# Effects of Thermal Radiation and Chemical Reaction on Hydromagnetic Fluid Flow in a Cylindrical Collapsible Tube with an Obstacle

Authors: Nictor Mwamba; Jeconia Okelo Abonyo; Kennedy Otieno Awuor
Journal: International Journal of Mathematics and Mathematical Sciences
 (2023)

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## Abstract

The aim of this research is to study the effects of thermal radiation and chemical reaction on hydromagnetic fluid flow in a cylindrical collapsible tube with an obstacle. The fluid flow is governed by continuity, momentum, energy, and concentration equations. Similarity transformation has been used to convert the obtained PDEs into ODEs. The collocation method has been used to numerically solve the ODEs. The method has been implemented in MATLAB using the bvp4c inbuilt function. The effects of the nondimensional parameters on velocity, temperature, and concentration have been presented graphically. Additionally, the skin-friction coefficient, the Nusselt number, and the Sherwood number have been discussed and are presented in a tabular form. The findings demonstrated that increasing the Reynolds number causes a rise in the fluid temperature and velocity. The fluid velocity decreases as the Hartmann number and the weight of the obstacle increase but increases with increasing Grashof numbers. The temperature of the fluid increases as the radiation parameter, or Eckert number, increases, but decreases as the Prandtl number increases. As the Soret number rises, so do the fluid’s temperature and concentration distribution. With an increase in the unsteadiness parameter, the fluid velocity and the concentration distribution decrease, whereas the opposite is seen in temperature. As the Schmidt number, the concentration Grashof number, and the chemical reaction parameter increase, the fluid’s concentration decreases. There is an increase in skin-friction coefficient with increasing Prandtl number, Eckert number, Soret number, thermal Grashof number, concentration Grashof number, thermal radiation parameter, Hartmann number, and unsteadiness parameter, while a decrease is observed with increasing Reynolds number. The Nusselt number increases with an increase in the Prandtl number, Eckert number, thermal radiation parameter, Hartmann number, and unsteadiness parameter. A slight decrease in the Nusselt number has been observed with increasing thermal Grashof number. The Sherwood number decreases with increasing Prandtl number, chemical reaction parameter, and thermal radiation parameter but increases with increasing Schmidt number, Eckert number, and Soret number. The research has the potential for a wide range of applications including but not limited to the medical field and other physical sciences.


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## Body

## 1. Introduction

In recent years, numerous researchers have conducted in-depth studies on the flow of fluids through collapsible tubes. This is due to its wide range of biological applications, including the movement of food in the gastrointestinal tract, the airflow in the lungs, and the blood flow in the vessels. Since blood vessels can constrict and` dilate through processes known as vasoconstriction and vasodilation, respectively, it is possible to think of them as collapsible tubes.Alsemiry et al. [1] investigated the numerical solution of blood flow and mass transport in an elastic tube with multiple stenoses. Kigo Mwangi et al. [2] analyzed an unsteady magnetohydrodynamic fluid flow in a collapsible tube. Bég et al. [3] investigated heat and mass transfer in a micropolar axisymmetric stagnation enclosing flow with variable thermal conductivity on a cylinder.Shaheen et al. [4] studied heat transmission in a Williamson fluid flow through a ciliated porous tube. Al-Kouz et al. [5] carried out the analysis of heat transfer and entropy generation of a hybrid magnetic nanofluid flow in a two-layered trapezoidal wavy enclosure with porous media. The unsteady flow of a Newtonian fluid through a cylindrical collapsible tube was studied by Phoebe et al. [6]. Fluid flow and heat transfer through a vertical cylindrical collapsible tube in the presence of a magnetic field and an obstacle were investigated by Chepkonga et al. [7]. Priyadharshini et al. [8] investigated the unsteady flow of collapsible tubes under transverse magnetohydrodynamic fluid. Ullah et al. [9] conducted a numerical study on a magnetized 2D hybrid nanofluid (GO + ZnO + blood) flow through a porous capillary.In many industrial and technical processes, thermal energy transfer is crucial to controlling excessive heat generation and for protecting the systems from overheating. There are three processes through which thermal energy is transferred: convection, conduction, and radiation. Convection occurs as a result of the heated substance moving in an accumulative motion, as opposed to conduction, which includes the transmission of heat through direct contact. Thermal radiation is the process by which electromagnetic radiation transfers thermal energy. Radiative heat transfer is critical in the treatment of cancer-affected tissues [10]. Loganathan et al. [11] investigated the dynamics of a nanofluid subjected to thermal radiation, free convection, and bioconvection over an electromagnetic surface heated by convection.In the flow of fluids, a temperature gradient may exist which can result in the phenomenon known as the Soret effect. This results in the movement of heated molecules from regions of higher temperature to regions of lower temperature. Heat and mass transfer flows involving chemical reactions are important in a variety of processes. A few of these processes include drying, geothermal pools, steel manufacturing system, oil recovery, and cooling mechanisms. Scholars have studied the transfer of heat and mass in various contexts for the reasons mentioned above and for many other reasons as well. Ferdows et al. [12] analyzed the thermal cooling performance of a convective nanofluid flowing across a moving extending surface. Alharbi et al. [13] investigated the effects of viscous dissipation and Coriolis on the heat and mass transfer of a 3D nonNewtonian fluid flow. Ali et al. [14] numerically simulated a bioconvective flow of a nanofluid over a stretching and heating surface. Thermal energy transfer characteristics during a hybrid nanofluid migration were studied by Rizk et al. [15]. The effects of mass transfer, thermal radiation, and Hall current on the Carreau fluid flow through a porous and stretching (shrinking) sheet was examined by Khan et al. [16]. Ali et al. [17] addressed the influence of a mixed convection flow and thermal radiation flow past a stretchable vertical surface on a hybrid nanofluid. Ali et al. [18] looked at the physical effects of a stretched sheet with convective heating and a Cross fluid containing carboxymethyl cellulose. Similar research studies have been performed by several authors [19–23].Hydromagnetics, also known as magnetohydrodynamics (MHD), is the study of the dynamics of electrically conducting fluids in magnetic fields. Examples of such fluids include electrolytes, plasmas, and liquid metals. MHD is used in a wide range of technical applications, including plasma physics, magnetic drug aiming, astronomy and astrophysics, and in electric power generation systems with high efficiency and low emissions. Ullah et al. [24] studied a magnetite-water-based nanofluid on an inclined rotating surface. The effects of thermal radiations and couple stress on the flow was considered. An analysis of the magnetized nanofluid flow in a permeable 3D tank was carried out by Shah et al. [25]. Reddy et al. [26] studied the chemical reaction impact on MHD natural convection flow through a porous medium past an exponentially stretching sheet in the presence of a heat source/sink and viscous dissipation. Prabhakar Reddy and Makinde [27] examined the unsteady hydromagnetic boundary layer flow of a reacting and radiating electrically conducting fluid past a slippery permeable vertical plate embedded in a porous medium. Elalamy et al. [28] analyzed the blood flow of MHD nonNewtonian nanofluid with heat transfer and slip effects. Ahmed et al. [29] conducted a numerical investigation of unsteady magnetohydrodynamics Eyring–Powell fluid in which thermal radiation and bioconvection were put into consideration. Salawu et al. [30] studied the flow of a magnetized hybrid Prandtl–Eyring nanofluid along the interior parabolic solar trough collector of an aircraft wing. Their focus was on thermal energy distribution, entropy generation, and fluid flow rate.Although numerous investigations have been conducted on various areas of fluid flow through collapsible tubes and heat and mass transfer, not much attention has been given to the study of the effects of thermal radiation and chemical reactions on hydromagnetic fluid flow in a collapsible tube with an obstacle. In the present study, the work of Chepkonga et al. [7] has been extended by incorporating the effects of thermal radiation and chemical reaction on a hydromagnetic fluid flow with an obstacle. Therefore, the objective of this research is to study the effects of thermal radiation and chemical reaction on hydromagnetic fluid flow through a cylindrical collapsible tube with an obstacle. The Lorentz force, weight of the obstacle, Joule heating, Soret, and the dissipation effects have been taken into account.


## 2. Mathematical Formulation

Figure1 shows a cylindrical geometric configuration with a coordinate system r,θ,z, where r is the radial distance, θ is the azimuthal angle, and z is the axial coordinate. The velocity components are uz,uθ, and ur in the z, angular, and r direction, respectively. B0 is a constant magnetic field. The tube’s wall is r=a01−α1t, where α1 is a constant of dimension T−1 which characterizes the unsteadiness in the flow field and a0 is the characteristic radius of the tube [31].Figure 1
Geometry of the collapsible tube.The fluid is assumed to be Newtonian, incompressible, and electrically conductive. There is no flow separation in the collapsible section of the tube and on the obstacle. The flow is laminar and unsteady. The tangential and radial directions of the flow are negligible, and no magnetic field is induced within the fluid. The governing equations are continuity, momentum, energy, and concentration equations, which are denoted as follows [7, 32, 33]:(1)∂uz∂z=0,(2)∂uz∂t=βgT−Tw+βcgC−Cw+μρ1r∂∂rr∂uz∂r−σB02uzρ+4πr13ρ1g3ρ,(3)∂T∂t+uz∂T∂z=κρCp∂2T∂r2+1r∂T∂r+∂2T∂z2+μρCp∂uz∂r2+σuz2B02ρCp−1Cp∂qr∂r,(4)∂C∂t+uz∂C∂z=D∂2C∂r2+1r∂C∂r+∂2C∂z2+DαTm∂2T∂r2+∂2T∂z2−krC−Cw,where β, βc, T, Tw, C, and Cw are the volumetric thermal expansion, volumetric concentration expansion, temperature, temperature at the wall, concentration, and concentration at the wall, respectively. And σ is the electrical conductivity, B0 is the magnetic field, ρ is the fluid density, ρ1 is the density of the obstacle, Cp is the specific heat capacity, D is the concentration of diffusion parameter, Tm is the mean temperature, α is the thermal diffusion ratio, kr is the chemical reaction parameter, r1 is the obstacle’s radius, and κ is the thermal conductivity.The following are the boundary conditions that have been considered:At the center line,(5)uz=U0,T=T0,C=C0,r=0.At the wall,(6)uz=0,T=TW,C=CW,r=at.The following similarity transformations obtained from other researchers [34–37] are used to convert the system of partial differential equations into nondimensional ordinary differential equations (ODEs):(7)uz=−Qz1δm+1fη,ωηδm+1=T−TWT0−TW,ωηcδm+1=C−CWC0−CW,where Q is the discharge, fη is the dimensionless velocity, ωη is the dimensionless temperature, and ωηc is the dimensionless concentration. δ is a time-dependent length scale, η=r/a0, and m is an arbitrary constant.By the Rosseland approximation, the radiative heat flux for the thermal radiationqr is given by [38](8)qr=−4σ13κ1∂T4∂r,where σ1 is the Stefan–Boltzmann constant and κ1 is the mean absorption coefficient. Let the temperature variations within the flow be used to express the term T4 as a function of the temperature [39]. This is achieved by disregarding higher-order terms and after expanding T4 in a Taylor series of T0.(9)T4=T04+T−T04T03+T−T022!12T02+…≅4TT03−3T04.Substituting the abovementioned transformations in equations (2)–(4) gives the following equations:(10)μρQa02z1δm+1f″η+μρQa02ηz1δm+1f′η+Qzm+1δm+2dδdtfη+σB02ρ−Qz1δm+1fη−βgT−TW−βcgC−CW−4πr13ρ1g3ρ=0,(11)−T0−TWm+1ωηδm+2dδdt=κρCp1a02T0−TWω″ηδm+1+1a02ηT0−TWω′ηδm+1+μρCpQ2a02z21δ2m+1f′η2+σB02ρCpQ2z21δ2m+1f2η+16σ1T033Cpκ11a02T0−TWω″ηδm+1,(12)−m+1ωηcδm+2dδdt=D1a02ω″ηcδm+1+1a02ηω′ηcδm+1+Dαa02TmT0−TWC0−CWω″ηδm+1−krC−CWC0−CW.We introduce the dimensionless variables and quantities listed as follows [7, 32]:(13)Re=QρμΓ=krρa02μ0Pr=CpμκEc=Q2Cp∆THa=a0BσμGr=gβT−Twa03ν2,Sc=μρDSr=DαTW−T0νTmCW−C0λ=ρδmμa0m−1dδdtRd=16σ1T033μCpκ1Grc=gβC−Cwa03ν2,where Gr is the thermal Grashof number, Pr is the Prandtl number, Rd is the thermal radiation parameter, Grc is the concentration Grashof number, Sr is the Soret number, and Sc is the Schmidt number. Ha is the Hartmann number, Re is the Reynolds number, Ec is the Eckert number, Γ is the chemical reaction parameter, and λ is the unsteadiness parameter. By substituting equation (13), in equations (10)–(12), the following set of ODEs is obtained with transformed boundary conditions:(14)f″η+1ηf′η+a0m+1m+1δm+1λfη−Ha2fη−zνωηGr+ωηcGrc−Rea02zδm+1Q24πr13ρ1g3ρ=0,(15)1Pr+Rdω″η+1Prηω′η+m+1a0m+1δm+1λωη+Ecz2δm+1f′η2+Ha2z2Ecδm+1f2η=0,(16)1Scω″ηcδm+1+1Scω′ηcηδm+1+a0m+1m+1δ2m+1λ−Γ1δm+1ωηc+Srω″ηδm+1=0.The transformed boundary conditions are as follows.At the center line,(17)f0=−zδm+1ω0=δm+1ω0c=δm+1,ifη=0.At the wall,(18)fη=0ωη=0ωηc=0,ifη=at,The skin-friction coefficient, the Nusselt number, and the Sherwood number are defined by the following relations, respectively [40]:(19)Cf=τwρU02,Nu=a0qwkT0−Tw,Sh=a0qmDC0−Cw,where τw is the skin shear stress on the surface, qw is the heat flux, and qm is the mass flux which is defined by the following equation:(20)τw=μ∂u∂rr=0,qw=−k∂T∂rr=0,qm=−D∂C∂rr=0.By substituting equations (7) and (20) in equation (19), the following quantities are obtained:(21)Cf=Re−1fη′,Nu=−ωη′δm+1,Sh=−ωηc′δm+1.


## 3. Numerical Solution

In order to find the numerical solutions for the obtained ODEs, the MATLAB boundary value problem solver function bvp4c is used. The bvp4c function in MATLAB is based on a collocation method that involves selecting a set of discrete points, or collocation points, in the interval of interest and requiring that the solution satisfy the differential equation at those points. The equations (14)–(16) are converted into a system of first-order ODEs as required by the bvp4c MATLAB function. Figure 2 shows the flowchart for the numerical scheme for bvp4c.Figure 2
Flowchart of numerical scheme (bvp4c).The book by Shampine et al. [41] contains additional information on the bvp4c procedure. The following conventions are used to reduce the ODEs from the second-order to the first-order:(22)fη=y1,f′η=y2,ωη=y3,ω′η=y4,ωηc=y5,ω′ηc=y6,y1′y2′y3′y4′y5′y6′=y2Ha2y1+zνy3Gr+y5Grc−1ηy2−a0m+1m+1δm+1λy1+Rea02zδm+1Q24πr13ρ1g3ρy4−1ηy4−Prm+1a0m+1δm+1λy3−PrEcz2δm+1y22−PrHa2z2Ecδm+1y1211+ρRdPry6ScΓ−a0m+1m+1δm+1λy5−y6η−ScSry4′.With the following boundary conditions at the center and at the wall, respectively, we get(23)y1=−zδm+1,y3=δm+1,y5=δm+1,y1=0,y3=0,y5=0.


## 4. Results and Discussion

The impacts of different dynamical parameters on the quantities of interest are studied using graphical plots and tables in this section.


### 4.1. Velocity Profiles

Figures3–7 are plotted to show the effects of Reynolds number Re, Grashof number Gr and Grc, Hartmann number Ha, chemical reaction parameter Γ, and unsteadiness parameter λ.Figure 3
Velocity profiles for the different values ofRe.Figure 4
Velocity profiles for the different values ofGr.Figure 5
Velocity profiles for the different values ofGrc.Figure 6
Velocity profiles for the different values ofHa.Figure 7
Velocity profiles for the different values ofλ.Figure3 shows that an increase in the value of the Reynolds number results in an increase in the primary fluid velocity. When the Reynolds number increases, the inertial forces in the fluid become dominant over the viscous forces that oppose the fluid motion. As a result, the momentum boundary layer becomes thinner, allowing the fluid to flow more easily. From a physical point of view, the fluid velocity reduces as you move from the center of the tube to its wall. This can be attributed to the high viscous forces at the wall that resists the fluid motion.Both Figures4 and 5 demonstrate that an increase in the Grashof numbers leads to an increase in velocity profiles. High values of the thermal Grashof number indicate a high-temperature gradient, resulting in a greater buoyancy force that drives the fluid motion. This effect enhances the fluid velocity while simultaneously thinning the momentum boundary layer. Due to higher temperatures at the center and lower temperatures at the wall, there is a greater buoyancy force acting on the fluid at the center compared to the force at the wall. The concentration Grashof number exhibits a similar behavior, as seen in Figure 5.From Figure6, it has been found that increasing the Hartmann number causes a decrease in the velocity profiles, and decreasing the Hartmann number causes an increase in the velocity profiles. Increasing the Hartmann number causes an increase in the Lorentz force which causes the fluid’s motion to slow down. Therefore, the Hartmann number reduces the fluid velocity while increasing the thickness of the boundary layer.Figure7 indicates that the velocity tends to reduce as the unsteadiness parameter increases. This is because the velocity has an inverse proportion to the time-dependent length scale, δ, which is directly proportional to the unsteadiness parameter λ. Furthermore, an increase in the unsteadiness parameter denotes that the boundary layer has pushed into the flow region, thus increasing the viscous drag forces and lowering the fluid velocity.According to Figure8, increasing the weight of the obstacle by increasing its density reduces fluid velocity. Increased density of the obstacle signifies an increase in the sphere’s weight pushing against the direction of the flow. This retards the velocity of the fluid. Physically, the highest velocity is between the obstacle and the tube’s wall.Figure 8
Velocity profiles for the different values ofρ1.


### 4.2. Temperature Profiles

Increasing the Reynolds number causes an increase in the fluid temperature as shown in Figure9. This is because increasing the Reynolds number leads to dominant inertia forces which leads to an increased fluid velocity. Apart from dissipation heating, if the magnetic field is present, increased fluid velocity results in an enhanced Joule heating that leads to temperature increase. Furthermore, the fluid temperature reduces towards the wall from the center. This is due to the difference in the fluid velocity and viscosity which affects the fluids’ internal heating abilities.Figure 9
Temperature profiles for the different values ofRe.In Figure10, the fluid temperature drops as the Prandtl number increases. This is because an increase in the Prandtl number indicates that thermal diffusivity has become less dominant. Higher values of the Prandtl number decrease the thermal boundary layer thickness. This leads to a rapid travel of heat, resulting in a fluid temperature decrease.Figure 10
Temperature profiles for the different values ofPr.The temperature profile shown in Figure11 implies that increasing the Hartmann number raises the fluid temperature. This is due to the Lorentz force, which is created when a magnetic field is applied to an electrically conducting fluid flow. The Lorentz force increases as the Hartmann number increases, raising the temperature of the fluid. The temperature drops towards the wall of the tube. This is due to the low velocities near the wall which results in a low Lorentz force.Figure 11
Temperature profiles for the different values ofHa.Figure12 shows that as the Eckert number increases, the fluid temperature also increases. This is because an increase in the Eckert number implies an increase in fluid velocity, which helps the fluid’s viscous dissipation. The velocity is highest at the tube’s center and so is the dissipation effect. The dissipation effect gets less noticeable at low velocity as you go closer to the wall. Also, the surface heat transfer rate is low, and the thickness of the thermal boundary layer grows.Figure 12
Temperature profiles for the different values ofEc.Figure13 shows that increasing the value of the radiation parameter causes an increase in the temperature profile of the flow. This is because increasing the radiation parameter releases heat energy into the fluid. This fits with the radiation parameter’s basic physical behavior. Growing values of radiation parameter also increases the thermal boundary layer which gives an increasing effect on temperature.Figure 13
Temperature profiles for the different values ofRd.Increasing the Soret number causes an increase in the fluid’s temperature as shown in Figure14. This is because increasing the Soret number implies that the temperature gradient is significant. Thus, more fluid particles will move from a region of higher temperature to regions of low temperature, thereby increasing the temperature of the fluid.Figure 14
Temperature profiles for the different values ofSr.According to Figure15, as the unsteadiness parameter increases, so does the temperature. From equations (7) and (13), it is seen that λ is directly proportional to the temperature of the fluid. As a result, increasing this parameter causes an increase in the temperature profile. Furthermore, the boundary layer tends to be at the tube’s center line, which is in accordance with the increase in the unsteadiness parameter. This increases the viscous dissipation effect which in turn produces thermal energy that increases the fluid’s temperature.Figure 15
Temperature profiles for the different values ofλ.


### 4.3. Concentration Profiles

The concentration field decreases with increasing Schmidt number. Increasing the Schmidt number implies that mass transfer is less efficient compared to momentum transfer.Therefore, the mass diffusion decreases and the concentration distribution decreases, so is its profile as shown in Figure16. Physically, the momentum diffusivity at the tube’s center is more than the momentum diffusivity towards the wall.Figure 16
Concentration profiles for the different values ofSc.It is found from Figure17 that the concentration of the fluid medium increases with an increase in the Soret number. Increasing the Soret numbers correspond to an increase in temperature gradients, which results in a higher convective flow. As a result, concentration distribution enhances. Since the temperature gradient is lessening as one approaches the wall, the Soret effect diminishes. Increasing the Soret number leads to a thinner concentration boundary layer.Figure 17
Concentration profiles for the different values ofSr.The concentration decreases as the unsteadiness parameter increases, as shown in Figure18. Increasing the unsteadiness parameter results in an increase in temperature which then increases the rate of a chemical reaction that leads to the reduction of concentration.Figure 18
Concentration profiles for the different values ofλ.From Figure19, it is seen that an increase in the value of the concentration Grashof number implies an increase in the concentration gradient. The Grashof number tends to boost the mass buoyancy effect. This causes an increase in the induced flow and, as a result, a decrease in concentration. Hence, the decrease in the concentration profile is shown in Figure 19. The mass buoyancy effect is strongest at the center of the tube due to the presence of a high concentration gradient. However, the mass buoyancy effect is less towards the wall.Figure 19
Concentration profiles for the different values ofGrc.Chemical molecular diffusivity decreases with increasing chemical reaction parameters due to the consumption of the species in the reaction. Therefore, it is found from Figure20 that the concentration field shrinks as the chemical reaction parameter increases.Figure 20
Concentration profiles for the different values ofΓ.


### 4.4. Effects of Variation of Parameters on Skin Friction

From Table1, it can be observed that when the Reynolds number is increased, the skin friction decreases. This is due to reduced viscous forces. There is an increase in the skin-friction coefficient with an increase in the Prandtl number and Eckert number. This can be attributed to the increased fluid momentum. An increase in the Soret number results in increased skin friction as it accelerates fluid motion. Furthermore, the thermal and concentration Grashof numbers, radiation parameters, and Hartmann numbers cause the skin friction to increase as they increase.Table 1
Skin-friction coefficient for various values of parameters.
PrEcScSrReΓλGrGcRdHaCf0.710.010.02110.10.2110.10.010.04581720.010.02110.10.2110.10.010.0461460.7110.02110.10.2110.10.010.0458390.710.011.5110.10.2110.10.010.0460010.710.010.02210.10.2110.10.010.045820.710.010.02150.10.2110.10.010.041530.710.010.02111.20.2110.10.010.0458170.710.010.02110.12110.10.010.0481950.710.010.02110.10.21010.10.010.1309260.710.010.02110.10.21100.10.010.0461670.710.010.02110.10.2111.50.010.0461670.710.010.02110.10.2110.11.20.050587


### 4.5. Effects of Variation of Parameters on Heat Transfer Rate

Table2 shows the observable effects of various parameters on the Nusselt number. Increasing the Prandtl number leads to an increase in the Nusselt number. The Nusselt number increases with an increase in the Eckert number, unsteadiness parameter, radiation parameter, and Hartmann number. These nondimension numbers increase the fluid temperature which results in enhanced heat transfer. There is little or no change in the Nusselt number when the Schmidt number, Soret number, chemical reaction parameter, Reynolds number, or concentration Grashof number is altered. However, the Nusselt number slightly decreases when the thermal Grashof number is increased.Table 2
Nusselt number for various values of parameters.
PrEcScSrReΓλGrGcRdHaNu0.710.010.02110.10.2110.10.01−7.48711520.010.02110.10.2110.10.01−7.8138610.7110.02110.10.2110.10.01−7.5044740.710.011.5110.10.2110.10.01−7.4871150.710.010.02210.10.2110.10.01−7.4871150.710.010.021100.10.2110.10.01−7.4871140.710.010.02111.20.2110.10.01−7.4871150.710.010.02110.12110.10.01−7.4871230.710.010.02110.10.21010.10.01−7.4871140.710.010.02110.10.21100.10.01−7.4871150.710.010.02110.10.2111.50.01−7.9646630.710.010.02110.10.2110.11.2−7.487117


### 4.6. Effects of Variation of Parameters on Mass Transfer Rate

It has been shown in Table3 that when the Prandtl number, chemical reaction parameter, or the radiation parameter is raised, the Sherwood number is reduced. On the other hand, the Sherwood number is enhanced when the Eckert, Schmit, or Soret number is raised. However, there is little or no effect of the Reynolds number, unsteadiness parameter, Grashof number, or Hartmann number on the Sherwood number.Table 3
Sherwood number for various values of parameters.
PrEcScSrReΓλGrGcRdHaSh0.710.010.02110.10.2110.10.01−0.0056220.010.02110.10.2110.10.01−0.0043680.7110.02110.10.2110.10.01−0.0056520.710.011.5110.10.2110.10.01−0.1525170.710.010.02210.10.2110.10.01−0.0076090.710.010.02150.10.2110.10.01−0.005620.710.010.02111.20.2110.10.01−0.0056130.710.010.02110.12110.10.01−0.005620.710.010.02110.10.21010.10.01−0.005620.710.010.02110.10.21100.10.01−0.005620.710.010.02110.10.2111.50.01−0.0037730.710.010.02110.10.2110.11.2−0.00562


### 4.7. Validation

Comparing the results obtained from the present study and that of the Odejide et al.’s [31], on the effects of the Reynolds number and the Prandtl number on the Nusselt number. It is clear from Table 4 that the two results are in close agreement.Table 4
Nusselt number for various values ofRe and Pr.
RePrOdejide et al. [31]Present10.71−1.5148−1.51951.50.71−0.98969−1.005720.71−0.1096−0.109511−1.5208−1.521811.5−1.5312−1.5361


## 4.1. Velocity Profiles

Figures3–7 are plotted to show the effects of Reynolds number Re, Grashof number Gr and Grc, Hartmann number Ha, chemical reaction parameter Γ, and unsteadiness parameter λ.Figure 3
Velocity profiles for the different values ofRe.Figure 4
Velocity profiles for the different values ofGr.Figure 5
Velocity profiles for the different values ofGrc.Figure 6
Velocity profiles for the different values ofHa.Figure 7
Velocity profiles for the different values ofλ.Figure3 shows that an increase in the value of the Reynolds number results in an increase in the primary fluid velocity. When the Reynolds number increases, the inertial forces in the fluid become dominant over the viscous forces that oppose the fluid motion. As a result, the momentum boundary layer becomes thinner, allowing the fluid to flow more easily. From a physical point of view, the fluid velocity reduces as you move from the center of the tube to its wall. This can be attributed to the high viscous forces at the wall that resists the fluid motion.Both Figures4 and 5 demonstrate that an increase in the Grashof numbers leads to an increase in velocity profiles. High values of the thermal Grashof number indicate a high-temperature gradient, resulting in a greater buoyancy force that drives the fluid motion. This effect enhances the fluid velocity while simultaneously thinning the momentum boundary layer. Due to higher temperatures at the center and lower temperatures at the wall, there is a greater buoyancy force acting on the fluid at the center compared to the force at the wall. The concentration Grashof number exhibits a similar behavior, as seen in Figure 5.From Figure6, it has been found that increasing the Hartmann number causes a decrease in the velocity profiles, and decreasing the Hartmann number causes an increase in the velocity profiles. Increasing the Hartmann number causes an increase in the Lorentz force which causes the fluid’s motion to slow down. Therefore, the Hartmann number reduces the fluid velocity while increasing the thickness of the boundary layer.Figure7 indicates that the velocity tends to reduce as the unsteadiness parameter increases. This is because the velocity has an inverse proportion to the time-dependent length scale, δ, which is directly proportional to the unsteadiness parameter λ. Furthermore, an increase in the unsteadiness parameter denotes that the boundary layer has pushed into the flow region, thus increasing the viscous drag forces and lowering the fluid velocity.According to Figure8, increasing the weight of the obstacle by increasing its density reduces fluid velocity. Increased density of the obstacle signifies an increase in the sphere’s weight pushing against the direction of the flow. This retards the velocity of the fluid. Physically, the highest velocity is between the obstacle and the tube’s wall.Figure 8
Velocity profiles for the different values ofρ1.


## 4.2. Temperature Profiles

Increasing the Reynolds number causes an increase in the fluid temperature as shown in Figure9. This is because increasing the Reynolds number leads to dominant inertia forces which leads to an increased fluid velocity. Apart from dissipation heating, if the magnetic field is present, increased fluid velocity results in an enhanced Joule heating that leads to temperature increase. Furthermore, the fluid temperature reduces towards the wall from the center. This is due to the difference in the fluid velocity and viscosity which affects the fluids’ internal heating abilities.Figure 9
Temperature profiles for the different values ofRe.In Figure10, the fluid temperature drops as the Prandtl number increases. This is because an increase in the Prandtl number indicates that thermal diffusivity has become less dominant. Higher values of the Prandtl number decrease the thermal boundary layer thickness. This leads to a rapid travel of heat, resulting in a fluid temperature decrease.Figure 10
Temperature profiles for the different values ofPr.The temperature profile shown in Figure11 implies that increasing the Hartmann number raises the fluid temperature. This is due to the Lorentz force, which is created when a magnetic field is applied to an electrically conducting fluid flow. The Lorentz force increases as the Hartmann number increases, raising the temperature of the fluid. The temperature drops towards the wall of the tube. This is due to the low velocities near the wall which results in a low Lorentz force.Figure 11
Temperature profiles for the different values ofHa.Figure12 shows that as the Eckert number increases, the fluid temperature also increases. This is because an increase in the Eckert number implies an increase in fluid velocity, which helps the fluid’s viscous dissipation. The velocity is highest at the tube’s center and so is the dissipation effect. The dissipation effect gets less noticeable at low velocity as you go closer to the wall. Also, the surface heat transfer rate is low, and the thickness of the thermal boundary layer grows.Figure 12
Temperature profiles for the different values ofEc.Figure13 shows that increasing the value of the radiation parameter causes an increase in the temperature profile of the flow. This is because increasing the radiation parameter releases heat energy into the fluid. This fits with the radiation parameter’s basic physical behavior. Growing values of radiation parameter also increases the thermal boundary layer which gives an increasing effect on temperature.Figure 13
Temperature profiles for the different values ofRd.Increasing the Soret number causes an increase in the fluid’s temperature as shown in Figure14. This is because increasing the Soret number implies that the temperature gradient is significant. Thus, more fluid particles will move from a region of higher temperature to regions of low temperature, thereby increasing the temperature of the fluid.Figure 14
Temperature profiles for the different values ofSr.According to Figure15, as the unsteadiness parameter increases, so does the temperature. From equations (7) and (13), it is seen that λ is directly proportional to the temperature of the fluid. As a result, increasing this parameter causes an increase in the temperature profile. Furthermore, the boundary layer tends to be at the tube’s center line, which is in accordance with the increase in the unsteadiness parameter. This increases the viscous dissipation effect which in turn produces thermal energy that increases the fluid’s temperature.Figure 15
Temperature profiles for the different values ofλ.


## 4.3. Concentration Profiles

The concentration field decreases with increasing Schmidt number. Increasing the Schmidt number implies that mass transfer is less efficient compared to momentum transfer.Therefore, the mass diffusion decreases and the concentration distribution decreases, so is its profile as shown in Figure16. Physically, the momentum diffusivity at the tube’s center is more than the momentum diffusivity towards the wall.Figure 16
Concentration profiles for the different values ofSc.It is found from Figure17 that the concentration of the fluid medium increases with an increase in the Soret number. Increasing the Soret numbers correspond to an increase in temperature gradients, which results in a higher convective flow. As a result, concentration distribution enhances. Since the temperature gradient is lessening as one approaches the wall, the Soret effect diminishes. Increasing the Soret number leads to a thinner concentration boundary layer.Figure 17
Concentration profiles for the different values ofSr.The concentration decreases as the unsteadiness parameter increases, as shown in Figure18. Increasing the unsteadiness parameter results in an increase in temperature which then increases the rate of a chemical reaction that leads to the reduction of concentration.Figure 18
Concentration profiles for the different values ofλ.From Figure19, it is seen that an increase in the value of the concentration Grashof number implies an increase in the concentration gradient. The Grashof number tends to boost the mass buoyancy effect. This causes an increase in the induced flow and, as a result, a decrease in concentration. Hence, the decrease in the concentration profile is shown in Figure 19. The mass buoyancy effect is strongest at the center of the tube due to the presence of a high concentration gradient. However, the mass buoyancy effect is less towards the wall.Figure 19
Concentration profiles for the different values ofGrc.Chemical molecular diffusivity decreases with increasing chemical reaction parameters due to the consumption of the species in the reaction. Therefore, it is found from Figure20 that the concentration field shrinks as the chemical reaction parameter increases.Figure 20
Concentration profiles for the different values ofΓ.


## 4.4. Effects of Variation of Parameters on Skin Friction

From Table1, it can be observed that when the Reynolds number is increased, the skin friction decreases. This is due to reduced viscous forces. There is an increase in the skin-friction coefficient with an increase in the Prandtl number and Eckert number. This can be attributed to the increased fluid momentum. An increase in the Soret number results in increased skin friction as it accelerates fluid motion. Furthermore, the thermal and concentration Grashof numbers, radiation parameters, and Hartmann numbers cause the skin friction to increase as they increase.Table 1
Skin-friction coefficient for various values of parameters.
PrEcScSrReΓλGrGcRdHaCf0.710.010.02110.10.2110.10.010.04581720.010.02110.10.2110.10.010.0461460.7110.02110.10.2110.10.010.0458390.710.011.5110.10.2110.10.010.0460010.710.010.02210.10.2110.10.010.045820.710.010.02150.10.2110.10.010.041530.710.010.02111.20.2110.10.010.0458170.710.010.02110.12110.10.010.0481950.710.010.02110.10.21010.10.010.1309260.710.010.02110.10.21100.10.010.0461670.710.010.02110.10.2111.50.010.0461670.710.010.02110.10.2110.11.20.050587


## 4.5. Effects of Variation of Parameters on Heat Transfer Rate

Table2 shows the observable effects of various parameters on the Nusselt number. Increasing the Prandtl number leads to an increase in the Nusselt number. The Nusselt number increases with an increase in the Eckert number, unsteadiness parameter, radiation parameter, and Hartmann number. These nondimension numbers increase the fluid temperature which results in enhanced heat transfer. There is little or no change in the Nusselt number when the Schmidt number, Soret number, chemical reaction parameter, Reynolds number, or concentration Grashof number is altered. However, the Nusselt number slightly decreases when the thermal Grashof number is increased.Table 2
Nusselt number for various values of parameters.
PrEcScSrReΓλGrGcRdHaNu0.710.010.02110.10.2110.10.01−7.48711520.010.02110.10.2110.10.01−7.8138610.7110.02110.10.2110.10.01−7.5044740.710.011.5110.10.2110.10.01−7.4871150.710.010.02210.10.2110.10.01−7.4871150.710.010.021100.10.2110.10.01−7.4871140.710.010.02111.20.2110.10.01−7.4871150.710.010.02110.12110.10.01−7.4871230.710.010.02110.10.21010.10.01−7.4871140.710.010.02110.10.21100.10.01−7.4871150.710.010.02110.10.2111.50.01−7.9646630.710.010.02110.10.2110.11.2−7.487117


## 4.6. Effects of Variation of Parameters on Mass Transfer Rate

It has been shown in Table3 that when the Prandtl number, chemical reaction parameter, or the radiation parameter is raised, the Sherwood number is reduced. On the other hand, the Sherwood number is enhanced when the Eckert, Schmit, or Soret number is raised. However, there is little or no effect of the Reynolds number, unsteadiness parameter, Grashof number, or Hartmann number on the Sherwood number.Table 3
Sherwood number for various values of parameters.
PrEcScSrReΓλGrGcRdHaSh0.710.010.02110.10.2110.10.01−0.0056220.010.02110.10.2110.10.01−0.0043680.7110.02110.10.2110.10.01−0.0056520.710.011.5110.10.2110.10.01−0.1525170.710.010.02210.10.2110.10.01−0.0076090.710.010.02150.10.2110.10.01−0.005620.710.010.02111.20.2110.10.01−0.0056130.710.010.02110.12110.10.01−0.005620.710.010.02110.10.21010.10.01−0.005620.710.010.02110.10.21100.10.01−0.005620.710.010.02110.10.2111.50.01−0.0037730.710.010.02110.10.2110.11.2−0.00562


## 4.7. Validation

Comparing the results obtained from the present study and that of the Odejide et al.’s [31], on the effects of the Reynolds number and the Prandtl number on the Nusselt number. It is clear from Table 4 that the two results are in close agreement.Table 4
Nusselt number for various values ofRe and Pr.
RePrOdejide et al. [31]Present10.71−1.5148−1.51951.50.71−0.98969−1.005720.71−0.1096−0.109511−1.5208−1.521811.5−1.5312−1.5361


## 5. Conclusion

This study has investigated the effects of thermal radiation and chemical reaction on hydromagnetic fluid flow through a cylindrical collapsible tube with an obstacle.The research study has developed a set of mathematical equations to model hydromagnetic fluid flow in a cylindrical collapsible tube with an obstacle. The developed equations were nonlinear partial differential equations which were later converted to nonlinear ordinary differential equations and solved by using the bvp4c in MATLAB. The results have shown that the equations can be used to predict fluid flow behavior under various conditions. The obtained model has significant implications for the field as it provides a framework for understanding and predicting hydromagnetic fluid flow in similar systems. However, it is important to note that the equations are based on certain assumptions and limitations.Furthermore, this study has determined the velocity, temperature, and concentration profiles of the flow for a conductive fluid in a cylindrical collapsible tube with an obstacle, in the presence of a magnetic field. The effects of various nondimensional parameters on velocity, temperature, and concentration have been determined and the results are presented graphically. Additionally, the skin-friction coefficient, the Nusselt number, and the Sherwood number have been obtained using Maple, and are presented in tabular form.The results showed that increasing the Reynolds number leads to an increased fluid velocity and temperature. While the fluid velocity decreases with increasing Hartmann and obstacle’s weight, it increases with rising Grashof numbers. The fluid temperature rises with an increase in the radiation parameter or Eckert number but drops with an increase in the Prandtl number. The temperature and concentration distribution of the fluid increases with increased Soret number. The fluid velocity and concentration distribution decreases with increasing unsteadiness parameter, while the opposite is observed with temperature. The concentration of the fluid decreases with increasing Schmidt number, concentration Grashof number, and chemical reaction parameter.The skin-friction coefficient increases when the Prandtl number, Eckert number, Soret number, thermal Grashof number, concentration Grashof number, thermal radiation parameter, Hartmann number, and unsteadiness parameter rise, whereas it decreases as the Reynolds number rises. Increases in the Prandtl number, Eckert number, thermal radiation parameter, Hartmann number, and unsteadiness parameter raise the Nusselt number. With the rising thermal Grashof number, the Nusselt number decreases slightly. The Sherwood number drops when the Prandtl number, chemical reaction parameter, and thermal radiation parameter rise, whereas it increases as the Schmidt number, Eckert number, and Soret number decrease.The results obtained in this research study have the potential to be applied in the medical field, i.e., understanding blood flow in blood vessels in the presence of fat deposits and also in the treatment of thrombosis. It can also be applied in targeted drug-aiming procedures. Additionally, the blood flow can be temporarily lowered by applying a magnetic field in order to minimize the excessive blood loss.Further work can be performed on a three-dimensional MHD fluid flow by considering a nonNewtonian turbulent flow with a varying magnetic field, incorporating both Soret and Dufour effects.



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*Source: 9991376-2023-03-03.xml*