$\rho $ density | |
$ {T}_{\infty } $ free stream temperature | |
$ {\sigma }_{1} $ Stefan–Boltzmann constant | |
$ \mu $ dynamical viscosity | |
$ T\, $ nanoparticle temperature | |
$ n $ power law index | |
$ K $ viscoelastic parameter | |
$ {Rd}\, $ radiation parameter | |
$ {Nt} $ thermophoresis parameter | |
$ {Sc} $ Schmidt number | |
$ {\beta }_{c} $ chemical reaction parameter | |
$ {Ec} $ Eckert number | |
$ {Q}_{1}\, $ power law lubricant | |
$ \kappa $ thermal conductivity | |
$ {c}_{f} $ heat capacity of fluid | |
$ {D}_{B} $ Brownian diffusion coefficient | |
$ {q}_{r} $ radioactive heat flux | |
$ ({\alpha }_{1},{\alpha }_{2}) $ normal stress moduli | |
$ B\, $ magnetic field | |
$ {{Re}}_{{r}}^{-1/2}{{Nu}}_{{r}} $ local Nusselt number | |
$ (r,\theta ,z) $ cylindrical coordinates | |
$ P $ pressure | |
$ {\boldsymbol{S}} $ stress tensor | |
$ k* $ mean absorption coefficient | |
$ C $ nanoparticle concentration | |
$ m* $ consistency index | |
$ M $ magnetic parameter | |
$ {\theta }_{w} $ temperature ratio | |
$ {Re} $ Reynolds number | |
$ {\Pr } $ Prandtl number | |
$ {Nb} $ Brownian motion parameter | |
$ \gamma \, $ slip parameter | |
$ {\rho }_{p} $ density of the particle | |
$ {c}_{p} $ heat capacity of the nanoparticle | |
$ {D}_{T} $ thermophoresis diffusion coefficient | |
$ \mu $ viscosity of nanofluid | |
$ ({A}_{1},{A}_{2}) $ kinematic tensors | |
$ E\, $ electric field | |
$ \sigma $ electric conductivity | |
$ {{Sh}}_{{r}}{{Re}}_{{r}}^{-1/2} $ local Sherwood number |
1. Introduction
2. Governing equations and flow configuration
Figure 1. Geometry of the problem. |
2.1. Boundary conditions
2.2. Dimensionless formulation
3. Solution methodology
Reduction to first order
Discretization
Linearization
Solution of tri-diagonal system
4. Results and discussion
4.1. Heat and mass transfer analysis
Figure 2. Temperature profile for (a) Eckert number $\mathrm{Ec}$, (b) thermophoresis parameter $\mathrm{Nt}$, (c) Brownian motion parameter $\mathrm{Nb}$ and (d) magnetic parameter $M$ with other parameters $\mathrm{Rd}=K=0.1,\,\Pr =1.0,\,{\beta }_{c}=1=\mathrm{Sc},$ and ${\theta }_{w}=2.0.$ |
Figure 3. Temperature profile for (a) Prandtl number $\Pr $ and (b) radiation parameter $\mathrm{Rd}$ with other parameters $M=K=0.1=\mathrm{Ec},\,\mathrm{Nt}\,=0.5,\,\beta =5.0,\,{\beta }_{c}=1=\mathrm{Sc}$ and $\mathrm{Nb}=0.5.$ |
Figure 4. Concentration profile for (a) chemical reaction parameter $\mathrm{Ec}$, (b) thermophoresis parameter $\mathrm{Nt}$, (c) Brownian motion parameter $\mathrm{Nb}$ and (d) magnetic parameter $M$ with other parameters $\mathrm{Rd}=K=0.1,\,\Pr =1.0=\mathrm{Sc},\,{\theta }_{w}=2.0$ and $\mathrm{Ec}=0.1.$ |
4.2. Analysis for engineering quantities
Table 1. Variation of ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Nu}}_{r}$ and ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Sh}}_{r}$ when ${\theta }_{w}=2.0.$ |
$K$ | $M$ | ${Nt}$ | ${Nb}$ | ${Ec}$ | ${Rd}$ | ${\Pr }$ | ${Sc}$ | ${\beta }_{c}$ | $\,\,\,{{Re}}_{r}^{-1/2}{{Nu}}_{r}$ | $\,\,\,{{Re}}_{r}^{-1/2}{{Sh}}_{r}$ | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
$\gamma =1.0$ | $\gamma =\infty $ | $\gamma =1.0$ | $\gamma =\infty $ | |||||||||
0.1 | 1.0 | 0.1 | 0.2 | 0.1 | 0.5 | 1.0 | 1.0 | 1.0 | $3.903702$ | $3.134812$ | $0.566838$ | $0.243775$ |
0.3 | $3.909849$ | $3.020867$ | $0.572327$ | $0.188560$ | ||||||||
0.5 | $3.983275$ | $3.042313$ | $0.579039$ | $0.145631$ | ||||||||
0.1 | $3.924066$ | $2.936786$ | $0.555653$ | $0.071709$ | ||||||||
1.0 | $3.983275$ | $3.042313$ | $0.579039$ | $0.145631$ | ||||||||
2.0 | $4.072229$ | $3.206671$ | $0.613841$ | $0.253723$ | ||||||||
0.0 | $4.125119$ | $3.248299$ | $0.763135$ | $0.365192$ | ||||||||
0.1 | $4.072229$ | $3.206671$ | $0.613841$ | $0.253723$ | ||||||||
0.2 | $4.020467$ | $3.165973$ | $0.461747$ | $0.136220$ | ||||||||
0.1 | $4.086230$ | $3.217587$ | $0.093059$ | $-0.178268$ | ||||||||
0.2 | $4.020467$ | $3.165973$ | $0.461747$ | $0.136220$ | ||||||||
0.3 | $3.956246$ | $3.115716$ | $0.572863$ | $0.223684$ | ||||||||
0.0 | $3.961857$ | $3.232958$ | $0.572302$ | $0.210579$ | ||||||||
0.1 | $3.956246$ | $3.115717$ | 0.572863 | 0.223684 | ||||||||
0.2 | $3.950637$ | $2.999367$ | $0.5734230$ | $0.236672$ | ||||||||
0.0 | $0.8584289$ | $0.471210$ | $0.437985$ | $0.246934$ | ||||||||
0.2 | $2.306986$ | $1.603666$ | $0.510880$ | $0.215718$ | ||||||||
0.4 | $3.449831$ | $2.565424$ | $0.556595$ | $0.229287$ | ||||||||
0.71 | $2.987213$ | $2.314769$ | $0.601913$ | $0.254733$ | ||||||||
1.0 | $3.449831$ | $2.565424$ | $0.556595$ | $0.229287$ | ||||||||
1.5 | $4.058356$ | $2.849868$ | $0.491956$ | $0.196987$ | ||||||||
0.71 | 0.71 | $2.994353$ | $2.320576$ | $0.465613$ | $0.221906$ | |||||||
1.0 | $2.987213$ | $2.314769$ | $0.601913$ | $0.254734$ | ||||||||
1.5 | $2.981865$ | $2.310136$ | $0.788335$ | $0.278144$ | ||||||||
0.1 | $2.981753$ | 2.3100000 | $1.163436$ | $0.823712$ | ||||||||
0.2 | $2.981771$ | $2.310016$ | $1.087679$ | $0.722929$ | ||||||||
0.5 | $2.981793$ | $2.310036$ | $1.008099$ | $0.613064$ |
Table 2. Variation of ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Nu}}_{r}$ and Sherwood number ${\mathrm{Re}}_{r}^{-1/2}{\mathrm{Sh}}_{r}$ when $\gamma =1.0.$ |
$K$ | $M$ | ${Nt}$ | ${Nb}$ | ${Ec}$ | ${Rd}$ | ${\Pr }$ | ${Sc}$ | ${\beta }_{c}$ | $\,\,\,{{Re}}_{r}^{-1/2}{{Nu}}_{r}$ | $\,\,\,{{Re}}_{r}^{-1/2}{{Sh}}_{r}$ | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|
${\theta }_{w}=1.0$ | ${\theta }_{w}=\infty $ | ${\theta }_{w}=1.0$ | ${\theta }_{w}=\infty $ | |||||||||
0.1 | 1.0 | 0.1 | 0.2 | 0.1 | 0.5 | 1.0 | 1.0 | 1.0 | $1.226403$ | $3.903702$ | $0.518212$ | $0.243775$ |
0.3 | $1.231105$ | $3.909849$ | $0.522688$ | $0.188560$ | ||||||||
0.5 | $1.236816$ | $3.983275$ | $0.528663$ | $0.145631$ | ||||||||
0.1 | $1.216886$ | $3.932879$ | $0.50589$ | $0.555003$ | ||||||||
1.0 | $1.236816$ | $3.983275$ | $0.528663$ | $0.145631$ | ||||||||
2.0 | $1.267015$ | $4.072229$ | $0.562444$ | $0.253723$ | ||||||||
0.0 | $1.294824$ | $4.125119$ | $0.763135$ | $0.365192$ | ||||||||
0.1 | $1.267015$ | $4.072229$ | $0.562444$ | $0.253723$ | ||||||||
0.2 | $1.240395$ | $4.020467$ | $0.36243$ | $0.136220$ | ||||||||
0.1 | $1.284422$ | $4.086230$ | $-0.161836$ | $-0.178268$ | ||||||||
0.2 | $1.240395$ | $4.020467$ | $0.36243$ | $0.136220$ | ||||||||
0.3 | $1.194590$ | $3.956246$ | $0.519961$ | 0.223684 | ||||||||
0.0 | $1.198536$ | $3.961857$ | $0.518421$ | $0.210579$ | ||||||||
0.1 | $1.194590$ | $3.956246$ | $0.519961$ | 0.223684 | ||||||||
0.2 | $1.190644$ | $3.950637$ | $0.521501$ | $0.236672$ | ||||||||
0.0 | $0.858429$ | $0.8584289$ | $0.437985$ | $0.246934$ | ||||||||
0.2 | $1.001098$ | $2.306986$ | $0.478334$ | $0.215718$ | ||||||||
0.4 | $1.130161$ | $3.449831$ | $0.508833$ | $0.229287$ | ||||||||
0.71 | $0.991195$ | $2.987213$ | $0.558264$ | $0.254733$ | ||||||||
1.0 | $1.130161$ | $3.449831$ | $0.508833$ | $0.229287$ | ||||||||
1.5 | $1.306614$ | $4.058356$ | $0.441824$ | $0.196987$ | ||||||||
0.71 | 0.71 | $1.001418$ | $2.994353$ | $0.415386$ | $0.221906$ | |||||||
1.0 | $0.991194$ | $2.987213$ | $0.558264$ | $0.254734$ | ||||||||
1.5 | $0.980300$ | $2.981865$ | $0.752819$ | $0.278144$ | ||||||||
0.1 | $0.976104$ | $2.981753$ | $1.140249$ | $0.823712$ | ||||||||
0.2 | $0.976497$ | $2.981771$ | $1.101776$ | $0.722929$ | ||||||||
0.5 | $0.977772$ | $2.981793$ | $0.980345$ | $0.613064$ |
4.3. Entropy generation analysis
Figure 5. Variation of (a) entropy generation number $Ns$ and (b) Bejan number $\mathrm{Be}$ against $K$ when $n=0.5,\,M=1={\Pr }={Sc}={Ec}={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Nt}=0.5\,$ and ${Nb}=0.3.$ |
Figure 6. Variation of (a) entropy generation number $Ns$ and (b) Bejan number $\mathrm{Be}$ against $M$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,={Ec}={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Nt}=0.5\,$ and ${Nb}=0.3.$ |
Figure 7. Variation of (a) entropy generation number $Ns$ and (b) Bejan number $\mathrm{Be}$ against $\mathrm{Ec}$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,=M={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Nt}=0.5\,$ and ${Nb}=0.3.$ |
Figure 8. Variation of (a) entropy generation number $Ns$ and (b) Bejan number $\mathrm{Be}$ against $\mathrm{Nt}$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,=M={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Ec}=0.1\,$ and ${Nb}=0.3.$ |
Figure 9. Variation of (a) entropy generation number $\mathrm{Ns}$ and (b) Bejan number $\mathrm{Be}$ against $\mathrm{Nb}$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,=M={\beta }_{c}$, ${Rd}=0.5,\,{\theta }_{w}=2.0,\,{Ec}=1.0\,$ and ${Nt}=1.0.$ |
Figure 10. Variation of (a) entropy generation number $\mathrm{Ns}$ and (b) Bejan number $\mathrm{Be}$ against $\mathrm{Rd}$ when $n=0.5,\,K=0.1,\,{\Pr }=1={Sc}\,=M={\beta }_{c}$, ${Nt}=0.1,\,{\theta }_{w}=2.0,\,{Ec}=0.1\,$ and ${Nb}=0.3.$ |
5. Main findings
• | The Brownian motion and magnetic force parameters reduce the concentration at the lubricated surface as well as at the rough surface. |
• | The magnitudes of the local Nusselt number and local Sherwood number are enhanced over a lubricated disk compared to a rough disk. |
• | The presence of the second grade parameter over the lubricated surface increases the local Nusselt number and Sherwood number, while on a traditional surface this behavior is contrary. |
• | A rise in entropy generation number is observed with increasing fraction between the layers of fluid and the Bejan number has large magnitude due to lubrication. |
• | A decrease in entropy generation number is observed for thermophoresis diffusion and Brownian motion when lubrication effects are dominant. |
• | With growth in thermal radiation phenomena, the entropy generation number and Bejan number increase. |