Connective and magnetic effects in a curved wavy channel with nanoparticles under different waveforms

Anum Tanveer; Zain Ul Abidin

doi:10.1515/jmbm-2022-0256

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Connective and magnetic effects in a curved wavy channel with nanoparticles under different waveforms

Anum Tanveer and Zain Ul Abidin

Published/Copyright: March 21, 2023

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From the journal Journal of the Mechanical Behavior of Materials Volume 32 Issue 1

Abstract

The present study describes the peristaltic flow of Jeffrey fluid with nanomaterial bounded under peristaltic waves in a curved channel. Silver (Ag) is the nanomaterial used for this purpose, and base fluid is water. The diversity of peristaltic waves is captured under four different wave profiles traveling along the curved channel. The consequences of heat generation and mass concentration are also taken. The problem is modeled under physical laws and then simplified using the lubrication technique. The obtained system is sketched in graphs directly using a numerical scheme. The physical outcomes of involved parameters on axial velocity, temperature variation, concentration profile, and streamline patterns are discussed in the last section.

Keywords: peristalsis; silver–water nanoparticles; different waveforms; mixed convection; magnetic field; connective conditions

1 Introduction

Nanofluid is a mixture of base fluid and nano-sized particles. Novel aim of nanofluid has a wide range of applications in heat exchangers, power construction, cooling process, chemical industry, magnetic sealants, nutriment industry, atmosphere engineering, nano-cryosurgery, car warmers, air conditioning, heat controller systems, energy storage, and many more. Nanofluids are suspensions of particles ranging in size from 1 to 100 nm plus a base fluid. First, a detailed study on nanofluids to enhance the thermophysical feature has been carried out by Choi and Eastma [1]. Some literature types about nanofluid flow are cited in previous studies [2–14].

Magnetohydrodynamics (MHD) with peristalsis plays a significant role in blood flow reduction. Magnetic field when applied at suitable intensity bears the capability to cure tumors in brain, detect various heart diseases, removal of conducting instruments from body, and regulation of blood flow in blood flow regimes. Moreover, the applications of MHD in various forms of peristaltic waves are found in cancer tumor treatment, drug conveyance, surgical operation, magnetotherapy, and magnetic resonance imaging. Recently, researchers have investigated MHD flows that require interaction between magnetic field and electrically conducting materials (see refs [15–25]).

Peristalsis occurs in human physiological systems such as food swallowing, blood circulation, chyme movement, ovum movement, spermatic movement, and urine transport. Peristalsis is the involuntary contraction and relaxation of a canal, which results in a wave that drives material down in an anterograde direction. It is a requirement in various body functions as well as in a variety of engineering and biomedical fields. Initially, Latham [26] published a theoretical model of the peristaltic flow, which was later verified experimentally by Shapiro et al. [27]. The peristaltic flow of Williamson fluid with a magnetic effect in a curved channel was explained by Rashid et al. [20]. Nawaz et al. [28] discussed the analysis of the peristaltic flow of Williamson fluid with radial magnetic field in curved channel. Some other useful studies relating peristaltic flow are found in previous studies [29–32].

In industrial and engineering processes, the combined effects of mass and heat transfer are found prominent. Thermal diffusion and chemical species are the important causes of such expansion in building, and they aid in a variety of technical advances such as DNA and artistic production, atomic protection, paper production, food development, and aerosol formation to diffusion. Motivated by the above utility, several researchers are interested in carrying out the research on heat and mass transport in fluid flows in recent years. Riaz et al. [33] explained the mass and heat transfer of the peristaltic flow of Eyring–Powell model through a compliant channel. The analysis of the peristaltic flow of Sisko fluid with heat and mass transfer effects in a curved channel was discussed by Asghar et al. [30]. Viadya et al. [34] explained the analysis of MHD peristaltic flow through a complaint porous channel with heat and mass transfer.

The current work explores the magnetic and connective effects on Jeffrey fluid flow in a curved channel under different waveforms. To catch the peristaltic effect, the four diverse wave profiles are used. Moreover, we have extended our analysis by taking the consequences of heat transfer and mass concentration. The phenomenon is modeled by applying the equations of continuity, momentum, energy, and concentration. The graphical results of considered parameter such as axial velocity, temperature variation, concentration, and streamline profile are physically examined. The major findings of this research are objectified in the last section.

2 Mathematical formulation

Here, we consider an incompressible viscoelastic Jeffrey fluid of width 2a with center O and radius N ^* bounded in curved channel. The velocity along axial component X ˆ 1 is V ˆ , and radial component N ˆ 1 is U ˆ . In the presence of a radial magnetic field, the fluid is flowing in curved channel. The magnetic field is defined as:

(1) B ˆ = N * B 0 N ˆ 1 + N * , 0 , 0 .

The following expression occurs in view of Ohm’s law:

(2) J ˆ ′ = σ ef ( V ˆ × B ˆ ) ,

where V ˆ is the velocity vector, σ ef is the effective electric conductivity of nanofluid, and J ˆ ′ is the current density. The following expression represents the current density as:

(3) J ˆ ′ = N * σ ef B 0 N ˆ 1 + N * U ˆ e ˆ Z .

Eq. (3) yields

(4) J ˆ ′ × B ˆ = − N * 2 σ ef B 0 2 ( N ˆ 1 + N * ) 2 e ˆ X ˆ 1 ,

where e ˆ X ˆ 1 is a vector in axial direction.

The geometry of various peristaltic waves is as follows [35]:

Sinusoidal wave:
(5) H ˆ ( X ˆ 1 , t ˆ ) = A + B sin 2 π λ ( X ˆ 1 − c t ˆ ) .
Square wave:
(6) H ˆ ( X ˆ 1 , t ˆ ) = A + B 4 π ∑ i = 1 ∞ ( − 1 ) i + 1 2 i − 1 cos 2 π λ ( 2 i − 1 ) ( X ˆ 1 − c t ˆ ) .
Trapezoidal wave:
(7) H ˆ ( X ˆ 1 , t ˆ ) = A + B 32 π 2 ∑ i = 1 ∞ ( − 1 ) i + 1 sin π 8 ( 2 i − 1 ) ( 2 i − 1 ) 2 sin 2 π λ ( 2 i − 1 ) ( X ˆ 1 − c t ˆ ) .
Sawtooth wave:

(8) H ˆ ( X ˆ 1 , t ˆ ) = A + B 8 π 3 ∑ i = 1 ∞ ( − 1 ) i + 1 ( 2 i − 1 ) 2 sin 2 π λ ( 2 i − 1 ) ( X ˆ 1 − c t ˆ ) ,

where H ˆ ( X ˆ 1 , t ˆ ) represents the displacement in the channel, λ the wavelength, A the half-width of the curved channel, B the amplitude of peristaltic wave, and c the wave speed. The Cauchy stress tensor τ for Jeffrey fluid is

(9) τ = − p ˆ I + S ,

and extra stress tensor is as follows [35]:

(10) S = μ ef 1 + λ 1 1 + λ 2 d d t A 1 ,

where μ ef denotes the effective viscosity, λ 1 the ratio of relaxation to retardation time, λ 2 the retardation time, and d A 1 d t the material time derivative. The constitutive equations are given as:

(11) ∂ ∂ N ˆ 1 ( N ˆ 1 + N * ) V ˆ + N * ∂ U ˆ ∂ X ˆ 1 = 0 ,

(12) ρ ef ∂ V ˆ ∂ t ˆ + V ˆ ∂ V ˆ ∂ N ˆ 1 + U ˆ N * N ˆ 1 + N * ∂ V ˆ ∂ X ˆ 1 − U ˆ 2 N ˆ 1 + N * ) = ∂ p ˆ ∂ N ˆ 1 + μ ef 1 + λ 1 N ˆ 1 + N * ∂ ∂ N ˆ 1 ( N ˆ 1 + N * ) ∂ V ˆ ∂ N ˆ 1 + N * N ˆ 1 + N * 2 ∂ 2 V ˆ ∂ X ˆ 1 2 − V ˆ ( N ˆ 1 + N * ) 2 − 2 N * ( N ˆ 1 + N * ) 2 ∂ U ˆ ∂ X ˆ 1 ,

(13) ρ ef ∂ U ˆ ∂ t ˆ + V ˆ ∂ U ˆ ∂ N ˆ 1 + U ˆ N * N ˆ 1 + N * ∂ U ˆ ∂ X ˆ 1 − U ˆ V ˆ N ˆ 1 + N * = − N * N ˆ 1 + N * ∂ p ˆ ∂ X ˆ 1 + μ ef 1 + λ 1 ( N ˆ 1 + N * ) ∂ ∂ N ˆ 1 ( N ˆ 1 + N * ) ∂ U ˆ ∂ N ˆ 1 + N * N ˆ 1 + N * 2 ∂ 2 U ˆ ∂ X ˆ 1 2 − U ˆ ( N ˆ 1 + N * ) 2 + 2 N * ( N ˆ 1 + N * ) 2 ∂ V ˆ ∂ X ˆ 1 − σ ef N * 2 B 0 2 U ˆ ( N ˆ 1 + N * ) 2 + g ( γ ) ef ( T − T 0 ) ,

(14) ( ρ c p ) ef ∂ T ∂ t ˆ + V ˆ ∂ T ∂ N ˆ 1 + U ˆ N * N ˆ 1 + N * ∂ T ∂ X ˆ 1 = κ ef ∂ 2 T ∂ N ˆ 1 2 + 1 N ˆ 1 + N * ∂ T ∂ N ˆ 1 + ∂ 2 T ∂ X ˆ 1 2 + σ ef N * 2 B 0 2 U ˆ 2 ( N ˆ 1 + N * ) 2 ,

(15) ∂ C ∂ t ˆ + V ˆ ∂ C ∂ N ˆ 1 + U ˆ N * N ˆ 1 + N * ∂ C ∂ X ˆ 1 = D 1 ∂ 2 C ∂ N ˆ 1 2 + 1 N ˆ 1 + N * ∂ C ∂ N ˆ 1 + ∂ 2 C ∂ X ˆ 1 2 + D 1 K t T m ∂ 2 T ∂ N ˆ 1 2 + 1 N ˆ 1 + N * ∂ T ∂ N ˆ 1 + ∂ 2 T ∂ X ˆ 1 2 .

The connective boundary conditions are as follows:

(16) U ˆ = 0 , N ˆ 1 = ± H ˆ ,

(17) κ f ∂ T ∂ N ˆ 1 ± d 1 ( T − T 0 ) = 0 , N ˆ 1 = ± H ˆ

(18) D 1 ∂ C ∂ N ˆ 1 ± d 2 ( C − C 0 ) = 0 , N ˆ 1 = ± H ˆ .

Here, ρ ef represents the effective density, ( ρ γ ) ef the effective thermal expansion, T the temperature, d 1 and d 2 the heat and mass transfer coefficient at upper and lower walls of channel, C the concentration, T 0 and C 0 the temperature and concentration at lower and upper walls, K t the thermal diffusion rate, ( ρ c p ) ef the effective heat capacity, T m the mean temperature, κ ef the effective thermal conductivity, and D 1 the coefficient of mass diffusivity. The expression for ρ ef , σ ef , ( ρ γ ) ef , μ ef , ( ρ c p ) ef , and κ ef read as follows:

(19) ρ ef = ( 1 − ϕ 1 ) ρ f + ϕ 1 ρ p , ( ρ c p ) ef = ( 1 − ϕ 1 ) ( ρ c p ) f + ϕ 1 ( ρ c p ) p ( ρ γ ) ef = ( 1 − ϕ 1 ) ( ρ γ ) f + ϕ 1 ( ρ γ ) p , κ ef κ f = κ p + ( n − 1 ) κ f − ( n − 1 ) ϕ 1 ( κ f − κ p ) κ p + ( n − 1 ) κ f − ϕ 1 ( κ f − κ p ) σ ef σ f = 1 + 3 σ p σ f − 1 ϕ 1 σ p σ f + 2 − σ p σ f − 1 ϕ 1 , μ ef = μ f ( 1 − ϕ ) 2 . 5 .

The thermophysical properties of nanomaterial and base fluid are shown in Table 1.

Table 1

Comparison between nanomaterial and base fluid.

	ρ ( k g − 3 )	c p ( J kg − 1 K − 1 )	K ( W m − 1 K − 1 )	γ 1 k × 10 − 6
H₂O	997.1	4,179	0.613	210
Ag	10,500	235	429	18.6

In the case of steady state, we neglect the impact of the Jeffery fluid parameter λ ₂. Using the following linear transformations:

(20) x ˆ 1 = X ˆ 1 − c t ˆ , n ˆ 1 = N ˆ 1 , v ˆ = V ˆ , u ˆ = U ˆ − c , p ˆ = p ˆ , T = T , C = C ,

and the dimensionless variables as:

(21) x 1 = 2 π x ˆ 1 α , n 1 = n ˆ 1 A , v = v ˆ c , u = u ˆ c , Re = Ac ρ f μ f , P = 2 π A 2 p ˆ α μ f c , h 1 = H ˆ A , δ = 2 π A α , θ = T − T 0 T 0 ,

ϕ = C − C 0 C 0 , Pr = μ f ( c p ) f κ f , Gr = gA 2 γ f ρ f T 0 c μ f , Ec = c 2 T 0 ( c p ) f , Br = EcPr , B 1 = d 1 A κ f , B 2 = d 2 A D 1 , Sc = μ f ρ f D 1 , Sr = ρ f D 1 K t T 0 μ f T m C 0 , K = N * A , M 2 = A 2 B 0 2 σ f μ f , v = δ K 1 n 1 + K 1 ∂ ψ ∂ x , u = − ∂ ψ ∂ n 1 .

Under long wavelength δ ≪ 1 and small Reynolds number Re → 0, we obtain the following simplified system of equations, which is attained

(22) ∂ P ∂ n 1 = 0 ,

(23) − K n 1 + K ∂ p ∂ x 1 − 1 ( 1 − ϕ 1 ) 2 . 5 ( 1 − λ 1 ) 1 ( n 1 + K ) ∂ ∂ n 1 ( n 1 + K ) ∂ 2 ψ ∂ n 1 2 + 1 ( n 1 + K ) 2 1 − ∂ ψ ∂ n 1 − A 1 M 2 K 2 ( n 1 + K ) 2 + Gr A 3 θ = 0 ,

(24) K 3 ∂ 2 θ ∂ n 1 2 + 1 n 1 + K ∂ θ ∂ n 1 + Br A 1 M 2 K 2 ( n 1 + K ) 2 1 − ∂ ψ ∂ n 1 2 = 0 ,

(25) ∂ 2 ϕ ∂ n 1 2 + 1 n 1 + K ∂ ϕ ∂ n 1 + ScSr ∂ 2 θ ∂ n 1 2 + 1 n 1 + K ∂ θ ∂ n 1 = 0 ,

where

A 3 = ( 1 − ϕ 1 ) ( ρ γ ) f + ϕ 1 ( ρ γ ) p , A 1 = 1 + 3 σ p σ f − 1 ϕ 1 σ p σ f + 2 − σ p σ f − 1 ϕ 1 ,

(26) K 3 = κ p + ( n − 1 ) κ f − ( n − 1 ) ϕ 1 ( κ f − κ p ) κ p + ( n − 1 ) κ f − ϕ 1 ( κ f − κ p ) .

Here, Re represents the Reynolds number, K the radius of curvature, M the Hartman number, Gr the Grash of number, Ec the Eckert number, δ the wave number, Sc the Schmidt number, κ the thermal conductivity, Br the Brinkman number, Pr the Prandtl number, Sr the Soret number, and B ₁ and B ₂ the heat and mass Biot number. The pressure term is eliminated from Eqs. (22) and (23), and one obtains:

(27) ∂ ∂ n 1 1 ( 1 − ϕ 1 ) 2 . 5 ( 1 − λ 1 ) 1 ( n 1 + K ) ∂ ∂ n 1 ( n 1 + K ) ∂ 2 ψ ∂ n 1 2 + 1 ( n 1 + K ) 2 1 − ∂ ψ ∂ n 1 + A 1 M 2 K 2 ( n 1 + K ) 2 − Gr A 3 θ = 0 .

The dimensionless boundary conditions under stream function are as follows:

(28) ψ [ n 1 ] = ± q 1 2 n 1 = ± h 1 ,

(29) ψ ′ [ n 1 ] = 0 , n 1 = ± h 1 ,

(30) θ ′ [ n 1 ] ± B 1 θ [ n 1 ] = 0 , n 1 = ± h 1 ,

(31) ϕ ′ [ n 1 ] ± B 1 ϕ [ n 1 ] = 0 , n 1 = ± h 1 ,

where q ₁ is the flow rate. The dimensionless peristaltic waves are as follows:

Sinusoidal wave:
(32) h 1 ( x ) = 1 + ϵ sin ( 2 π x ) .
Square wave:
(33) h 1 ( x ) = 1 + ϵ 4 π ∑ i = 1 ∞ ( − 1 ) i + 1 2 i − 1 cos ( ( 2 i − 1 ) ( 2 π x ) ) .
Trapezoidal wave:
(34) h 1 ( x ) = 1 + ϵ 32 π 2 ∑ i = 1 ∞ ( − 1 ) i + 1 sin π 8 ( 2 i − 1 ) ( 2 i − 1 ) 2 sin ( ( 2 i − 1 ) ( 2 π x ) ) .
Sawtooth wave:

(35) h 1 ( x ) = 1 + ϵ 8 π 3 ∑ i = 1 ∞ ( − 1 ) i + 1 ( 2 i − 1 ) 2 sin ( ( 2 i − 1 ) ( 2 π x ) ) .

3 Discussion

This section discusses the results of axial velocity u, temperature variation θ, concentration ϕ , and streamline patterns are discussed physically under different involved parameters. The graphical results are obtained using a numerical method NDSolve in Mathematica.

3.1 Velocity profile

Here, we represent the graphical results of the Hartman number M, the viscoelastic parameter λ ₁, the radius of curvature K, and the Grashof number Gr on velocity profile. The velocity profile bends toward the center of channel for larger values of the Hartman number M (Figure 1(a) and (b)). Lorentz force is responsible for this impact of a magnetic field. In Figure 2(a) and (b), the velocity profile rises for increasing the value of viscoelastic parameter λ ₁. Similar behavior is shown for the radius of curvature K, the Grashof number Gr, volume fraction of nanomaterial ϕ 1 , and nanomaterials A ₃ and A ₁ (Figures 3–7(a) and (b)). This increasing impact of the Grashof number acts as opposing force in the treatment of diseases such as migraine headaches, cancer, and depression.

$Figure 1 (a and b) Velocity profile for λ 1 = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sr = 0.5, A 1 = 2, Sc = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Gr = 0.4, K = 3, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 1

(a and b) Velocity profile for λ ₁ = 0.5, ϕ 1 = 0.1, Sr = 0.5, A ₁ = 2, Sc = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Gr = 0.4, K = 3, ϵ = 0 . 2 , B 1 = 0 . 4 , and B 2 = 0 . 2 .

$Figure 2 (a and b) Velocity profile for M = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sr = 0.5, A 1 = 2, Sc = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Gr = 0.4, K = 3, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 2

(a and b) Velocity profile for M = 0.5, ϕ 1 = 0.1, Sr = 0.5, A ₁ = 2, Sc = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Gr = 0.4, K = 3, ϵ = 0 . 2 , B 1 = 0 . 4 , and B 2 = 0 . 2 .

$Figure 3 (a and b) Velocity profile for M = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sr = 0.5, A 1 = 2, Sc = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Gr = 0.4, λ 1 = 0.5, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 3

(a and b) Velocity profile for M = 0.5, ϕ 1 = 0.1, Sr = 0.5, A ₁ = 2, Sc = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Gr = 0.4, λ ₁ = 0.5, ϵ = 0 . 2 , B 1 = 0 . 4 , and B 2 = 0 . 2 .

$Figure 4 (a and b) Velocity profile for λ 1 = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sc= 0.5, A 1 = 2, Sr= 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Gr = 0.4, M = 0.5, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 4

(a and b) Velocity profile for λ ₁ = 0.5, ϕ 1 = 0.1, Sc= 0.5, A ₁ = 2, Sr= 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Gr = 0.4, M = 0.5, ϵ = 0 . 2 , B 1 = 0 . 4 , and B 2 = 0 . 2 .

$Figure 5 (a and b) Velocity profile for λ 1 = 0.5, Sr = 0.5, A 1 = 2, Sc = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Gr = 0.4, M = 0.5, K = 3 , ϵ = 0 . 2 K=3,\epsilon =0.2 , B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 5

(a and b) Velocity profile for λ ₁ = 0.5, Sr = 0.5, A ₁ = 2, Sc = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Gr = 0.4, M = 0.5, K = 3 , ϵ = 0 . 2 , B 1 = 0 . 4 , and B 2 = 0 . 2 .

$Figure 6 (a and b) Velocity profile for λ 1 = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sc = 0.5, A 1 = 2, Sr = 0.5, K = 3, Br = 0.5, K 3 = 2, Gr = 0.4, M = 0.5, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 6

(a and b) Velocity profile for λ ₁ = 0.5, ϕ 1 = 0.1, Sc = 0.5, A ₁ = 2, Sr = 0.5, K = 3, Br = 0.5, K ₃ = 2, Gr = 0.4, M = 0.5, ϵ = 0 . 2 , B 1 = 0 . 4 , and B 2 = 0 . 2 .

$Figure 7 (a and b) Velocity profile for λ 1 = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sc = 0.5, K = 3, Sr = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Gr = 0.4, M = 0.5, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 7

(a and b) Velocity profile for λ ₁ = 0.5, ϕ 1 = 0.1, Sc = 0.5, K = 3, Sr = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Gr = 0.4, M = 0.5, ϵ = 0 . 2 , B 1 = 0 . 4 , and B 2 = 0 . 2 .

3.2 Temperature profile

The temperature profile reduces for increasing values of the Hartmann number M as shown in Figure 8(a) and (b). Since greater values of the Hartmann number M produce a strong magnetic force that generates current in motor via fluid heat, so magnetic force acts as retarding force that induces temperature reduction. However, decreasing behavior is shown for the Brinkman number Br for different types of wave profiles (Figure 9(a) and (b)). An increase in the values of curvature K affects the temperature positively, i.e., temperatures rise in K (Figure 10(a) and (b)). The Grashof number Gr enhances θ due to gravitational agitation as seen from drawn results in Figures 11 and 12(a) and (b). Nanomaterial coefficients ϕ 1 and A ₁ cause temperature enhancement as shown in Figure 13(a) and (b). The heat transfer Biot number B ₁ produces heat near the channel walls; also, it enhances the thermal conductivity of fluid, and so the temperature rises in all forms of considered waves (Figure 14(a) and (b)).

$Figure 8 (a and b) Temperature profile for λ 1 = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sc = 0.5, A 1 = 2, Sr = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Gr = 0.4, K = 3, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 4 {B}_{1}=4 , and B 2 = 5 {B}_{2}=5 .$

Figure 8

(a and b) Temperature profile for λ ₁ = 0.5, ϕ 1 = 0.1, Sc = 0.5, A ₁ = 2, Sr = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Gr = 0.4, K = 3, ϵ = 0 . 2 , B 1 = 4 , and B 2 = 5 .

$Figure 9 (a and b) Temperature profile for λ 1 = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sr = 0.5, A 1 = 2, A 3 = 1, K = 3, K 3 = 2, Sc = 0.5, Gr = 0.4, M = 0.2, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 4 {B}_{1}=4 , and B 2 = 5 {B}_{2}=5 .$

Figure 9

(a and b) Temperature profile for λ ₁ = 0.5, ϕ 1 = 0.1, Sr = 0.5, A ₁ = 2, A ₃ = 1, K = 3, K ₃ = 2, Sc = 0.5, Gr = 0.4, M = 0.2, ϵ = 0 . 2 , B 1 = 4 , and B 2 = 5 .

$Figure 10 (a and b) Temperature profile for λ 1 = 0.5, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, A 3 = 1, M = 0.2, Sr = 0.5, K 3 = 2, Sc = 0.5, Gr = 0.4, ϵ = 0 . 2 \epsilon =0.2 , Br = 0.5, B 1 = 4 {B}_{1}=4 , and B 2 = 5 {B}_{2}=5 .$

Figure 10

(a and b) Temperature profile for λ ₁ = 0.5, ϕ 1 = 0.1, A ₁ = 2, A ₃ = 1, M = 0.2, Sr = 0.5, K ₃ = 2, Sc = 0.5, Gr = 0.4, ϵ = 0 . 2 , Br = 0.5, B 1 = 4 , and B 2 = 5 .

$Figure 11 (a and b) Temperature profile for λ 1 = 0.5, M = 0.2, Sc= 0.5, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sr = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, K = 3, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 4 {B}_{1}=4 , and B 2 = 5 {B}_{2}=5 .$

Figure 11

(a and b) Temperature profile for λ ₁ = 0.5, M = 0.2, Sc= 0.5, ϕ 1 = 0.1, A ₁ = 2, Sr = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, K = 3, ϵ = 0 . 2 , B 1 = 4 , and B 2 = 5 .

$Figure 12 (a and b) Temperature profile for λ 1 = 0.5, M = 0.2, Sc = 0.5, Gr {\rm{Gr}} = 0.4, A 1 = 2, Sr = 0.5 A 3 = 1, Br = 0.5, K 3 = 2, K = 3, ϵ = 0 . 2 \epsilon =0.2 , B 1 = 4 {B}_{1}=4 , and B 2 = 5 {B}_{2}=5 .$

Figure 12

(a and b) Temperature profile for λ ₁ = 0.5, M = 0.2, Sc = 0.5, Gr = 0.4, A ₁ = 2, Sr = 0.5 A ₃ = 1, Br = 0.5, K ₃ = 2, K = 3, ϵ = 0 . 2 , B 1 = 4 , and B 2 = 5 .

$Figure 13 (a and b) Temperature profile for λ 1 = 0.5, M = 0.2, Sc = 0.5, ϕ 1 {\phi }_{1} = 0.1, Sr = 0.5 A 3 = 1, Br = 0.5, K 3 = 2, K = 3, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0.4, B 1 = 4 {B}_{1}=4 , and B 2 = 5 {B}_{2}=5 .$

Figure 13

(a and b) Temperature profile for λ ₁ = 0.5, M = 0.2, Sc = 0.5, ϕ 1 = 0.1, Sr = 0.5 A ₃ = 1, Br = 0.5, K ₃ = 2, K = 3, ϵ = 0 . 2 , Gr = 0.4, B 1 = 4 , and B 2 = 5 .

$Figure 14 (a and b) Temperature profile for λ 1 = 0.5, M = 0.2, Sc = 0.5, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sr = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, K = 3, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0 . 4 {\rm{Gr}}=0.4 , and B 2 = 5 {B}_{2}=5 .$

Figure 14

(a and b) Temperature profile for λ ₁ = 0.5, M = 0.2, Sc = 0.5, ϕ 1 = 0.1, A ₁ = 2, Sr = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, K = 3, ϵ = 0 . 2 , Gr = 0 . 4 , and B 2 = 5 .

3.3 Concentration profile

In this subsection, the concentration profile φ affected by the Schmidt number Sc, the Soret number Sr, the radius of curvature K, and the mass transfer Biot number B ₂ is physically discussed via Figures 15–18(a) and (b). Figure 15(a) and (b) shows that the concentration profile decreases with increase in the value of Sc. If the value of Sc increases, the thermal diffusion decreases and the less diffused species prevent the fluid from being thick enough to boost the concentration distribution, and thus, the concentration falls. The Soret number reduces the concentration as seen from captured results of Figure 16(a) and (b). It is due to the fact that increasing values of Sr reduces the viscosity fluid; thus, the less viscous fluid particles cause this concentration drop. Curvature and mass transfer Biot number bear decreasing response toward concentration as noted from Figures 17–18(a) and (b).

$Figure 15 (a and b) Concentration profile for λ 1 = 0.5, M = 0.2, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sr = 0.5 A 3 = 1, Br = 0.5, K 3 = 2, K = 3, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0 . 4 {\rm{Gr}}=0.4 , B 1 = 2 , and B 2 = 1 {B}_{1}=2,{\rm{and}}{B}_{2}=1 .$

Figure 15

(a and b) Concentration profile for λ ₁ = 0.5, M = 0.2, ϕ 1 = 0.1, A ₁ = 2, Sr = 0.5 A ₃ = 1, Br = 0.5, K ₃ = 2, K = 3, ϵ = 0 . 2 , Gr = 0 . 4 , B 1 = 2 , and B 2 = 1 .

$Figure 16 (a and b) Concentration profile for λ 1 = 0.5, M = 0.2, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sc = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, K = 3, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0 . 4 {\rm{Gr}}=0.4 , B 1 = 2 , and B 2 = 1 {B}_{1}=2,{\rm{and}}{B}_{2}=1 .$

Figure 16

(a and b) Concentration profile for λ ₁ = 0.5, M = 0.2, ϕ 1 = 0.1, A ₁ = 2, Sc = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, K = 3, ϵ = 0 . 2 , Gr = 0 . 4 , B 1 = 2 , and B 2 = 1 .

$Figure 17 (a and b) Concentration profile for λ 1 = 0.5, M = 0.2, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sr = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Sc = 0.5, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0 . 4 {\rm{Gr}}=0.4 , B 1 = 2 , and B 2 = 1 {B}_{1}=2,{\rm{and}}{B}_{2}=1 .$

Figure 17

(a and b) Concentration profile for λ ₁ = 0.5, M = 0.2, ϕ 1 = 0.1, A ₁ = 2, Sr = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Sc = 0.5, ϵ = 0 . 2 , Gr = 0 . 4 , B 1 = 2 , and B 2 = 1 .

$Figure 18 (a and b) Concentration profile for λ 1 = 0.5, M = 0.2, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sr = 0.5, A 3 = 1, Br = 0.5, K 3 = 2, Sc = 0.5, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0 . 4 {\rm{Gr}}=0.4 , K = 3, and B 1 = 4 {B}_{1}=4 .$

Figure 18

(a and b) Concentration profile for λ ₁ = 0.5, M = 0.2, ϕ 1 = 0.1, A ₁ = 2, Sr = 0.5, A ₃ = 1, Br = 0.5, K ₃ = 2, Sc = 0.5, ϵ = 0 . 2 , Gr = 0 . 4 , K = 3, and B 1 = 4 .

3.4 Streamlines

Figures 19–21(a)–(h) describe the streamline patterns for different values of the Hartmann number M, the viscoelastic parameter λ ₁, and the radius of curvature K in a curved channel under different waveforms. Figure 19(a)–(f) shows that bolus size enlarges for sinusoidal, square, and trapezoidal waves for an increase in M. The shrinking in bolus size for sawtooth wave is noticed for increase in M (Figure 19(g) and (h)). However, with an increase in the values of K, the bolus size reduces for sinusoidal and sawtooth waves (Figure 20(a, b, e, and f)). The bolus size enlarges for square and trapezoidal waves for rise in the values of K (Figure 20(c, d, g, and h)). Similar behavior is shown for λ ₁, i.e., bolus shrinking in size for sinusoidal and trapezoidal waves, and enlarges in size for square and sawtooth waves (Figure 21(a)–(h)).

$Figure 19 (a–h) Streamline profile for (M = 1,1.5) at λ 1 = 0.5, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sr = 0.5 A 3 = 1, Br = 0.5, K 3 = 3, Sc = 0.5, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0 . 4 {\rm{Gr}}=0.4 , K = 2, B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 . {B}_{2}=0.2.$

Figure 19

(a–h) Streamline profile for (M = 1,1.5) at λ ₁ = 0.5, ϕ 1 = 0.1, A ₁ = 2, Sr = 0.5 A ₃ = 1, Br = 0.5, K ₃ = 3, Sc = 0.5, ϵ = 0 . 2 , Gr = 0 . 4 , K = 2, B 1 = 0 . 4 , and B 2 = 0 . 2 .

$Figure 20 (a–h) Streamline profile for (K = 2,3) at λ 1 = 0.5, M = 1, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sr = 0.5 A 3 = 1, Br = 0.5, K 3 = 3, Sc = 0.5, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0 . 4 {\rm{Gr}}=0.4 , K = 2, B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 20

(a–h) Streamline profile for (K = 2,3) at λ ₁ = 0.5, M = 1, ϕ 1 = 0.1, A ₁ = 2, Sr = 0.5 A ₃ = 1, Br = 0.5, K ₃ = 3, Sc = 0.5, ϵ = 0 . 2 , Gr = 0 . 4 , K = 2, B 1 = 0 . 4 , and B 2 = 0 . 2 .

$Figure 21 (a–h) Streamline profile for (λ 1 = 0.5,1.5) at M = 1, ϕ 1 {\phi }_{1} = 0.1, A 1 = 2, Sr = 0.5 A 3 = 1, Br =0.5, K 3 = 3, Sc = 0.5, ϵ = 0 . 2 \epsilon =0.2 , Gr = 0 . 4 {\rm{Gr}}=0.4 , K = 2, B 1 = 0 . 4 {B}_{1}=0.4 , and B 2 = 0 . 2 {B}_{2}=0.2 .$

Figure 21

(a–h) Streamline profile for (λ ₁ = 0.5,1.5) at M = 1, ϕ 1 = 0.1, A ₁ = 2, Sr = 0.5 A ₃ = 1, Br =0.5, K ₃ = 3, Sc = 0.5, ϵ = 0 . 2 , Gr = 0 . 4 , K = 2, B 1 = 0 . 4 , and B 2 = 0 . 2 .

4 Conclusions

In this article, the physical impacts of peristalsis on viscoelastic Jeffrey fluid with nanomaterial in a curved channel are carried out for magnetic field, connective conditions, and mixed convection. The study reveals that the viscoelastic parameter due to its viscous and elastic properties increases the velocity profile. The Hartman number leaves similar impact on velocity and temperature. Also, mixed convection effects on u and θ are found to be same. The consideration of different waveforms does not alter the curvature effect. The Biot number increases the temperature and reduces the concentration. The bolus shrinking is observed with increase in M in sawtooth wave, K for sinusoidal and sawtooth waves, and λ ₁ for sinusoidal and trapezoidal waves, whereas the bolus size enlarges with increase in M for sinusoidal, square, and trapezoidal waves, K for trapezoidal and square waves, and λ ₁ for sawtooth and square waves.

Funding information: The authors received no specific funding for this research work.
Author contributions: AT conceived the presented idea. AT contributed to providing a necessary guide in mathematical formation and its graphical description. ZUA developed the theory and performed the calculations. Both the authors contributed to finalizing the manuscript writing.
Conflict of interest: The authors declare no conflict of interest with anyone.

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Received: 2022-07-16

Revised: 2022-08-25

Accepted: 2022-09-05

Published Online: 2023-03-21

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

Articles in the same Issue

https://doi.org/10.1515/jmbm-2022-0256

Keywords for this article

peristalsis; silver–water nanoparticles; different waveforms; mixed convection; magnetic field; connective conditions

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