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

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 c1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{1}$$\end{document} (thermal controlling parameter) for diverging (converging). It is also decreased against Pr (Prandtle number). For c1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}_{1}=0$$\end{document}, the temperature profiles may be exactly determined from the governing equations and the rate of heat transfer at the upper wall is θ′(1)=m(1+m2)tan-1m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta^{\prime } (1) = \frac{m}{{(1 + m^{2} )\tan^{ - 1} m}}$$\end{document}. 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.

is maintained via a source or sink at origin where the walls of a channel are situated at θ = ±a . The past models are strictly presented in polar coordinates and concerned with purely radial flows. The idea of radial flows given in [23] is commonly propagated in literature, whereas, they formulate and present well-known converging/diverging flow problems. The investigation of flow in converging/diverging channel/tube is further elaborated, however, consequences of other physical effects are studied on flow characteristic in Rehman et al. [24]. Ramesh and Devakar [25] used different methods to explore flow behavior in such channels. A common approach is used in all these research papers, however, it is strictly based on utilization of similarity transformation, which converts the equation of motion into ODE's. Viscous flow in a converging/diverging channel is studied in Turkyilmazoglu [26], moreover, he analyzed Jeffery-Hammel flows for stretching (shrinking) walls of the channel. Heat transfer inside converging/diverging channel of stretching (shrinking) walls is examined in the presence of viscous dissipation effects. The problem of fluid flow and heat transfer in converging channel is studied by Turkyilmazoglu [27]. He gives exact (closed form) multiple solutions to the momentum and energy equation associated with momentum and thermal slip boundary conditions. The exact multiple solutions and numerical results of the modeled problem are exactly matched for a small value of the parameters. The flow problem is strictly depend upon the appropriate coordinate system and geometry of the problem, so the right choice for choosing the proper coordinate system decides on the bases of flow problem see [28]. Many research articles are available on a purely radial flow in a converging/diverging channel for both Newtonian and non-Newtonian fluids. Makinde [30] presented a compact model of channel flow and he found the numerical and perturbation solutions of simulated problem, however, he investigated an incompressible viscous nanofluid in four different types of channels (divergent, convergent, locally constricted and wavy). Note he solved a system of ODE's with the perturbation series method, furthermore, he found accurate results for heat and mass transfer with special cases in the channels. Makinde [31] also examined the steady flow of incompressible viscous fluid inside a diverging symmetrical channel, furthermore, he presented the Taylor series solution to the modeled problem and all the field variables are evaluated and computed accurately using this method.
The previous problems of converging and diverging flow are simulated in polar coordinates system, whereas, we demonstrate the fluid flow and heat transfer problems in converging (diverging) channels of rectangular plan walls. The investigations of this paper have not been discussed in the open literature and a new problem of rectangular channels is simulated, whereas, we solved the momentum and energy equations for a converging (diverging) channel of rectangular walls. A set of appropriate transformation is formed for the stream function and temperature variables which reduces the Navier-Stokes and energy equations into ODE's and the final system of equations is solved with the regular perturbation method. A numerical method is also employed for confirmation and validation of the approximate analytical solutions. The numerical technique used here is the finite difference method, which is based on polynomial collocation with four Lobatto points. It is observed that the velocity and temperature profiles are charged significantly with the slope m of the upper wall and characteristic number Re. The skin friction coefficient and heat transfer from the upper wall are graphed against different parameters. Further, the classical work of Millsaps and Pohlhausen [18] is also recovered from the result of the current model, whereas, the two solutions are exactly matched with each other.

Formulation of the problem
Consider a rectangular converging/diverging channel of heated inclined plane walls with variable gap h(x) between them. The upper (lower) wall has slope m (− m). The gap between walls is 2 h(x) and the constant gap is a 0 when x = 0. A line is drawn at the center of the channel which is equidistance from the upper and lower walls and representing the x-axis whereas the y-axis is normal to it. Note that the walls of a channel are equally heated and have a variable temperature. A steady flow of an incompressible viscous fluid is maintained in a two dimensional channel of inclined plane walls. The velocity vector has decomposed into two orthogonal components i.e. the axial velocity (u) in x-direction and the normal velocity (v) in the y-direction.
Here, we considered a problem of converging (diverging) flow in a rectangular channel whose upper (lower) wall is situated at y = mx + a 0 (y = − mx − a 0 ) where m is the slope of upper (lower) wall and 2a 0 is entrance (for diverging flow)/exist (for converging flow) channel's height. The upper wall of the channel has variable temperature T w (x) = T 0 + T 1 (a 0 + mx) c 1 and the fluid at the center has uniform temperature T 0 . The parameter m = 0 is representing the flow and heat transfer between parallel walls and for that choice of m, the model problem is exactly reduced to the well-known Poiseuille model of one dimensional flow and heat transfer between parallel plates. Here we assumed similar flows and the velocity vector has normal and axial components whereas the classical Jeffery-Hammel model is equipped with only a radial component of velocity. The fluid attached to the plates has the velocity of solid sheet and would behave like walls. The axial velocity u(x; y) is maximum i.e. U(x) at the mid of channel or at y = 0. The normal velocity component and vorticity function will be zero at mid (y = 0) of channel due to symmetry conditions. Moreover, the center line as a reference stream line. The following equations are used for governing the incompressible viscous flow and heat transfer in the converging/diverging channel (Fig. 1).
Continuity equation: The component of Navier-Stokes equations: The energy equation: The boundary conditions for the flow problem are: where in Eq. (7) the vorticity (ξ) is defined and ν, ρ, p are kinematic viscosity, density and pressure, respectively. In view of the stream function formulation the velocity components are Now defining the stream function (ψ) in term of such that The vorticity equations is formed by eliminating pressure term between Eqs. (2) and (3).
where the vorticity function ξ is defined as: , The transformations in Eqs. (7) and (9) are used and converted the energy Eq. (4) and vorticity Eq. (10) into the following ODE's: are used for Prandtl number and Reynolds number, respectively.

Perturbation solution
Perturbation solution of the modeled problem is given below. The perturbation solution is obtained for a small value of m, where m is representing the slope of the upper wall of the channel. The unknown quantities f and θ are expanded in terms of m as: By substituting the expressions for f and θ from Eqs. (15)-(16) into (12)- (14) and equate like powers of m on both sides of them, we obtained the zeroth, first, second and third order systems. The zero, first, second and third order systems are solved for f 0 , θ 0 , f 1 , θ 1 , f 2 , θ 2 , f 3 , θ 3 and we get the following solution. The final perturbation solution is obtained when we substitute the zeroth, first, second and third order solutions from Eqs. (17)-(24) into (15)- (16). The unknown pressure is calculated from Eqs. Substituting values of g ′ from Eqs. (26) into (25), we get the following expression for g (the representative of pressure).

Numerical solution
The numerical solution of Eqs. (12)-(14) is obtained with the help of finite difference method, which is based on polynomial collocation with four Lobatto points. The scheme is developed for solution of boundary value problem, whereas, produced Cebeci and Keller is produced it and the code is established with the name bvp4c in MATLAB. The (21)  governing equations (continuity, momentum and energy) are transformed into a simplest system of boundary values ODE's with the help of similarity transformation discussed in Khan Marwat et al. [29]. The problem is formulated for the upper half channel and characterized by slop m of the upper wall of the channel, the Reynold's number, Prandtl number, and a temperature controlling parameter c 1 . The following procedure is established to distinguish between different physical situations. Note that the mathematical model is presented for upper half of the channel. The modeled problem is classified into the following case on the bases of parameters and their numerical values: • m > 0 and Re > 0 (Re < 0) corresponds to the diverging (converging) flow in the upper half of the channel.

Results and discussion
Note Hence, it is likely that not all profiles at large Re* > 0 are shown over here. In Fig. 2 the stream wise velocities are graphed for m=0, moderate, and high Reynolds number (Re* > 0 and Re* < 0). The velocity profiles are highly sensitive to Re* for this set of parameters  Figs. 3 and 4, respectively. The favorable pressure rises against in both of these two figures, whereas, it is increased nonlinearly against for each Re. In Fig. 3 (Fig. 4) it is decreased with the increasing of Re, however, for large Re > o (i.e. fast diverging flow) apposing pressure appears at the center of the channel. Note that, for values of Re between two and three, the pressure is apposing at the  center of the channel, whereas, it is saveable at wall. Furthermore, beyond the values of Re ≥ 3, the pressure at center and wall is apposing. For high apposing pressure occurs for Re > 10, therefore, the flow separation point appear in such situation. In Fig. 5, we graphed the streamlines for different values of Re. In this figure, we have shown the perturbation solutions in Eq. (15) for two different values of Re i.e. Re = 5; − 5 and m = 0:3. It is observed from this figure that the behavior of streamlines is identical at the center of channel for this set of parameters value, therefore, it is concluded that the Reynolds number (Re) has minor effects on the streamlines near the center of the channel, whereas, slight variation in its profile is observed. In Figs. 6 and 7, temperature profiles are graphed against for Re* > 0 (Re* < 0), different c 1 (both positive and negative values are taken). In Fig. 6, the temperature profiles are graphed against for diverging flow and different c 1 . The temperature profiles are decreased with increasing c 1 < 0 and increased for decreasing c 1 > 0. The profiles for c 1 < 0 are concave downward whereas for c 1 > 0, they are concave upward. The variation in temperature against c 1 is prominent in case of converging channel flows. It is observed from Fig. 7 that for decreasing values of c 1 > 0, the temperature profile are decreased and the profiles are concave downward. The variation in against c 1 > 0, is very weak. Similarly for increasing values of c 1 < 0, the profiles are increasing and any small changes in c 1 , creates significant changes in the profiles of θ. Note that for c 1 = 0, the exact solution of Eq. (13) satisfying the relevant boundary conditions in Eq. (14) is θ(η) = tan −1 (mη) tan −1 m and it tends to θ 0 (η) = η (i.e., zeroth order solution for small m in Eq. (18) in the limiting case when m is being approached to zero. The profiles in Figs. 8 and 9 are representing the dimensionless temperature distribution against the similarity variable (η) in the upper channel for different values of m. Heat transfer in converging and diverging channels is examined in Figs. 8 and 9, respectively. The temperature profiles in either case are increased with increasing of m for converging and diverging channel in the upper channel. For m = 0, the velocity profiles are linearly changed with while for increasing m, the profiles are uniformly increased. The variation in temperature distribution for converging and diverging flows is totally different. The temperature has linear profiles for m = 0 and exactly same predications are obtained from the zeroth order perturbation solution for θ 0 in Eq. (18). The shear stress (τ) at the flow region is calculated by the formulaτ = µ ∂u ∂y + ∂v ∂x . The dimensionless shear stress or the skin friction coefficient (C f ) at upper sheet is defined by C f = µ τ h µU . where τ h represents shear stress at the upper wall of the channel. By taking the definition of shear stress and substituting the values of velocity components from Eqs. (8)-(9), we get the final candidate for skin friction coefficient at the upper wall as= f ′ (1)(1 − m) . The heat flux (q) at the upper wall of the channel is calculated by the Fourier's law i.e.q = −k∇T = −k ∂T ∂x , ∂T ∂y , where k and T are already defined. In view of the transformation employed in Eq. (9), we get q w = q y=h = −k �T h(x) (m(c 1 − (1)), θ ′ (1)) , the ratio of q w and ∆T is known as heat transfer coefficient (ς ) and defined by (1)) . Now non-dimensionalized ς by k h(x) , gives the Nusselt number ( N u h ) and written as The amount of thermal energy or heat flux θ ′ (1) is evaluated and graphed against c 1 in Fig. 10 for different values of Pr. It is observed from this figure that with the increasing of Pr, the value of θ ′ (1) increases and the different profiles are intersecting each other at c 1 = 0. For c 1 = 0, Eq. (13) is reduced to−2m2ηθ ′ − (1 + m2η2)θ ′′ = 0 , which has an exact solution of the form θ(η) = tan −1 (mη) tan −1 m . This solution also satisfies the relevant boundary condition in Eq. (14). The amount of heat transfer at the upper plate is evaluated from this exact solution and computed numerically, which has the  Fig. 12, it is noticed that the value of θ ′ (1) is decreased with the increase of Re* for Pr > 0. But reverse phenomenon occurs in Fig. 13 and we observed that the value of ′ (1) is increased with the decreasing of Re* for Pr (upper half of the channel and converging flow). From these two figures i.e. Figures 12 and 13, we observed that, for large positive values of Re*, value of Pr > 1, the differences between profiles are more significant. However, for all values of Re* < 0 and the values of Pr in the interval 0 < Pr < 1.4, the gaps between profiles are more prominent. Moreover, for small values of Pr, the gaps between profiles are negligibly small and these observations are also noted here. For c 1 = 0, Pr = 0, Eq. (13) reduced to the form: In Fig. 14, the skin friction coefficient is graphed against Re* for different m and it is also graphed against m in Fig. 15 for different Re*. Note that the skin friction coefficient is increased with the increasing of both Re* and m. In Fig. 14, it is varied linearly against Re* for all values of m, moreover, the non-linear variation in its behavior is depicted in Fig. 15, whereas, the profiles are started at negative values and vanished at zero asymptotically. Moreover, negative values are representing a dragging force on the fluid and positive sign to correspond opposite behavior. It behaves asymptotically against m for different Re* and reaches zero for large values of m.

Comparison of present simulations with previously published results
We compared the present results with the published work of Khan Marwat [29] and Millsaps and Pohlhausen [18] in this section. The numerical solutions of Eq. (12) and relevant boundary conditions in Eq. (14) is graphed in Fig. 16 for f ′ (η), different value of Re* = mRe and m = 0.1. These profiles are also reported in Khan Marwat [29]. Moreover, the profiles in Fig. 17 are graphed from the numerical solutions of F ′′ (η) + 2ReαF (η)F ′ (η) + 4α 2 F ′ (η) = 0 with boundary conditions F (1) = F (− 1) = 0, F (0) = 1 and this problem is reported in Millsaps and Pohlhausen [18], where, F (η) is representing the radial velocity and η = r/α , r is the radial coordinate and α is representing the inclination of the channel's wall. Note that the equations in Khan Marwat [29], Millsaps and Pohlhausen [18] are different, but they are presenting the flow inside a converging and diverging channel. The profiles in Figs. 16 and 17 are exactly same and they are obtained from the solution of two different problems. In Figs. 16 and 17, we compared the present results with the published work and found the same pattern in these two figures.

Conclusion
The following interesting remarks are solicited from the solutions of the modeled problem where the simulated model describes the flow and heat transfer in a channel of inclined plane walls. The modeled equations are simplified and solved with numerical and analytical means. The two solutions are exactly matched for small value of the parameters of the current modeled problem. The governing equations are simulated for the upper half of the channel and generalized their solutions for the lower half of the channel. However, a solution network is produced for the whole channel. For m = 0 (i.e. parallel walls channel), we exactly obtained the classical Poiseuille profiles for the velocity and temperature. Later on, a modified Reynolds number is defined by Re* = mRe and for different values of Re* and m = 0, we obtained the similar profiles as shown in Fig. 2. Effects of c 1 (a thermal controlling parameter) are seen on the temperature profiles for both converging and diverging flows. The temperature profiles behaved differently for these two different channel models. In case of diverging flow, the temperature profiles are decreased for − 0:9 ≤ c 1 ≤ 2:3, whereas, for converging flows, the temperature profiles are increased for − 3 ≤ c 1 ≤ 0:9. Moreover, the temperature profiles are also graphed in both upper and lower halves of the channel for converging/diverging flows. It is confirmed that the profiles are similar. Besides that skin friction coefficient and heat transfer coefficient at the upper wall of the converging/diverging channel are also calculated and graphed. For zero-value of any of these parameters i.e., either Pr = 0, or m = 0, or c 1 = 0, or Re* = 0, heat is purely transferred by conduction.