adopts the form as follows:

(22) q ̅ r = − 16 σ * ( T 1 − T 0 ) 3 3 K * ∂ T ¯ ∂ Y ¯ .

2.5 Electrical potential distribution

The electric potential inside the microfluidic pump is depicted by the Poisson equation [13] given as follows:

(23) ∂ 2 Φ ̅ ∂ X ¯ 2 + ∂ 2 Φ ̅ ∂ Y ¯ 2 = − ρ e ϵ ϵ 0 ,

where Φ ̅ is the electric potential function, ρ _e stands for the total charge density, ϵ denotes medium permittivity, and ϵ ₀ indicates permittivity of free space. The total charge density for binary fluid is defined as follows:

(24) ρ e = ez ( n ̅ + − n ̅ − ) = n ο e − ze Φ ̅ k b T ave − n ο e ze Φ ̅ k b T ave ,

where n _o describes the average number of positive n ̅ + and counter positive n ̅ − charges in volume concentration, z corresponds to the valence number of ions, e denotes the electric charge, k _b is the Boltzmann constant, and T _ave is the local absolute temperature of the electrolytic solution. The intensity of nano bits in Eqs. (14)–(20) is intended to be homogeneous, which implies that the concentration gradient inside the fluid is insignificant and the Peclet number for this stream is adequately low. This supposition rationalizes the dissemination of ionic intensity.

For a symmetric electrolyte, the net charge density for the ionic charges may be calculated as follows:

(25) ρ e = − 2 n ο ze sinh ze Φ ̅ k b T ave ,

where T _ave represents the local absolute temperature and k _b is the mean number of electrolytes. The Debye–Hückel linearization (on letting wall zeta potential ≤25 mV) applies as follows:

(26) sinh ze Φ ̅ k b T ave ≅ ze Φ ̅ k b T ave .

Using Eqs. (24) and (26) in Eq. (23), one finds

(27) ∂ 2 Φ ̅ ∂ X ¯ 2 + ∂ 2 Φ ̅ ∂ Y ¯ 2 = 2 n ο z 2 e 2 k b T ave ϵ ϵ ο Φ ̅ .

2.6 Moving frame transformations and normalization of governing equations

To obtain the analytical solutions of the intended boundary value problem, it is convenient to transform the governing equations from the fixed (unsteady) frame into moving (steady) frame. The interconnection between the two frames is defined by the following set of transformations:

(28) x ¯ = X ¯ − c t ̅ , y ¯ = Y ¯ , u ¯ = U ¯ − c , v ¯ = V ¯ , p ¯ ( x , y ) = P ¯ ( X ¯ , Y ¯ , t ̅ ) .

Furthermore, let us introduce the following non-dimensional quantities:

(29) x = x ¯ λ , y = y ¯ d 1 , u = u ¯ c , v = λ v ¯ d 1 c , a = a 1 d 1 , b = a 2 d 1 , t = c t ¯ d 1 , p = p ¯ d 1 2 μ f c λ , Φ = Φ ̅ ζ , θ = T ¯ − T m T 1 − T 2 , η = C ¯ − C m C 1 − C 2 , S = S ¯ d 1 μ f c , β = d 1 λ , H = H ¯ d 1 , d = d 2 d 1 ,

where u and v are velocity components along x and y directions, respectively, p represents the fluid pressure, β represents the wave number, and θ indicates the fluid temperature, and η indicates the mass concentration.

Let us introduce the stream function Ψ as follows:

(30) u = ∂ Ψ ∂ y , v = − ∂ Ψ ∂ x ,

and non-dimensionalizing Eqs. (5)–(9) and (27) with the aid of Eqs. (28)–(30) and then employing long wavelength and small inertia approximations, the resulting equations are given as follows:

(31) ∂ p ∂ x = μ hnf μ f ∂ 3 Ψ ∂ y 3 + ( n − 1 ) We 2 2 ∂ ∂ y ∂ 2 Ψ ∂ y 2 3 + σ hnf σ f Ha 2 ∂ Ψ ∂ y + U HS ∂ 2 Φ ∂ y 2 + k hnf k f g r θ + ρ hnf ρ f g c η ,

(32) ∂ p ∂ y = 0 ,

(33) k hnf k f + R n ∂ 2 θ ∂ y 2 + N t Pr ∂ θ ∂ y 2 + μ hnf μ f EcPr ∂ 2 Ψ ∂ y 2 2 + n − 1 2 We 2 ∂ 2 Ψ ∂ y 2 4 + σ hnf σ f S p + PrEcHa 2 ∂ Ψ ∂ y 2 + N b Pr ∂ θ ∂ y ∂ η ∂ y = 0 ,

(34) ( ρ C P ) hnf ( ρ C P ) f ∂ 2 η ∂ y 2 + N t N b ∂ 2 θ ∂ y 2 − γ 1 ( θ Ω − θ + 1 ) e − E 1 + ( Ω − 1 ) θ η = 0 ,

(35) ∂ 2 Φ ∂ y 2 = K 2 Φ .

Cross-differentiation of Eqs. (31) and (32) results as follows:

(36) μ hnf μ f ∂ 4 Ψ ∂ y 4 + ( n − 1 ) We 2 2 ∂ 2 ∂ y 2 ∂ 2 Ψ ∂ y 2 3 + σ hnf σ f Ha 2 ∂ 2 Ψ ∂ y 2 + k hnf k f g r ∂ θ ∂ y + ρ hnf ρ f g c ∂ η ∂ y + U HS ∂ 3 Φ ∂ y 3 = 0 .

The corresponding dimensionless boundary conditions are listed as follows:

(37) Ψ = F 2 , ∂ Ψ ∂ y + μ hnf μ f L ∂ 2 Ψ ∂ y 2 + n − 1 2 We 2 ∂ 2 Ψ ∂ y 2 3 = − 1 − 2 π α ε β cos ( 2 π x ) 1 − 2 π α ε β cos ( 2 π x ) , Φ = θ = η = 0 at y = H 1 = 1 + a ε cos ( 2 π x ) ,

(38) Ψ = − F 2 , ∂ Ψ ∂ y − μ hnf μ f L ∂ 2 Ψ ∂ y 2 + n − 1 2 We 2 ∂ 2 Ψ ∂ y 2 3 = − 1 − 2 π α ε β cos ( 2 π x ) 1 − 2 π α ε β cos ( 2 π x ) , θ = Φ = η = 1 , at y = H 2 = − d + b ε cos ( 2 π x + ϕ ) ,

where We is the Weissenberg number, Ha is the Hartmann number, Ec is the Eckert number, Pr represents the Prandtl number, U _HS stands for the Helmholtz–Smoluchowski velocity parameter, R _n is the radiation number, g _r denotes the thermal Grashof number, g _c symbolizes the mass Grashof number, E is the activation energy parameter, N _t is the thermophoresis parameter, N _b is the parameter for Brownian motion, S _p is the joule heating parameter, K signifies the electroosmotic parameter, and L stands for the velocity slip parameter and are defined, respectively, as follows:

(39) We = Γ c d 1 , Ha = σ f μ f B 0 d 1 , Ec = c 2 ( C P ) f ( T 1 − T 0 ) , Pr = μ f ( C P ) f k f , U HS = − E x ϵ ϵ ο ζ c μ f , R n = 16 σ * ( ∆ T ̅ ) 3 3 C P μ f k f * , g r = g β t ( T 1 − T 0 ) d 1 2 c ν f , g c = g β c ( C 1 − C 0 ) d 1 2 c ν f , E = E r k r T 0 , γ 1 = d 1 2 ρ f ω 2 μ f , N t = D T χ ( T 1 − T 0 ) T 0 ν f , N b = χ D B ( C 1 − C 0 ) ν f , S p = σ f E x 2 d 1 2 ∆ T ̅ k f , K = d 1 ze 2 n ο ϵ ϵ ο k b T ave , L = μ f L * d 1 .

The correlation between the mean flow rates in laboratory (Q) and wave (F) frames is defined as follows:

(40) F = ∫ H 2 H 1 ∂ Ψ ∂ y d y = Ψ ( H 1 ) − Ψ ( H 2 ) , Q = F + d + 1 .

3 Entropy analysis

In cilia-assisted motion of hybrid nano-Carreau liquid, the major sources of entropy generation are considered as radiated heat transfer, fluid friction, Joule heating, and mass transfer. Hence, the entropy representation in the microchannel is indicated as follows [51,52]:

(41) S gen ‴ = 1 T 0 2 k hnf ( ∇ T ̅ ) 2 + 16 σ * ( T 1 − T 0 ) 3 3 K * ( ∇ T ̅ ) 2 + μ hnf T 0 [ S ⋅ ∇ V ] + σ nf T 0 ( B 0 2 U ¯ 2 + E x 2 ) + D B C 0 ∂ C ¯ ∂ X ¯ 2 + ∂ C ¯ ∂ Y ¯ 2 + D B T 0 ∂ T ¯ ∂ X ¯ ∂ C ¯ ∂ X ¯ + ∂ T ¯ ∂ Y ¯ ∂ C ¯ ∂ Y ¯ .

Incorporating Eqs. (28)–(30) in Eq. (41) and after applying lubrication approximation, the representation for total entropy generation number is reported as follows:

(42) N G = L 4 + R n P r ∂ θ ∂ y 2 ︸ N H + L 2 Br τ ∂ 2 Ψ ∂ y 2 2 + n − 1 2 We 2 ∂ 2 Ψ ∂ y 2 2 ︸ N F + L 3 BrHa 2 τ ∂ Ψ ∂ y + 1 2 + L 3 τ S p ︸ N J + Z 1 2 Z 2 τ 2 ∂ η ∂ y 2 + Z 3 v ∂ η ∂ y ∂ θ ∂ y ︸ N C

where τ = ∆T/T ₀ represents the temperature difference number taken as one. The first term, N _H, in Eq. (42) evolves due to the radiation irreversibility instigated by radiative heat flux and the second term, N _F, contributes to fluid friction irreversibility, the third term, N _J, signifies irreversibility owing to Joule heating, and the last term, N _C, proposes mass transfer irreversibility. The Bejan number is the ratio of the heat transfer irreversibility to the total irreversibility and is expressed as follows:

(43) Be = N H N H + N F + N J + N C .

The values of the Bejan number Be ranges between 0 and 1 inclusively and describes the ascendancy between heat transfer irreversibility and other irreversibilities. The interpretation of Be equal to 1 is that the heat transfer irreversibility governs irreversibilities due to other factors. Be = 0 leads that other irreversibilities dominate the heat transfer irreversibility. Whereas Be = 0.5 suggests that the entropy generation rates produced by the irreversibilities due to heat transfer, mass transfer, fluid friction, and Joule heating are equal.

4 Numerical approach, results, and discussion

Eqs. (31)–(36) with boundary condition Eqs. (37) and (38) are highly nonlinear coupled fourth-order differential equations. The numerical solutions of their equations are calculated with the shooting method under NDSolve numerical package of computational software Mathematica. This section exhibits the graphical visualizations and physical explanation of numerical solutions for double diffusive pumping motion through an electroosmotic microfluidic pump with symmetry-breaking flexible walls. Thermally radiated hybrid nanomaterial is considered as working fluid in the pump. The rheological characteristics of the contemplated fluid are examined by letting the mixture of Cu and CuO nano particles with 4% ratio scattered in blood (Carreau liquid). The structure of deferred nano grains is considered to be of platelet-shape by assigning the value 5.7 to the shape parameter S. We have provided different visualizations of dynamically or thermodynamically significant quantities for pertinent parameters of concern through Figures 2–9. For this parametric investigation, some parameters, like the fluid Prandtl number is kept to be 1, the Eckert number is taken as 0.05, the axial distance x and the cilia length parameter ε are considered as 0.5, wave number and parameter for the measurement of eccentricity of cilia elliptical motion path are considered as 0.2, and width ratios are taken as a = b = 0.3 and d = 1.1.

Figure 2

Velocity distribution u for different values of (a) momentum slip parameter L, (b) electroosmotic parameter K, (c) thermal Grashof number g _r, (d) solutal Grashof number g _c, and (e) Helmholtz–Smoluchowski velocity parameter U _HS.

Figure 3

Temperature distribution θ(y) for different values of (a) thermophoresis parameter N _t, (b) Brownian motion parameter N _b, (c) thermal radiation number R _n, and (d) electroosmotic parameter K.

Figure 4

Concentration distribution η(y) for different values of (a) thermophoresis parameter N _t, (b) Brownian motion parameter N _b, (c) thermal radiation number R _n, and (d) activation energy input parameter E.

Figure 5

Entropy generation number N _S profiles for different values of (a) electroosmotic parameter K, (b) momentum slip parameter L, (c) thermal radiation number R _n, (d) Joule heating parameter S _p, and (e) activation energy input parameter E.

Figure 6

The Bejan number Be profiles for different values of (a) electroosmotic parameter K, (b) momentum slip parameter L, (c) thermal radiation number R _n, (d) Joule heating parameter S _p, and (e) activation energy input parameter E.

Figure 7

Influence of electroosmotic parameter (K) on trapping when Ha = 1, g _r = g _c = 0.5, U _HS = 1, n = 0.2, We = 0.01, and L = 0.1 for (a) K = 2 and (b) K = 2.5.

Figure 8

Influence of Hartmann number (Ha) on trapping when K = 1, g _r = g _c = 0.5, U _HS = 1, We = 0.01, and L = 0.1 for (a) Ha = 0.5 and (b) Ha = 1.5.

Figure 9

Influence of phase difference (ϕ) on trapping when Ha = K = 1, g _r = g _c = 0.5, We = 0.01, and L = 0.1 for (a) ϕ = 0 and (b) ϕ = π/3 .

4.1 Axial velocity distribution

The effects of slip parameter (L), electroosmotic parameter (K), thermal Grashof number (g _r), mass Grashof number (g _c), and the Helmholtz–Smoluchowski velocity (U _HS) parameter on axial fluid velocity distribution u(y) can be viewed through Figure 2(a–e). Figure 2(a) reveals that axial velocity augments for the elevated values of L in the vicinity of the pump walls. But an impediment in fluid flow is noticed close to channel center when large values of L are taken into consideration. This behavior is influenced by the existence of partial slip at the pump walls for which the liquid flow is more rapid near the walls as compared to pump core part. Thus, the impact of slippage on liquid motion is remarkable as it diminishes the frictional force at the pump surface and propels the liquid stream appositely. Figure 2(b) demonstrates that the elevated values of K propose an escalation in liquid motion in pump upper space and induce a deceleration in its lower part. This conduct is interpreted as the gain in Debye width (small electroosmotic parameter) creates a low electric potential and hence both acceleration and deceleration in liquid motion can be combined in microfluidic pump. Figure 2(c) demonstrates the significance of g _r on axial velocity profile. The thermal Grashof number identifies the comparative impact of buoyancy forces over viscous forces. For the values of g _r greater than one, direct the buoyancy forces effects over viscous forces. Consequently, a significant reinforcement is consequent with fluid motion in pump lower space (a drop in pump upper zone) for higher values of g _r. Moreover, it can be substantiated that buoyancy force supports the liquid motion along the microfluidic pump. Large values of g _c generate reduction and escalation in fluid flow in the pump upper and lower parts, respectively (Figure 2(d)). In micropumps with symmetry-breaking walls, this behavior is anticipated as the concentration gradients hold direct and inverse relationship with buoyancy and viscous forces, respectively. From Figure 2(e), it is noticed that introduction of U _HS velocity in the fluid stream direction enhances fluid velocity in the vicinity of the pump upper wall and impedes in lower space.

4.2 Temperature distribution

Thermophoresis is an interesting trend in nano-liquids where various solid nano-scaled granules offer different responses to the temperature gradient. The higher values of the thermophoresis parameter (N _t) heat up the working fluid due to significant temperature differences and intense temperature gradients as shown in Figure 3(a). The micro-mixing process is intermittent motion exhibited by nano granules in the nanofluid that appears due to their random constant collision with nearby fast-moving granules. These haphazard collisions convert molecular kinetic energy into thermal energy. Therefore, the liquid temperature lifts. This fact has been viewed in Figure 3(b) which elaborates that the large values of the Brownian motion parameter (N _b) suggest an enhancement in the temperature distribution of nanofluid. The thermal radiation parameter (R _n) characterizes the comparative effect of heat transfer caused by conduction to thermal radiation heat transfer. Figure 3(c) indicates that the elevated value of R _n plays a considerable role in cooling process as the heat radiation impacts keep an opposite relation with thermal conduction. A substantial augmentation in liquid temperature is described for elevated values of electroosmotic parameter (K). Hence, it can be established that the thickening of Debye width plays an essential part in cooling process by reducing the temperature of nanofluids (Figure 3(d)).

4.3 Concentration distribution

Figure 4(a and b) describes the concentration distribution for various values of N _t and N _b. It is noted that concentration distribution appears parabolic in shape and indicates a decline for high values of N _t and N _b. This deterioration is justified as the dispersion of nanofluid commences due to considerable heat convection. Since large values of R _n weaken heat conduction, consequently a bulk of fluid assembles near the channel core part as seen through Figure 4(c). It is also observed that the radiated hybrid nanofluid can propagate over extended distances than conventional liquid and thus deliver a promising medium for drug supply and biomimicry. Figure 4(d) exhibits the effect of activation energy parameter (E) on concentration profile. The critical energy input desired to commence a reaction is labeled as activation energy. Thus, lower temperature and elevated activation energy induce a drop-in reaction rate constant which ultimately enhances mass concentration.

4.4 Entropy generation number and the Bejan number

Figures 5(a–e) and 6(a–e) represent the effects of electroosmotic parameter (K), velocity slip parameter (L), radiation parameter (R _n), the Joule heating parameter (S _p), and activation energy parameter (E) on entropy generation number (N _S) and the Bejan number (Be). From Figure 5(a), it is noted that for the elevated values of K, entropy generation is insignificantly close to pump center. The large values of K enhance entropy production near the lower (hot) wall, whereas, lessen the entropy formation near the upper (cold) wall. Looking at the Bejan number profile in Figure 6(a), one may notice that at lower wall, heat transfer irreversibility is weak and at the upper cold wall, heat transfer irreversibility is stronger. This might have occurred, as evident from the temperature profile (Figure 3(d)), because of lower temperature differences at the lower wall and higher temperature differences at upper wall. Entropy generation in the pump can be minimized by escalating the slippage effects on the pump surface as shown in Figure 5(b). Also, near the pump center, entropy formation is minimal. For small values of L, entropy production is more visible near the colder wall (due to effective convection) than the heated wall. From Figure 6(b), it can be stated that the heat transfer irreversibility dominates over the irreversibilities due to other factors. This behavior is quite expected since all frictional forces become less as the momentum slip augments. Thus, the entropy is only controlled by heat transfer effects and total entropy drops down. Figure 5(c) reveals that the rising values of R _n induce a diminution in heat transfer rate in the pump upper portion and propel the adequate amount of heat far from the system. Therefore, total entropy generated in the pump upper space reduces. It is also noted that near the heated wall, entropy in the channel rises with an increase in R _n. The Bejan number sketched in Figure 6(c) establishes that the liquid friction irreversibility directs the flow system for high values of R _n. This conduct is expected as the conduction at the lower wall becomes destabilized for large values of R _n. Near the pump upper space, irreversibility due to heat transfer undertakes for elevated values of R _n. An increment in S _p values proposes a deceleration in entropy formation near the lower surface and a surge in the pump upper part (Figure 5(d)). Figure 6(d) demonstrates that Be increases with an increase in S _p near the colder wall, but an opposite situation is viewed close to the heated wall. It can be witnessed from Figure 5(e) that disarray in the pump mediates with the provision of activation energy input. This comportment can be ascribed with the supremacy of heat transfer irreversibility over the irreversibilities because of fluid friction, ohmic heating, and mass transfer with the provision of activation energy input. From Figure 6(e), it can be quantified that influence of E on the Bejan number is insignificant in the vicinity of the pump center and colder space. But, close to the heated wall, high activation energy strengthens fluid friction and other irreversibilities over the irreversibility of heat transfer.

4.5 Trapping

An interesting trend linked with electroosmotic ciliary transport is trapping. This phenomenon arises at large amplitude ratio and at constant flow rate. In the moving frame, a group of closed streamlines of metachronal waves can be seen in the distended portion. This group of closed streamlines under specific situations split to trap a fluid mass named as bolus. This bolus moves with the same speed as that of the metachronal wave.

The impact of different parameters of interest like electroosmotic parameter (K), the Hartmann number (M), and phase difference (ϕ) on trapping are numerically evaluated. A rapid decline in the size of the confined bolus in the lower and upper spaces of the channel can be seen through Figure 7 for the elevated values of K. This conduct is anticipated as the increase in Debye thickness causes weak EDL and consequently a bulk of fluid flow occurs. Figure 8 demonstrates that the size of the trapped bolus decreases in the channel upper space and increases in the channel lower part when the elevated values of Ha are selected. This reflects the reduction and proliferation in fluid flow rate in the upper and lower portions of the channel, respectively. Figure 9 depicts that the small values of ϕ hasten fluid flow rate more precipitously rather than its elevated values. It is also seen that for the elevated values of ϕ, the confined bolus shifts toward left and reduces in size.

5 Summary and conclusion

A parametric analysis on cilia-assisted EOF of hybrid nanofluids in asymmetric microchannel is presented. The effect of electric and magnetic fields, slip parameter, Grashof numbers, thermophoresis parameter, Brownian motion, and thermal radiation on axial velocity, temperature, concentration, entropy generation, Bejan number, and streamline patterns are discussed. Entropy analysis is also carried out with the activation energy input. The numerical results are simulated using NDSolve numerical package. Based on the discussion, the key findings of the present model are noted as follows:

The axial velocity profile varies significantly with the change in the magnitudes of slip parameter, Grashof number, and electric field; however, very minor changes are noted with EDL thickness.
The temperature profile enlarges with the thermophoresis parameter, Brownian motion, thermal radiation, and electroosmotic parameter.
The concentration profile shrinks with thermophoresis parameter, Brownian motion, and thermal radiation; however, it expands with energy parameter.
The entropy generation depends on the electric field, EDL thickness, thermal radiation, joule heating, and activation energy.
The involvement of momentum slip is to increase the heat transfer irreversibility; however, overall entropy of the system decreases with slip.
The trapping phenomena is highly affected by various physical quantities like Harman number, electroosmotic parameter, Weissenberg number, and phase difference.

The present study may be applicable to design the cilia-assisted electroosmotic pumps to transport the hybrid fluids/drugs in more complex network where various physical mechanisms affect the fluid flow. Future study may focus on other combination of nanoparticles and other rheological effects for wide range of applications.

Acknowledgements

The authors would like to acknowledge the support provided by the Deanship of Research (DR) at the Prince Mohammad Bin Fahd University for supporting this work.

Funding information: No funding is available for this research.
Author contributions: S.M.: mathematical modeling, supervision, and writing – reviewing and editing. N.S.: conceptualization, analysis, methodology, software, and writing – original draft preparation. D.T.: comparison and validation.
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: There is no associated data with this manuscript. All the data is reported.

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Received: 2022-09-19

Revised: 2023-02-11

Accepted: 2023-03-07

Published Online: 2023-04-12

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0287

Keywords for this article

thermal analysis; electric double layer; metachronal waves; hybrid nanofluid; activation energy; magnetic field; symmetry breaking walls

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