Self-Similar Solution of Radial Stagnation Point Flow and Heat Transfer of a Viscous, Compressible Fluid Impinging on a Rotating Cylinder

Abstract

In this study, the radial stagnation point flow of strain rate $\bar{k}$ impinging on a cylinder rotating at constant angular velocity ω and its heat transfer are investigated. Reduction in the Navier–Stokes equations and energy equation to primary nonlinear ordinary differential equation systems is obtained by use of appropriate transformations when the angular velocity and wall temperature or wall heat flux all are constant. The impinging free stream is steady and normal to the surface from all sides, and the range of Reynolds number variation ($Re = \bar{k}a^{2} /2\upsilon$) is 0.1–1000 in which a and υ are cylinder radius and kinematic viscosity, respectively. Flow results are presented for selected values of compressibility factor and different values of Prandtl numbers along with shear stress and Nusselt number. For all values of Reynolds numbers and surface temperature or surface heat flux, as compressibility factor increases the radial velocity field, the heat transfer coefficient and the wall shear stress increase, whereas the angular velocity field decreases. The rotating movement of the cylinder does not have any effect on the radial component of the velocity, but its increase increases the angular component of the fluid velocity field and the surface shear stress.

Appendix

Details of derivation of Eqs. (11), (12), (13) and (17) are presented as follows:

$$\begin{aligned} \eta = &\, \frac{2}{{a^{2} }}\int\limits_{a}^{r} {\frac{\rho r}{{\rho_{\infty } }}{\text{d}}r} \Rightarrow \frac{{{\text{d}}\eta }}{{{\text{d}}r}} = \frac{2\rho r}{{a^{2} \rho_{\infty } }} \to \frac{{{\text{d}}\eta }}{{{\text{d}}r}} = \frac{2r}{{a^{2} }}c(\eta ) \to 2r{\text{d}}r \\ = &\, a^{2} \frac{{{\text{d}}\eta }}{c(\eta )}\mathop{\longrightarrow}\limits{{\int {} }}\int_{a}^{r} {2r{\text{d}}r = a^{2} \int_{0}^{\eta } {\frac{{{\text{d}}\eta }}{c(\eta )}} } \to r^{2} - a^{2} \\ = &\, a^{2} \int_{0}^{\eta } {\frac{{{\text{d}}\eta }}{c(\eta )}} \to r^{2} = a^{2} \left[ {1 + \int_{0}^{\eta } {\frac{{{\text{d}}\eta }}{c(\eta )}} } \right] \\ \end{aligned}$$

With definition: $\varGamma (\eta ) = \left[ {1 + \int_{0}^{\eta } {\frac{{{\text{d}}\eta }}{c(\eta )}} } \right]$, we have: $r^{2} = a^{2} \varGamma (\eta )$.

• 1. To derive pressure:

By use of non-dimensional pressure as:

$$p = \frac{P}{{\rho_{\infty } \bar{k}^{2} a^{2} }} \Rightarrow P = \rho_{\infty } \bar{k}^{2} a^{2} p$$

Start from r-momentum:

$$u\frac{\partial (\rho u)}{\partial r} - \frac{{\rho v^{2} }}{r} + w\frac{\partial (\rho u)}{\partial z} = - \frac{\partial P}{\partial r} + \upsilon \left\{ {\frac{1}{r}\frac{\partial }{\partial r}\left[ {r\frac{\partial (\rho u)}{\partial r}} \right] - \frac{(\rho u)}{{r^{2} }} + \frac{{\partial^{2} (\rho u)}}{{\partial z^{2} }}} \right\}$$

But:

$$\begin{aligned} \rho u = &\, - \frac{{\bar{k}a^{2} }}{r}\rho_{\infty } f(\eta ) \Rightarrow \frac{\partial (\rho u)}{\partial r} = \frac{{\bar{k}a^{2} }}{{r^{2} }}\rho_{\infty } f - 2\bar{k}\rho_{\infty } cf^{\prime} \Rightarrow r\frac{\partial (\rho u)}{\partial r} \\ = &\, \frac{{\bar{k}a^{2} }}{r}\rho_{\infty } f - 2\bar{k}\rho_{\infty } cf^{\prime}r \Rightarrow \frac{\partial }{\partial \,r}\left[ {r\frac{\partial (\rho u)}{\partial r}} \right] \\ = &\, - \frac{{\bar{k}a^{2} }}{{r^{2} }}\rho_{\infty } f + \underline{{\frac{{\bar{k}a^{2} }}{r}\rho_{\infty } f^{\prime}(\eta )\frac{2r}{{a^{2} }}c}} - 2\bar{k}\rho_{\infty } (cf^{\prime})^{\prime}\frac{{2r^{2} }}{{a^{2} }}c\underline{{ - 2\bar{k}\rho_{\infty } cf^{\prime}}} \Rightarrow \frac{1}{r}\frac{\partial }{\partial \,r}\left[ {r\frac{\partial \,(\rho \,u)}{\partial \,r}} \right] \\ = & - \frac{{\bar{k}a^{2} }}{{r^{3} }}\rho_{\infty } \,f - \frac{{4\bar{k}r}}{{a^{2} }}\rho_{\infty } c(cf^{\prime})^{\prime} \\ \end{aligned}$$

Then r-momentum is:

$$\begin{aligned} & - \frac{{\bar{k}a^{2} }}{r}\frac{{\rho_{\infty } }}{\rho }f(\eta )\left[ {\frac{{\bar{k}a^{2} }}{{r^{2} }}\rho_{\infty } f(\eta ) - 2\bar{k}\rho_{\infty } cf^{\prime}} \right] - \frac{{\bar{k}^{2} a^{4} }}{{r^{3} }}\rho_{\infty } \frac{{G^{2} }}{c} \\ & \quad = - \rho_{\infty } \bar{k}^{2} a^{2} \frac{\partial \,p}{\partial \,\eta }\frac{2r}{{a^{2} }}c + \upsilon \left[ {\underline{{ - \frac{{\bar{k}a^{2} }}{{r^{3} }}\rho_{\infty } \,f}} - \frac{{4\bar{k}r}}{{a^{2} }}\rho_{\infty } c(cf^{\prime})^{\prime} + \underline{{\frac{{\bar{k}a^{2} }}{{r^{3} }}\rho_{\infty } \,f}} } \right] \\ \end{aligned}$$

After omitting underlined phrases:

$$- \frac{{\bar{k}^{2} a^{4} }}{{r^{3} }}\frac{{\rho_{\infty } }}{\rho }\rho_{\infty } f^{2} + \frac{{2\bar{k}^{2} a^{2} }}{r}\frac{{\rho_{\infty } }}{\rho }\rho_{\infty } ff^{\prime} - \frac{{\bar{k}^{2} a^{4} }}{{r^{3} }}\rho_{\infty } \frac{{G^{2} }}{c} = - 2\rho_{\infty } \bar{k}^{2} cr\frac{\partial \,p}{\partial \,\eta } - \frac{{4\bar{k}\upsilon }}{{a^{2} }}rc\rho_{\infty } (cf^{\prime})^{\prime}$$

Dividing by $2\bar{k}^{2} r\rho_{\infty }$ and using $r^{2} = a^{2} \varGamma (\eta )$,

$$\begin{aligned} - \frac{1}{2}\frac{{a^{4} }}{{r^{4} }}\frac{{f^{2} }}{c} + \frac{{a^{2} }}{{r^{2} }}\frac{{ff^{\prime}}}{c} - \frac{{a^{4} }}{{2\,r^{4} }}\frac{{G^{2} }}{c} = & - c\frac{\partial \,p}{\partial \,\eta } - \frac{2\upsilon }{{\bar{k}a^{2} }}c(cf^{\prime})^{\prime}\mathop{\longrightarrow}\limits{{r^{2} = a^{2} \varGamma (\eta )}}\frac{\partial p}{\partial \,\eta } = \frac{1}{2}\left( {\frac{f}{\varGamma c}} \right)^{2} \\ & - \frac{{ff^{\prime}}}{{\varGamma \,c^{2} }} + \frac{1}{2}\left( {\frac{G}{\varGamma c}} \right)^{2} - \frac{1}{Re}(cf^{\prime})^{\prime} \\ \end{aligned}$$

Integrating this, we have:

$$p - p_{0} = \int_{0}^{\eta } {\left[ {\frac{1}{2}\left( {\frac{f}{\varGamma \,c}} \right)^{2} - \frac{{ff^{\prime}}}{{\varGamma \,c^{2} }} + \frac{1}{2}\left( {\frac{G}{\varGamma c}} \right)^{2} - \frac{1}{Re}(cf^{\prime})^{\prime}} \right]{\text{d}}\eta + c_{1} (Z)}$$

Here, $c_{1}$ is a function of $z$ which will be calculated by use of z-momentum as

$$\,u\frac{\partial \,(\rho \,w)}{\partial \,r} + \,w\frac{\partial \,(\rho \,w)}{\partial \,z} = - \frac{\partial P}{\partial \,z}$$

But at $r \to \infty:\rho \,w = 2\,\bar{k}\,z\rho_{\infty } \Rightarrow \left\{ \begin{aligned} \frac{\partial \,(\rho \,w)}{\partial \,r} = 0 \hfill \\ \frac{\partial \,(\rho \,w)}{\partial \,z} = 2\,\bar{k}\rho_{\infty } \hfill \\ \end{aligned} \right.,\quad \frac{\partial \,P}{\partial \,z} = \rho_{\infty } \bar{k}^{2} a^{2} \frac{\partial \,p}{\partial \,z}$.

Then:

$$- \rho_{\infty } \bar{k}^{2} a^{2} \frac{{{\text{d}}c_{1} (z)}}{{{\text{d}}\,z}} = 2\bar{k}z(2\bar{k}\rho_{\infty } ) \Rightarrow \frac{{{\text{d}}c_{1} (z)}}{{{\text{d}}\,z}} = - \frac{4z}{{a^{2} }} \Rightarrow c_{1} (z) = - 2\left( {\frac{z}{a}} \right)^{2}$$

This gives pressure as:

$$p - p_{0} = \int_{0}^{\eta } {\left[ {\frac{1}{2}\left[ {\frac{f}{\varGamma \,c}} \right]^{2} - \frac{{ff^{\prime}}}{{\varGamma c^{2} }} + \frac{1}{2}\left( {\frac{G}{\varGamma c}} \right)^{2} - \frac{1}{Re}(cf^{\prime})^{\prime}} \right]{\text{d}}\eta - 2\left( {\frac{z}{a}} \right)^{2} }$$

• 2. To derive (11):

Start from z-momentum:

$$u\frac{\partial (\rho w)}{\partial r} + w\frac{\partial (\rho w)}{\partial z} = - \frac{\partial P}{\partial z} + \upsilon \left\{ {\frac{1}{r}\frac{\partial }{\partial r}\left[ {r\frac{\partial (\rho w)}{\partial r}} \right] + \frac{{\partial^{2} (\rho w)}}{{\partial z^{2} }}} \right\}$$

But:

$$\begin{aligned} \frac{\partial (\rho \,w)}{\partial \,r} = &\, \frac{\partial (\rho \,w)}{\partial \eta }\frac{\partial \eta }{\partial \,r} = (2\bar{k}c^{\prime}f^{\prime}\,z + 2\bar{k}cf^{\prime\prime}\,z)\rho_{\infty } \frac{2r}{{a^{2} }}c \Rightarrow r\frac{\partial (\rho \,w)}{\partial \,r} \\ =& \,(2\bar{k}cc^{\prime}f^{\prime}\,z + 2\bar{k}c^{2} f^{\prime\prime}\,z)\rho_{\infty } \frac{{2r^{2} }}{{a^{2} }} \Rightarrow \frac{\partial }{\partial \,r}\left[ {r\frac{\partial (\rho \,w)}{\partial \,r}} \right] \\ = & \,\left[ {2\bar{k}cc^{\prime}f^{\prime}\,z + 2\bar{k}c^{2} f^{\prime\prime}\,z} \right]\rho_{\infty } \frac{4r}{{a^{2} }} + \left[ {2\bar{k}(c^{\prime})^{2} cf^{\prime}\,z + 2\bar{k}c^{2} c^{\prime\prime}f^{\prime}\,z + 6\bar{k}c^{2} c^{\prime}f^{\prime\prime}\,z} \right. \\ & + \left. {2\bar{k}c^{3} f^{\prime\prime\prime}\,z} \right]\rho_{\infty } \frac{{4r^{3} }}{{a^{4} }} \Rightarrow \frac{1}{r}\frac{\partial }{\partial \,r}\left[ {r\frac{\partial (\rho \,w)}{\partial \,r}} \right] \\ = & \left[ {2\bar{k}cc^{\prime}f^{\prime}\,z + 2\bar{k}c^{2} f^{\prime\prime}\,z} \right]\rho_{\infty } \frac{4}{{a^{2} }} + \left[ {2\bar{k}(c^{\prime})^{2} cf^{\prime}\,z + 2\bar{k}c^{2} c^{\prime\prime}f^{\prime}\,z + 6\bar{k}c^{2} c^{\prime}f^{\prime\prime}\,z} \right. \\ & + \left. {2\bar{k}c^{3} f^{\prime\prime\prime}\,z} \right]\rho_{\infty } \frac{4}{{a^{2} }}\varGamma (\eta ) \\ \end{aligned}$$

Substitute in z-momentum:

$$\begin{aligned} & - \frac{{\bar{k}a^{2} }}{r}\frac{{\rho_{\infty } }}{\rho }f(2\bar{k}c^{\prime}f^{\prime}z + 2\bar{k}cf^{\prime\prime}\,z)\rho_{\infty } \frac{2r}{{a^{2} }}\frac{\rho }{{\rho_{\infty } }} + \frac{1}{\rho }(2\bar{k}\rho_{\infty } cf^{\prime}\,z)2\bar{k}\rho_{\infty } \frac{\rho }{{\rho_{\infty } }}f^{\prime} = \upsilon \left\{ {\left[ {2Kcc^{\prime}f\,z + 2Kc^{2} f^{\prime\prime}\,z} \right]\rho_{\infty } \frac{4}{{a^{2} }} + \left[ {2\bar{k}(c^{\prime})^{2} cf^{\prime}\,z + 2\bar{k}c^{2} c^{\prime\prime}f^{\prime}\,z + 6\bar{k}c^{2} c^{\prime}f^{\prime\prime}\,z} \right.} \right. \\ & + \left. {\left. {2\bar{k}c^{3} f^{\prime\prime\prime}\,z} \right]\rho_{\infty } \frac{4}{{a^{2} }}\varGamma (\eta )} \right\} - \rho_{\infty } \bar{k}^{2} a^{2} \left( { - \frac{4z}{{a^{2} }}} \right) \\ \end{aligned}$$

Equating the coefficient of z, the equation for $f$ is:

$$\begin{aligned} & - 4\bar{k}^{2} c^{\prime}f\,f^{\prime} - 4\bar{k}^{2} cf\,f^{\prime\prime} + 4\bar{k}^{2} c(f^{\prime})^{2} \\ & \quad = \frac{{8\bar{k}\upsilon }}{{a^{2} }}cc^{\prime}f^{\prime} + \frac{{8\bar{k}\upsilon }}{{a^{2} }}c^{2} f^{\prime\prime} + \frac{{8\bar{k}\upsilon }}{{a^{2} }}(c^{\prime})^{2} c\varGamma \,f^{\prime} \\ & \quad \quad + \frac{{8\bar{k}\upsilon }}{{a^{2} }}c^{2} c^{\prime\prime}\varGamma \,f^{\prime} + \frac{{24\bar{k}\upsilon }}{{a^{2} }}c^{2} c^{\prime}\,f^{\prime\prime} + \frac{{8\bar{k}\upsilon }}{{a^{2} }}c^{3} \varGamma \,f^{\prime\prime\prime} + 4\bar{k}^{2} \\ \end{aligned}$$

• 3. To derive (12):

Start from φ-momentum:

$$u\frac{\partial (\rho v)}{\partial r} + \frac{\rho uv}{r} + w\frac{\partial (\rho v)}{\partial z} = \upsilon \left\{ {\frac{1}{r}\frac{\partial }{\partial r}\left[ {r\frac{\partial (\rho v)}{\partial r}} \right] - \frac{(\rho v)}{{r^{2} }} + \frac{{\partial^{2} (\rho v)}}{{\partial z^{2} }}} \right\}$$

But:

$$\frac{\partial \,(\rho \,v)}{\partial \,r} = - \bar{k}\frac{{a^{2} }}{{r^{2} }}\rho_{\infty } G(\eta ) + \bar{k}\frac{{a^{2} }}{r}\rho_{\infty } G^{\prime}(\eta )\frac{2r}{{a^{2} }}C(\eta )$$

And

$$\frac{1}{r}\frac{\partial }{\partial r}\left[ {r\frac{\partial (\rho v)}{\partial r}} \right] = \frac{{\bar{k}a^{2} }}{{r^{3} }}\rho_{\infty } G(\eta ) + \frac{{4\bar{k}r}}{{a^{2} }}\rho_{\infty } cc^{\prime } G^{\prime } + \frac{{4\bar{k}r}}{{a^{2} }}\rho_{\infty } c^{2} G^{\prime \prime }$$

Then φ-momentum is:

$$\begin{aligned} & - \frac{{\bar{k}a^{2} }}{r}\frac{f}{c}\left( { - \frac{{\bar{k}a^{2} }}{{r^{2} }}\rho_{\infty } G + 2\bar{k}\rho_{\infty } cG^{\prime}} \right) + \frac{1}{r}\left( { - \frac{{\bar{k}a^{2} }}{r}\rho_{\infty } f\frac{{\bar{k}a^{2} }}{r}\frac{G}{c}} \right) \\ & \quad = \upsilon \left( {\underline{{\frac{{\bar{k}a^{2} }}{{r^{3} }}\rho_{\infty } G}} + \frac{{4\bar{k}r}}{{a^{2} }}\rho_{\infty } cc^{\prime}G^{\prime} + \frac{{4\bar{k}r}}{{a^{2} }}\rho_{\infty } c^{2} G^{\prime\prime} - \underline{{\frac{{\bar{k}a^{2} }}{{r^{3} }}\rho_{\infty } G}} } \right) \\ \end{aligned}$$

After omitting underlined phrases and dividing by $\frac{{4\bar{k}\upsilon \rho_{\infty } }}{r}$:

$$- \frac{{\bar{k}a^{2} }}{2\,\upsilon }f\,G^{\prime} = \frac{{r{}^{2}}}{{a^{2} }}cc^{\prime}G^{\prime} + \frac{{r{}^{2}}}{{a^{2} }}c^{2} G^{\prime\prime}$$

To derive energy equation, Eq. (17):

Consider a change in variable as: $\frac{{T(\eta ) - T_{\infty } }}{{T_{\text{w}} - T_{\infty } }} = \theta (\eta )$ or $\theta (\eta ) = \frac{{T(\eta ) - T_{\infty } }}{{\frac{{a\,q_{\text{w}} }}{2k}}}$. Then energy equation can be written as:

$$\rho \,u\frac{\partial \,\theta }{\partial \,r} + \rho \,w\frac{\partial \,\theta }{\partial \,z} = \frac{\mu }{\Pr }\frac{1}{r}\frac{\partial }{\partial \,r}\left( {r\frac{\partial \,\theta }{\partial \,r}} \right)$$

Using chain rule:

$$\begin{aligned} \frac{\partial \,\theta }{\partial \,r} = &\, \frac{\partial \,\theta }{\partial \,\eta }\frac{\partial \,\eta }{\partial \,r} = \frac{2r}{{a^{2} }}c\theta^{\prime},\quad \frac{\partial \,\theta }{\partial \,z} = 0 \\ r\frac{\partial \,\theta }{\partial \,r} = &\, \frac{{2r^{2} }}{{a^{2} }}c\theta^{\prime} \Rightarrow \frac{\partial }{\partial \,r}\left( {r\frac{\partial \,\theta }{\partial \,r}} \right) \\ = &\, \frac{4r}{{a^{2} }}c\theta^{\prime} + (c^{\prime}\theta^{\prime} + c\theta^{\prime\prime})\frac{{4r^{3} }}{{a^{4} }}c \Rightarrow \frac{1}{r}\left[ {\frac{\partial }{\partial \,r}\left( {r\frac{\partial \,\theta }{\partial \,r}} \right)} \right] \\ = &\, \frac{4}{{a^{2} }}c\theta^{\prime} + (c^{\prime}\theta^{\prime} + c\theta^{\prime\prime})\frac{{4r^{2} }}{{a^{4} }}c\mathop{\longrightarrow}\limits{{r^{2} = a^{2} \varGamma (\eta )}}\frac{1}{r}\left[ {\frac{\partial }{\partial \,r}\left( {r\frac{\partial \,\theta }{\partial \,r}} \right)} \right] \\ = &\, \frac{4}{{a^{2} }}c\theta^{\prime} + \frac{4\varGamma }{{a^{2} }}c(c^{\prime}\theta^{\prime} + c\theta^{\prime\prime}) \\ \end{aligned}$$

By substitution:

$$- \frac{{\bar{k}a^{2} }}{r}\rho_{\infty } f\frac{2r}{{a^{2} }}\frac{\rho }{{\rho_{\infty } }}\theta^{\prime} = \frac{\mu }{Pr}\frac{4}{{a^{2} }}\left( {c\theta^{\prime} + \varGamma \,cc^{\prime}\,\theta^{\prime} + \varGamma \,c^{2} \,\theta^{\prime\prime}} \right)$$

Divide by $2\bar{k}\rho$ and since, $1/Re = 2\nu /\bar{k}a^{2}$

$$\frac{1}{Re \cdot Pr}\left( {c^{2} \varGamma \theta^{\prime\prime} + \varGamma \,cc^{\prime}\theta^{\prime} + c\theta^{\prime}} \right) + \,f\theta^{\prime} = 0$$