Natural Convection of a Micropolar Fluid Between Two Vertical Walls with Newtonian Heating/Cooling and Heat Source/Sink

Abstract

The aim of this paper is to investigate the natural convection of a micropolar fluid flow in two vertical walls with the Newtonian heating/cooling on one of its walls. The governing linear differential equations with their appropriate boundary conditions of the considered model are changed first into non-dimensional differential equations and boundary conditions by using the dimensionless parameters and variables. Analytic solutions of the non-dimensional differential equations have been obtained one by one for several cases of source or sink parameter. To obtain the influence of the Biot number and other physical parameters, the numerical results of the velocity, temperature, and microrotational velocity are finally shown in the graphs and table. It is found that the effect of the Newtonian heating is to increase the velocity, microrotational velocity, and rate of volume flow, while in the case of the Newtonian cooling, velocity, microrotational velocity, and rate of volume flow have decreasing tendency.

Appendix

$$\begin{aligned} & k_{1} = \cos \sqrt s ,\;\;k_{2} = \sin \sqrt s ,\;\;k_{3} = \frac{\sqrt s }{{(k_{2} Bi + \sqrt s k_{1} )}}, \hfill \\ & k_{4} = \frac{Bi}{{\left( {k_{2} Bi + \sqrt s k_{1} } \right)}},\;\;k_{5} = \frac{2BR}{1 + R},\;\;k_{6} = \frac{1}{2 + R},\;\;k_{7} = k_{5} k_{6} , \hfill \\ & k_{8} = - \frac{{\sqrt s k_{3} k_{7} }}{{s\left( {s + k_{5} } \right)}},\;\;k_{9} = \frac{{\sqrt s k_{4} k_{7} }}{{s\left( {s + k_{5} } \right)}},\;\;k_{10} = \frac{2 + R}{2BR}, \hfill \\ & k_{11} = k_{10} k_{5} - 2,\;\;k_{12} = 2k_{8} + sk_{10} k_{8} ,\;\;k_{13} = 2k_{9} + sk_{10} k_{9} , \hfill \\ & k_{14} = \frac{{k_{11} }}{{\sqrt {k_{5} } }},\;\;k_{15} = - \frac{{k_{12} }}{\sqrt s },\;\;k_{16} = \frac{{k_{13} }}{\sqrt s },\;\;k_{17} = \sinh \left( {\sqrt {k_{5} } } \right), \hfill \\ & k_{18} = \cosh \left( {\sqrt {k_{5} } } \right),\;\;k_{19} = k_{14} k_{17} ,\;\;k_{20} = k_{14} k_{18} ,\;\;k_{21} = k_{15} k_{1} + k_{16} k_{2} , \hfill \\ & k_{22} = - \left( {k_{8} k_{2} + k_{9} k_{1} } \right),\;\;k_{23} = - \frac{{k_{22} + k_{9} }}{{k_{18} - 1}},\;\;k_{24} = - \frac{{k_{17} }}{{k_{18} - 1}}, \hfill \\ & k_{25} = k_{23} \left( {k_{19} + 2} \right) + k_{21} - 2k_{9} ),\;\;k_{26} = k_{24} \left( {k_{19} + 2} \right) + k_{20} , \hfill \\ & k_{27} = \frac{{k_{15} - k_{25} }}{{k_{26} - k_{14} }},\;\;k_{28} = k_{23} + k_{27} k_{24} ,\;\;k_{29} = k_{9} - k_{28} , \hfill \\ & k_{30} = - k_{15} - k_{14} k_{27} ,\;\;k_{31} = \frac{{k_{14} k_{28} }}{{\sqrt {k_{5} } }},\;\;k_{32} = \frac{{k_{14} k_{27} }}{{\sqrt {k_{5} } }},\;\;k_{33} = \frac{{ - k_{16} }}{\sqrt s }, \hfill \\ & k_{34} = \frac{{k_{15} }}{\sqrt s },\;\;k_{35} = - k_{29} + k_{30} - k_{31} + k_{33} , \hfill \\ \end{aligned}$$

$$\begin{aligned} & A_{1} = \frac{Bi}{Bi + 1},\;\;A_{2} = \frac{1}{Bi + 1},\;\;A_{3} = \frac{2BR}{1 + R}, \\ & A_{4} = \frac{1}{2 + R},\;\;A_{5} = \frac{{A_{4} A_{1} }}{2},\;\;A_{6} = A_{2} A_{4} , \\ & A_{7} = \frac{2 + R}{2BR},\;\;A_{8} = A_{7} A_{3} - 2,\;\;A_{9} = 2A_{5} A_{7} , \\ & A_{10} = \frac{{A_{8} }}{{\sqrt {A_{3} } }},\;\;A_{11} = \frac{2}{{3A_{5} }},\;\;A_{12} = \sinh \sqrt {A_{3} } , \\ & A_{13} = \cosh \sqrt {A_{3} } ,\;\;A_{14} = A_{9} - A_{6} - A_{11} ,\;\;A_{15} = A_{10} A_{12} , \\ & A_{16} = A_{10} A_{13} ,\;\;A_{17} = A_{5} + A_{6} ,\;\;A_{18} = - \frac{{A_{17} }}{{A_{13} - 1}}, \\ & A_{19} = - \frac{{A_{12} }}{{A_{13} - 1}},\;\;A_{20} = A_{16} - A_{10} + \left( {A_{15} + 2} \right)A_{19} , \\ & A_{21} = A_{14} + (A_{15} + 2)A_{18} ,\;\;A_{22} = - \frac{{A_{21} }}{{A_{20} }}, \\ & A_{23} = A_{18} + A_{19} A_{22} ,\;\;A_{24} = - A_{23} ,\;\;A_{25} = - A_{10} A_{22} , \\ & A_{26} = \frac{{A_{23} }}{{\sqrt {A_{3} } }},\;\;A_{27} = \frac{{A_{22} }}{{\sqrt {A_{3} } }},\;\;A_{28} = A_{24} + \frac{{A_{5} }}{3} + \frac{{A_{6} }}{2} - \frac{{A_{22} }}{{\sqrt {A_{3} } }}, \\ \end{aligned}$$

$$\begin{aligned} & p_{1} = \cosh \sqrt s i,\;\;p_{2} = \sinh \sqrt {si} ,\;\;p_{3} = \frac{{\sqrt {si} }}{{\left( {p_{2} Bi + \sqrt s i p_{1} } \right)}}, \\ & p_{4} = \frac{Bi}{{\left( {p_{2} Bi + \sqrt s i p_{1} } \right)}},\;\;k_{5} = \frac{2BR}{1 + R} = p_{5} ,\;\;p_{6} = \frac{1}{2 + R} = k_{6} , \\ & p_{7} = p_{5} p_{6} ,\;\;p_{8} = - \frac{{\sqrt {si} \, p_{3} p_{7} }}{{si\left( {si + p_{5} } \right)}},\;\;p_{9} = \frac{{\sqrt {si} \, p_{4} p_{7} }}{{si\left( {si + p_{5} } \right)}}, \\ & p_{10} = \frac{2 + R}{2BR} = k_{10} ,\;\;p_{11} = p_{10} p_{5} - 2,\;\;p_{12} = - 2p_{8} + si \, p_{10} p_{8} , \\ & p_{13} = - 2p_{9} + si\,p_{10} p_{9} ,\;\;p_{14} = \frac{{p_{11} }}{{\sqrt {p_{5} } }},\;\;p_{15} = \frac{{p_{12} }}{{\sqrt {si} }}, \\ & p_{16} = \frac{{p_{13} }}{{\sqrt {si} }},\;\;p_{17} = \sinh \left( {\sqrt {p_{5} } } \right),\;\;p_{18} = \cosh \left( {\sqrt {p_{5} } } \right), \\ & p_{19} = p_{14} p_{17} ,\;\;p_{20} = p_{14} p_{18} ,\;\;p_{21} = p_{15} p_{1} + p_{16} p_{2} , \\ & p_{22} = \left( {p_{8} p_{2} + p_{9} p_{1} } \right),\;\;p_{23} = \frac{{ - p_{22} + p_{9} }}{{p_{18} - 1}},\;\;p_{24} = \frac{{p_{17} }}{{1 - p_{18} }}, \\ & p_{25} = p_{14} p_{17} + 2,\;\;p_{26} = p_{21} + 2p_{9} ,\;\;p_{27} = p_{25} p_{24} + p_{20} , \\ & p_{28} = p_{26} + p_{23} p_{25} ,\;\;p_{29} = \frac{{p_{15} - p_{28} }}{{p_{27} - p_{14} }},\;\;p_{30} = p_{23} + p_{29} p_{24} , \\ & p_{31} = p_{30} - p_{9} ,\;\;p_{33} = \frac{{p_{14} p_{30} }}{{\sqrt {p_{5} } }},\;\;p_{34} = \frac{{p_{14} p_{29} }}{{\sqrt {p_{5} } }}, \\ & p_{35} = \frac{{p_{16} }}{{\sqrt {si} }},\;\;p_{36} = \frac{{p_{15} }}{{\sqrt {si} }},\;\;p_{37} = - p_{31} + p_{32} - p_{33} + p_{35} , \\ \end{aligned}$$