Flow and heat transfer in a rectangular converging (diverging) channel: new formulation

Abstract

In this paper, a model problem of viscous flow and heat transfer in a rectangular converging (diverging) channel has been investigated. The governing equations are presented in Cartesian Coordinates and consequently they are simplified and solved with perturbation and numerical methods. Initially, symmetrical solutions of the boundary value problem are found for the upper half of the channel. Later on, these solutions are extended to the lower half and then to the whole channel. The numerical and perturbation solutions are compared and exactly matched with each other for a small value of the parameters involved in the problem. It is also confirmed that the solutions for the converging/diverging channel are independent of the sign of m (the slope). Moreover, the skin friction coefficient and heat transfer at the upper wall are calculated and graphed against the existing parameters in different figures. It is observed that the heat transfer at walls is decreased (increased) with increasing $${c}_{1}$$ c 1 (thermal controlling parameter) for diverging (converging). It is also decreased against Pr (Prandtle number). For $${c}_{1}=0$$ c 1 = 0 , the temperature profiles may be exactly determined from the governing equations and the rate of heat transfer at the upper wall is $$\theta^{\prime } (1) = \frac{m}{{(1 + m^{2} )\tan^{ - 1} m}}$$ θ ′ ( 1 ) = m ( 1 + m 2 ) tan - 1 m . It is confirmed that the skin friction coefficient behaves linearly against Re* (modified Reynolds number) and it is increased with increasing of Re* (changed from negative to positive). Moreover, it is increased asymptotically against m and converges to a constant value i.e. zero.

Double-diffusive convection in Jeffery–Hamel flow

“…( 11 – 14 ). We exactly found the same observations as reported in Laila et al 25 and Marwat et al 26 .

Figure 5 The skin friction coefficient ( ) is plotted against modified Reynolds number for .

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“…For , the profile of has reached to zero value at . The dimensionless velocity profiles plotted against the similarity variable for modified Reynold’s number , are compared with the published work of Millsaps and Pohlhausen given in White 24 , Laila et al 25 and Marwat et al 26 and exactly the same results have obtained. The thermal and solutal Grashof numbers have assisted and opposed the flows inside a channel depending upon the values of these two Grashof numbers, nature of the fluid and type of the channel, and its walls inclination.…”

“…Note that the profiles in ( a ) and ( b ) are published in the Fig. 4 and 16 of R. Laila et al 25 and Marwat et al 26 , respectively. …”

“…Besides that, new variables are formed which relate the previous work of the same nature in Polar Coordinates to the new simulation in Cartesian Coordinates. The new simulations are compared with the classical work of Millsaps and Pohlhausen presented in White 24 , Laila et al 25 and Marwat et al 26 , whereas, these results have been retrieved exactly. It is hoped that the present simulations perfectly narrates real world problems, frequently used in such special circumstances.…”

“… The axial velocity is plotted against the similarity variable for different values of modified Reynolds number and the results are compared with published work of ( a ) Laila et al 25 and ( b ) Marwat et al 26 . The profiles in ( c ) showed that curves of are exactly matched with these two published papers.…”