1 Introduction

Over the years, there have been rising demand in understanding analysis of heat transfer in a channel owing to its unending cooling or heating applications in automotive industries, aerospace, biochemical processing industries, and home appliances, resulting from the unstable climatic conditions and global warming. Based on the quest in solving the aforementioned applications, a lot of research works have been credited to understand mixed convection flow formation in a vertical channel [1], [2], [3], [4], [5], [6].

Despite these contributions, the works above presented a far from real life assumption in order to obtain a closed-form solution. It is known that physical property such as viscosity of fluid changes significantly with temperature. For gases, viscosity increases with increase in temperature, while for liquid it decreases as temperature increases. However, most of the existing analytical studies assume constant fluid viscosity. Accurate prediction for flow formation and heat transfer can be achieved by considering variation of physical properties with temperature [7], [8], particularly fluid viscosity. Shome and Jensen [9] studied a simultaneously developing laminar flow and heat transfer, with variable viscosity. Other related works on mixed convection flow formation in a channel with temperature-dependent viscosity can be seen in [10], [11], [12], [13], [14], [15], [16], [17], [18].

The study of mass transfer is of great importance in engineering, especially for physical procedures that comprise diffusive and convective transport of chemical species within physical system. It is also applicable in medical field for purification of blood in the kidneys and liver. In industries, the knowledge of mass transfer is applied in distillation of alcohol [19]. To this credit, several published works have been carried out [20], [21], [22].

Flow reversal is a situation where fluid flows in opposite direction to the direction of flow at the region close to the walls. The analysis of flow reversal in mixed convection flow has received great attention in the recent past. Sparrow et al. [23] presented an experiment to justify the occurrence of reverse flow. Ostrach [24], Cebeci et al. [25] provided an approximation analysis for parallel-streamline, bidirectional shear flow in mixed convection. Aung and Worku [26], [27] gave analytical solution for mixed convection flow in a vertical channel with asymmetric heating of the walls for developing and fully developed flow, respectively. In these works, they gave conditions under which flow reversal arises. Other related articles on mixed convection flow can be found in [28], [29], [30].

Despite the above contributions, no research work has been carried out to present exact solution to heat and mass transfer of mixed convection flow in a vertical channel with temperature-dependent viscosity and wall asymmetric temperatures and concentrations. Hence, the objective of this present article is to solve the above stated problem. The governing momentum, energy, and concentration equations are derived and solved exactly. For clarity and justification of accuracy, lines graphs and tables are presented to see the role of pertinent parameters on flow formation and heat and mass transfer.

2 Mathematical Modelling

A steady laminar fully developed mixed convection flow of an incompressible viscous fluid in a vertical channel of width L with concentration effect and temperature-dependent viscosity is considered in this article. The flow is set up through combined effect of constant pressure gradient, wall temperature difference, and corresponding concentration difference. The wall (y = L) is assumed to be heated with temperature T₂ and corresponding concentration raised to c₂ greater than the surrounding fluids having temperature and concentration T₀ and c₀, respectively, while the wall (y = 0) is heated to a temperature (T₁) with concentration c₁ such that T2>T1 and c2>c1. Due to this temperature and concentration differences as well as constantly applied pressure gradient in the direction parallel to flow direction, convection current is set up. A schematic geometry of the problem under investigation is shown in Figure 1, where x-axis is the flow direction parallel to the gravitational acceleration g but in the opposite direction. The viscosity is assumed to be exponentially increasing with temperature [8]. For convergence, b > 0 whenever the temperature difference is positive, and negative otherwise.

Figure 1:

Schematic diagram of the problem.

Following the above assumptions and neglecting the viscous dissipation term (which is usual assumption for laminar, steady flow formation) in the energy equation, the mathematical models governing the flow formation heat transfer and mass transfer are the momentum, energy, and concentration equations, which are, respectively, presented in dimensional form as

where the temperature-dependent viscosity is captured by [8]:

Subject to the following boundary conditions:

On introducing the following dimensionless parameters (Aung and Worku [26]) to (1–5),

Using the dimensionless parameters in (6), (1–5) become

The solution of (8, 9) with boundary conditions (10) is given by

On substituting (11, 12) into (7) and using the no slip boundary conditions (10), the exact solution for dimensionless fluid velocity is obtained as

where Ei′s and Ai′s for i=1,2,3,… are constants defined in the Appendix.

The dimensionless constant favourable pressure gradient for fully developed flow in a vertical channel is obtained from

On solving for dPdX in (14), the expression for dimensionless pressure gradient is obtained as

3 Results and Discussion

This article is devoted to understanding the role of concentration and temperature-dependent viscosity effect on flow formation in a vertical channel with asymmetric wall heating and concentration. The exact solutions obtained for concentration profile, temperature distribution, velocity profile, skin friction, bulk temperature, and Nusselt number are graphically represented in order to have clearer understanding on the role of various pertinent parameters. Throughout this article, the viscosity variation parameter (δ) has been taken over the range −2≤δ≤2 to capture different scenarios: δ < 0 represents case when temperature of fluid is greater than that of the wall (Y=0), and δ > 0 corresponds to the situation when wall temperature is greater than fluid temperature. It is good to state that δ → 0 matches with the case of constant viscosity. Generally, one can assume δ > 0 to represent liquid and δ < 0 to represent gases based on their temperature dependency on viscosity. The ratio of Grashof number due to concentration and temperature (N) is selected over reasonable values of N=0, 0.5, 1.0, 1.5 to capture cases when the Grashof number due to mass transfer is absent, lesser, equal to, and greater than Grashof number due to temperature, respectively. In addition, wall ambient temperature difference ratio (rt) and wall ambient concentration difference ratio (rc) are taken over 0≤rt≤1.0, 0≤rc≤1.0. For purely symmetric heating and symmetric wall concentration, rt=rc=1, also rt=rc=0 for purely asymmetric heating and concentration. In addition, there is a mixed convection parameter (−150≤GrRe≤150) so that it can clearly show the occurrence of flow reversal at the walls.

Figure 2 displays dimensionless fluid velocity for different values of viscosity variation parameter (δ) and N at fixed value of rt,rc , and GrRe. It is found that fluid velocity is an increasing function of δ and N at the region close to the wall Y = 1, whereas the reverse case is found at the region around Y = 0. This can be attributed to the fact that for positive values of δ, the variable viscosity term with exponentially decaying nature decreases steadily, which in turn increases flow formation continually. On the other hand, for equilibrium between the buoyancy force and pressure gradient, points of inflexion are noticed around the centre of the channel whose position is strictly dependent on δ. It is interesting to find that the presence of N > 0 leads to the occurrence of reverse flow formation at the wall Y = 0.

Figure 2:

Velocity profile for different values of N and δ at r_t = 0.5, r_c = 0.5, Gr/Re = 50.

Figure 3:

Velocity profile for different values of r_t and δ at r_c = 0.5, N = 0.5, Gr/Re = 50.

Figure 3, on the other hand, exhibits the role of wall-ambient temperature difference ratio (rt) and wall ambient concentration difference ratio (rc), respectively, for fixed value of GrRe and N. From Figure 3, fluid velocity increases with increase in r_t at the region close to the wall Y = 0, whereas the opposite situation is found close to the wall Y = 1 for δ > 0. On the other hand, for δ < 0, a complete reverse scenario is observed. This is expected as δ < 0 leads to increase in viscosity of the fluid, which in turn has a corresponding decrease in velocity.

The role of r_c, on the other hand, in Figure 4 is realized to be a corresponding decreasing function of fluid velocity as the region around Y = 1, while the converse result is obtained for the region Y = 0. This happens because concentration is known to be directly proportional to temperature, therefore an increasing or decreasing function of velocity, no matter the value of δ.

Figure 4:

Velocity profile for different values of δ and r_c at r_t = 0.5, N = 0.5, Gr/Re = 50.

Table 1 represents the critical value of GrRe and the interval of no-reverse flow. It is obvious from this table that the critical value of GrRe decreases with increase in δ but increases with increase in r_c. In addition, for positive value of δ = 2.0, the interval of no-flow reversal is found to be lowest. This could be due to the exponential decaying nature of viscosity with temperature.

Figure 5 depicts the combined role of ratio of Grashof number (N) and mixed convection parameter GrRe on dimensionless fluid velocity at the fixed value of rt,rc, and N. It is found that increasing GrRe increases fluid velocity around the wall Y = 1 and the reverse at the wall with wall ambient temperature difference ratio (Y=0). A point of inflexion is noticed at the centre of the channel, which is independent on the value of N. It is good to mention that when (T1−T0)≫(c1−c0), N → 0, and the work reduces to mixed convection flow in the absence of concentration effect. In addition, T1=T0, implies no temperature difference at the walls, hence no natural convection (GrRe=0), and therefore fluid velocity is independent of N and GrRe. This means the only driving force for flow formation is the constant pressure gradient. In addition, it is evidence that reverse flow formation is enhanced by the combined effect of N and GrRe.

Figure 5:

Velocity profile for different values of N and Gr/Re at r_t = 0.5, r_c = 0.5, δ = 2.0.

Figures 6 and 7 present the role of pertinent parameters such as rt,rc,N,GrRe , and δ on skin friction at the wall with ambient temperature difference ratio (Y=0). Figure 6 reveals that the role of viscosity variation parameter (δ) and r_t is to increase skin friction when δ > 0, but reduces skin friction otherwise (δ<0). It is also found that the maximum skin friction at this wall is attained when symmetric case is considered (rt→1). This can be attributed to the fact that as the heat supplied at the channel wall is increased there is a rise in the kinetic energy of molecule and collisions in the fluid, thereby increasing the velocity and hence the skin friction. Figure 7, on the other hand, shows the effect of mixed convection parameter (GrRe) and wall ambient concentration difference ratio (rc) on skin friction at the wall Y = 0. It is evident from this figure that for N = 0, skin friction is independent of r_c. Also, when GrRe=0, skin friction is independent of N as well as buoyancy parameter; hence, the only force responsible for this friction is the constant pressure gradient applied in the direction parallel to flow direction. Likewise, skin friction increases with increase in r_c for GrRe>0 and decreases otherwise. In addition, the effect of N is to either decrease or increase skin friction as GrRe>0 or GrRe<0, respectively. Similar trends for skin friction at the wall with uniform heating (Y=1) are also found. But for brevity, they are not reported in this present article.

Figure 6:

Skin friction for different values of δ, r_t, and N at r_c = 0.5, Gr/Re = 50.

Figure 7:

Skin friction for different values of r_c, Gr/Re and N at r_t = 0.5, δ = 2.0.

To understand the amount of heat added to the system as a result of flow parameters, Figure 8 depicts the total heat added to the system (H) as a function of r_t, r_c, N, GrRe, and δ. This figure discloses that the total heat supplied to the system increases with increase in r_t and N. As expected, as rt→1, the amount of heat supplied is independent of δ or N. This is due to the fact that as the heating on the walls attains equality every point in the channel attains saturation, and hence no heat enters the system anymore.

Figure 8:

Total heat added to the system for different values of δ, r_t, and N at r_c = 0.5, Gr/Re = 50.

Figure 9:

Total species rate added to the fluid for different values of δ, r_t, and N at r_c = 0.5, Gr/Re = 50.

Similarly, the amount of species added to the system due to unequal wall concentration is presented in Figure 9 for different values of δ, r_t, and N. One can infer that the role of temperature-dependent viscosity is to increase or decrease the total amount of species in the fluid as δ > 0 or δ < 0 accordingly as rt→1. In addition, N is seen to be an increasing function of Φ. Figure 10 consequently exhibits the total species rate added to the fluid as a function of r_c. As expected, Φ is an increasing function of r_c until a symmetric wall concentration is attained regardless of the viscosity variation parameter. This figure justifies that for symmetric wall concentration the total species rate is uniform and independent of any governing parameters.

Figure 10:

Total species rate added to the fluid for different values of δ, r_c, and N at r_t = 0.5, Gr/Re = 50.

Understanding heat transfer is one of the major analyses in fluid mechanics, and this is owed to the fact that there is always a need to increase or decrease temperature in a system. Figures 11 and 12 show the rate of heat transfer represented by Nusselt number at the wall with ambient temperature difference ratio (Y=0) for different values of viscosity variation parameter (δ), rt,rc, and N. From Figure 11, heat transfer is seen to decrease with increase in r_t in the absence of variable viscosity δ → 0 and δ=−2.0 but increases for δ = 2.0. The aforementioned decrease can be attributed to the fact that as rt→1, the temperature difference between the fluid and the wall tends to become zero, and hence no or little heat is transferred as seen from the figure. On the other hand, one can conclude that the presence of temperature-dependent viscosity leads to an enhancement in heat transfer in the system.

Figure 11:

Nusselt number for different values of δ, r_t, and N at r_c = 0.5, Gr/Re = 50 (Y = 0).

Figure 12 illustrates Nusselt number as a decreasing function of N for GrRe>0, and the contrary otherwise. In addition, for GrRe=0, a uniform heat transfer is found, which is independent of any governing parameter. A careful study of this figure suggests that the highest heat transfer is attained when asymmetric wall concentration with negative GrRe is considered.

Figure 12:

Nusselt number for different values of r_c, and Gr/Re at r_t = 0.5, δ = 2.0 (Y = 0).

For accuracy check, Table 2 computes numerical comparison of fluid velocity of the present work with those of Aung and Worku [26] by relaxing certain parameters in the present article (absence of temperature-dependent viscosity, buoyancy ratio parameter). This comparison gives an excellent agreement.

Table 3 presents numerical values for rate of heat transfer and skin friction at the walls. It is found that skin friction and rate of heat transfer attain the peak for symmetric wall heating.

4 Conclusion

This article presents exact solutions for flow formation of mixed convection flow in a vertical channel with temperature-dependent viscosity as well as mass transfer subject to asymmetric wall heating. The closed-form solutions that are obtained after modelling the momentum, energy, and concentration equations are a function of various pertinent parameters. Based on the results obtained in the form of line graphs and tables, the following conclusions can be drawn:

1. The role of temperature-dependent viscosity is to increase fluid velocity, skin friction, total amount of heat supplied to the system, total amount of species supplied to the system, Nusselt number, and reverse flow formation.

2. The presence of buoyancy ratio parameter is sufficient to induce flow reversal at the walls even at insignificant values of mixed convection parameter.

3. Highest heat transfer as well as skin friction is achieved when asymmetric wall concentration with negative GrRe is considered.

4. Reverse flow formation can be minimized by initiating symmetric wall heating and concentration.