# 分数阶Oldroyd-B流体在垂直圆柱外的流动和传热Heat and Mass Transfer of the Generalized Fractional Oldroyd-B Fluid past a Moving Vertical Cylinder

Abstract: This paper studies the heat and mass transfer of an incompressible viscous radiative generalized Oldroyd-B fluid past a moving vertical cylinder. The Caputo fractional derivative operator is in-troduced to describe the constitutive relationship of Oldroyd-B fluid. This partial differential system including convection terms is solved by implicit finite difference scheme of Crank-Nicoson type combined with L1-algorithm.The effects of different physical parameters on the velocity and temperature are illustrated and discussed in detail. The results indicate that these new physical parameters have dramatic influence on the velocity distribution, but little on the temperature and concentration distribution.

1. 引言

2. 物理模型

$\left(1+{\lambda }_{1}^{\alpha }\frac{{D}^{\alpha }}{D{t}^{\alpha }}\right)S=\mu \left(1+{\lambda }_{2}^{\beta }\frac{{D}^{\beta }}{D{t}^{\beta }}\right){A}_{1},\text{\hspace{0.17em}}\left(0<\beta \le \alpha <1\right),$ (1)

$\left(1+{\lambda }_{1}^{\alpha }{D}_{t}^{\alpha }\right){S}_{xr}=\mu \left(1+{\lambda }_{2}^{\beta }{D}_{t}^{\beta }\right)\frac{\partial u}{\partial r},$ (2)

${D}_{t}^{\alpha }f\left(t\right)=\frac{1}{\Gamma \left(1-\alpha \right)}{\int }_{0}^{t}{\left(t-\eta \right)}^{-\alpha }{f}^{\prime }\left(\eta \right)\text{d}\eta ,\text{\hspace{0.17em}}0<\alpha <1,$ (3)

$\frac{\partial \left(ru\right)}{\partial x}+\frac{\partial \left(rv\right)}{\partial r}=0,$ (4)

$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}+{\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha +1}u}{\partial {t}^{\alpha +1}}+{\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(v\frac{\partial u}{\partial r}\right)+{\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(u\frac{\partial u}{\partial x}\right)-g\beta \left(T-{T}_{\infty }\right)\\ -g{\beta }^{*}\left(C-{C}_{\infty }\right)-g\beta {\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(T-{T}_{\infty }\right)-g{\beta }^{*}{\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(C-{C}_{\infty }\right)\\ =\frac{\nu }{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{\nu }{r}{\lambda }_{2}^{\beta }\frac{{\partial }^{\beta }}{\partial {t}^{\beta }}\left(\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)\right),\end{array}$ (5)

$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{k}{\rho {C}_{p}}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\frac{16\delta {T}_{\infty }^{3}}{3\rho {C}_{p}{K}^{*}}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right),$ (6)

$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial r}=\frac{D}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right).$ (7)

$t\le 0:u\left(x,r,t\right)=0,v\left(x,r,t\right)=0,T\left(x,r,t\right)={T}_{\infty },C\left(x,r,t\right)={C}_{\infty };$

$t\ge 0:u\left(x,{r}_{0},t\right)={u}_{0},v\left(x,{r}_{0},t\right)=0,T\left(x,{r}_{0},t\right)={T}_{W},C\left(x,{r}_{0},t\right)={C}_{W},$

$u\left(0,r,t\right)=0,v\left(0,r,t\right)=0,T\left(0,r,t\right)={T}_{\infty },C\left(0,r,t\right)={C}_{\infty },r\ge {r}_{0},$

$u\left(x,r,t\right)\to 0,T\left(x,r,t\right)\to {T}_{\infty },C\left(x,r,t\right)\to {C}_{\infty },r\to \infty .$

${u}^{*}=\frac{u}{{u}_{0}},{r}^{*}=\frac{r}{{r}_{0}},{x}^{*}=\frac{x\upsilon }{{u}_{0}{r}_{0}^{2}},{v}^{*}=\frac{\upsilon {r}_{0}}{v},{t}^{*}=\frac{t\upsilon }{{r}_{0}^{2}},\theta =\frac{T-{T}_{\infty }}{{T}_{W}-{T}_{\infty }},{C}^{*}=\frac{C-{C}_{\infty }}{{C}_{W}-{C}_{\infty }}.$

$\frac{\partial \left(ru\right)}{\partial x}+\frac{\partial \left(rv\right)}{\partial r}=0,$ (8)

$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}+{\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha +1}u}{\partial {t}^{\alpha +1}}+{\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(v\frac{\partial u}{\partial r}\right)+{\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(u\frac{\partial u}{\partial x}\right)-g\beta \left(T-{T}_{\infty }\right)\\ -g{\beta }^{*}\left(C-{C}_{\infty }\right)-g\beta {\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(T-{T}_{\infty }\right)-g{\beta }^{*}{\lambda }_{1}^{\alpha }\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(C-{C}_{\infty }\right)\\ =\frac{\nu }{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{\nu }{r}{\lambda }_{2}^{\beta }\frac{{\partial }^{\beta }}{\partial {t}^{\beta }}\left(\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)\right),\end{array}$ (9)

$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{k}{\rho {C}_{p}}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\frac{16\delta {T}_{\infty }^{3}}{3\rho {C}_{p}{K}^{*}}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right),$ (10)

$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial r}=\frac{D}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right),$ (11)

$t\le 0:u\left(x,r,t\right)=0,v\left(x,r,t\right)=0,T\left(x,r,t\right)=0,C\left(x,r,t\right)=0;$

$t\ge 0:u\left(x,1,t\right)=1,v\left(x,1,t\right)=0,T\left(x,1,t\right)=1,C\left(x,1,t\right)=1,$

$u\left(0,r,t\right)=0,v\left(0,r,t\right)=0,T\left(0,r,t\right)=0,C\left(0,r,t\right)=0,r\ge 1,$

$u\left(x,r,t\right)\to 0,T\left(x,r,t\right)\to 0,C\left(x,r,t\right)\to 0,r\to \infty .$

3. 数值计算方法

3.1. 离散化方法

$\frac{{\partial }^{\alpha }u\left({x}_{i},{t}_{k+1}\right)}{\partial {t}^{\alpha }}=\frac{\Delta {t}^{-\alpha }}{\Gamma \left(2-\alpha \right)}\left({u}_{i,j}^{k+1}+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right){u}_{i,j}^{k-s+1}\right)+O\left(t\right),$

${b}_{s}={\left(1+s\right)}^{1-\alpha }-{s}^{1-\alpha }.$

${\frac{\partial u}{\partial t}|}_{t={t}_{k+\frac{1}{2}}}=\frac{{u}_{i.j}^{k+1}+{u}_{i,j}^{k}}{\Delta t}+O\left(\Delta t\right),$

${v\frac{\partial u}{\partial r}|}_{t={t}_{k+\frac{1}{2}}}={v}_{i,j}^{k}\frac{{u}_{i,j+1}^{k+1}-{u}_{i,j-1}^{k+1}+{u}_{i,j+1}^{k}-{u}_{i,j-1}^{k}}{4\Delta r}+O\left(\Delta r\right),$

${u\frac{\partial u}{\partial x}|}_{t={t}_{k+\frac{1}{2}}}={u}_{i,j}^{k}\frac{{u}_{i,j}^{k+1}-{u}_{i-1,j}^{k+1}+{u}_{i,j}^{k}-{u}_{i-1,j}^{k}}{2\Delta x}+O\left(\Delta x\right).$

$\frac{{\partial }^{\alpha +1}u}{\partial {t}^{\alpha +1}}\approx \frac{\Delta {t}^{-\alpha -1}}{\Gamma \left(2-\alpha \right)}\left({u}_{i,j}^{k+1}+{u}_{i,j}^{k}+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right)\left({u}_{i,j}^{k-s+1}+{u}_{i,j}^{k-s}\right)\right),$

$\begin{array}{c}\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(v\frac{\partial u}{\partial x}\right)\approx \frac{\Delta {t}^{-\alpha }}{4\Delta r\Gamma \left(2-\alpha \right)}\left({v}_{i,j}^{k}\left({u}_{i,j+1}^{k+1}-{u}_{i,j-1}^{k+1}+{u}_{i,j+1}^{k}-{u}_{i,j-1}^{k}\right)\begin{array}{c}\\ \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right){v}_{i,j}^{k-s}\left({u}_{i,j+1}^{k-s+1}-{u}_{i,j-1}^{k-s+1}+{u}_{i,j+1}^{k-s}-{u}_{i,j-1}^{k-s}\right)\right),\end{array}$

$\begin{array}{c}\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(u\frac{\partial u}{\partial x}\right)\approx \frac{\Delta {t}^{-\alpha }}{2\Delta x\Gamma \left(2-\alpha \right)}\left({u}_{i,j}^{k}\left({u}_{i,j}^{k+1}-{u}_{i-1,j}^{k+1}+{u}_{i,j}^{k}-{u}_{i-1,j}^{k}\right)\begin{array}{c}\\ \end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right){u}_{i,j}^{k-s}\left({u}_{i,j}^{k-s+1}-{u}_{i-1,j}^{k-s+1}+{u}_{i,j}^{k-s}-{u}_{i-1,j}^{k-s}\right)\right).\end{array}$

$\begin{array}{l}\left(-\frac{-{v}_{i,j}^{k}}{4\Delta r}-\frac{1+\frac{4}{3N}}{2Pr\cdot \Delta {r}^{2}}+\frac{1+\frac{4}{3N}}{4Pr\cdot \left(1+\left(j-1\right)\Delta r\right)\Delta r}\right){\theta }_{i,j-1}^{k+1}+\left(\frac{1}{\Delta t}+\frac{{u}_{i,j}^{k}}{2\Delta x}+\frac{1+\frac{4}{3N}}{Pr\cdot \Delta {r}^{2}}\right){\theta }_{i,j}^{k+1}\\ +\left(\frac{{v}_{i,j}^{k}}{4\Delta r}-\frac{1+\frac{4}{3N}}{2Pr\cdot \Delta {r}^{2}}-\frac{1+\frac{4}{3N}}{4Pr\cdot \left(1+\left(j-1\right)\Delta r\right)\Delta r}\right){\theta }_{i,j+1}^{k+1}\\ =\frac{\left(1+\frac{4}{3N}\right)\left({\theta }_{i,j-1}^{k+1}-2{\theta }_{i,j}^{k+1}+{\theta }_{i,j+1}^{k+1}\right)}{2Pr\cdot \Delta {r}^{2}}+\frac{\left(1+\frac{4}{3N}\right)\left({\theta }_{i,j+1}^{k}-{\theta }_{i,j-1}^{k}\right)}{4Pr\cdot \left(1+\left(j-1\right)\Delta r\right)\Delta r}+\frac{{\theta }_{i,j}^{k}}{\Delta t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{u}_{i,j}^{k}\left({\theta }_{i,j}^{k}-{\theta }_{i-1,j}^{k}\right)}{2\Delta x}-\frac{{v}_{i,j}^{k}\left({\theta }_{i,j+1}^{k}-{\theta }_{i-1,j}^{k}\right)}{}+\frac{{u}_{i,j}^{k}{\theta }_{i-1,j}^{k+1}}{},\end{array}$ (12)

$\begin{array}{l}\left(\frac{1}{\Delta t}+\frac{{u}_{i,j}^{k}}{2\Delta x}+\frac{1}{Sc\cdot \Delta {r}^{2}}\right){C}_{i,j}^{k+1}+\left(\frac{{v}_{i,j}^{k}}{4\Delta r}-\frac{1}{2Sc\cdot \Delta {r}^{2}}-\frac{1}{4Sc\cdot \left(1+\left(j-1\right)\Delta r\right)\Delta r}\right){C}_{i,j+1}^{k+1}\\ +\left(-\frac{-{v}_{i,j}^{k}}{4\Delta r}-\frac{1}{2Sc\cdot \Delta {r}^{2}}+\frac{1}{4Sc\cdot \left(1+\left(j-1\right)\Delta r\right)\Delta r}\right){C}_{i,j-1}^{k+1}\\ =\frac{\left(1+\frac{4}{3N}\right)\left({C}_{i,j-1}^{k}-2{C}_{i,j}^{k}+{C}_{i,j+1}^{k}\right)}{2Sc\cdot \Delta {r}^{2}}+\frac{{C}_{i,j+1}^{k}-{C}_{i,j-1}^{k}}{4Sc\cdot \left(1+\left(j-1\right)\Delta r\right)\Delta r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{C}_{i,j}^{k}}{\Delta t}-\frac{{u}_{i,j}^{k}\left({C}_{i,j}^{k}-{C}_{i-1,j}^{k}\right)}{2\Delta x}-\frac{{v}_{i,j}^{k}\left({C}_{i,j+1}^{k}-{C}_{i-1,j}^{k}\right)}{4\Delta r}+\frac{{u}_{i,j}^{k}{C}_{i-1,j}^{k+1}}{2\Delta x},\end{array}$ (13)

$\begin{array}{l}\left(\frac{{v}_{i,j}^{k}}{4\Delta r}+\frac{{r}_{1}{v}_{i,j}^{k}}{4\Delta r}-\frac{1}{4\left(1+\left(j-1\right)\Delta r\right)\Delta r}-\frac{1}{2\Delta {r}^{2}}-\frac{{r}_{2}}{4\left(1+\left(j-1\right)\Delta r\right)\Delta r}-\frac{{r}_{2}}{2\Delta {r}^{2}}\right){u}_{i,j+1}^{k+1}\\ +\left(\frac{1}{\Delta t}+\frac{{u}_{i,j}^{k}}{2\Delta x}+\frac{{r}_{1}}{\Delta t}+\frac{{r}_{1}{u}_{i,j}^{k}}{2\Delta x}+\frac{1}{\Delta {r}^{2}}+\frac{{r}_{2}}{\Delta {r}^{2}}\right){u}_{i,j}^{k+1}\\ +\left(-\frac{{v}_{i,j}^{k}}{4\Delta r}-\frac{{r}_{1}{v}_{i,j}^{k}}{4\Delta r}+\frac{1}{4\left(1+\left(j-1\right)\Delta r\right)\Delta r}-\frac{1}{2\Delta {r}^{2}}+\frac{{r}_{2}}{4\left(1+\left(j-1\right)\Delta r\right)\Delta r}-\frac{{r}_{2}}{2\Delta {r}^{2}}\right){u}_{i,j-1}^{k+1}\\ =\frac{{u}_{i,j+1}^{k}-{u}_{i,j-1}^{k}}{4\left(1+\left(j-1\right)\Delta r\right)\Delta r}+\frac{{u}_{i,j-1}^{k}-2{u}_{i,j}^{k}+{u}_{i,j+1}^{k}}{2\Delta {r}^{2}}+\frac{{r}_{2}{A}_{6}}{4\left(1+\left(j-1\right)\Delta r\right)\Delta r}+\frac{{r}_{2}{A}_{7}}{2\Delta {r}^{2}}+\frac{{u}_{i,j}^{k}}{\Delta t}-\frac{{v}_{i,j}^{k}\left({u}_{i,j+1}^{k}-{u}_{i,j-1}^{k}\right)}{4\Delta r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{u}_{i,j}^{k}\left({u}_{i-1,j}^{k+1}-{u}_{i,j}^{k}+{u}_{i-1,j}^{k}\right)}{2\Delta x}-\frac{{r}_{1}{A}_{1}}{\Delta t}-\frac{{r}_{1}{A}_{2}}{4\Delta r}-\frac{{r}_{1}{A}_{3}}{2\Delta x}+Gr\frac{{\theta }_{i,j}^{k+1}+{\theta }_{i,j}^{k}}{2}+Gc\frac{{C}_{i,j}^{k+1}+{C}_{i,j}^{k}}{2}+Gr\frac{{r}_{1}{A}_{4}}{2}+Gc\frac{{r}_{1}{A}_{5}}{2},\end{array}$ (14)

$\begin{array}{l}{v}_{i,j}^{k+1}=\frac{{v}_{i,j-1}^{k+1}-{v}_{i,j}^{k}+{v}_{i,j-1}^{k}}{2\Delta r}\frac{1}{\left(\frac{1}{2\Delta r}+\frac{1}{1+\left(j-1\right)\Delta r}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{u}_{i,j-1}^{k+1}-{u}_{i-1,j-1}^{k+1}+{u}_{i,j}^{k+1}-{u}_{i-1,j}^{k+1}+{u}_{i,j-1}^{k}-{u}_{i-1,j-1}^{k}+{u}_{i,j}^{k}-{u}_{i-1,j}^{k}}{4\Delta x}\frac{1}{\left(\frac{1}{2\Delta r}+\frac{1}{1+\left(j-1\right)\Delta r}\right)},\end{array}$ (15)

${r}_{1}=\frac{{\lambda }_{1}^{\alpha }\Delta {t}^{-\alpha }}{\Gamma \left(2-\alpha \right)},$

${r}_{2}=\frac{{\lambda }_{2}^{\beta }\Delta {t}^{-\beta }}{\Gamma \left(2-\beta \right)},$

${A}_{1}=-{u}_{i,j}^{k}+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right)\left({u}_{i,j}^{k-s+1}-{u}_{i,j}^{k-s}\right),$

${A}_{2}={v}_{i,j}^{k}\left({u}_{i,j+1}^{k}-{u}_{i,j-1}^{k}\right)+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right){v}_{i,j}^{k-s}\left({u}_{i,j+1}^{k-s+1}-{u}_{i,j-1}^{k-s+1}+{u}_{i,j}^{k-s}-{u}_{i,j-1}^{k-s}\right),$

${A}_{3}={u}_{i,j}^{k}\left(-{u}_{i-1,j}^{k+1}+{u}_{i,j}^{k}-{u}_{i-1,j}^{k}+{u}_{i,j}^{k+1}\right)+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right){u}_{i,j}^{k-s}\left({u}_{i,j+1}^{k-s+1}-{u}_{i,j-1}^{k-s+1}+{u}_{i,j}^{k-s}-{u}_{i,j-1}^{k-s}\right),$

${A}_{4}={\theta }_{i,j}^{k+1}+{\theta }_{i,j}^{k}+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right)\left({\theta }_{i,j}^{k-s+1}-{\theta }_{i,j}^{k-s}\right),$

${A}_{5}={C}_{i,j}^{k+1}+{C}_{i,j}^{k}+\underset{s=1}{\overset{k}{\sum }}\left({b}_{s}-{b}_{s-1}\right)\left({C}_{i,j}^{k-s+1}-{C}_{i,j}^{k-s}\right),$

${A}_{6}={u}_{i,j+1}^{k}-{u}_{i,j-1}^{k}+\underset{s=1}{\overset{k}{\sum }}\left({\beta }_{s}-{\beta }_{s-1}\right)\left({u}_{i,j+1}^{k-s+1}-{u}_{i,j-1}^{k-s+1}+{u}_{i,j+1}^{k-s}-{u}_{i,j-1}^{k-s}\right),$

${A}_{7}={u}_{i,j+1}^{k}-2{u}_{i,j}^{k}+{u}_{i,j-1}^{k}+\underset{s=1}{\overset{k}{\sum }}\left({\beta }_{s}-{\beta }_{s-1}\right)\left({u}_{i,j+1}^{k-s+1}-2{u}_{i,j}^{k-s+1}+{u}_{i,j-1}^{k-s+1}+{u}_{i,j+1}^{k-s}-2{u}_{i,j}^{k-s}+{u}_{i,j-1}^{k-s}\right).$

3.2. 求解过程

Figure 1. Velocity distribution with increasing time

Figure 2. Velocity distribution for different mesh sizes

4. 分析讨论

4.1. 分数阶导数参数a的影响

Figure 3. Velocity distribution for different α

Figure 4. Concentration distribution for different α

4.2. 分数阶导数参数b的影响

Figure 5. Temperature distribution for different α

Figure 6. Velocity distribution for different β

4.3. 松弛时间参数l1和迟滞时间参数l2的影响

Figure 7. Concentration distribution for different β

Figure 8. Temperature distribution for different β

Figure 9. Velocity distribution for different λ1

Figure 10. Concentration distribution for different λ1

5. 结论

Figure 11. Temperature distribution for different λ1

Figure 12. Temperature distribution for different λ2

Figure 13. Concentration distribution for different λ2

Figure 14. Temperature distribution for different λ2

The Fundamental Research Funds for the Central Universities (No.FRF-BR-17-004B)。

[1] Deka, R.K. and Paul, A. (2012) Transient Free Convection Flow past an Infinite Moving Vertical Cylinder in a Stably Stratified Fluid. Journal of Heat Transfer, 134, Article ID: 042503.
https://doi.org/10.1115/1.4005205

[2] Mukhopadhyay, S. (2012) Mixed Convection Boundary Layer Flow along a Stretching Cylinder in Porous Medium. Journal of Petroleum Science and Engineering, 96, 73-78.
https://doi.org/10.1016/j.petrol.2012.08.006

[3] Takhar, H.S., Chamkha, A.J. and Nath, G. (2000) Combined Heat and Mass Transfer along a Vertical Moving Cylinder with a Free Stream. Heat and Mass Transfer, 36, 237-246.
https://doi.org/10.1007/s002310050391

[4] Datta, P., Anilkumar, D., Roy, S. and Mahanti, N.C. (2006) Effect of Non-Uniform Slot Injection (Suction) on a Forced Flow over a Slender Cylinder. International Journal of Heat and Mass Transfer, 49, 2366-2371.
https://doi.org/10.1016/j.ijheatmasstransfer.2005.10.044

[5] Kumari, M. and Nath, G. (2004) Mixed Convection Boundary Layer Flow over a Thin Vertical Cylinder with Localized Injection/Suction and Cooling/Heating. International Journal of Heat and Mass Transfer, 47, 969-976.
https://doi.org/10.1016/j.ijheatmasstransfer.2003.08.014

[6] Ishak, A., Nazar, R. and Pop, I. (2008) Uniform Suction/Blowing Effect on Flow and Heat Transfer due to a Stretching Cylinder. Applied Mathematical Modelling, 32, 2059-2066.
https://doi.org/10.1016/j.apm.2007.06.036

[7] Ganesan, P. and Rani, H.P. (2000) On Diffusion of Chemically Reactive Species in Convective Flow along a Vertical Cylinder. Chemical Engineering and Processing: Process Intensification, 39, 93-105.
https://doi.org/10.1016/S0255-2701(99)00018-5

[8] Ganesan, P. and Loganathan, P. (2001) Unsteady Natural Convective Flow past a Moving Vertical Cylinder with Heat and Mass Transfer. Heat and Mass Transfer, 37, 59-65.
https://doi.org/10.1007/s002310000128

[9] Machireddy, G.R. (2013) Chemically Reactive Species and Radiation Effects on MHD Convective Flow past a Moving Vertical Cylinder. Ain Shams Engineering Journal, 4, 879-888.
https://doi.org/10.1016/j.asej.2013.04.003

[10] Rani, H.P. (2003) Transient Natural Convection along a Vertical Clyinder with Variable Surface Temperature and Mass Diffusion. Heat and Mass Transfer, 40, 67-73.
https://doi.org/10.1007/s00231-002-0372-1

[11] Ganesan, P. and Rani, H.P. (1998) Transient Natural Convection along Vertical Cylinder with Heat and Mass Transfer. Heat and Mass Transfer, 33, 449-455.
https://doi.org/10.1007/s002310050214

[12] Ganesan, P.G. and Rani, P.H. (2000) Unsteady Free Convection MHD Flow Past a Vertical Cylinder with Heat and Mass Transfer. International Journal of Thermal Sciences, 39, 265-272.
https://doi.org/10.1016/S1290-0729(00)00244-1

[13] Ganesan, P. and Loganathan, P. (2003) Magnetic Field Effect on a Moving Vertical Cylinder with Constant Heat Flux. Heat and Mass Transfer, 39, 381-386.
https://doi.org/10.1007/s00231-002-0383-y

[14] Ganesan, P. and Loganathan, P. (2002) Heat and Mass Flux Effects on a Mov-ing Vertical Cylinder with Chemically Reactive Species Diffusion. Journal of Engineering Physics and Thermophysics, 75, 899-909.
https://doi.org/10.1023/A:1020367102891

[15] Ganesan, P. and Rani, H.P. (2000) Transient Natural Convection Flow over Vertical Cylinder with Variable Surface Temperatures. Forschung im Ingenieurwesen, 66, 11-16.
https://doi.org/10.1007/s100100000033

[16] Khan, M., Ali, S.H. and Qi, H. (2009) Exact Solutions of Starting Flows for a Fractional Burgers’ Fluid between Coaxial Cylinders. Nonlinear Analysis: Real World Applications, 10, 1775-1783.
https://doi.org/10.1016/j.nonrwa.2008.02.015

[17] Qi, H. and Jin, H. (2006) Unsteady Rotating Flows of a Viscoelastic Fluid with the Fractional Maxwell Model between Coaxial Cylinders. Acta Mechanica Sinica, 22, 301-305.
https://doi.org/10.1007/s10409-006-0013-x

[18] Qi, H. and Jin, H. (2009) Unsteady Helical Flows of a Generalized Oldroyd-B Fluid with Fractional Derivative. Nonlinear Analysis: Real World Applications, 10, 2700-2708.
https://doi.org/10.1016/j.nonrwa.2008.07.008

[19] Povstenko, Y.Z. (2008) Fractional Radial Diffusion in a Cylinder. Journal of Molecular Liquids, 137, 46-50.
https://doi.org/10.1016/j.molliq.2007.03.006

[20] Mahmood, A., Parveen, S., Ara, A. and Khan, N.A. (2009) Exact Analytic Solutions for the Unsteady Flow of a Non-Newtonian Fluid between Two Cylinders with Fractional Derivative Model. Communica-tions in Nonlinear Science and Numerical Simulation, 14, 3309-3319.
https://doi.org/10.1016/j.cnsns.2009.01.017

[21] Mahmood, A. and Rubab, Q. (2008) Exact Solutions for a Rotational Flow of Generalized Second Grade Fluids through a Circular Cylinder.

[22] Fetecau, C., Mahmood, A., Fetecau, C. and Vieru, D. (2008) Some Exact Solutions for the Helical Flow of a Generalized Oldroyd-B Fluid in a Circular Cylinder. Computers and Mathematics with Applications, 56, 3096-3108.
https://doi.org/10.1016/j.camwa.2008.07.003

[23] Tong, D., Zhang, X. and Zhang, X. (2009) Unsteady Helical Flows of a Ge-neralized Oldroyd-B Fluid. Journal of Non-Newtonian Fluid Mechanics, 156, 75-83.
https://doi.org/10.1016/j.jnnfm.2008.07.004

[24] Zhao, J., Zheng, L., Zhang, X. and Liu, F. (2016) Convection Heat and Mass Transfer of Fractional MHD Maxwell Fluid in a Porous Medium with Soret and Dufour Effects. International Journal of Heat and Mass Transfer, 103, 203-210.
https://doi.org/10.1016/j.ijheatmasstransfer.2016.07.057

[25] Zhao, J., Zheng, L., Zhang, X., Liu, F. and Chen, X. (2017) Unsteady Natural Convection Heat Transfer Past a Vertical Flat Plate Embedded in a Porous Medium Saturated with Fractional Oldroyd-B Fluid. Journal of Heat Transfer, 139, Article ID: 012501.
https://doi.org/10.1115/1.4034546

[26] Zhao, J., Zheng, L., Zhang, X. and Liu, F. (2016) Unsteady Natural Convection Boundary Layer Heat Transfer of Fractional Maxwell Viscoelastic Fluid over a Vertical Plate. International Journal of Heat and Mass Transfer, 97, 760-766.
https://doi.org/10.1016/j.ijheatmasstransfer.2016.02.059

[27] Cao, Z., Zhao, J., Wang, Z., Liu, F. and Zheng, L. (2016) MHD Flow and Heat Transfer of Fractional Maxwell Viscoelastic Nanofluid over a Moving Plate. Journal of Molecular Liquids, 222, 1121-1127.
https://doi.org/10.1016/j.molliq.2016.08.012

[28] Khan, M., Hayat, T. and Asghar, S. (2006) Exact Solution for MHD Flow of a Generalized Oldroyd-B Fluid with Modified Darcy’s Law. International Journal of Engineering Science, 44, 333-339.
https://doi.org/10.1016/j.ijengsci.2005.12.004

[29] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego, 78-85.

[30] Liu, F., Zhuang, P., Anh, V., Turner, I. and Burrage, K. (2007) Stability and Convergence of the Difference Me-thods for the Space—Time Fractional Advection—Diffusion Equation. Applied Mathematics and Computation, 191, 12-20.
https://doi.org/10.1016/j.amc.2006.08.162

[31] Carnahan, B., Luther, H.A. and Wilkes, J.O. (1969) Applied Numerical Methods. John Wiley and Sons, Inc., Hoboken.

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