Hybrid Effects of Thermal and Concentration Convection on Peristaltic Flow of Fourth Grade Nanofluids in an Inclined Tapered Channel:Applications of Double-Diffusivity

Safia Akramand Alia Razia

MCS,National University of Sciences and Technology,Islamabad,Pakistan

ABSTRACT This article brings into focus the hybrid effects of thermal and concentration convection on peristaltic pumping of fourth grade nanofluids in an inclined tapered channel.First,the brief mathematical modelling of the fourth grade nanofluids is provided along with thermal and concentration convection.The Lubrication method is used to simplify the partial differential equations which are tremendously nonlinear.Further, analytical technique is applied to solve the differential equations that are strongly nonlinear in nature,and exact solutions of temperature,volume fraction of nanoparticles,and concentration are studied.Numerical and graphical findings manifest the influence of various physical flow-quantity parameters.It is observed that the nanoparticle fraction decreases because of the increasing values of Brownian motion parameter and Dufour parameter,whereas the behaviour of nanoparticle fraction is quite opposite for thermophoresis parameter.It is also noted that the temperature profile decreases with increasing Brownian motion parameter values and rises with Dufour parameter values.Moreover,the concentration profile ascends with increasing thermophoresis parameter and Soret parameter values.

KEYWORDS Nanofluids; thermal and concentration convection; peristaltic flow; inclined tapered channel; fourth grade fluid

1 Introduction

Fluid transport with the help of peristaltic waves is frist studied by Latham [1].Since then it has been the central domain of research interest in physiological and mechanical situation.Peristaltic pumping is a mean or device for pumping fluids.It carries the fluid from lower pressure to higher pressure along the tube through contraction wave.This process occurs in many physiological mechanisms.For example, food movement from esophagus via stomach to intestine, urine excretion of a bladder through kidneys, movement of sperms and ova in male and female (fallopian tube) reproductive system respectively and chyme movement of gastrointestinal tract.Furthermore, peristaltic action relates to lump transfer in lymphatic vessels, blood flow in minute arteries and veins, and bile conduct through bile duct.There are many practical utilities of peristaltic pumping in biomechanical systems.Moreover, to pump any corroding material, roller and finger pumps are utilized to avoid the direct contact with the surface.The initial mathematical models of peristalsis were presented by Shapiro et al.[2] and Fung et al.[3].They acquired it through their working on a sinusoidal wave in endlessly long symmetrical channel or tube.In later studies, the focus was to further explore the peristaltic action for Newtonian and non-Newtonian fluids in diverse situations.Numerous experimental, analytical, and numerical models have been discussed in this regard.Though much work is available on the subject but few recent research are mentioned by the studies given at [4-10].

The field of practical application of nanofluids in industry and engineering has renowned the interest of the researchers.These applications are related to photodynamic therapy, the lotus effect for self-cleaning surfaces, primary cellular level of biological organisms, membranes for filtering on size or charge (e.g., for desalination), shrimps snapping along with beetle wings super-hydrophobic process, use of charged polymers for lubrication, nano porous materials for size exclusion chromatography, molecular motors, drug transfer, neuro electronic interfaces, cancer diagnostics and therapies, protein engineering, machines for cell repair and light casting on molecular motor cells such as kinesis and charged filtration in the kidney basal membrane,etc.[11].Choi et al.[12] coined the word nanofluid which designates to the fluid containing nano-sized particles for conventional heat transfer.Moreover, the liquid contains ultrafine particles which are below 50 nm diameter.These particles could be deducted with metals like (Cu, Al),nitrides (SiN), oxides (Al2O2), or in non-metals namely nanofibers, graphite, carbon nanotubes and droplets.Masuda et al.[13] propagated that nanofluids can be used in advanced nuclear system because they have the characteristics to enhance the thermal conductivity.An analytical model was followed by Buongiorno et al.[14,15] which is based on the nanofluids flow.The model implied convective transport in nanofluids with Brownian diffusion and thermophoresis.It is revealed in his studies that Brownian diffusion and thermophoresis were significant nanoparticle or base-fluid slip process.This also explains abnormal convective heat transfer enhancement in nanofluids.The concept of nanofluids both in peristaltic and non-peristaltic flow are mentioned in [16-30].

A fluid dynamics phenomenon, termed as double diffusive convection, is a convection directed by two different density gradients having different rates of diffusion.Fluids convection is propelled by density variation under the influence of gravity.Such density variation can occur due to the gradients in fluid composition or differences in temperature through thermal expansion.Compositional and thermal gradients usually diffuse over time which hampers the ability to conduct the convection.Therefore, that gradient requires in the flow areas to carry on convection.We can find out an example of double diffusive convection in oceanography.Here salt and heat concentration dwell with various gradients and diffuse at varying rates.Influence of cold water, such as from iceberg, can affect these variables.Akbar et al.[31] has examined peristaltic flow in nanofluid with double diffusive natural convection.Further theoretical works on double diffusion are mentioned in [32-40].

Limited work has been traced in literature review on inclined tapered channel with double diffusive convection on peristaltic flow of nanofluids.Hence, inclined tapered channel on peristalsis is considered with double diffusive convection flow for the current study by taking non-Newtonian nanofluid.

2 Formulation and Methodology

Let’s consider the fourth-grade peristaltic transport in tapered channel with 2d width.The sinusoidal wave propagates at constant velocitycalong channel walls.At upper and lower walls the temperature, solute and concentration of nanoparticles isT0,C0,Θ0andT1,C1,Θ1respectively.In addition, we also assume that the channel is inclined at an angleα.For 2-dimensional and directional flow the field of velocity is characterized asThe flow geometry of the physical model is now described in Fig.1.The walls of tapered channel that are at lower leveland upper levelare represented in a fixed frame of reference as

Against fourth grade fluid, the stress tensor is described by [5]

whereμrefers to constant viscosity,refers for material constants,refers to transpose anddenotes Rivilin-Ericksen tensors.

Figure 1: Flow geometry of the physical model

The equation of continuity, momentum, temperature, fraction of nanoparticles and the solute concentration of an incompressible fluid for two-dimensional cases is given as

In the above equationsρf,g,ρf0,ρp,T,C,Θ,DB,DT,DCT,Ds,DTC,βC,βT,ε,(ρc)p,(ρc)frefers to base fluid density, gravity acceleration, fluid density atT0, particles density, temperature, concentration, nanoparticle volume fraction, Brownian diffusion coefficient, thermophoretic diffusion coefficient, soret diffusively, solutal diffusively, Dufour diffusively, volumetrically solutal expansion coefficient of a fluid, volumetrically thermal expansion coefficient of a fluid, thermal conductivity,nanoparticle heat capacity and fluid heat capacity, respectively.

Defining the subsequent dimensionless quantities

In above dimensionless quantities Pr ,δ,Re,Grc,GrF,GrT,Le,Nb,Ln,Nt,NCT,NTC,θ,Ωandγrepresenting Prandtl number, wave number, Reynolds number, solutal Grashof number, nanoparticle Grashof number, thermal Grashof number, Lewis number, parameter of Brownian motion,nanofluid Lewis number, parameter of thermophoresis, parameter of Soret, parameter of Dufour,dimensionless temperature, solutal concentration and fraction nanoparticle, respectively.

By using Eq.(6), Eq.(13) is automatically satisfied and Eqs.(7)-(12) for stream functionψ,temperatureθ, nanoparticle fractionγand solute concentrationΩin wave frame becomes

Now using supposition of long wavelength and low number of Reynolds, the Eqs.(13)-(17)becomes

Eliminate pressure from Eqs.(18) and (19) we get

where

andΓ=κ2+κ3is Deborah number.

In wave frame the boundary conditions regarding stream functionΨ, temperatureθ, fraction of nanoparticleΩand solute concentrationγare described as:

3 Exact Solution

The exact solution of the volume fraction of nanoparticles which satisfies the relevant condition (28) is described as

The exact solution of the solutal (species) concentration which satisfies the relevant condition(29) is described as

The exact solution of temperature which satisfies the relevant condition (27) is described as

where

4 Analytical Technique

The differential Eqs.(19) and (24) are non-linear so it is difficult to find exact solutions to these equations.So regular perturbation technique is used for finding the solutions of Eqs.(19)and (24).Expand nowΨ,pandFas

With assistance from Eqs.(34)-(36) into Eqs.(19), (24) and (26) combining like powers ofΓ,we get the following system as follows:

System of orderΓ0

System of orderΓ1

4.1 Solution for Zeroth Order System

Solution of Eq.(37) that satisfies the boundary conditions (39)-(42) is described as follows:

The pressure gradient for this order is described as

whereconstants are used for simplifying equations and are described in Appendix.The remaining constantsL1,L2,L3andL4are determined using boundary conditions (39)-(42) and are described in Appendix.

4.2 Solution for First Order System

Using solution zero-order (49) into (43), the solution of Eq.(43) which satisfies the boundary conditions (45)-(48) is described as follows:

The pressure gradient for this order is described as

whereconstants are used for simplifying equations and are described in Appendix.The remaining constantsL5,L6,L7andL8are determined using boundary conditions (45)-(48) and are described in Appendix.

Now for small parameterΓ, summarizing the perturbation results we have

DefiningF=F0+ΓF1and usingF0=F-ΓF1and then ignoring terms larger thanO(Γ)the results obtained by Eq.(53) to Eq.(55) showing up tillΓ.

For average pressure increase the non-dimensional expression is given as follows:

5 Different Wave Forms

The expression (in non-dimensional form) for the considered wave forms is defined as follows:

1.Multisinusoidal wave

2.Triangular wave

3.Trapezoidal wave

6 Graphical Outcomes

To interpret the results quantitatively we consider the instantaneous volume flow rateF(x,t)periodic in (x-t) [5] as

F(x,t)=Q+asin[2π(x-t)+φ]+bsin[2π(x-t)]

hereQis average time of flow through a single wave cycle and

To observe the graphical outcomes of concentration, temperature, nanoparticle fraction, pressure gradient, pressure rise and streamlines Figs.2-11 are displayed.The temperature profile effect is plotted for the different values ofNbandNTCin Figs.2a and 2b.It is seen in Fig.2a that temperature profile behaviour decreases with increasingNbvalues.This is because temperature exhibits a direct relationship withNb.In Fig.2b the temperature profile show opposite effect as compared withNb.Here temperature effects increases with increasingNTCvalues.Fig.3 shows the impact ofNtandNCTon concentration profile.It is shown in Figs.3a and 3b that concentration profile increases with increasingNtandNCTvalues.This is due to the direct relationship of concentration withNtandNCT.To view the impact of nanoparticle fraction onNb,NtandNTCFigs.4a-4c are plotted.It is shown in Fig.4 that behavior of nanoparticle fraction decreases because of the increasing values ofNbandNTC(see Figs.4a and 4c), whereas the behaviour of nanoparticle fraction is quite opposite forNt.In this case nanoparticle fraction increases due to the increasingNtvalues.

Figure 2: (a) Profile of temperature (θ) for various Nb values (sinusoidal wave).(b) Temperature (θ) profile for various NTC values (sinusoidal wave)

Figure 3: (a) Profile of Solutal concentration (γ) for various Nt values (sinusoidal wave).(b) Solutal concentration (γ) profile for various NCT values (sinusoidal wave)

Figure 4: (a) Profile of Nanoparticle fraction (Ω) for various Nb values (sinusoidal wave).(b) Nanoparticle fraction (Ω) profile for various Nt values (sinusoidal wave).(c) Nanoparticle fraction (Ω) profile for various NTC values (sinusoidal wave)

To study the graphical results of pressure, rise Figs.5a-5d are plotted.As it turns out in Figs.5a and 5b that pressure rise decreases within the regions whereΔp ＞0,Q ＜0 (retrograde pumping),Δp ＞0,Q ＞0 (peristaltic pumping) andΔp=0 (free pumping), whereas behavior is opposite over the regions whereΔp ＜0,Q ＞0 (copumping region).Here pressure rise enhanced by increasing values ofmandΓ.Figs.5c and 5d show the pressure rise actions for the various Re andNtvalues.From these figures we can see that pressure rise increases in all pumping regions (peristaltic, retrograde, copumping and free) by increasing values of Re andNt.Graphical behavior of pressure gradient is illustrated in Fig.6a-6c.It is represented in Fig.6a that whenxε[0,0.7] the pressure gradient decreases by increasingNbvalues, while the pressure gradient behaviour is quite opposite whenxε[0.7,1].Here pressure gradient increases because of increasing values ofNb.Fig.6b indicates that whenxε[0.7,1] the value of the pressure gradient decreases because of increasing values ofΓ.It shows up in Fig.6c that the behavior of pressure gradient decreases due to the increasing values ofFr.The pressure gradient behaviour for the various wave forms is shown in Figs.7a-7d.From these figures it is seen that trapezoidal waves are found to have maximum pressure gradient.

Figure 5: (a) Pressure rise over one wavelength (Δp) against volume flow rate (Q) for various m values (sinusoidal wave).(b) Pressure rise over one wavelength (Δp) against volume flow rate (Q)for various Γ values (sinusoidal wave).(c) Pressure rise over one wavelength (Δp) against volume flow rate (Q) for various Re values (sinusoidal wave).(d) Pressure rise over one wavelength (Δp)against volume flow rate (Q) for various Nt values (sinusoidal wave)

Figure 6: (a) Pressure gradient (∂p/∂x) against axial distance (x) for various Nb values (sinusoidal wave).(b) Pressure gradient (∂p/∂x) against axial distance (x) for various Γ values(sinusoidal wave).(c) Pressure gradient (∂p/∂x) against axial distance (x) for various Fr values(sinusoidal wave)

Figure 7: (a-d): Pressure gradient (dp/dx) against axial distance (x) for different wave shapes

For studying the phenomenon of trapping Figs.8-11 are plotted.Streamlines for the different values ofNCTandmare displayed in Figs.8 and 9.It shows up in Figs.8 and 9 that the size and number of the trapped bolus are increased by rising values ofNCTandm.Fig.10 is plotted to observe the streamlines pattern of the streamlines for differentNbvalues.Number of trapping bolus is observed to decrease by increasingNbvalues.Patterns of streamlines for various wave types are shown in Figs.11a and 11b.

Figure 8: Streamlines of NCT

Figure 9: Streamlines of m

Figure 10: Streamlines of Nb

Figure 11: (a-d): Streamlines of different wave shapes

7 Concluding Remarks

This article highlights the hybrid effects of thermal and concentration convection on peristaltic pumping of fourth grade nanofluids in an inclined tapered channel.The mathematical modelling of the fourth grade nanofluids is given along with thermal and concentration convection.Analytical technique is used to solve the differential equations that are strongly nonlinear in nature.Exact solutions of temperature, volume fraction of nanoparticles, and concentration are explored.The key finding can be encapsulated as follows:

· The temperature profile behaviour decreases with increasingNbvalues and increases with increasingNTCvalues.

· The concentration profile increases with increasingNtandNCTvalues.

· The behaviour of nanoparticle fraction decreases because of the increasing values ofNbandNTC, whereas the behavior of nanoparticle fraction is quite opposite forNt.

· The behaviour of pressure gradient decreases due to the increasing values ofFr.

· The size and number of the trapped bolus are increased by rising values ofNCTandm.

Funding Statement:The authors received no specific funding for this study.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.

Appendix