DOI: 10.1007/s13367-021-0029-6
Hall and ion-slip currents’ role in transportation dynamics of ionic Casson hybrid nano-liquid in a microchannel via electroosmosis and peristalsis
Sanatan Das1,*, Bhola Nath Barman1 and Rabindra Nath Jana2
1Department of Mathematics, University of Gour Banga, Malda 732 103, India
2Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India (Reveibed May 3, 2021; final revision received July 27, 2021; accepted September 8, 2021) This article intends to conduct an analytical simulation for the electroosmosis modulated peristaltic transport of ionic hybrid nano-liquid with Casson model through a symmetric vertical microchannel occupying a homogeneous porous material in the existence of the dominant magnetic field, Hall, and ion-slip currents.
The hybrid nano-liquid is acquired by the suspension of silver and silicon dioxide nanoparticles into pure water. The wall slip and convective heating impacts are imposed. The Casson fluid (CF) model is adopted to mimic the rheological behaviour accounting for hybrid nano-liquid. Darcy’s law is applied to evaluate the impact of a porous medium. The Poisson-Boltzmann equation is engaged to accommodate the electric double layer (EDL) in the microchannel. Assumptions of low Reynolds number (LRN), long wavelength (LWL), and Debye-Hückel linearization (DHL) are undertaken to simplify the normalized constitutive equa- tions. Closed-form solutions for the linearized dimensionless resulting equations are achieved by ND-solve code in Mathematica. For a comprehensive physical investigation of the problem under simulation, several graphs are furnished to evaluate the role of emerging thermal and physical parameters in developing the flow patterns and thermal characteristics. Outcomes envisage that Hall, ion-slip, and electro-osmotic param- eters have a marked impact on the velocity of the ionic liquid. A decrement in the EDL thickness corre- sponds to an augmentation in the axial velocity profile in the locality of the channel walls. An increment in radiation parameter results in a demotion in the temperature profile. The pressure gradient is elevated with higher Hall and ion-slip parameters, thermal Grashof number, and electro-osmotic parameter, whereas it is dropped due to higher estimates of Hartmann number. The trapping phenomena under the flow factors are also outlined in brief. The bolus formation is deeply affected by Hall, ion-slip, and electro-osmotic parameters. Outcomes achieved here are expected to shed light on the design and analysis of electro- osmotic pumps, microchannel devices, water filtration and purification processes, DNA analyzers, nanoscale electro-fluid thruster designs in-space propulsion, and many more.
Keywords: Electroosmosis, peristalsis, hybrid nano-liquid, Casson fluid model, Hall and ion slip currents, microfluidic channel
1. Introduction
Nanotechnology has started in the nineteenth century and rapidly evolved. Nanofluid is one of the most effica- cious discoveries of nanotechnology. Fluids with nanopar- ticles (1 nm-100 nm) suspended in them are named nanofluids having unique properties that enhance thermal conductivity. The concept of nanofluid was first reported by Choi (1995) in 1995. Nanofluids are used in a broad spectrum of emergent applications in all fields of sciences, such as solar collectors, nuclear/atomic reactors, automo- biles, micro-electronics, neuroelectronic interfaces, micro- fluidic systems, indicative testing, pharmaceutical, and biological processes, etc. Out of all types of nanoparticles, silver nanoparticles (Ag NPs) seem to have attracted the most interest in their potential applications. The availabil- ity of Ag NPs has ensured rapid adoption in the medical
sector. Ag NPs are used in a variety of critical applications such as anti-microbial processes (air and water purifica- tions), optical (solar cell), and conductive (touch screen, LCDs) processes, etc.
On the other hand, hybrid nanofluid is an advanced cat- egory of nanofluids made by dispersing nanoparticles' hybrid nature into the conventional heat transfer fluid.
Hybrid nanofluids offer better thermal performance than nanofluids. It is expected to use hybrid nanofluids for sim- ilar applications with nanofluids. Considering the emer- gent uses of hybrid nanofluids, many researchers have focused on the flow and thermal characteristics of hybrid nanofluids adopting various nanoscale models. Sarkar et al. (2015) performed a critical review on hybrid nanofluid flow to expose recent developments and applications in the field of thermal, medical, and industrial engineering.
Minea (2017) reported the thermal performance of hybrid nanofluids using Al2O3, TiO2 and SiO2 nanoparticles.
They concluded that all the thermophysical properties and
*Corresponding author; E-mail: tutusanasd@yahoo.co.in
thermal conductivity of hybrid nanofluids augment by 12%. Prakash et al. (2020) have examined the electro- osmotic flow of hybrid nano-liquids through an asymmet- ric microchannel vis peristalsis. They reported that the hybrid nanofluid was more efficient in heat transfer per- formance for similar nanofluids. Tripathi et al. (2020) numerically studied the electroosmosis-induced peristaltic pumping of couple stress hybrid nanofluids through a microchannel with Joule heating. Their results envisaged that as the couple stress parameter enlarges, the axial velocity increases in the core part of the channel for hybrid metallic nanofluids. Some recent articles related to hybrid nanofluid flow can be seen through the references (Sadaf et al., 2019; Ellahi et al., 2019; Akram et al., 2020;
Ali et al., 2021; Tripathi et al., 2020) and few references therein.
The peristalsis is of immense importance in biomechan- ics, biophysics, physiology, and bio-engineering due to its enormous mechanical and biological utilities. A wave motion initiated and regulated by symmetric contraction and expansion of flexible boundaries resulting in a push of the content within the flexible track is referred to as peri- stalsis. Due to its continuously increasing importance, this mechanism appeared in many folds. Therefore, it has been extensively studied with an even broader impact. This nat- ural transportation phenomenon often encounters in the functioning of urine transport, sperm transport, chyme motion, reptilian breathing, human speech (laryngeal pho- nation), robotic endoscopy, intestinal physiology, heart- lung machines, blood pumps in dialysis, diabetic pumps, finger and roller pumps, pharmacological drug delivery systems (Slawinsk and Terry, 2014). From the onset, it has attracted the special attention of researchers after one of the first reporting by Latham (1966). Akbar et al.(2016) analyzed the MHD peristaltic transport of nanofluids through a vertical channel with slip conditions. They observed that the pressure for Ag nanofluid is minimum.
Akram et al. (2021) have elucidated the electro-osmosis augmented MHD peristaltic transport of single-walled car- bon nanotubes (SWCNTs) suspension in aqueous media.
Outcomes revealed that a reduction in EDL thickness intensifies the nanofluid velocity as well as temperature.
Several mathematical models for the peristaltic motion have also received ample attention (Ranjit et al., 2019;
Sharma et al., 2019; Prakash et al., 2020; Prakash and Tripathi, 2020; Tripathi et al., 2021).
Electro-magneto-hydrodynamics (EMHD) has become attractive research attention in recent years owing to its diverse applications in the fields of biomechanics and engineering. EMHD has an imperative role in microfluidic devices, micro-electro-mechanical systems (MEMS), chemical devices, biomedical devices, biochemical anal- ysis, biomedical diagnostic instruments, separation pro- cess, reverse osmosis, etc. The flow actuation mechanism
is used in the micropump to create a pressure gradient using a pumping device. Such devices need frequent maintenance and use moving parts to create flow. When a charged solid surface is connected with water or an aque- ous solution, negative charges are formed on this surface.
The positive ions in the liquid will then be attracted to that surface, and the negative ions will repel from it. This cre- ates a thin layer with an unbalanced charge known as the electrical double layer (EDL). When there is an electric field parallel to the solid surface, the positively charged EDL will move in the direction of the electric field.
Because of this, the bulk liquid motion is generated via the viscous effect. This phenomenon is called the electro- osmotic flow (EOF), a very lucrative micro pumping mechanism for transporting a small amount of liquid in microchannel/capillaries. EOF possesses more advantages over conventional pressure-driven transport through microchannels. Electroosmosis has a broader spectrum of nano-and microscale technological applications like ion- exchange membrane designs, biochip fabrication, micro- bial fuel cells, non-absorbent polymer injection systems, corrosion mitigation in civil engineering, nano-bot propul- sion for medical utility, etc. The EOF was first elucidated by Reuss (1809) in 1809. Later on, Wiedemann (1852) has given the mathematical theory for the electro-osmotic flow. Considering the challenging demands and benefits of EOF devices, many researchers have focused their attention on the EOF subject to different flow assumptions with various fluid models. The dual occurrence of elec- troosmosis and peristalsis is critical in many industrial, engineering and biological analyses. Tripathi et al. (2016) inspected the unsteady electroosmosis and peristalsis- induced motion of charged liquid through a microchannel.
They reported that an increase in electrical field parameter causes an increase in maximum time-averaged flow rate.
Shit et al. (2016) developed a mathematical model for the electro-magnetohydrodynamic flow of biofluid with the couple-stress fluid model through a microchannel propa- gated by the peristaltic wave. Their result envisaged that the formation of the trapping bolus is strongly affected by electro-osmotic parameter and magnetic field strength.
The EOF of aqueous nanofluids via microchannel induced by peristalsis was studied analytically by Tripathi et al.
(2018). It was observed that the pressure difference rises with adding the electric field in the flow direction. Chaube et al. (2018) conducted an analytical study on the EOF of micropolar fluids via a microchannel driven by peristaltic pumping. The results showed that peristaltic pumping might alter by applying external electric fields. Jayavel et al. (2019) examined the EOF of pseudoplastic aqueous nano-liquids through a peristaltic microchannel. Prakash et al. (2019) analyzed the peristalsis-driven EMHD flow of intrauterine fluid flow. The electro-magnetohydrody- namic flow of an aqueous solution induced by peristaltic
waves in a non-Darcy porous m edium was analyzed by Noreen et al. (2019). Prakash et al. (2020) have simulated the EMHD double-diffusive convection of ionic nanoflu- ids through a peristaltic asymmetric microchannel.
Recently, Akram et al. (2020) have examined the elec- troosmosis-regulated peristaltic flow of nanofluids. Some more critical relevant studies can be seen through (Ranjit et al., 2019; Tripathi et al., 2017; Ramesh et al., 2019;
Prakash et al., 2019; Abo-Elkhair et al., 2019; Narla et al., 2020) and the references therein.
In the presence of an applied magnetic field, a partially ionized liquid encounters forces of more than one kind.
Subjected to a strong magnetic field, Hall current is induced due to the collision of electrons and ion-slip cur- rent due to the collision of ions. It is experimentally tes- tified that due to the implementation of a dominating magnetic field, Hall and ion slip currents are contrary to the force. These electromagnetic phenomena (Hall and ion slip currents) have a significant role in mechanics, bio- mechanics, and engineering, for example, Hall accelera- tors, Hall actuators, Hall effect sensors, magneto-meters, spacecraft propulsion, coils refrigeration, electric proces- sors, etc. Abbasi et al. (2016) inspected the impacts of Hall currents on the MHD peristaltic transport of nano- fluid in the presence of slip phenomena and Ohmic heat- ing. They reported that the presence of Hall currents lessens the changes due to an applied magnetic field in the state of nanofluid. Eldabe et al. (2016) explored the sig- nificance of Hall currents on the MHD peristaltic flow and heat and mass transfer of Williamson fluid in a symmetric channel via a porous medium under the influence of vis- cous dissipation and Joule heating. Outcomes revealed that the temperature of Williamson fluid decreases with increasing Hall parameter. Hayat et al. (2016a, 2016b) explained the influences of Hall and ion-slip currents on the mixed convection peristaltic motion of nanofluid in a channel. It was observed that the fluid velocity boosts and temperature drops for higher values of Hall and ion-slip parameters. The impacts of Hall and ion-slip currents on the peristaltic motion of couple stress fluid was exposed by Hayat et al. (2019). They concluded that an increase in Hall and ion-slip parameters lower the fluid temperature.
Other related works in this direction are cited in Refs.
(Hayat et al., 2007; Abo-Eldahab et al., 2010; Asghar et al., 2014; Hayat et al., 2014; Hayat et al., 2017; Abbasi et al., 2019; Asha and Sunitha, 2020).
In many industrial, bio-engineering, and physiological utilizations, non-Newtonian materials have received immense importance over Newtonian materials. In gen- eral, biofluids, industrial fluids, and chemicals are exam- ples of non-Newtonian fluids (NNF). The intricate flow physics of biofluids or industrial fluids can not be cap- tured accurately using Newtonian viscous fluid models.
Most of the non-Newtonian fluids with their various prop-
erties cannot be described by a single functional relation between shear stress and shear rate. Consequently, several NNF models like Casson, Sisko, couple stress, Ellis-Sisko, Ostwald-de-Waele, Carreau, Peek-McLean-Williamson, Reiner-Phillippoff, Herschel-Bulkley, etc. have been pro- posed by researchers. The Casson fluid (CF) model is the simplest model for describing the rheological conditions for transporting non-Newtonian fluids. This model was first explained by Casson (1959) for depicting the intricate rheological features of pigment-oil suspension. This model is more accurate and compact at both zero shear rate and infinite shear rate. CF exhibits yield stress. Jelly, tomato sauce, honey, soup, concentrated juices, shampoos, blood, physiological fluids are excellent examples of CF.
Akbar (2015) examined the MHD peristaltic flow involv- ing the Casson fluid model. Kattamreddy et al. (2018) investigated a Casson fluid's MHD electro-osmotic peri- staltic motion through a rotating asymmetric microchannel under the impacts of slip condition, viscous dissipation, and Joule heating. They observed that the electro-osmotic force causes a substantial elevation in the profile of Cas- son fluid velocity, temperature, and heat transfer rate.
Moatimid et al. (2019) studied the electro-osmotic flow of a non-Newtonian nanofluid through a microchannel due to peristaltic waves. Farooq et al. (2019) illustrated the per- istaltic transport of Casson material with thermal and vis- cous dissipation. Several other essential studies on the peristaltic transport with non-Newtonian fluid models under different configurations are cited in Refs. (Berli and Olivares, 2008; Chaube et al., 2018; Sadek and Pinho, 2019; Prakash et al., 2019; Nadeem et al., 2020).
Numerous experimental, numerical and analytical stud- ies have been conducted to better understand fluid flow situations through porous media, which has a complex internal structure pattern. The studies are largely based on experimental work performed by Darcy (1856) for an iso- thermal creeping flow. The application of Darcy’s law is restricted to cases in which the flow is laminar or stream- lined. Some of the applications of the study of flow through porous media are combustion processes, heat exchanger applications, solar energy collections, drying problems (vegetable, grains, ceramic, wood, and brick), flow through the different organs of the animal body (lungs and kidneys), problems involving nuclear energy (heat removal from the pebble-bed atomic reactors), seep- age of water through the river bed, migration of pollutants into the soil, aquifer and chemical reactors, and petroleum recovery processes. Because of this, many researchers have focused their study on the flows through the porous medium. Eldesoky and Mousa (2010) explored the peri- staltic movement of a Maxwellian fluid in a cylindrical tube filled with a homogeneous porous medium. Tanveer et al. (2017) reported the consequences of presented the nanofluid flow with the Carreau-Yasuda model in a curved
channel saturated by porous materials. The impact of porosity of the porous medium and an inclined magnetic field on the peristaltic motion of couple stress fluid flow through an inclined asymmetric channel is explored by Ramesh (2016). Tanveer and Malik (2021) scrutinized the impact of porosity on the MHD peristaltic flow of Ree- Eyring nanofluid in a curved geometry. Noreen et al.
(2021) have investigated the EOF flow of nanofluid through an asymmetric microfluidic channel via peristal- sis. The porosity decays the temperature throughout the microchannel.
Thermal radiation (TR) is the dominant mode of heat transmission in sunlight and is the most complex mode of thermal transportation. TR has earned special attention due to its significant role in drying technology, solar ponds, nuclear power plants, engine cooling procedures, furnaces, medical treatment, etc. Nanofluids' radiative (optical) properties are crucial in their efficient implemen- tation in solar power technology, energy systems, and bio- mechanics Said et al. (2013). Nanofluid flow with TR has many submissions in several innovative products like pho- tocatalysts, organic contaminants with low concentration, electrical energy production, and environmental resto- ration due to its strong oxidizing power and long-term sta- bility (Borges et al., 2016). Du and Tang (2015) elaborated on the transmission and scattering characteristics of nano- fluids. Bég et al. (2016) simulated thermal convection and radiation heat transmission via an annular porous medium as a solar energy absorber. Prakash et al.(2018) examined the impact of thermal radiation on the electroosmosis modulated peristaltic transport of ionic nano-liquids in a microchannel. They showed that the pressure rise enhances with increasing the radiation effects. Dissimilar delibera- tions of thermal radiation can be seen in references (Shit et al., 2016; Prakash et al., 2018; Prakash et al., 2019).
After reviewing the literature on electro-osmotic flow (EOF), it is identified that the electro-osmotic peristaltic transport of non-Newtonian hybrid nano-liquid through a microchannel under a high-powered electromagnetic field force has not been deliberated so far. The current study aims to examine the dual impact of Hall and ion-slip cur- rents on the EOF of non-Newtonian hybrid nano-liquid with Casson fluid model through a peristaltic symmetric vertical microchannel filled by a porous medium. The slip condition and thermal radiation effects are also outlined.
The Casson rheological nanofluid model is employed. An axial electrical field and transverse magnetic field are imposed. The charge distribution in the electric double layer (EDL) is assumed to follow the Boltzmann distri- bution. The surface potential is considered low enough to allow the Debye-Hückel linearization approximation. The governing equations are simplified under reliable LRN and LWL approximations. Exact solutions for the perti- nent physical variables of the resulting equations are
acquired utilizing the Mathematical built-in function DSolve. The consequences of diverse fluid dynamic parameters of interest on flow quantities are deliberated and featured through graphs. This disquisition is signifi- cant from engineering and bioengineering points of view.
From these points of view, this non-Newtonian rheologi- cal hybrid nanofluid simulation is relevant to electro- osmotic transport phenomena in pharmacology and nano- drug delivery systems.
2. Mathematical Formulation and Solution 2.1. Casson fluid model
The rheological equation representing an isotropic and incompressible flow of CF is as follows (Akbar et al., 2015; Moatimid et al., 2019):
(1)
where and are the shear rate and strain rate tensors respectively, b the dynam ic viscosity, b the plastic dynamic viscosity of the non-Newtonian fluid, 0 the yield stress for the non-Newtonian fluid, , and c the critical value of based on the non-Newtonian model. For
, equation (1) gives
(2) where is known as the Casson fluid param- eter which takes small value. Furthermore, the constitutive equation (2) tends to the ordinary Newtonian fluid when
.
2.2. Geometric model
Consider a laminar two-dimensional electro-osmotic flow (EOF) of an incompressible ionic non-Newtonian Casson hybrid nano-liquid through a vertical symmetric m icrochannel filled. The channel is filled with a hom o- geneous Darcian porous medium. The hybrid nano-liquid (Ag-SiO2/H2O) is consists of water (H2O) and dissimilar nanoparticles (Ag and SiO2 NPs). In order to the mathe- matical formation of the model problem, a coordinate frame , is chosen in which is assumed along the axial direction, and -axis along the transverse direction.
The origin is located at the channel centerline, and the right-hand side and left-hand side wall boundaries are located at , respectively. The flow is induced by the propagation of sinusoidal waves moving along the channel walls with the constant speed c of small amplitude a and wavelength . The flow model is sketched in Fig. 1.
An external electric field is applied along the axial direc-
0
0
2( ) , >
2
2( ) , <
2
b ij c
ij
b ij c
c
e e
ij eij
ij ij
e e
< c
(1 1)2 ,
ij b eij
0
2
b c
(x y) x
y
= y h
tion, which induces the EOF in the microchannel. A uni- form magnetic field of intensity B0 is also imposed transversely to the axial direction of the flow. Since the impacts of Hall and ion-slip currents are anticipated, the magnetic Reynolds number is assumed to be small (Cowl- ing, 1957). The microchannel walls are heated with a con- stant temperature T0 and carry a surface charge balanced by an equal and the opposite charge in the liquid main- tained through the distribution of ions.
The walls of the symmetric microchannel are described in the following mathematical form as (Noreen et al., 2019; Akram et al., 2020):
(3) where d is half width of the microchannel, a the wave amplitude, c the velocity of the wave, and the wave- length, t the time. The channel’s half width is small com- pared with the wavelength ( ).
2.3. Magnetohydrodynamics
The generalized Ohm’s law that serves MHD effect in the presence of Hall and ion slip currents can be described as (Cowling, 1957; Hayat et al., 2007; Hayat et al., 2019):
(4) where , , , , hnf, , are respectively the velocity vector, the magnetic field, the electrokinetic force, the current density vector, the effective electric con- ductivity of hybrid nanofluid, Hall parameter and ion slip parameter.
The Maxwell equations in the flow dynamics are stated
as (Hayat et al., 2019)
(5) In the view of the above assumptions, equation (4) yields
(6) (7) where and are the velocity components along and , respectively, and the current density com- ponents, and the electric filed components in the radial and axial directions in the laboratory frame, respec- tively.
Since induced magnetic field is neglected, thus
gives . In the absence of external electric field, one has [solving for and from equations (6) and (7)]
(8)
where .
2.4. Electrohydrodynamics
The well known Poisson-Boltzmann equation is employed to describe the electrical potential distribution across the microchannel. According to it, the electric potential in the microchannel is given by (Tripathi et al., 2016; Ramesh and Prakash, 2019; Akram et al., 2020):
(9) in which 0 the dielectric permittivity of the medium, and
e the net ionic charge density in a unit volume of the elec- trolyte due to the presence of the EDL and it is defined as:
(10) where e is the net electronic charge, z the valence of ions, and are the number of densities of cations and anions in the electrolytic solution, respectively and are defined by the Boltzmann distribution considering no EDL overlap (Noreen et al., 2019 Tripathi et al., 2016)
(11) where n0 is the average number of cations and anions in the electrolytic solution (which is independent of surface electro-chemistry), KB the Boltzmann constant, and Ta the average temperature of the electrolytic solution. In the microchannel, there is no gradient of ionic concentration in the axial direction that implies the distribution of ionic concentration is valid (Noreen et al., 2019).
Inserting (11) into (10), the electric charge density e is
= ( , ) = [ sin2 ( )],
y h x t d a z ct
d
0 02
= hnf( ) e( ) e i[( ) ],
J E q B J B J B B
B B
q B E
J
e i
= B, = 0, = 0.
E B J
t
(1 e i)JxeJy=hnf(Exv B 0), (1 e i)JyeJx=hnf(Eyu B 0),
u v x
y Jx Jy
Ex Ey
= B
E t
= 0 E
Jx Jy
0 0
2 2 2 2
= hnf ( ), = hnf ( ),
x e e y e e
e e e e
B B
J v u J v u
e= 1 e i
2 0
= e
= ( ),
e ez n n
n n
( )
= 0 ,
ez K TB a
n n e
Fig. 1. (Color online) Physical model of flow configuration.
rewritten as:
(12) In view of (9) and (12), the Poisson-Boltzmann equation is simplified in the form:
(13) In order to acquire analytical solution, this equation is further simplified via lubrication and Debye-Hückel lin- earization estimations in the subsection 2.8.
2.5. Governing equations
In the fixed frame of reference ( , ), the governing equations under the foregoing assumptions and on using (8) take the following form (Akrab et al., 2016;Noreen et al., 2019; Akram et al., 2020):
(14)
(15)
(16)
(17) where T is the fluid temperature, the fluid pressure, g the acceleration due to gravity, the axially applied electric field, the heat source, K the permeability of porous medium, qr the radiative heat flux, is the den- sity of hybrid nanofluid, the dynam ic viscosity of hybrid nanofluid, the thermal extension coefficient of hybrid nanofluid, the heat capacity of hybrid nanofluid, and khnf the thermal conductivity of hybrid nanofluid.
The radiation energy per unit time from a black body is proportional to the fourth power of the absolute tempera- ture and can be expressed following the Rosseland approximation as (Prakash et al., 2019; Prakash et al., 2019; Prakash et al., 2019):
(18)
where * is the Stefan-Boltzman constant and k* the Ros- seland mean absorption coefficient. The radiative heat flux is assumed to be small in the -direction.
It is adopted that the temperature difference within the flow domain is adequately small. Expanding the term T4 about T0 in a Taylor series and ignoring higher order terms in the first order in , the radiative heat flux can be estimated in the form
(19) The boundary conditions for the fluid velocity, tempera- ture and electric potential at the microchannel walls are given by (Noreen et al., 2019; Akram et al., 2020)
(20)
where * is the velocity slip coefficient, h* the heat trans- fer coefficient, and the thermal potential.
The slip flow phenomenon became significant due to its marked impact on the flow dynamics of polymer solu- tions, molten polymer, and non-Newtonian fluids (Akbar et al., 2016). It has crucaial a variety of applications in polishing internal cavities and artificial cardiac valves, blood pumping, polymer technology, etc. The convec- tively warmed wall boundary condition is very relevant due to its extensive implementations in technology and physiology (Ali et al., 2021).
2.6. Thermophysical properties of hybrid nanofluid The thermophysical properties of hybrid nanofluid are assumed to be constant except the density variation in the buoyancy force, which is estimated based on the Bouss- inesq approximation. The thermophysical properties of water, silver nanoparticles, and silicon dioxide nanoparti- cles which are adopted to use in this study, are tabulated in Table 1 and Table 2.
2.7. Governing equations in wave frame
The microchannel walls are propagated by peristaltic waves having the speed c and wavelength . The flow phenomenon is inherently unsteady in the laboratory frame of reference ( , ). The flow becomes steady in the wave frame of reference ( , ). The suitable trans- formations from the laboratory frame to the wave frame are defined as (Akbar et al., 2016; Prakash et al., 2019):
(21)
= 2 0 sinh( ),
e
B a
n ez ez
K T
2 0
0
=2 sinh( ).
B a
n ez ez
K T
x y
u v= 0,
x y
2 2
2 2 0
( ) =
(1 1)( ) ( ) ( )
hnf
hnf hnf
u u u v u
t x y
p u u g T T
x x y
02
2 2
(1 1) ( ) ,
hnf hnf
e e e x
e e
u B u v E
K
2 2
2 2
( ) = (1 1)( )
hnf v u v v v p hnf v v
t x y y x y
02
2 2
(1 1) ( ),
hnf hnf
e e
e e
v B u v
K
2 2
2 2 0
( cp hnf) ( T u T v T) =khnf( T T) ( qr qr) Q,
t x y x y x y
p
Ex
Q0
hnf
hnf
hnf
(cp hnf)
4 4
= ,
r 3 T
q k y
x
(T T 0)
= 16 .
r 3 T
q k y
0, 0, 0 at = 0 (central line of symmetry),
u T y
y y y
0 0
(1 1) = 0, = ( ), =
at = ( , ),
hnf u hnf T
u k h T T
y y
y h x t
0
x y
x y
= , = , = , = , = ,
( , ) = ( , , ).
x x ct y y u u c v v p x y p x y t
In view of (21), governing equations (14)-(17) can be put in the following form:
(22)
(23)
(24)
(25)
In the wave frame ( , ), the boundary conditions can be written as:
(26)
2.8. Dimensional analysis and approximations In order to normalize the governing equations, the fol- lowing non-dim ensional variables are introduced as u v = 0,
x y
2 2
2 2
[ ( ) ] = (1 1)( )
hnf c u u c u v u p hnf u u
x x y x x y
0 1
( )hnfg T T( ) hnf (1 )(u c) K
02
2hnf 2[ (e ) e ] e x,
e e
B u c v E
2 2
2 2
[ ( ) ] = (1 1)( )
hnf c v u c v v v p hnf v v
x x y y x y
02
2 2
(1 1) [ ( ) ) ],
hnf hnf
e e
e e
v B u c v
K
2 2
2 2 0
( ) [ ( ) ] =
( ) ( ) ,
p hnf
r r
hnf
T T T
c c u c v
x x y
T T q q
k Q
x y x y
x y
0, 0, 0 at 0,
u T T y
y y
0 0
(1 1) , ( ),
= at = ( )
hnf u hnf T
u c k h T T
y y
y h x
Table 1. Expressions for thermophysical properties for nanoliquid and hybrid nanoliquid (Sadaf et al., 2019).
Properties Nanoliquid (Ag-H2O) Hybrid nanoliquid (Ag-SiO2/H2O) Density
Viscosity
Thermal expansion coefficient Heat capacity Electrical conductivity
, Thermal
conductivity ,
where 1 and 2 are the solid volume fraction of nanoparticles (1 corresponds to the solid volume fraction of silver nanoparticles and
2 the solid volume fraction of silicon dioxide nanoparticle). SiO2-nanoparticles are dispersed into water to form a SiO2/H2O nanofluid, and then Ag-nanoparticles are dissolved into the nanofluid mixture to produce a homogeneous mixture of Ag-SiO2/H2O hybrid nano- fluid. The suffices s1, s2, f, nf, hnf denote the solid nanoparticles of silver (Ag), solid nanoparticles of silicon dioxide (SiO2), base fluid (blood), nanofluid and hybrid nanofluid, respectively. Special case, if , substantially, hybrid nanofluid is reduced to the pure water.
1 1
= (1 )
nf f s
hnf = (12)[(1 1) f 1 s1] 2 s2
1 2.5
=(1 )
f nf
= 1 2.5 2 2.5
(1 ) (1 )
f hnf
2 1
() = (1nf )( )f ( )s () = (1hnf 2)[(1 1)( )f
1 1
(cp nf) = (1 )( cp f) ( cp s) (cp hnf) = (12)[(1 1)( cp f) 1( cp s) ]1 2( cp s)2
1 1
3( 1)
= [1 ],
( 2) ( 1)
=
nf f
s f
2 2 2
2 2 2
2 2 ( )
= [ ]
2 ( )
s bf bf s
hnf bf
s bf bf s
1 1 1
1 1 1
2 2 ( )
= [ ]
2 ( )
s f f s
bf f
s f f s
1 1
2 2 ( )
= [ ]
2 ( )
s f f s
nf f
s f f s
k k k k
k k
k k k k
2 22 2
2 2
2 2 ( )
= [ ]
2 ( )
s bf bf s
hnf bf
s bf bf s
k k k k
k k
k k k k
1 1 1
1 1 1
2 2 ( )
= [ ]
2 ( )
s f f s
bf f
s f f s
k k k k
k k
k k k k
1= = 02
Table 2. Thermophysical properties of water, silver (Ag) and sil- icon oxide (SiO2) nanoparticles (Akram et al., 2020; Akbar et al., 2016).
Properties Water/base fluid Ag SiO2
(kg/m3) 997.1 10500 3970
cp(J/kg K) 4179 235 765
k (W/m K) 0.613 429 36
×10-5 (1/K) 21 1.89 0.63
(S/m) 5.5×10-6 3.6×10-7 4×105
(Akbar et al., 2016; Jayavel et al., 2019; Akram et al., 2020):
(27) where x is non-dimensional axial coordinate, y non- dimensional transverse coordinate, u and v non-dimen- sional axial and transverse velocity components, wave number, non-dimensional electric potential, 0 the ther- mal potential. amplitude ratio or occlusion of wall dis- placement parameter, h non-dimensional distance, the dimensionless temperature, and p dimensionless pressure.
The electric potential and ion distributions are de-cou- pled from the Navier- Stokes equations (i.e. independent of fluid velocity). On employing the Debye-Hückel lin- earization approximation, the Poisson-Boltzmann equation (13) can be linearized by . This approximation is valid for many applications when an aqueous solution pH value is near neutral and the magnitude of zeta potential is less than 25 mV (Kirby and Hasselbrink, 2004).
Therefore, the analytical tractability of the Poisson- Boltzmann equation is possible. The non-dimensional lin- earized Poisson-Boltzmann equation is of the form:
(28)
where is the electro-osmotic param-
eter (inverse of EDL thickness) or Debye-Hückel param- eter and is Debye length or characteristic thickness of electrical double layer (EDL). Electric double layer arises at the interface where the fluid solution and material attain opposite charges. Under the imposition of an elec- trical field, the electrical double layer is mobilized by the resulting Coulomb force.
On the use of (27), equations (22)-(25) can be put in the non-dimensional form as follows:
(29)
(30)
(31)
(32)
where the Reynolds number (the ratio of inertial force to the viscous force), is the squared Hatmann number (the ratio of magnetic force to the vis- cous force), the thermal Grashof num- ber (the ratio of the buoyancy to the viscous force), the Darcy number, the radiation parameter, the Prandtl number (the ratio of the momentum diffusivity to the thermal diffusivity), the heat source parameter, the Helmholtz-Smoluchowski velocity or maximum electro- osmotic velocity. A negative value of Uhs implies that electro-osmotic velocity is in the direction of peristaltic pumping whereas a positive value means that electro- osmotic velocity is in opposite direction to that of peri- staltic pumping. Furthermore
(33)
The steam function in the wave frame of reference (obeying the Cauchy-Riemann equations) satisfying (29) can be given as (Noreen et al., 2019; Akram et al., 2020) (34) In pursuance of (34) and adopting the assumptions of LRN (i.e. ) and LWL (i.e. ) and ignoring the terms of order and higher, equations (24)-(26) reduce to the following form:
(35)
(36)
(37)
0 0
, , , , , , B n,
x y u v d K T
x y u v
d c d ez
0 2 0
, , , ,
f
a h h T T p d p
d d T c
ε
sinh
2 2
2
2 2
( ) = ,
x y
0 0
= 2 =
B a D
n d
d ez K T
ε
1
D
u v= 0,
x y
2 2
1 2 2 2
( u u) = p (1 1)( u u)
Rex u v x
x y x x y
2 2
2 3
4 2 2
(1 1)( 1) [ ( 1)e e] hs ,
e e
x x M
x Gr u u v U
Da
2 2
2
1 2 2 2
( v v) = p (1 1)( v v)
Re x u v x
x y y x y
2 3 2
2 2
(1 1) [ (e 1) e ],
e e
x v x M u v
Da
2 2 2 2
5( ) =x6( 2 2) ( 2) ,
Rex u v Ra
x y Pr x y xy y
=
f
Re cd
2 2
0
2= f
f
M B d
2 0
( )
= f
f
Gr gd T c
= K2
Da d
03
= 16
3 ( p f) Ra T
k c
( )
= p f
f
Pr c k
0 2 0
= ( p f) Q d
T c
Uhs= 0c 0fEx
1 2
1 (1 2)[(1 1) 1 s] 2 s ,
f f
x
2 2.5 2.5 3
1 2
1 , ,
(1 ) (1 )
hnf f
x x
1 2
4 2 1 1 2
( ) ( )
(1 )[(1 ) ] ,
( ) ( )
s s
f f
x
1 2
5 2 1 1 2 6
( ) ( )
(1 )[(1 ) ] ,
( ) ( )
p s p s hnf
p f p f f
c c k
x x
c c k
, .
u v
y x
0
Re 1
2 2
2 ,
y
3 2
1 2
1 3 2 2
3 2
[(1 1) ]( 1)
,
e
e e
hs
p x x x M
x y Da y
x Gr U
p 0,
y
