Peristaltic Flow of a Nanofluid under the Effect of Hall Current and Porous Medium

Abstract

The problem of peristaltic flow of an incompressible viscous electrically conducting nanofluid in a vertical asymmetric channel through a porous medium is investigated by taking the Hall effects into account. The governing equations are formulated and simplified under the assumptions of long wavelength and low Reynolds number. The solutions for temperature and nanoparticle profiles are obtained by using the homotopy perturbation method (HPM) and closed form solutions for stream function and pressure gradient are developed. Finally, the effects of various emerging parameters on the physical quantities of interest are plotted and discussed.

1. Introduction

The peristaltic flow problems have attracted the attention of many researchers because of its increasing importance, specially in engineering, physiology, and biological systems. This is due to its many applications in real life such as urine transport from kidney to bladder through the ureter; transport of lymph in the lymphatic vessels; swallowing food through the esophagus; the movement of chyme in the gastrointestinal tract; ovum movement in the fallopian tube; vasomotion of small blood vessels such as venules, capillaries, and arterioles; and also in sanitary fluid transportation of corrosive fluids. Many modern mechanical devices have been designed on the principal of the peristaltic pumping for transporting fluids without internal moving parts, for example, the blood pump in the heart-lung machine and the peristaltic transport of noxious fluid in nuclear industry. For more details in this direction see, for example, [1–22].

The study of magnetohydrodynamics flows with Hall currents has important engineering applications in problems of magnetohydrodynamic generators and of Hall accelerators as well as in flight magnetohydrodynamics. Siddiqui et al. [23], studied effects of Hall current and heat transfer on MHD flow of a Burgers fluid due to a pull of eccentric rotating disks. Hall effects on peristaltic flow of a Maxwell fluid in a porous medium have been studied by Hayat et al. [24]. Hayat et al. [25], studied effects of Hall current and heat transfer on rotating flow of a second grade fluid through a porous medium. The effect of Hall currents on interaction of pulsatile and peristaltic transport induced flows of a particle fluid suspension that had been examined by Gad [26]. Recently the analytical solution for Hall and ion-slip effects on mixed convection flow of couple stress fluid between parallel disks is considered by Srinivasacharya and Kaladhar [27]. Narayana et al. [28] studied the effects of Hall current and radiation absorption on MHD micropolar fluid in a rotating system.

Nanofluid is a fluid containing nanometer-sized particles, called nanoparticles. These fluids are engineered colloidal suspensions of nanoparticles in a base fluid. The nanoparticles used in nanofluids are typically made of metals (Al, Cu), oxides (Al₂O₃), carbides (SiC), nitrides (AlN, SiN), or nonmetals (Graphite, carbon nanotubes) and the base fluid is usually a conductive fluid, such as water or ethylene glycol. Other base fluids are oil and other lubricants, biofluids, and polymer solutions. Nanoparticles range in diameter between 1 and 100 nm. Nanofluids commonly contain up to a 5% volume fraction of nanoparticles to ensure effective heat transfer enhancements. Nanofluids have novel properties that make them potentially useful in many applications in heat transfer, including microelectronics, fuel cells, pharmaceutical processes, and hybrid-powered engines. In engineering devices it has been widely used for engine cooling/vehicle thermal management, domestic refrigerator, chiller, heat exchanger, and nuclear reactor; in grinding, in machining, in space, defense, and ships; and in boiler flue gas temperature reduction.

Nowadays, there is a continuous focus of the researchers in the flow analysis of nanofluids because of, in addition to, the above mentioned applications these fluids enhance thermal conductivity of the base fluid enormously, which is beyond the explanation of any existing theory. They are also very stable and have no additional problems, such as sedimentation, erosion, and additional pressure drop. Choi [29] was the first who initiated this nanofluid technology. A detailed analysis of nanofluids was discussed by Buongiorno [30]. Khan and Pop [31] considered the boundary layer flow of a nanofluid passing a stretching sheet. Recently, Mustafa et al. [32] examined the influence of wall properties on the peristaltic flow of a nanofluid. The endoscopic effects on peristaltic flow of a nanofluid are introduced by Akbar and Nadeem [33]. The same authors Akbar and Nadeem studied the peristaltic flow of a Phan-Thien-Tanner nanofluid in a diverging tube [34]. Akbar et al. [35, 36] studied peristaltic flow of a nanofluid with slip effects and peristaltic flow of a nanofluid in a nonuniform tube.

To the best of the author’s knowledge, no attempt is available in the literature which deals with the peristalsis of nanofluid including the Hall effect through a porous medium. Even the study of peristaltic flow of an electrically conducting nanofluid in a porous medium is not available. Also, the study of peristaltic flow of a nanofluid through a porous medium with Hall effects has not been made yet. The present study fills the gap in these directions.

The aim of this work is to investigate the peristaltic flow of an electrically conducting nanofluid in a vertical asymmetric channel through a porous medium with constant permeability. Modified Darcy’s law for a nanofluid including the Hall current has been used for the modeling. In fact, the Hall effect is important when the Hall parameter is high. This happens when the magnetic field is strong. In most cases, the Hall term has been ignored in applying Ohm’s law as it has no marked effects for small and moderate values of the magnetic field. However, the current trend in the application of magnetohydrodynamics is towards a strong magnetic field, so that the influence of electromagnetic force is noticeable. Under these conditions, the Hall current is important and it has marked effects on the magnitude and direction of the current density and consequently on the magnetic-force term. Therefore, it is important to study the effect of the Hall current on the flow. The governing equations of motion, energy, and nanoparticles for nanofluids have been reduced under the assumption of long wavelength and low Reynolds number. The reduced equations are then solved analytically via homotopy perturbation method [37–39]. The physical behaviors of emerging parameters are discussed through graphs. At the end the trapping phenomenon is discussed.

2. Problem Formulation

Consider the peristaltic flow of an incompressible viscous electrically conducting nanofluid in a two-dimensional vertical asymmetric channel of width 𝑑₁+𝑑₂ through a porous medium. Asymmetry in the channel is produced by assuming the peristaltic wave trains propagating with constant speed 𝑐 along the walls (𝐻₁ is the right hand side wall and 𝐻₂ is the left hand side wall) to have different amplitudes and phases as shown in Figure 1. The shapes of the channel walls are represented as 𝑌=𝐻₁(𝑋,𝑡)=𝑑₁+𝑎₁cos2𝜋𝜆[𝑋−𝑐𝑡],rightwall,𝑌=𝐻₂(𝑋,𝑡)=−𝑑₂−𝑏₁cos2𝜋𝜆[𝑋−𝑐𝑡]+𝜙,leftwall, (1) where 𝑎₁, 𝑏₁ are the amplitudes of the right and left waves, 𝜆 is the wave length, and 𝜙 is the phase difference. Further, 𝑑₁, 𝑑₂, 𝑎₁, 𝑏₁, and 𝜙 satisfy the following inequality: 𝑎²₁+𝑏²₁+2𝑎₁𝑏₁cos𝜙≤𝑑₁+𝑑₂², (2) so that the walls will not intersect with each other. A uniform magnetic field with magnetic flux density vector 𝐁=(0,0,𝐵₀) is applied; the induced magnetic field is neglected by taking a very small magnetic Reynolds number. The expression for the current density 𝐉 including the Hall effect and neglecting ion-slip and thermoelectric effects is given by [40] 1𝐉=𝜎𝐄+𝐕∧𝐁−𝑒𝑛_𝑒(𝐉∧𝐁), (3) where 𝐉 is the current density, 𝜎 is the electric conductivity of the fluid, 𝐕 is the velocity vector, and 𝑚=𝜎𝐵₀/𝑒𝑛_𝑒 is the Hall parameter. It is also assumed that 𝐄=0 (since there is no applied polarization voltage), 𝑒 is the electric charge and 𝑛_𝑒 is the number density of electrons. The heat transfer and nanoparticle processes are maintained by considering temperatures 𝑇₀, 𝑇₁ and nanoparticle phenomena 𝐶₀, 𝐶₁ to the right and left sides walls of the channel, respectively. The fundamental equations governing this model [41] together with the generalized Ohm’s law taking the effects of Hall currents and Maxwell’s equations into account are 𝜌∇⋅𝐕=0,_𝑓𝑑𝐕𝑑𝑡=−∇𝑃+𝜇∇²𝜇𝐕−𝑘₁𝐕+𝐉×𝐁+𝐟,𝜌𝑐_𝑓𝑑𝑇𝑑𝑡=𝑘∇²𝑇+𝜌𝑐_𝑝𝐷_𝐵𝐷∇𝐶∇𝑇+^𝑇𝑇₀∇𝑇∇𝑇,𝑑𝐶𝑑𝑡=𝐷_𝐵∇²𝐷𝐶+^𝑇𝑇₀∇²𝑇, (4) where 𝑃 is the pressure, 𝑘₁ is the permeability parameter, 𝜇 is the dynamic viscosity coefficient, 𝐟 is the body forces, ∇² is the Laplacian operator, 𝑑/𝑑𝑡 is the material derivative, 𝑡 is the time, 𝜌_𝑓 is the density of the fluid, 𝜌_𝑝 is the density of the particle, 𝑇 is the temperature of the fluid, 𝑐 is the volumetric volume expansion, 𝐶 is the nanoparticle phenomena, the ambient values of 𝑇 and 𝐶 as 𝑦 tends to ℎ₁ are denoted by 𝑇₀ and 𝐶₀, 𝐷_𝐵 is the Brownian diffusion coefficient, and 𝐷_𝑇 is the thermophoretic diffusion coefficient.

Figure 1 

Geometry of the problem.

The equations governing the two-dimensional motion of this model are 𝜕𝑈+𝜕𝑋𝜕𝑉𝜌𝜕𝑌=0,_𝑓𝜕𝑈𝜕𝑡+𝑈𝜕𝑈𝜕𝑋+𝑉𝜕𝑈𝜕𝑌=−𝜕𝑃𝜕𝜕𝑋+𝜇²𝑈𝜕𝑋²+𝜕²𝑈𝜕𝑌²−𝜇𝑘₁𝑈+𝜌_𝑓𝑔𝛼𝑇−𝑇₀+𝜌_𝑓𝑔𝛼𝐶−𝐶₀+𝜎𝐵²₀1+𝑚²(),𝜌𝑚𝑉−𝑈_𝑓𝜕𝑉𝜕𝑡+𝑈𝜕𝑉𝜕𝑋+𝑉𝜕𝑉𝜕𝑌=−𝜕𝑃𝜕𝜕𝑌+𝜇²𝑉𝜕𝑋²+𝜕²𝑉𝜕𝑌²−𝜇𝑘₁𝑉−𝜎𝐵²₀1+𝑚²(𝑚𝑈+𝑉),𝜕𝑇𝜕𝑡+𝑈𝜕𝑇𝜕𝑋+𝑉𝜕𝑇=𝑘𝜕𝑌𝜌𝑐_𝑓𝜕²𝑇𝜕𝑋²+𝜕²𝑇𝜕𝑌²+𝜌𝑐_𝑝𝜌𝑐_𝑓×𝐷_𝐵𝜕𝐶𝜕𝑋𝜕𝑇+𝜕𝑋𝜕𝐶𝜕𝑌𝜕𝑇+𝐷𝜕𝑌_𝑇𝑇₀𝜕𝑇𝜕𝑋²+𝜕𝑇𝜕𝑌²,𝜕𝐶𝜕𝑡+𝑈𝜕𝐶𝜕𝑋+𝑉𝜕𝐶𝜕𝑌=𝐷_𝐵𝜕²𝐶𝜕𝑋²+𝜕²𝐶𝜕𝑌²+𝐷_𝑇𝑇₀𝜕²𝑇𝜕𝑋²+𝜕²𝑇𝜕𝑌², (5) where 𝑈 and 𝑉 are the velocity components in the laboratory frame (𝑋,𝑌). Following Shapiro et al. [3] we introduce a wave frame of reference (𝑥,𝑦) moving with velocity 𝑐₁ in which the motion becomes independent of time when the channel is an integral multiple of the wavelength and the pressure difference at the the ends of the channel is a constant. The transformation from the laboratory frame of reference (𝑋,𝑌) to the wave frame of reference (𝑥,𝑦) is given by 𝑥=𝑋−𝑐₁𝑡,𝑦=𝑌,𝑢=𝑈−𝑐₁,𝑣=𝑉,𝑝(𝑥)=𝑃(𝑥,𝑡), (6) where (𝑢,𝑣), 𝑝 and (𝑈,𝑉), 𝑃 are the velocity components and pressure in the wave and laboratory frames of reference, respectively. The pressure 𝑝(𝑥) remains a constant across any axial station of the channel under the assumption that the wavelength is large and the curvature effects are negligible.

The following are nondimensional variables: 𝑥^∗=𝑥𝜆,𝑦^∗=𝑦𝑑₁,𝑢^∗=𝑢𝑐₁,𝑣^∗=𝑣𝑐₁,𝑑𝛿=¹𝜆,𝑝^∗=𝑝𝑑²₁𝑐₁𝜆𝜇,𝑡^∗=𝑡𝜆,ℎ₁=ℎ₁𝑑₁,ℎ₂=ℎ₂𝑑₁𝑑,𝑑=²𝑑₁,𝑎𝑎=¹𝑑₁𝑏,𝑏=¹𝑑₁𝜌,Re=^𝑓𝑑₁𝑐₁𝜇𝑘,𝑠=¹𝑑²₁,𝜃=𝑇−𝑇₀𝑇₁−𝑇₀,Ω=𝐶−𝐶₀𝐶₁−𝐶₀𝑘,𝛼=𝜌𝑐_𝑓,𝑁_𝑏=𝜌𝑐_𝑝𝐷_𝐵𝑐₁−𝑐₀𝜌𝑐_𝑓𝛼𝜈,Pr=𝛼,𝑁_𝑡=𝜌𝑐_𝑝𝐷_𝑇𝑇₁−𝑇₀𝜌𝑐_𝑓𝛼𝑇₀,𝐺_𝑟=𝛼𝑔𝑇₁−𝑇₀𝑑²₁𝜈𝑐₁,𝐵_𝑟=𝛼𝑔𝐶₁−𝐶₀𝑑²₁𝜈𝑐₁,𝑀=𝜎𝐵²₀𝑑²₁𝜇𝜇,𝜈=𝜌_𝑓, (7) where Re is the Reynolds number, 𝛿 is the dimensionless wave number, 𝑠 is the porosity parameter, Pr is the Prandtl number, 𝑁_𝑏 is the Brownian motion parameter, 𝑁_𝑡 is the thermophoresis parameter, 𝐺_𝑟 is the local temperature Grashof number, 𝐵_𝑟 is the local nanoparticle Grashof number, and 𝑀 is the magnetic parameter. Using (6) and (7) together with the stream function Ψ defined by 𝑢=𝜕Ψ𝜕𝑦,𝑣=−𝛿𝜕Ψ𝜕𝑥 (8) in (5), adopting the long wave length and low Reynolds number assumptions, we get the following: 𝜕𝑝=𝜕𝜕𝑥³Ψ𝜕𝑦³−Γ²𝜕Ψ𝜕𝑦+𝐺_𝑟𝜃+𝐵_𝑟Ω−Γ²,𝜕𝑝𝜕𝜕𝑦=0,0=²𝜃𝜕𝑦²+𝑁_𝑏𝜕𝜃𝜕𝑦𝜕Ω𝜕𝑦+𝑁_𝑡𝜕𝜃𝜕𝑦²,𝜕0=²Ω𝜕𝑦²+𝑁_𝑡𝑁_𝑏𝜕²𝜃𝜕𝑦². (9)(10)(11)(12) Eliminating the pressure gradient between (9) and (10) we get 𝜕⁴Ψ𝜕𝑦⁴−Γ²𝜕²Ψ𝜕𝑦²+𝐺_𝑟𝜕𝜃𝜕𝑦+𝐵_𝑟𝜕Ω𝜕𝑦=0, (13) where Γ²=1𝑠+𝑀1+𝑚². (14)

The corresponding boundary conditions are 𝐹Ψ=2,𝜕Ψ𝜕𝑦=−1,𝜃=0,Ω=0at𝑦=ℎ₁,𝐹Ψ=−2,𝜕Ψ𝜕𝑦=−1,𝜃=1,Ω=1at𝑦=ℎ₂, (15) where ℎ₁=1+𝑎cos2𝜋𝑥,ℎ₂=−𝑑−𝑏cos2𝜋𝑥+𝜙, (16) and 𝑎, 𝑏, 𝑑, and 𝜙 satisfy the relation 𝑎²+𝑏²+2𝑎𝑏cos𝜙≤(1+𝑑)². (17) And, 𝐹 is the dimensionless average flux in the wave frame defined by 𝐹=^ℎ1_ℎ^(𝑥)_2(𝑥)𝜕Ψℎ𝜕𝑦𝑑𝑦=Ψ₁(𝑥)−Ψℎ₂(𝑥). (18)

3. Method of Solution

3.1. Homotopy Perturbation Method

The homotopy perturbation method is a combination of the perturbation method and the homotopy method which eliminates the drawbacks of the traditional perturbation methods while keeping all their advantages. On the basis of the homotopy perturbation method [37–39] we can write (11) and (12) as follows: 𝐻𝑞,𝜃=1−𝑞𝐿(𝜃)−𝐿𝜃₁₀+𝑞𝐿(𝜃)+𝑁_𝑏𝜕𝜃𝜕𝑦𝜕Ω𝜕𝑦+𝑁_𝑡𝜕𝜃𝜕𝑦²,𝐻𝑞,Ω=1−𝑞𝐿(Ω)−𝐿Ω₁₀𝐿(Ω)+𝑁+𝑞^𝑡𝑁_𝑏𝜕²𝜃𝜕𝑦². (19) Or 𝐻𝑞,𝜃=𝐿(𝜃)−𝐿𝜃₁₀+𝑞𝐿𝜃₁₀+𝑞𝑁_𝑏𝜕𝜃𝜕𝑦𝜕Ω𝜕𝑦+𝑁_𝑡𝜕𝜃𝜕𝑦²,𝐻𝑞,Ω=𝐿(Ω)−𝐿Ω₁₀+𝑞𝐿Ω₁₀𝑁+𝑞^𝑡𝑁_𝑏𝜕²𝜃𝜕𝑦², (20) with 𝐿=𝜕²/𝜕𝑦² as the linear operator, the initial approximations 𝜃₁₀ and Ω₁₀ can be defined as 𝜃₁₀𝑥,𝑦=𝑦−ℎ₁ℎ₁−ℎ₂=Ω₁₀𝑥,𝑦. (21) And assume that 𝜃𝑦,𝑞=𝜃₀+𝑞𝜃₁+𝑞²𝜃₂Ω+⋯,𝑦,𝑞=Ω₀+𝑞Ω₁+𝑞²Ω₂+⋯. (22) The solution of temperature and nanoparticle phenomenon (for 𝑞=1) are constructed by adopting the same procedure as done by [37–39] 𝜃=𝐴𝑥,𝑦¹𝑁_𝑏+2𝑁_𝑡6ℎ₂−ℎ₁𝑦³−𝐴₁𝑁_𝑏+2𝑁_𝑡ℎ₁+ℎ₂+2ℎ₂−ℎ₁4ℎ₂−ℎ₁𝑦²+𝐴₁𝑁_𝑏+2𝑁_𝑡ℎ²₁+4ℎ₁ℎ₂+ℎ²₂+6ℎ²₂−ℎ²₁×ℎ+1212₂−ℎ₁⁻¹𝑦−𝐴₁ℎ₁ℎ₂𝑁_𝑏+2𝑁_𝑡ℎ₁+ℎ₂+6ℎ₂−ℎ₁+12ℎ₁12ℎ₂−ℎ₁⁻¹,Ω=𝐴𝑥,𝑦¹𝑁_𝑡2𝑁_𝑏𝑦²+2𝑁_𝑏−𝐴₁𝑁_𝑡ℎ²₂−ℎ²₁2𝑁_𝑏ℎ₂−ℎ₁𝑦+𝐴₁𝑁_𝑡ℎ₁ℎ₂ℎ₂−ℎ₁−2ℎ₁𝑁_𝑏2𝑁_𝑏ℎ₂−ℎ₁, (23) where 𝐴₁=𝑁_𝑏+𝑁_𝑡ℎ₂−ℎ₁². (24) Substituting (23) in (13), the exact solution for velocity is Ψ𝑥,𝑦=𝐶₁₁+𝐶₁₂𝑦+𝐶₁₃coshΓ𝑦+𝐶₁₄−𝐴sinhΓ𝑦⁹12Γ²𝑦⁴−𝐴₁₀6Γ²𝑦³−𝐴₉Γ⁴𝑦²−𝐴₁₁2Γ²+𝐴₁₀Γ⁴𝐴𝑦−¹¹Γ⁴+2𝐴₉Γ⁶. (25) And the pressure gradient is 𝜕𝑃=𝐴𝜕𝑥²𝐺_𝑟+𝐴₉3𝑦³+𝐴₃𝐺_𝑟+𝐴₆𝐵_𝑟+𝐴₁₀2𝑦²+𝐴₄𝐺_𝑟+𝐴₇𝐵_𝑟𝑦+𝐴₅𝐺_𝑟+𝐴₈𝐵_𝑟+𝐴₁₁2−Γ²𝐶₁₂+1. (26) The dimensionless pressure rise Δ𝑃 is given by Δ𝑃=¹₀𝜕𝑃𝜕𝑥𝑑𝑥, (27) where 𝐴₂=𝐴₁𝑁_𝑏+2𝑁_𝑡6ℎ₂−ℎ₁𝐴₃,𝐴=−¹𝑁_𝑏+2𝑁_𝑡ℎ₁+ℎ₂+2ℎ₂−ℎ₁4ℎ₂−ℎ₁𝐴₄,=𝐴₁𝑁_𝑏+2𝑁_𝑡ℎ²₁+4ℎ₁ℎ₂+ℎ²₂+6ℎ²₂−ℎ²₁+1212ℎ₂−ℎ₁,𝐴₅𝐴=−¹ℎ₁ℎ₂𝑁_𝑏+2𝑁_𝑡ℎ₁+ℎ₂+6ℎ₂−ℎ₁+12ℎ₁12ℎ₂−ℎ₁,𝐴₆=𝑁_𝑡𝑁_𝑏+𝑁_𝑡2𝑁_𝑏ℎ₂−ℎ₁²,𝐴₇=2𝑁_𝑏−𝑁_𝑡𝐴₁ℎ²₂−ℎ²₁2𝑁_𝑏ℎ₂−ℎ₁,𝐴₈=2𝑁_𝑡𝐴₁ℎ₁ℎ₂ℎ₂−ℎ₁−2ℎ₁𝑁_𝑏2𝑁_𝑏ℎ₂−ℎ₁𝐴₉,=−𝐺_𝑟𝐴₁𝑁_𝑏+2𝑁_𝑡ℎ²₁+4ℎ₁ℎ₂+ℎ²₂+6ℎ²₂−ℎ²₁×4ℎ+12₂−ℎ₁⁻¹,𝐴₁₀=𝐺_𝑟𝐴₁𝑁_𝑏+2𝑁_𝑡ℎ₁+ℎ₂+2ℎ₂−ℎ₁2ℎ₂−ℎ₁−𝐵_𝑟𝐴₁𝑁_𝑡𝑁_𝑏,𝐴₁₁=−𝐺_𝑟𝐴₁𝑁_𝑏+2𝑁_𝑡ℎ²₁+4ℎ₁ℎ₂+ℎ²₂+6ℎ²₂−ℎ²₁×ℎ+1212₂−ℎ₁⁻¹−𝐵_𝑟2𝑁_𝑏−𝑁_𝑡𝐴₁ℎ²₂−ℎ²₁2𝑁_𝑏ℎ₂−ℎ₁, (28)(29)(30)(31)(32)(33)(34)(35)(36)(37)𝐶^{𝐶11=−csch12Γℎ1−ℎ2×26𝐴11Γℎ2cosh1−ℎ2Γℎ−Γsinh1−ℎ2ℎ1Γℎ+Γsinh1−ℎ2ℎ2ℎ−2+Γ1+ℎ2×3Γℎ2cosh1−ℎ2Γℎ+𝐹Γsinh1−ℎ2Γ−23+𝐴10ΓΓℎcosh1−ℎ2Γℎ+Γsinh1−ℎ2×ℎ2ℎ+221−Γ2Γℎsinh1−ℎ2ℎ22Γℎ+2Γ2cosh1−ℎ2)ℎ+12Γℎ+3sinh1−ℎ2×ℎ1+ℎ2×Γℎ3sinh1−ℎ2×Γℎ+Γcosh1−ℎ2+2×ℎ2Γ2+𝐴9×Γ4×Γℎcosh1−ℎ2Γℎ+Γsinh1−ℎ2ℎ2+3×ℎ41−4Γ3Γℎsinh1−ℎ2Γℎ+Γcosh1−ℎ2ℎ2×ℎ31+12Γ2×Γℎcosh1−ℎ2Γℎ+Γsinh1−ℎ2×ℎ2+1×ℎ21×Γ−Γ4Γℎsinh1−ℎ2×ℎ42+4Γ3Γℎ×cosh1−ℎ2ℎ32+12Γ2Γℎ×sinh1−ℎ2×ℎ22Γℎ+48Γ×cosh1−ℎ2ℎ2Γℎ+48sinh1−ℎ2×ℎ1+Γ4×Γℎcosh1−ℎ2+3×ℎ42+4Γ3Γℎ×sinh1−ℎ2×ℎ32+24Γ2×cosh212Γℎ1−ℎ2×ℎ22×Γℎ+48cosh1−ℎ2Γℎ−1+48Γsinh1−ℎ2×ℎ2×24Γ61−2sinh2Γℎ1−ℎ21+Γcosh2Γℎ1−ℎ2ℎ11−Γcosh2Γℎ1−ℎ2ℎ2−1,𝐶121=−−12𝐹cosh2Γℎ1−ℎ2Γ51−24sinh2Γℎ1−ℎ2Γ41−cosh2Γℎ1−ℎ2𝐴9ℎ41Γ31+cosh2Γℎ1−ℎ2𝐴9ℎ42Γ31+4sinh2Γℎ1−ℎ2×𝐴9ℎ31Γ21+4sinh2Γℎ1−ℎ2𝐴9ℎ32Γ2−6𝐴111−2sinh2Γℎ1−ℎ21+Γcosh2Γℎ1−ℎ2ℎ11−Γcosh2Γℎ1−ℎ2ℎ2Γ21−12cosh2Γℎ1−ℎ2×𝐴9ℎ211Γ+12cosh2Γℎ1−ℎ2×𝐴9ℎ221Γ+24sinh2Γℎ1−ℎ2×𝐴9ℎ11+24sinh2Γℎ1−ℎ2𝐴9ℎ2+2𝐴10−Γ31cosh2Γℎ1−ℎ2×ℎ31+3Γ21sinh2Γℎ1−ℎ2ℎ211−6Γcosh2Γℎ1−ℎ2ℎ1+Γ31cosh2Γℎ1−ℎ2ℎ32+3Γ21sinh2Γℎ1−ℎ2ℎ221+12sinh2Γℎ1−ℎ21+6Γcosh2Γℎ1−ℎ2ℎ2×12Γ41−2sinh2Γℎ1−ℎ21+Γcosh2Γℎ1−ℎ2ℎ11−Γcosh2Γℎ1−ℎ2ℎ2−1,𝐶13=csch12Γℎ1−ℎ2×12𝐹coshΓℎ1−coshΓℎ2Γ5+coshΓℎ1+3coshΓℎ2𝐴9ℎ41Γ3+3coshΓℎ1+coshΓℎ2𝐴9ℎ42Γ32+2sinhΓℎ1−sinhΓℎ2𝐴9×+Γ2coshΓℎ1+coshΓℎ2𝐴10×ℎ32Γ2+2ℎ31×ΓcoshΓℎ1+2coshΓℎ2𝐴10−2𝐴9sinhΓℎ1−sinhΓℎ2+ΓcoshΓℎ2ℎ2Γ22+6coshΓℎ1+coshΓℎ2𝐴9+ΓsinhΓℎ1−sinhΓℎ2𝐴10×ℎ22Γ−6ℎ21×Γ𝐴10sinhΓℎ1−sinhΓℎ2+ΓcoshΓℎ2ℎ2−2coshΓℎ1+coshΓℎ2𝐴9×Γ×Γ−125coshΓℎ1−coshΓℎ2−2sinhΓℎ1−sinhΓℎ2×𝐴9ℎ2+2ℎ1×6coshΓℎ1−coshΓℎ2Γ5−3coshΓℎ1𝐴10ℎ22Γ3−2𝐴9×Γ3coshΓℎ1ℎ32×+6ΓcoshΓℎ1+coshΓℎ2×ℎ2+6sinhΓℎ1−sinhΓℎ2×24Γ51−2sinh2Γℎ1−ℎ21+Γcosh2Γℎ1−ℎ2ℎ11−Γcosh2Γℎ1−ℎ2ℎ2−1,}₁₄1=−csch2Γℎ₁−ℎ₂×2−coshΓℎ₁+coshΓℎ₂+ΓsinhΓℎ₂ℎ₁−ΓsinhΓℎ₂ℎ₂×3𝐴₁₀ℎ²₁−ℎ²₂Γ²+2𝐴₉×Γ²ℎ³₁+6ℎ₁−ℎ₂Γ²ℎ²₂+6+ΓsinhΓℎ₁−sinhΓℎ₂×12𝐹Γ⁴−12ℎ₂Γ⁴+𝐴₉ℎ⁴₁Γ²+3𝐴₉ℎ⁴₂Γ²+2𝐴₁₀ℎ³₁Γ²+4𝐴₁₀ℎ³₂Γ²+12𝐴₉ℎ²₁+12𝐴₉ℎ²₂+2ℎ₁×6Γ⁴−3𝐴₁₀ℎ²₂Γ²−2𝐴₉ℎ₂Γ²ℎ²₂×+624Γ⁵1−2sinh2Γℎ₁−ℎ₂1+Γcosh2Γℎ₁−ℎ₂ℎ₁1−Γcosh2Γℎ₁−ℎ₂ℎ₂⁻¹. (38)(39)(40)(41)

When calculating the constants 𝐶₁₁, 𝐶₁₂, 𝐶₁₃, and 𝐶₁₄ we used the mathematica program.

4. Results and Discussion

In this section we have studied the velocity, pressure gradient, pressure rise, temperature, and nanoparticle phenomena graphically. The effects of Hall parameter 𝑚, porosity parameter 𝑠, thermophoresis parameter 𝑁_𝑡, and local temperature Grashof number 𝐺_𝑟 on the velocity profile have been displayed in Figure 2. Figures 2(a) and 2(b) declare that velocity profile increases with an increase in the value of each of 𝑚 and 𝑠, Figure 2(c) announces that the velocity near the channel walls are not similar in view of 𝑁_𝑡, and the velocity decreases by increasing 𝑁_𝑡. In Figure 2(d) it is found that the velocity field increases in the region 𝑦∈(−0.8,−0.16) and it decreases in the region 𝑦∈(−0.16,0.45) with an increase in 𝐺_𝑟. Figures 3(a) to 3(d) examine the pressure gradient for different values of Hall parameter 𝑚, porosity parameter 𝑠, amplitude of right wave 𝑎, and local nanoparticle Grashof number 𝐵_𝑟. It is observed that the magnitude of pressure gradient decreases in view of an increase in 𝑚 and 𝑠 Figures 3(a) and 3(b). The situation is reversed in Figures 3(c) and 3(d), the magnitude of pressure gradient increases with an increase in 𝑎 and 𝐵_𝑟. The pressure rise versus volume flow rate is shown in Figures 4(a) to 4(d). It is noticed from Figures 4(a) and 4(c) that in in the peristaltic pumping region (Δ𝑃>0), the pressure rise increases with an increase in Hall parameter 𝑚 and porosity parameter 𝑠, while decreases in the augmented pumping region (Δ𝑃<0). Free pumping region holds when (Δ𝑃=0). It is also noticed that the pressure rise is an increasing function of the increase in local temperature Grashof number 𝐺_𝑟 (Figure 4(b)); moreover, by increasing the value of Brownian motion parameter 𝑁_𝑏, the pressure rise is a decreasing function (Figure 4(d)). Figures 5 and 6 demonstrate the variation of temperature profile and nanoparticle phenomena for different values of thermophoresis parameter 𝑁_𝑡 and Brownian motion parameter 𝑁_𝑏. Here the temperature profile decreases when 𝑁_𝑡 and 𝑁_𝑏 are increased and nanoparticle phenomenon decreases by increasing 𝑁_𝑡, while it increases when 𝑁_𝑏 is increased.

(a)

(b)

(c)

(d)

Figure 2 

Variations of velocity profile 𝑈 with 𝑦 for various values of Hall parameter 𝑚 (panel a), porosity parameter 𝑠 (panel b), thermophoresis parameter 𝑁𝑡 (panel c), and local temperature Grashof number 𝐺𝑟 (panel d). The other parameters chosen are 𝐹=2, 𝑥=0.1, 𝑑=1, 𝑀=1.2, 𝑎=0.1, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑠=0.5, 𝜙=0.2, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panel a); 𝐹=2, 𝑥=0.1, 𝑑=1, 𝑀=1.2, 𝑎=0.1, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑚=1, 𝜙=0.2, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panel b); 𝐹=2, 𝑥=0.1, 𝑑=1, 𝑀=1.2, 𝑎=0.1, 𝑏=0.5, 𝑁𝑏=0.8, 𝑚=1, 𝑠=0.5, 𝜙=0.2, 𝐺𝑟=0.5 and 𝐵𝑟=0.5 (panel c); and 𝐹=2, 𝑥=0.1, 𝑑=1, 𝑀=1.2, 𝑎=0.1, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑚=1, 𝑠=0.5, 𝜙=0.2 and 𝐵𝑟=0.5 (panel d).

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(b)

(c)

(d)

Figure 3 

Variations of the axial pressure gradient 𝑑𝑝/𝑑𝑥 with 𝑥 for various values of Hall parameter 𝑚 (panel a), porosity parameter 𝑠 (panel b), amplitude of right wave 𝑎 (panel c), and local nanoparticle Grashof number 𝐵𝑟 (panel d). The other parameters chosen are 𝐹=−2, 𝑦=0.1, 𝑑=1, 𝑀=1.2, 𝑎=0.1, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑠=0.5, 𝜙=0.2, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panel a); 𝐹=−2, 𝑦=0.1, 𝑑=1, 𝑀=1.2, 𝑎=0.1, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑚=1, 𝜙=0.2, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panel b); 𝐹=−2, 𝑦=0.1, 𝑑=1, 𝑀=1.2, 𝑁𝑡=0.5, 𝑏=0.5, 𝑁𝑏=0.8, 𝑚=1, 𝑠=0.5, 𝜙=0.2, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panel c); and 𝐹=−2, 𝑦=0.1, 𝑑=1, 𝑀=1.2, 𝑎=0.1, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑚=1, 𝑠=0.5, 𝜙=0.2, and 𝐺𝑟=0.5 (panel d).

(a)

(b)

(c)

(d)

Figure 4 

Variations of pressure rise Δ𝑃 versus flow rate 𝑄 for different values of Hall parameter 𝑚 (panel a), local temperature Grashof number 𝐺𝑟 (panel b), porosity parameter 𝑠 (panel c), and the Brownian motion parameter 𝑁𝑏 (panel d). The other parameters chosen are 𝑑=1, 𝑦=1, 𝑀=1.2, 𝑎=0.3, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑠=0.5, 𝜙=0.2, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panel a); 𝑑=1, 𝑦=1, 𝑀=1.2, 𝑎=0.3, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑠=0.5, 𝜙=0.2, 𝑚=1, and 𝐵𝑟=0.5 (panel b); 𝑑=1, 𝑦=1, 𝑀=1.2, 𝑎=0.3, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑚=1, 𝜙=0.2, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panel c); and 𝑑=1, 𝑦=1, 𝑀=1.2, 𝑎=0.3, 𝑏=0.5, 𝑚=1, 𝑁𝑡=0.5, 𝑠=0.5, 𝜙=0.2, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panel d).

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Figure 5 

Variations of temperature profile Θ(𝑥,𝑦) with 𝑦 for different values of thermophoresis parameter 𝑁𝑡 (panel a) and the Brownian motion parameter 𝑁𝑏 (panel b). The other parameters chosen are 𝑑=1, 𝑥=0.05, 𝑎=0.1, 𝑏=0.5, 𝑁𝑏=0.8, and 𝜙=0.2 (panel a); 𝑑=1, 𝑥=0.05, 𝑎=0.3, 𝑏=0.5, 𝑁𝑡=1.5, and 𝜙=0.2 (panel b).

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(b)

Figure 6 

Variations of nanoparticle phenomena Ω(𝑥,𝑦) with 𝑦 for different values of thermophoresis parameter 𝑁𝑡 (panel a) and the Brownian motion parameter 𝑁𝑏 (panel b). The other parameters chosen are 𝑑=1, 𝑥=0.25, 𝑎=0.3, 𝑏=0.5, 𝑁𝑏=1.5, and 𝜙=0.2 (panel a); 𝑑=1, 𝑥=0.25, 𝑎=0.3, 𝑏=0.5, 𝑁𝑡=1.5, and 𝜙=0.2 (panel b).

5. Trapping Phenomenon

The formation of an internally circulating bolus of fluid is called trapping. The trapped bolus moves along with the wave. The effects of Hall parameter 𝑚 and Brownian motion parameter 𝑁_𝑏 on trapping can be observed through Figure 7, where the number of trapping bolus increases with an increase in 𝑚 and 𝑁_𝑏 in the left half of the channel. Figure 8 depicts that increasing the local temperature Grashof number 𝐺_𝑟 leads to a decrease in the number of trapping bolus in the left half of the channel but it increases in both the left and right halves of the channel with an increase in the porosity parameter 𝑠.

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Figure 7 

Streamlines for two different values of Hall parameter 𝑚 (panels a, b) and two different values of the Brownian motion parameter 𝑁𝑏 (panels c, d). The other parameters chosen are 𝑄=2, 𝑑=1, 𝑀=1.2, 𝑎=0.3, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑠=0.5, 𝜙=0.02, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panels a, b); 𝑄=2, 𝑑=1, 𝑀=1.2, 𝑎=0.3, 𝑏=0.5, 𝑚=1, 𝑁𝑡=0.5, 𝑠=0.5, 𝜙=0.02, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panels c, d).

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(b)

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(d)

Figure 8 

Streamlines for two different values of local temperature Grashof number 𝐺𝑟 (panels a, b) and two different values of porosity parameter 𝑠 (panels c, d). The other parameters chosen are 𝑄=2, 𝑑=1, 𝑀=1.2, 𝑎=0.3, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑠=0.5, 𝜙=0.02, 𝑚=1, and 𝐵𝑟=0.5 (panels a, b); 𝑄=2, 𝑑=1, 𝑀=1.2, 𝑎=0.3, 𝑏=0.5, 𝑁𝑏=0.8, 𝑁𝑡=0.5, 𝑚=1, 𝜙=0.02, 𝐺𝑟=0.5, and 𝐵𝑟=0.5 (panels c, d).

6. Conclusions

The effects of Hall current and porous media on peristaltic transport of a nanofluid are investigated. The main results are summarized as follows.(1)Effects of Hall 𝑚 and porosity 𝑠 parameters are similar on velocity profile but thermophoresis parameter 𝑁_𝑡 has opposite effect.(2)The velocity near the channel walls is not similar in view of the local temperature Grashof number 𝐺_𝑟.(3)The magnitude of pressure gradient decreases by increasing Hall 𝑚 and porosity 𝑠 parameters. However it increases with an increase in amplitude of right wave 𝑎.(4)The behaviors of Hall parameter 𝑚 and porosity parameter 𝑠 on the pressure rise are similar.(5)The behaviors of local temperature Grashof number 𝐺_𝑟 and Brownian motion parameter 𝑁_𝑏 on the pressure rise are opposite.(6)Temperature profile decreases when thermophoresis parameter 𝑁_𝑡 and Brownian motion parameter 𝑁_𝑏 are increased.(7)Nanoparticle phenomenon decreases by increasing thermophoresis parameter 𝑁_𝑡 while it increases by increasing Brownian motion parameter 𝑁_𝑏.(8)The number of trapping bolus increases in the left half of the channel due to an increase in Hall parameter 𝑚 and Brownian motion parameter 𝑁_𝑏. On the other hand it decreases in the left half of the channel with an increase in local temperature Grashof number 𝐺_𝑟.(9)Increasing the magnitude of porosity parameter 𝑠 led to an increase in the number of trapping bolus in both halves of the channel.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This project was supported by King Saud University, Deanship of Scientific Research, College of Science, Research center.