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Mathematical modelling and exact solution of steady fully developed mixed convection flow in a vertical micro-porous-annulus

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Abstract

This work analyzes the behaviors of fully developed mixed convection flow of an incompressible and viscous fluid in a vertical micro-porous-annulus (MPA) taking into account the velocity slip and temperature jump at the outer surface of inner porous cylinder and inner surface of outer porous cylinder. The present study explores the effects of the suction/injection, mixed convection, the Knudsen number, Prandtl number and the radius ratio on the MPA hydrodynamic and thermal behaviours. It is interesting to remark that as suction/injection on the MPA increases, the fluid velocity and temperature is enhanced. In addition, the rate of heat transfer at outer surface of inner porous cylinder decrease with increase in Knudsen number for suction at the inner porous cylinder and simultaneous injection at the outer porous cylinder while the result is just contrast at inner surface of outer porous cylinder.

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Abbreviations

\(C_p =\) :

Specific heat at constant pressure

\(D_h =\) :

Hdraulic diameter \([{2( {r_2 -r_1 })}]\)

\(F_v =\) :

Tangential momentum accommodation coefficient

\(F_t =\) :

Thermal accommodation coefficient

\(g=\) :

Gravitational acceleratioin

\(Gr=\) :

Grashof number

\(GR=\) :

Mixed convection parameter \(( {Gr/Re})\)

\(h=\) :

Convective heat transfer coefficient

\(k=\) :

Thermal conductivity

\(Kn=\) :

Knudsen number

\(Nu_0 =\) :

Rate of heat transfer at the outer surface of the inner porous cylinder

\(Nu_1 =\) :

Rate of heat transfer at the inner surface of the outer porous cylinder

\(P=\) :

Pressure difference

\(Pr=\) :

Prandtl number

\(Re=\) :

Reynolds number

\(r^*=\) :

Ratio of radius \((r_1 /r_2 )\)

\(r_1 =\) :

Radius of the inner porous cylinder

\(r_2 =\) :

Radius of the outer porous cylinder

\(S =\) :

Suction/injection parameter

\(T=\) :

Fluid temperature

\(T_0 =\) :

Reference fluid temperature

\(u=\) :

Axial velocity

\(U=\) :

Dimensionless axial velocity

\(V_0 =\) :

Constant suction/injection velocity

\(z,r=\) :

Axial and radial coordinate, respectively

\(Z,R=\) :

Dimensionless axial and radial coordinate, respectively

\(\alpha =\) :

Thermal diffusivity

\(\beta =\) :

Thermal expansion coefficient

\(\beta _t ,\beta _\nu =\) :

Dimensionless variables

\(\gamma =\) :

Ratio of specific heats (\(C_p /C_v )\)

\(\lambda =\) :

Molecular mean free path

\(\mu =\) :

Dynamic viscosity

\(\theta =\) :

Dimensionless temperature

\(\rho _0 =\) :

Density

\(\nu =\) :

Fluid kinematic viscosity

\(1=\) :

Value on inner cylinder

\(2=\) :

Value on outer cylinder

\(b=\) :

Bulk value

\(s_1 =\) :

Fluid properties on inner cylinder

\(s_2 =\) :

Fluid properties on outer cylinder

\(m=\) :

Mean value

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Authors and Affiliations

  1. Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

    Basant K. Jha & Babatunde Aina

Authors
  1. Basant K. Jha
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  2. Babatunde Aina
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Corresponding author

Correspondence to Babatunde Aina.

Appendix

Appendix

The constants used in the work are as follows:

$$\begin{aligned}&b_1 =( {{r^*} })^{S\Pr }-2\beta _T Kn( {1-{r^*} })S\Pr ( {{r^*} })^{(S\Pr -1)}, \quad b2=1+2\beta _T Kn( {1-{r^*} })S\Pr ,\\&b3=-\frac{1}{(b2-b1)}, \\&b_4 =\frac{b_1 }{( {b_2 -b_1 })} \quad b_5 =\frac{1}{2}-\frac{{r^*} ^2}{2}, \quad b_6 =\frac{1}{( {S\Pr +2})}-\frac{( {{r^*} })^{( {S\Pr +2})}}{( {S\Pr +2})}, \quad C_7 =\frac{C_0 }{( {4-2S})}, \\&C_8 =\frac{C_1 }{( {S \Pr +2})^2-S( {S \Pr +2})}, \quad F_1 =-\frac{1}{A}\frac{Gr}{\hbox {Re}}C_7 , \quad F_2 =-\frac{1}{A}\frac{Gr}{\hbox {Re}}C_8 , \quad F_3 =\frac{C_7 }{A},\\&F_4 =\frac{1}{2}-\frac{{r^*} ^2}{2}, \\&F_5 =\frac{1}{( {S+2})}-\frac{( {{r^*} })^{( {S+2})}}{( {S+2})}, \quad F_6 =\frac{F_1 }{4}-\frac{F_1 {r^*} ^4}{4}+\frac{F_2 }{( {S\Pr +4})}-\frac{F_2 ( {{r^*} })^{( {S\Pr +4})}}{( {S\Pr +4})} \\&F_7 =\frac{F_3 }{4}( {1-{r^*} ^4}), \quad F_8 =\frac{1}{2}( {1-{r^*} ^2}), \quad F_9 =F_1 +\frac{F_3 F_8 }{F_7 }-\frac{F_3 F_6 }{F_7 }, \quad F_{10} =F_9 +F_2 , \\&F_{11} =-\frac{F_3 F_4 }{F_7 }, \\&F_{12} =-\frac{F_3 F_5 }{F_7 }, \quad F_{13} ={r^*} ^S, \quad F_{14} =F_9 {r^*} ^2+F_2 {r^*} ^{(S\Pr +2)}, \quad F_{15} =-\frac{F_3 F_4 {r^*} ^2}{F_7 }, \\&\quad F_{16} =-\frac{F_3 F_5 {r^*} ^2}{F_7 }, \\&F_{17} =2F_9 +F_2 ( {S\Pr +2}), \quad F_{18} =-\frac{2F_3 F_4 }{F_7 }, \quad .F_{19} =-\frac{2F_3 F_5 }{F_7 }, \quad F_{20} =S{r^*} ^{( {S-1})}, \\&F_{21} =2F_9 {r^*} +F_2 ( {S\Pr +2}){r^*} ^{( {S\Pr +1})}, \quad F_{22} =-\frac{2F_3 F_4 {r^*} }{F_7 }, \quad F_{23} =-\frac{2F_3 F_5 {r^*} }{F_7 }, \\&F_{24} =F_{14} -2\beta _v Kn( {1-{r^*} })F_{21} , \quad F_{25} =1+F_{15} -2\beta _v Kn( {1-{r^*} })F_{22} \\&F_{26} =F_{13} +F_{16} -2\beta _v Kn( {1-{r^*} })F_{20} -2\beta _v Kn( {1-{r^*} })F_{23}, \\&F_{27} =F_{10} +2\beta _v Kn(1-r^*) F_{17,} \\&F_{28} =1+F_{11} +2\beta _v Kn( {1-{r^*} })F_{18},\\&F_{29} =1+F_{12} +2\beta _v Kn( {1-{r^*} })S+2\beta _v Kn( {1-{r^*} }) F_{19}, \\&F_{30} =F_{29} -\frac{F_{26} F_{28} }{F_{25}}, \quad {F_{31}}=\frac{F_{24}F_{28}}{F_{25}}-F_{27}, \quad F_{32} =C_0 C_2 ,F_{33} =C_0 C_3 , \\&F_{34} =C_0 F_9 -\frac{F_3 F_4 C_0 C_2 }{F_7 }-\frac{F_3 F_5 C_0 C_3 }{F_7 },\\&F_{35} =C_0 F_2 +C_1 F_9 -\frac{F_3 F_4 C_1 C_2 }{F_7 }-\frac{F_3 F_5 C_1 C_3 }{F_7 }, \\&{F_{36}}=C_1 C_2 , \; {F_{37}}=C_1 C_3 , \; {F_{38}}=C_1 {F_{2}}, \quad F_{39} =\frac{2}{( {1-/{r^*}^2})}, \quad F_{40} =\frac{F_{32} }{2}-\frac{F_{32} {r^*}^2}{2}, \\&F_{41} =\frac{F_{33} }{( {S\!+\!2})}-\frac{F_{33} {r^*}^{( {S+2})}}{( {S\!+\!2})}, \; F_{42} =\frac{F_{34} }{4}-\frac{F_{34} {r^*} ^4}{4}, \; F_{43} =\frac{F_{35} }{( {S\Pr \!+4})}-\frac{F_{35} {r^*} ^{( {S \Pr +4})}}{({S \Pr \!+4})}, \\&F_{44} =\frac{F_{36} }{( {S \Pr +2})}-\frac{F_{36} {r^*} ^{( {S \Pr +2})}}{( {S\Pr +2})}, \quad F_{45} =\frac{F_{37} }{( {S \Pr +S+2})}-\frac{F_{37} {r^*} ^{( {S \Pr +S+2})}}{( {S\Pr +S+2})},\\&F_{46} =\frac{F_{38} }{( {2S \Pr +4})}-\frac{F_{38} {r^*} ^{( {2S\Pr +4})}}{( {2S \Pr +4})}, \\&F_{47} =F_{40} +F_{41} +F_{42} +F_{43} +F_{44} +F_{45} +F_{46} . \end{aligned}$$

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Jha, B.K., Aina, B. Mathematical modelling and exact solution of steady fully developed mixed convection flow in a vertical micro-porous-annulus. Afr. Mat. 26, 1199–1213 (2015). https://doi.org/10.1007/s13370-014-0277-4

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  • DOI: https://doi.org/10.1007/s13370-014-0277-4

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