Convectively driven flow past an infinite moving vertical cylinder with thermal and mass stratification

Abstract

An analysis is performed to study the unsteady, incompressible, one-dimensional, free convective flow over an infinite moving vertical cylinder under combined buoyancy effects of heat and mass transfer with thermal and mass stratifications. Laplace transform technique is adopted for finding solutions for velocity, temperature and concentration with unit Prandtl and Schmidt numbers. Solutions of unsteady state for larger times are compared with the solutions of steady state. Velocity, temperature and concentration profiles are analysed for various sets of physical parameters. Skin friction, Nusselt number and Sherwood number are shown graphically. It has been found that the thermal as well as mass stratification affects the flow appreciably.

Appendix

We have the Laplace transformation of U as

$$ \begin{array}{rll} \bar{U} &=&\frac{1}{2}\left\{ {\frac{K_0 \!\left( {R\sqrt {p+iM} } \right)}{pK_0\! \left( {\sqrt {p+iM} } \right)}+\frac{K_0\! \left( {R\sqrt {p-iM} } \right)}{pK_0\! \left( {\sqrt {p-iM} } \right)}} \right\}\\[8pt] &&-\,\frac{{\rm Gr}+{\rm Gc}}{2iM}\left\{ {\frac{K_0\! \left( {R\sqrt {p+iM} } \right)}{pK_0 \!\left( {\sqrt {p+iM} } \right)}-\frac{K_0\! \left( {R\sqrt {p-iM} } \right)}{pK_0\! \left( {\sqrt {p-iM} } \right)}} \right\} \end{array} $$

which then can be expressed in terms of inverse Laplace transform as

$$ U=\frac{1}{2}\left( {1+\frac{{\rm Gr}+{\rm Gc}}{iM}} \right){\rm e}^{iM}I_1 +\frac{1}{2}\left( {1-\frac{{\rm Gr}+{\rm Gc}}{iM}} \right){\rm e}^{-iM}I_2, $$

where

$$ I_1 =L^{-1}\left\{ {\frac{K_0\! \left( {R\sqrt p } \right)}{\left( {p+iM} \right)\!K_0\! \left( {\sqrt p } \right)}} \right\} $$

and

$$ I_2 =L^{-1}\left\{ {\frac{K_0\! \left( {R\sqrt p } \right)}{\left( {p-iM} \right)\!K_0 \!\left( {\sqrt p } \right)}} \right\}. $$

Now, we apply complex inversion formula for determining I ₁ and I ₂ as follows:

$$ I_1 =\frac{1}{2\pi i}\int_{\gamma ^{\prime} -i\infty }^{\gamma ^{\prime} +i\infty } {{\rm e}^{pt}\frac{K_0\! \left( {R\sqrt p } \right)}{\left( {p+iM} \right)\!K_0\!\left( {\sqrt p } \right)}{\rm d}p} =\sum {{\rm sum}\;{\rm of}\;{\rm residues}} , $$

where the integrand has a branch point at p = 0 and a simple pole at p = − iM.

Now $K_0( {\sqrt p })$ does not have zero at any point in the real and imaginary axes, if the branch cut is made along the negative real axis. To obtain I ₁, we use the adjoining Bromwich contour (figure 20). Therefore, the line integral in I ₁ may be replaced by the limit of the sum of the integrals over FE, ED, DC, CB and BA as S ₁ → ∞ and S ₀ →0.

Figure 20

Bromwich contour of integration.

Here, the particular form of the contour integral has been chosen because the values along the paths DC, BA and FE approach zero as S ₁ → ∞ and S ₀ →0.

Following Carslaw and Jaeger [20, 22], along the paths CB and ED we choose, p = V ²e^iπ and p = V ²e^− iπ, respectively.

Therefore, on the path CB,

$$ I_1 \left( {{\rm along}\;{\rm CB}} \right)=\frac{1}{\pi i}\int_0^\infty {{\rm e}^{-V^2t}\frac{J_0\!\left( {RV} \right)-iY_0\! \left( {RV} \right)}{\left( {V^2-iM} \right)\left\{ {J_0\! \left( V \right)-iY_0\! \left( V \right)} \right\}}V{\rm d}V} $$

and on the path ED,

$$ I_1 \left( {{\rm along}\;{\rm ED}} \right)=\frac{1}{\pi i}\int_0^\infty {{\rm e}^{-V^2t}\frac{J_0\! \left( {RV} \right)+iY_0 \!\left( {RV} \right)}{\left( {V^2-iM} \right)\left\{ {J_0 \!\left( V \right)+iY_0\! \left( V \right)} \right\}}V{\rm d}V} . $$

Adding the above two integrals, we get

$$ I_1 \left( {{\rm along}\;{\rm CB}+{\rm ED}} \right)=\frac{2}{\pi }\int_0^\infty {\left\{ {\frac{{\rm e}^{-V^2t}}{V^2-iM}\Gamma\! \left( {R,V} \right)\!V} \right\}{\rm d}V} , $$

where

$$ \Gamma\! \left( {R,V} \right)=\frac{J_0\! \left( {RV} \right)\!Y_0\! \left( V \right)-Y_0\! \left( {RV} \right)\!J_0\! \left( V \right)}{J_0^2\! \left( V \right)+Y_0^2\! \left( V \right)}. $$

Also, the residue of the integrand I ₁ at the pole $p=-iM\;{\rm is}={\rm e}^{-iM}[ K_0 ( {R\sqrt {-iM} } ) / {K_0 ( {\sqrt {-iM} } )} ]$.

Thus, from the theory of residues we have,

$$ I_1 =\frac{2}{\pi }\int_0^\infty {\left\{ {\frac{{\rm e}^{-V^2t}}{V^2-iM}\Gamma\! \left( {R,V} \right)\!V} \right\}{\rm d}V} +{\rm e}^{-iM}\frac{K_0 \!\left( {R\sqrt {-iM} } \right)}{K_0 \!\left( {\sqrt {-iM} } \right)}. $$

Similarly,

$$ I_2 =\frac{2}{\pi }\int_0^\infty {\left\{ {\frac{{\rm e}^{-V^2t}}{V^2+iM}\Gamma \!\left( {R,V}\! \right)\!V} \right\}{\rm d}V} +{\rm e}^{iM}\frac{K_0 \big( {R\sqrt {iM} } \big)}{K_0 \big( {\sqrt {iM} } \big)}. $$

Finally, we have,

$$ \begin{array}{rll} U&=&\frac{1}{2}\left\{ {\frac{K_0 \!\left( {R\sqrt {-iM} } \right)}{K_0 \!\left( {\sqrt {-iM} } \right)}+\frac{K_0 \big( {R\sqrt {iM} } \big)}{K_0 \big( {\sqrt {iM} } \big)}} \right\}\\ &&+\,\frac{{\rm Gr}+{\rm Gc}}{2Mi}\left\{ {\frac{K_0\! \left( {R\sqrt {-iM} } \right)}{K_0 \!\left( {\sqrt {-iM} } \right)}-\frac{K_0 \big( {R\sqrt {iM} } \big)}{K_0 \big( {\sqrt {iM} } \big)}} \right\}\\ &&+\,\frac{2\left( {{\rm Gr}+{\rm Gc}} \right)}{\pi M}\int_0^\infty {\frac{{\rm e}^{-V^2t}\!\left\{ {V^2\sin \!\left( {Mt} \right)+M\cos \!\left( {Mt} \right)} \right\}}{V^4+M^2}\Gamma \!\left( {R,V} \right)\!V{\rm d}V} \\ &&+\,\frac{2}{\pi }\int_0^\infty {\frac{{\rm e}^{-V^2t}\left\{ {V^2\cos\! \left( {Mt} \right)-M\sin\! \left( {Mt} \right)} \right\}}{V^4+M^2}\Gamma\! \left( {R,V} \right)\!V{\rm d}V} . \end{array} $$