Blood flow analysis in narrow channel with activation energy and nonlinear thermal radiation

Anum Tanveer; Zain Ul Abidin

doi:10.1515/jmbm-2022-0278

Article Open Access

Blood flow analysis in narrow channel with activation energy and nonlinear thermal radiation

Anum Tanveer and Zain Ul Abidin

Published/Copyright: July 19, 2023

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

Abstract

Blood flow in narrow channels such as veins and arteries is the major topic of interest here. The Casson fluid with its shear-thinning attribute serves as the blood model. Owing to the arterial walls, the channel is configured curved in shape. The activation energy and nonlinear thermal radiation aspects are highlighted. The channel boundaries are flexible with peristaltic wave travelling along the channel. The mathematical description of the problem is developed under physical laws and then simplified using the lubrication technique. The obtained system is then sketched in graphs directly using the numerical scheme NDSolve in Mathematica software. The physical interpretation of parameters on axial velocity, temperature profile, concentration, and streamline pattern is discussed in the last section.

Keywords: blood flow; nonlinear thermal radiation; viscous dissipation; activation energy; wall properties; magnetic field

1 Introduction

The basic smallest quantity of energy needed by the reactants for a chemical reaction to arise is called activation energy. The term activation energy was first given by Svante Arrhenius, a Swedish scientist in 1989. A detailed review of activation energy was studied by Bestman [1]. Activation energy plays a significant role in the field of oil emulsion mechanics, geothermal or oil reservoir engineering, and chemical engineering. Presently, some studies have been made related to activation energy [2–12].

Due to outstanding features in the field of technical and industrial sciences, non-Newtonian fluids are primarily used in mathematical models for explaining realistic flow problems. One of these fluids, invented by Casson [13] in 1959, is known as the Casson fluid. He developed this model for printing ink oil suspensions. The Casson fluid model can also be used in other fluid models with alike properties, such as honey, jelly, and ketchup. The Casson fluid model can be used to characterise the blood flow in the capillary vessels due to the occurrence of fibrinogens, red corpuscles, and proteins [14]. The Casson fluid also plays an important role in blood dialysers and oxygenators. Some other useful studies relating to the Casson fluid flow are presented in refs. [15–21].

Peristalsis is one of the most important aspects of human physiology that can be seen in the human tabular functioning organs and blood flow in narrow arterial walls. Peristalsis is the relaxation and involuntary constriction of a canal that produces a wave to form, which forces the material to fall in an anterograde direction. It is a necessary feature of a number of engineering and biomedical sciences. Peristaltic flow is a type of fluid transport that occurs when a wave’s advanced area of shortening and development with a combination of the length of a distensible channel transports the fluid in the wave’s propagation direction. Initially, Latham [22] published a theoretical model of the peristaltic flow, and later, Shapiro et al. [23] confirmed it through its experiment. Further studies on the peristaltic flow can be seen in refs. [24–29].

Thermal radiation has an important application in heat exchangers, solar power plants, nuclear reactors, furnace design, power and solar technology, etc. Electromagnetic radiation causes radiative heat transfer. Radiation does not need a medium to work, but it is affected by factors such as temperature, surface properties, and geometric structure of the material absorbing or emitting heat. It is self-evident that when the temperature difference between two bodies is high or low, the radiation transmission between them is reduced. In industry, designing a structure with fluid flow and small temperature differences within a structure is laborious. Due to this laborious, researchers’ recently proposed nonlinear thermal radiation, which differs from linear thermal radiation in that it has an additional parameter. Refs. [30–32] cover some useful studies with nonlinear thermal radiation.

In the view of activation energy, the current work explores the peristaltic flow of the Casson fluid model in a curved channel. The phenomenon is modelled by utilising the equations of continuity, momentum, energy, and concentration. The physical outcomes of involved parameters on axial velocity, temperature variation, concentration, and streamline profile are discussed through graph using a numerical scheme NDSolve in MATHEMATICA software. The objectives of this research are discussed in the last section.

2 Mathematical problem

Here, we consider the incompressible Casson fluid flowing in a curved channel of half width e ˆ and centre O. The radius of curved channel is N ^*. The channel walls are under pressure variation in terms of wall’s flexibility properties. The sinusoidal waves propagating along the walls with velocity c generate the flow. Here, x ˆ 1 is the axial component of channel with velocity v ˆ and n ˆ 1 is radial component with velocity u ˆ . In addition, we consider the flow in the radial direction only subject to the magnetic field. Eq. (1) shows the shape of the curved channel with peristalsis:

(1) n ˆ 1 = ± h ˆ ( x ˆ 1 , t ˆ ) = ± e ˆ + d s in 2 π λ ( x ˆ 1 − c t ˆ ) ,

where t ˆ is the time, λ is the wavelength, e ˆ is separation of channel, d is the wave amplitude, and ± h ˆ represents the lower and upper wall displacement. In the radial direction, the fluid is electrically conducting due to the applied magnetic field B ˆ . The magnetic field strength is given as follows:

(2) B ˆ = B 0 n ˆ 1 + N * , 0,0 .

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

(3) J ˆ × B ˆ = 0 , − σ B 0 u ˆ ( n ˆ 1 + N * ) 2 , 0 .

The constitutive equations are given as follows:

Continuity equation:

(4) ∂ v ˆ ∂ n ˆ 1 + N * n ˆ 1 + N * ∂ u ˆ ∂ x ˆ 1 + v ˆ n ˆ 1 + N * = 0 .

Component of equations of momentum in axial and radial directions are as follows:

(5) ρ d v ˆ d t ̅ − u ˆ 2 n ˆ 1 + N * = ∂ p ̅ ∂ n ˆ 1 + 1 n ˆ 1 + N * ∂ ∂ n ˆ 1 { ( n ˆ 1 + N * ) S ˆ n ˆ 1 n ˆ 1 } + N * n ˆ 1 + N * ∂ S ˆ x ˆ 1 n ˆ 1 ∂ x ˆ 1 − S ˆ x ˆ 1 x ˆ 1 n ˆ 1 + N * ,

(6) ρ d u ˆ d t ˆ − u ˆ v ˆ n ˆ 1 + N * = − N * n ˆ 1 + N * ∂ p ̅ ∂ x ˆ 1 + 1 ( n ˆ 1 + N * ) 2 ∂ ∂ n ˆ 1 { ( n ˆ 1 + N * ) 2 S ˆ n ˆ 1 x ˆ 1 } + N * n ˆ 1 + N * ∂ S ˆ x ˆ 1 x ˆ 1 ∂ x ˆ 1 − σ B 0 2 u ˆ ( n ˆ 1 + N * ) 2 .

Equation of energy with viscous dissipation and nonlinear radiation effects reads:

(7) ρ c p d T d t ˆ = κ 1 ∂ 2 T ∂ n ˆ 1 2 + 1 n ˆ 1 + N * ∂ T ∂ n ˆ 1 + ∂ 2 T ∂ x ˆ 1 2 + ( S ˆ x ˆ 1 x ˆ 1 − S ˆ n ˆ 1 n ˆ 1 ) ∂ v ˆ ∂ n ˆ 1 + S ˆ x ˆ 1 n ˆ 1 ∂ u ˆ ∂ n ˆ 1 + N * n ˆ 1 + N * ∂ v ˆ ∂ x ˆ 1 − u ˆ n ˆ 1 + N * − 1 n ˆ 1 + N * ∂ ∂ n ˆ 1 { ( n ˆ 1 + N * ) q n ˆ 1 } .

Rosseland [33] gives the definition for nonlinear thermal radiation as follows:

q n ˆ 1 = − 16 σ 1 * T 3 3 k 1 * ∂ T ∂ n ˆ 1 .

Equation of concentration with activation energy reads:

(8) d C d t ˆ = D ∂ 2 C ∂ n ˆ 1 2 + 1 n ˆ 1 + N * ∂ C ∂ n ˆ 1 + ∂ 2 C ∂ x ˆ 1 2 − k n 2 ( C − C 0 ) T T 0 n exp − E a k 1 T .

The no-slip and flexible wall conditions can be written as follows:

(9) u ˆ = 0 , n ˆ 1 = ± h ˆ ,

(10) N * n ˆ 1 + N * − τ 1 ∂ 3 ∂ x ˆ 1 3 + m 1 * ∂ 3 ∂ x ˆ 1 ∂ t ˆ 2 + d 1 ′ ∂ 2 ∂ x ˆ 1 ∂ t ˆ = − ρ d u ˆ d t ˆ + u ˆ v ˆ n ˆ 1 + N * 1 ( n ˆ 1 + N * ) 2 ∂ ∂ n ˆ 1 { ( n ˆ 1 + N * ) 2 S ˆ n ˆ 1 x ˆ 1 } + N * n ˆ 1 + N * ∂ S ˆ x ˆ 1 x ˆ 1 ∂ x ˆ 1 − σ B 0 2 u ˆ ( n ˆ 1 + N * ) 2 n ˆ 1 = ± h , ˆ

(11) T = T 1 , T = T 0 , n ˆ 1 = ± h ˆ ,

(12) C = C 1 , C = C 0 , n ˆ 1 = ± h ˆ .

In the aforementioned equations, d d t ̅ = ∂ ∂ t ̅ + v ˆ ∂ ∂ n ˆ 1 + u ˆ N * n ˆ 1 + N * ∂ ∂ x ˆ 1 represents the material time derivative, D the mass diffusivity coefficient, N* the curvature parameter, ρ the density, τ ₁ the elastic tension in membrane, k ₁ the Boltzmann constant, m 1 * the mass per unit area, T the temperature, σ the electrically conductivity, c _pthe specific heat, d 1 ′ the viscous damping coefficient, C the concentration, E _athe activation energy, n the fitted rate constant, k n the chemical reaction rate, T ₀ the temperature at the lower wall, C ₀ the concentration at the lower wall, κ ₁ the thermal conductivity.

The rheological equation for the Casson fluid model is given by [34]:

(13) τ ij = 2 μ B 1 + p z 2 π e ij , π > π c , 2 μ B 1 + p z 2 π e ij , π c > π ,

where μ B 1 is plastic dynamic velocity, p z the yield stress fluid, π the component of deformation rate of self-product, and π c is the self-product value. The Casson fluid model gives the component S ˆ n ˆ 1 n ˆ 1 , S ˆ n ˆ 1 x ˆ 1 , S ˆ x ˆ 1 n ˆ 1 , and S ˆ x ˆ 1 x ˆ 1 through:

(14) S ˆ = μ 1 + 1 α A 1 ,

where µ is the fluid viscosity, α is the Casson fluid parameter, and A 1 is the first Rivlin–Erickson tensor. The components S ˆ n ˆ 1 n ˆ 1 , S ˆ n ˆ 1 x ˆ 1 , S ˆ x ˆ 1 n ˆ 1 , and S ˆ x ˆ 1 x ˆ 1 are given as follows:

(15) S ˆ n ˆ 1 n ˆ 1 = 2 μ 1 + 1 α ∂ v ˆ ∂ x ˆ 1 ,

(16) S ˆ x ˆ 1 n ˆ 1 = S ˆ n ˆ 1 x ˆ 1 = μ 1 + 1 α ∂ u ˆ ∂ n ˆ 1 + N * n ˆ 1 + N * ∂ v ˆ ∂ x ˆ 1 − u ˆ n ˆ 1 + N * ,

(17) S ˆ x ˆ 1 x ˆ 1 = 2 μ 1 + 1 α N * n ˆ 1 + N * ∂ u ˆ ∂ x ˆ 1 + v ˆ n ˆ 1 + N * .

The dimensionless variables are as follows:

t = c t ˆ λ , x = x ˆ 1 λ , n 1 = n ˆ 1 e ˆ , v = v ̅ c , u = u ̅ c , Re = e ˆ c ν , p = e ˆ 2 p ̅ λ μ c , h = h ˆ e ˆ , S ij = e ˆ μ c S ˆ ij , δ = e ˆ λ ,

θ = T − T 0 T 1 − T 0 , ϕ = C − C 0 C 1 − C 0 , Pr = μ c p κ 1 , Rd = 16 σ 1 * T 0 3 3 κ 1 k 1 * , Ec = c 2 c p ( T − T 0 ) , Br = EcPr , Sc = ν D ,

β = T 1 − T 0 T 0 , θ w = T 1 T 0 , k = N * e ˆ , H 2 = σ B 0 2 μ , v = δ k n + k ∂ ψ ∂ x , u = − ∂ ψ ∂ n , K * = e ˆ 2 k n 2 ν , ϵ = d e ˆ ,

(18) E 1 = − e ˆ 3 τ 1 λ 3 μ c , E 2 = e ˆ 3 m 1 * c λ 3 μ , E 3 = e ˆ 3 d 1 ′ λ 3 μ c , E 4 = E a k 1 T 0 ,

where p is the pressure, Re the Reynolds number, T ₁ the temperature at upper wall, ν the kinematic viscosity, C ₁ the concentration at upper wall, H the Hartman number, k the radius of curvature, Pr the Prandtl number, δ the wave number, Br the Brinkman number, E ₁ , E _2, and E ₃ the elasticity parameters, Sc the Schmidt number, the amplitude ratio parameter, Ec the Eckert number, Rd the thermal radiation parameter, ϵ the ampltude ratio parameter, K ^∗ the dimensionless reaction parameter, β the temperature difference parameter, θ w the temperature parameter, and E ₄ the non-dimensionless activation energy.

By implication of non-dimensional variables in the considered system under large wavelength δ << 1 and small Reynolds number Re → 0, we obtain the following simplified system of equations:

(19) ∂ p ∂ n = 0 ,

(20) − k ∂ p ∂ x + 1 ( n + K ) ∂ ∂ n ( ( n + k ) 2 S nx ) + H 2 n + k ∂ ψ ∂ n = 0 ,

(21) ∂ 2 θ ∂ n 2 + 1 n + K ∂ θ ∂ n + Rd n + k ∂ ∂ n ( θ ( θ w − 1 ) + 1 ) 3 ( n + k ) ∂ θ ∂ n + Br 1 n + k ∂ ψ ∂ n − ∂ 2 ψ ∂ n 2 S nx = 0 ,

(22) ∂ 2 ϕ ∂ n 2 + 1 n + K ∂ ϕ ∂ n − K * Sc ϕ ( 1 + β θ ) n e − E 4 1 + β θ = 0 ,

where

(23) S nx = S xn = ( α + 1 ) 1 n + k ∂ ψ ∂ n − ∂ 2 ψ ∂ n 2 .

Eliminating the pressure term from Eqs. (19) and (20), we obtain

(24) ∂ ∂ n 1 ( n + K ) ∂ ∂ n ( ( n + k ) 2 S nx ) + H 2 n + k ∂ ψ ∂ n = 0 ,

(25) ∂ 2 θ ∂ n 2 + 1 n + K ∂ θ ∂ n + Rd n + k ∂ ∂ n ( θ ( θ w − 1 ) + 1 ) 3 ( n + k ) ∂ θ ∂ n + Br ( 1 n + k ∂ ψ ∂ n − ∂ 2 ψ ∂ n 2 ) S nx = 0 ,

(26) ∂ 2 ϕ ∂ n 2 + 1 n + K ∂ ϕ ∂ n − K * Sc ϕ ( 1 + β θ ) n e − E 4 1 + β θ = 0 .

The dimensionless boundary conditions under the stream function are as follows:

(27) ψ ′ [ n ] = 0 , n = ± h ,

(28) θ = 1 0 , n = ± h ,

(29) ϕ = 1 0 , n = ± h ,

(30) 1 ( n + K ) ∂ ∂ n ( ( n + k ) 2 S nx ) + H 2 n + k ψ ′ [ n ] = L n = ± h .

Here,

(31) L = K 1 E 1 ∂ 3 ∂ x 3 + E 2 ∂ 3 ∂ x ∂ t 2 + E 3 ∂ 2 ∂ x ∂ t h .

The dimensionless wave shape is given by

(32) h = 1 + ϵ s in ( 2 π ( x − t ) ) .

3 Discussion

This section describes the effects of involved parameters on the peristaltic flow of the Casson fluid with activation energy, nonlinear thermal radiation, and viscous dissipation in a curved channel. Their behaviour on axial velocity u, temperature variation θ, concentration φ, and streamlines are physically discussed in this section.

3.1 Velocity profile

Here, we represent the graphical results of the Hartman number, the Casson fluid parameter, and elasticity parameter on the velocity profile. In Figure 1(a), we see that the increase in the values of Hartman number H reduces the velocity profile u. The magnetic field applied in the direction of flow behaves as a retarding force and is used to regulate the flow. This opposing character of the magnetic field is useful in treating diseases such as joint disorders, cancer, migraine headaches, and depression. Similar behaviour for the Casson fluid parameter α is shown in Figure 1(b). Figure 1(c) indicates that the velocity profile increases with wall elastic parameters E ₁ and E ₂ and reduces with damped constant E ₃. The recorded outcomes are in comparison to blood vessels in which the blood velocity increases with elasticity E ₁ or mass expansion per unit area E ₂. However, damping E ₃ increases the blood flow capacity and decreases the velocity within the channel. Similar results are observed in studies [11,12].

$Figure 1 (a–c) Velocity profile at α = 0 . 1 , k = 3 , E 1 = 0 . 01 , E 2 = 0 . 02 , \alpha =0.1,\hspace{ .3em}k=3,\hspace{ .3em}{E}_{1}=0.01,\hspace{ .3em}{E}_{2}=0.02, E 3 = 0 . 1 , {E}_{3}=0.1, x = 0 . 2 , and t = 0 . 1 . x=0.2,{\rm{and}}t=0.1.$

Figure 1

(a–c) Velocity profile at α = 0 . 1 , k = 3 , E 1 = 0 . 01 , E 2 = 0 . 02 , E 3 = 0 . 1 , x = 0 . 2 , and t = 0 . 1 .

3.2 Temperature profile

The behaviour of Hartmann number H on temperature θ is decreasing as shown in Figure 2(a). Since greater values of H produces a strong magnetic force that generates current in motor via fluid heat, the magnetic force acts as retarding force that induces temperature reduction. Similar behaviour is shown for the Casson fluid parameter α and radiation parameter Rd (see Figure 2(b) and (c)). Radiation converts thermal energy into electromagnetic energy, and it causes heat decay. In Figure 2(d), the temperature profile increases for increasing the values of the Brinkman number Br. In Figure 2(e), for increasing the value of Prandtl number Pr, a very small variation in temperature is detected. The results captured in Figure 2(f) display the dual response of temperature parameter θ _won θ. These results are considered to be very similar to refs. [11,12].

$Figure 2 (a–f) Temperature profile at Rd = 0 . 2 , k = 3 , {\rm{Rd}}=0.2,\hspace{ .3em}k=3, Br = 2 , θ w = 1 . 2 , E 1 = 0 . 01 , {{\rm{Br}}=2,{\theta }_{w}=1.2,{\rm{\ }}E}_{1}=0.01, E 2 = 0 . 02 , E 3 = 0 . 1 , x = 0 . 2 , and t = 0 . 1 . {E}_{2}=0.02,\hspace{ .3em}{E}_{3}=0.1,\hspace{ .3em}x=0.2,{\rm{and}}t=0.1.$

Figure 2

(a–f) Temperature profile at Rd = 0 . 2 , k = 3 , Br = 2 , θ w = 1 . 2 , E 1 = 0 . 01 , E 2 = 0 . 02 , E 3 = 0 . 1 , x = 0 . 2 , and t = 0 . 1 .

3.3 Concentration profile

This section describes the concentration variation φ effected by fitted rate constant n, temperature difference parameter β, electrically conductivity σ, Schmidt number Sc, and activation energy E ₄ (Figure 3(a)–(e)). Figure 3a shows that the concentration profile reduces with the increase in the value of Schmidt number Sc. The ratio of the momentum diffusion rate to the mass diffusion rate is mathematically expressed by Schmidt number Sc and the mass diffusion rate is low for larger Schmidt number Sc, which decreases concentration. Similar behaviour is captured for fitted rate constant n, temperature difference parameter β, and electrically conductivity σ (Figure 3(b)–(d)). The concentration profile increases with the increasing values of activation energy E ₄ (Figure 3(e)). The modified Arrhenius function degrades as E ₄ is increased. Eventually, this encourages reproductive chemical reactions, which result in an increase in concentration. The results are similar to the previous studies [11,12].

$Figure 3 (a–e) Concentration profile at Rd = 0 . 2 , k = 3 , {\rm{Rd}}=0.2,\hspace{ .3em}k=3, Br = 2 , θ w = 1 . 2 , E 1 = 0 . 01 , {{\rm{Br}}=2,{\theta }_{w}=1.2,{\rm{\ }}E}_{1}=0.01, E 2 = 0 . 02 , E 3 = 0 . 1 , x = 0 . 2 , and t = 0 . 1 . {E}_{2}=0.02,\hspace{ .3em}{E}_{3}=0.1,\hspace{ .3em}x=0.2,{\rm{and}}t=0.1.$

Figure 3

(a–e) Concentration profile at Rd = 0 . 2 , k = 3 , Br = 2 , θ w = 1 . 2 , E 1 = 0 . 01 , E 2 = 0 . 02 , E 3 = 0 . 1 , x = 0 . 2 , and t = 0 . 1 .

3.4 Streamline profile

Figures 4–9(a) and (b) describe the streamlines for diverse values of the Casson fluid parameter α, elasticity parameters E ₁, E _2, and E _3, and radius of curvature parameter k in curved channel. Figure 4(a) and (b) shows that an increment in the values of α causes a shrinking in the size of the bolus. Similar behaviour is captured for E ₁ and E ₂ (Figures 5, 6(a) and (b)). However, the bolus size enlarges for E ₃, H, and k (Figures 7, 9(a) and (b)).

$Figure 4 Streamlines for the Casson fluid parameter: (a) α \alpha = 0.1 and (b) α \alpha = 0.3.$

Figure 4

Streamlines for the Casson fluid parameter: (a) α = 0.1 and (b) α = 0.3.

$Figure 5 Streamlines for elastic parameter E 1 {E}_{1} : (a) E 1 {E}_{1} = 0.1 and (b) E 1 = 0.3 . {E}_{1}=0.3.$

Figure 5

Streamlines for elastic parameter E 1 : (a) E 1 = 0.1 and (b) E 1 = 0.3 .

$Figure 6 Streamlines for mass charactrising parameter E 2 {E}_{2} (a) E 2 {E}_{2} = 0.2 and (b) E 2 = 0.4 . {E}_{2}=0.4.$

Figure 6

Streamlines for mass charactrising parameter E 2 (a) E 2 = 0.2 and (b) E 2 = 0.4 .

$Figure 7 Streamlines for damping parameter E 3 {E}_{3} (a) E 3 {E}_{3} = 0.1, (b) E 3 = 0.3 . {E}_{3}=0.3.$

Figure 7

Streamlines for damping parameter E 3 (a) E 3 = 0.1, (b) E 3 = 0.3 .

Figure 8

Streamlines for Hartman number H (a) H = 0.3, (b) H = 0.6.

Figure 9

Streamlines for curvature parameter k (a) k = 3, (b) k = 6 .

4 Conclusions

In this article, the peristaltic flow of the Casson fluid in a curved channel is carried out for activation energy, nonlinear thermal radiation, and viscous dissipation. The main outcomes are as follows:

The reduction in flow velocity is noticed with the Hartman number H as it behaves as a retarding force to flow.
Behaviour of the Casson fluid parameter α on velocity and temperature profile is found to increase.
Thermal radiation Rd reduces the temperature profile, while the Brinkman number Br rises the temperature profile.
Concentration profile increases as we increase the effect of activation energy.
Reduction in φ is noticed with an increase in β, σ, Sc, and n.
The bolus shrinking is observed for an increase in α, E ₁, and E ₂. However, the bolus size enlarges for E ₃, H, and k.

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

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Received: 2022-08-12

Revised: 2022-11-08

Accepted: 2023-02-03

Published Online: 2023-07-19

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

Articles in the same Issue

https://doi.org/10.1515/jmbm-2022-0278

Keywords for this article

blood flow; nonlinear thermal radiation; viscous dissipation; activation energy; wall properties; magnetic field

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