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Temperature and Concentration Stratification Effects in Mixed Convection Flow of an Oldroyd-B Fluid with Thermal Radiation and Chemical Reaction.

Hayat T, Muhammad T, Shehzad SA, Alsaedi A - PLoS ONE (2015)

Bottom Line: Both temperature and concentration stratification effects are considered.Graphs are plotted to examine the impacts of physical parameters on the non-dimensional temperature and concentration distributions.The local Nusselt number and the local Sherwood number are computed and analyzed numerically.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad, 44000, Pakistan; Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia.

ABSTRACT
This research addresses the mixed convection flow of an Oldroyd-B fluid in a doubly stratified surface. Both temperature and concentration stratification effects are considered. Thermal radiation and chemical reaction effects are accounted. The governing nonlinear boundary layer equations are converted to coupled nonlinear ordinary differential equations using appropriate transformations. Resulting nonlinear systems are solved for the convergent series solutions. Graphs are plotted to examine the impacts of physical parameters on the non-dimensional temperature and concentration distributions. The local Nusselt number and the local Sherwood number are computed and analyzed numerically.

No MeSH data available.


Related in: MedlinePlus

Temperature distribution function θ(η) and concentration distribution function ϕ(η) when ε2 = 0.0, 0.1, 0.2 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc and ε1 = 0.3.
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pone.0127646.g005: Temperature distribution function θ(η) and concentration distribution function ϕ(η) when ε2 = 0.0, 0.1, 0.2 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc and ε1 = 0.3.

Mentions: This section presents the impacts of various emerging parameters including Deborah number in terms of relaxation time β1, Deborah number in terms of retardation time β2, Prandtl number Pr, thermal radiation parameter Rd, Schmidt number Sc, chemical reaction parameter γ, thermal stratification parameter ε1 and solutal stratification parameter ε2 on the dimensionless temperature profile θ(η) and concentration profile ϕ(η). This purpose is achieved through the plots in the Figs 2–9. Fig 2 is plotted to examine the effects of Deborah number β1 on the temperature profile θ(η) and concentration profile ϕ(η) when β1 = 0.0, 0.25, 0.50 and β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc, ε1 = 0.3 = ε2. Fig 2 examined that the temperature profile θ(η) and concentration profile ϕ(η) are enhanced when we use the larger values of Deborah number β1. Since Deborah number β1 has dependence on the relaxation time [4]. Larger values of Deborah number β1 implies to higher relaxation time. It is well known fact that the larger relaxation time fluids have higher temperature and concentration and smaller relaxation time fluids possess lower temperature and concentration. In view of this argument, both temperature profile θ(η) and concentration profile ϕ(η) are enhanced via larger Deborah number β1. The influence of Deborah number β2 on the dimensionless temperature and concentration fields when β2 = 0.0, 0.25, 0.50 and β1 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc, ε1 = 0.3 = ε2 is studied in Fig 3. Fig 3 clearly depicts that the temperature θ(η) and concentration ϕ(η) are decreasing functions of Deborah number β2 [11]. Here the Deborah number β2 is dependent on the retardation time. When we increase the values of Deborah number β2, the retardation time is increased. Such increase in retardation time is responsible for the reduction in the temperature θ(η) and concentration ϕ(η). Here it is interesting to mention that β1 = 0 = β2 correspond to viscous fluid case and β2 = 0 shows the Maxwellian fluid flow situation. From experimental point of view, it is quite obvious that the values of β2 are not much than the values of β1. Influence of thermal stratification parameter ε1 on the temperature θ(η) and concentration ϕ(η) is shown in Fig 4 when ε1 = 0.0, 0.1, 0.2 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc, ε2 = 0.3. Here the temperature and thermal boundary layer thickness are decreased while concentration and its related boundary layer thickness are increased when we increase in thermal stratification parameter. When the thermal stratification effect is taken into account, the effective temperature difference between the surface and the ambient fluid is decreased while opposite behavior is observed for concentration profile [28]. Influence of solutal stratification parameter ε2 on the temperature profile θ(η) and concentration profile ϕ(η) is shown in Fig 5 when ε2 = 0.0, 0.1, 0.2 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc and ε1 = 0.3. The temperature profile is enhanced while the concentration profile is reduced with an increase in solutal stratification parameter [29]. Influence of Prandtl number on the temperature profile is shown in Fig 6 when Pr = 0.5, 0.75, 1.0, 1.25 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Sc = 1.0, ε1 = 0.3 = ε2. The temperature and thermal layer thickness are reduced for the increasing values of Prandtl number. Physically larger Prandtl fluids possess lower thermal diffusivity and smaller Prandtl fluids have higher thermal diffusivity. This change in thermal diffusivity causes a reduction in the temperature and thermal boundary layer thickness. Basically Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. In heat transfer, Prandtl number is used to control the thicknesses of momentum and thermal boundary layers. Fig 7 is plotted to examine the change in temperature profile when Rd = 0.0, 0.3, 0.6, 1.0 and β1 = β2 = 0.2 = γ, λ = 0.1 = N, Pr = 1.0 = Sc, ε1 = 0.3 = ε2. Fig 7 describes that the temperature and thermal boundary layer thickness are enhanced with an increase in the thermal radiation parameter. Larger values of thermal radiation parameter provide more heat to working fluid that shows an enhancement in the temperature and thermal boundary layer thickness [20]. Influence of Schmidt number on the concentration field is shown in Fig 8 when Sc = 0.5, 0.75, 1.0, 1.25 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0, ε1 = 0.3 = ε2. It is clearly observed that the concentration and its related boundary layer thickness are decreasing functions of Schmidt number. Schmidt number is inversely proportional to the diffusion coefficient. Hence an increase in Schmidt number corresponds to a smaller diffusion coefficient. Such smaller diffusion coefficient creates a reduction in the concentration field. Fig 9 is plotted to investigate the effects of chemical reaction parameter when γ = 0.0, 0.3, 0.6, 1.0 and β1 = β2 = 0.2 = Rd, λ = 0.1 = N, Pr = 1.0 = Sc, ε1 = 0.3 = ε2. It is noticed from Fig 9 that the concentration and its associated boundary layer thickness are decreasing functions of chemical reaction parameter. Chemical reaction increases the rate of interfacial mass transfer. The reaction reduces the local concentration, thus increasing the concentration gradient and its flux. As a result, concentration of the chemical species in the boundary layer decreases with an increase in chemical reaction parameter. Tables 2 and 3 show the numerical values of the local Nusselt and the local Sherwood numbers for different values of β1, β2, λ, N, Pr, Sc, Rd, γ, ε1 and ε2. The values of local Nusselt and the local Sherwood numbers are decreased by increasing ε1 and ε2 while these values are increased for the larger λ and N. Table 4 is computed to validate the present results with the previous published results in a limiting sense. Here we compared our results for a Maxwell fluid case. From this Table, we examined that the present series solutions have good agreement with the numerical solutions of Megahed [42] in limiting sense.


Temperature and Concentration Stratification Effects in Mixed Convection Flow of an Oldroyd-B Fluid with Thermal Radiation and Chemical Reaction.

Hayat T, Muhammad T, Shehzad SA, Alsaedi A - PLoS ONE (2015)

Temperature distribution function θ(η) and concentration distribution function ϕ(η) when ε2 = 0.0, 0.1, 0.2 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc and ε1 = 0.3.
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Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4478041&req=5

pone.0127646.g005: Temperature distribution function θ(η) and concentration distribution function ϕ(η) when ε2 = 0.0, 0.1, 0.2 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc and ε1 = 0.3.
Mentions: This section presents the impacts of various emerging parameters including Deborah number in terms of relaxation time β1, Deborah number in terms of retardation time β2, Prandtl number Pr, thermal radiation parameter Rd, Schmidt number Sc, chemical reaction parameter γ, thermal stratification parameter ε1 and solutal stratification parameter ε2 on the dimensionless temperature profile θ(η) and concentration profile ϕ(η). This purpose is achieved through the plots in the Figs 2–9. Fig 2 is plotted to examine the effects of Deborah number β1 on the temperature profile θ(η) and concentration profile ϕ(η) when β1 = 0.0, 0.25, 0.50 and β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc, ε1 = 0.3 = ε2. Fig 2 examined that the temperature profile θ(η) and concentration profile ϕ(η) are enhanced when we use the larger values of Deborah number β1. Since Deborah number β1 has dependence on the relaxation time [4]. Larger values of Deborah number β1 implies to higher relaxation time. It is well known fact that the larger relaxation time fluids have higher temperature and concentration and smaller relaxation time fluids possess lower temperature and concentration. In view of this argument, both temperature profile θ(η) and concentration profile ϕ(η) are enhanced via larger Deborah number β1. The influence of Deborah number β2 on the dimensionless temperature and concentration fields when β2 = 0.0, 0.25, 0.50 and β1 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc, ε1 = 0.3 = ε2 is studied in Fig 3. Fig 3 clearly depicts that the temperature θ(η) and concentration ϕ(η) are decreasing functions of Deborah number β2 [11]. Here the Deborah number β2 is dependent on the retardation time. When we increase the values of Deborah number β2, the retardation time is increased. Such increase in retardation time is responsible for the reduction in the temperature θ(η) and concentration ϕ(η). Here it is interesting to mention that β1 = 0 = β2 correspond to viscous fluid case and β2 = 0 shows the Maxwellian fluid flow situation. From experimental point of view, it is quite obvious that the values of β2 are not much than the values of β1. Influence of thermal stratification parameter ε1 on the temperature θ(η) and concentration ϕ(η) is shown in Fig 4 when ε1 = 0.0, 0.1, 0.2 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc, ε2 = 0.3. Here the temperature and thermal boundary layer thickness are decreased while concentration and its related boundary layer thickness are increased when we increase in thermal stratification parameter. When the thermal stratification effect is taken into account, the effective temperature difference between the surface and the ambient fluid is decreased while opposite behavior is observed for concentration profile [28]. Influence of solutal stratification parameter ε2 on the temperature profile θ(η) and concentration profile ϕ(η) is shown in Fig 5 when ε2 = 0.0, 0.1, 0.2 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0 = Sc and ε1 = 0.3. The temperature profile is enhanced while the concentration profile is reduced with an increase in solutal stratification parameter [29]. Influence of Prandtl number on the temperature profile is shown in Fig 6 when Pr = 0.5, 0.75, 1.0, 1.25 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Sc = 1.0, ε1 = 0.3 = ε2. The temperature and thermal layer thickness are reduced for the increasing values of Prandtl number. Physically larger Prandtl fluids possess lower thermal diffusivity and smaller Prandtl fluids have higher thermal diffusivity. This change in thermal diffusivity causes a reduction in the temperature and thermal boundary layer thickness. Basically Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. In heat transfer, Prandtl number is used to control the thicknesses of momentum and thermal boundary layers. Fig 7 is plotted to examine the change in temperature profile when Rd = 0.0, 0.3, 0.6, 1.0 and β1 = β2 = 0.2 = γ, λ = 0.1 = N, Pr = 1.0 = Sc, ε1 = 0.3 = ε2. Fig 7 describes that the temperature and thermal boundary layer thickness are enhanced with an increase in the thermal radiation parameter. Larger values of thermal radiation parameter provide more heat to working fluid that shows an enhancement in the temperature and thermal boundary layer thickness [20]. Influence of Schmidt number on the concentration field is shown in Fig 8 when Sc = 0.5, 0.75, 1.0, 1.25 and β1 = β2 = 0.2 = γ = Rd, λ = 0.1 = N, Pr = 1.0, ε1 = 0.3 = ε2. It is clearly observed that the concentration and its related boundary layer thickness are decreasing functions of Schmidt number. Schmidt number is inversely proportional to the diffusion coefficient. Hence an increase in Schmidt number corresponds to a smaller diffusion coefficient. Such smaller diffusion coefficient creates a reduction in the concentration field. Fig 9 is plotted to investigate the effects of chemical reaction parameter when γ = 0.0, 0.3, 0.6, 1.0 and β1 = β2 = 0.2 = Rd, λ = 0.1 = N, Pr = 1.0 = Sc, ε1 = 0.3 = ε2. It is noticed from Fig 9 that the concentration and its associated boundary layer thickness are decreasing functions of chemical reaction parameter. Chemical reaction increases the rate of interfacial mass transfer. The reaction reduces the local concentration, thus increasing the concentration gradient and its flux. As a result, concentration of the chemical species in the boundary layer decreases with an increase in chemical reaction parameter. Tables 2 and 3 show the numerical values of the local Nusselt and the local Sherwood numbers for different values of β1, β2, λ, N, Pr, Sc, Rd, γ, ε1 and ε2. The values of local Nusselt and the local Sherwood numbers are decreased by increasing ε1 and ε2 while these values are increased for the larger λ and N. Table 4 is computed to validate the present results with the previous published results in a limiting sense. Here we compared our results for a Maxwell fluid case. From this Table, we examined that the present series solutions have good agreement with the numerical solutions of Megahed [42] in limiting sense.

Bottom Line: Both temperature and concentration stratification effects are considered.Graphs are plotted to examine the impacts of physical parameters on the non-dimensional temperature and concentration distributions.The local Nusselt number and the local Sherwood number are computed and analyzed numerically.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad, 44000, Pakistan; Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia.

ABSTRACT
This research addresses the mixed convection flow of an Oldroyd-B fluid in a doubly stratified surface. Both temperature and concentration stratification effects are considered. Thermal radiation and chemical reaction effects are accounted. The governing nonlinear boundary layer equations are converted to coupled nonlinear ordinary differential equations using appropriate transformations. Resulting nonlinear systems are solved for the convergent series solutions. Graphs are plotted to examine the impacts of physical parameters on the non-dimensional temperature and concentration distributions. The local Nusselt number and the local Sherwood number are computed and analyzed numerically.

No MeSH data available.


Related in: MedlinePlus