Couette flow and heat transfer of heat-generating/absorbing fluid in a rotating channel in presence of viscous dissipation

Abstract

Theoretical investigation is carried out on flow formation and heat transfer characteristics of viscous incompressible and heat-generating/absorbing fluid in a horizontal rotating channel in presence of viscous dissipation. Exact solution is obtained for two cases of interest: when the fluid considered is heat generating and also when the fluid is heat absorbing. Interesting effect of some governing parameters on fluid temperature, rate of heat transfer and critical Eckert number are identified and discussed using line graphs and tables for both cases. It is found out that the rate of heat transfer decreases with increase in heat source causing an increase in fluid temperature while the contrast is true for heat sink. In addition, critical Eckert number decreases with heat source while it increases with heat sink.

Keywords:

• Couette flow
• heat absorption
• heat generation
• rotation
• viscous dissipation

1. Introduction

Couette flow in presence of rotation remains an active area of research as it occurs naturally in several atmospheric science problems, like formation of galaxies and in oceanic circulation. Studies relating to these flows are also important in several start-up processes encountered in engineering and technological processes like geophysics, plasma physics, nuclear energy systems, astronautical propulsion technology, fluid engineering amongst others. Schlichting (1979) presented an analysis on flow of viscous incompressible fluid when one of the boundaries is set into sudden motion describing how the boundary layer spreads within the flow field. Vidyanidhi and Nigam (1967) investigated the effect of constant pressure gradient on flow of viscous incompressible fluid in a revolving channel. Jana and Datta (1977) examined the effect of rotation on flow of viscous incompressible fluid between two infinite parallel walls with heat transfer characteristics, and found out that the primary component of fluid velocity is directly proportional to increase in rotation parameter whereas secondary component is strongly influenced by the value of the rotation parameter. Seth, Jana, and Maiti (1982) presented an analysis on time-dependent flow of viscous incompressible fluid in a revolving channel in presence of an applied magnetic field. Berker (1979) investigated the effect of rotation on time-dependent flow of viscous incompressible fluid in the vicinity between two parallel disks. Guria, Jana, and Ghosh (2006) studied on hydrodynamic time-dependent flow of viscous fluid incompressible in a revolving system when one of the channel is set into sudden motion. They recorded that secondary component of fluid velocity decreases about the fixed wall when the rotation parameter becomes large. Other relevant works include: Erdogan (1995), Parter and Rajagopal (1984), Lai, Rajagopal, and Szeri (1984) and Greenspan (1990).

On the other hand, heat generation/absorption effects become relevant in many processes involving chemical reaction and dissociating fluids, since the flow and thermal fields can be greatly influenced by the heat source/sink term. Jha, Samaila, and Ajibade (2012) examined the effect of heat generation/absorption on flow formation in a parallel plate channel with ramped temperatures. In a related work, Jha and Ajibade (2009) examined the effect of heat source/sink on natural convection flow of viscous incompressible fluid between two vertical porous plates with periodic heat input. Recently, Jha and Aina (2016) gave the exact solution for heat-generating/absorbing fluid in a cylinder with time periodic boundary condition in presence of magnetic field. They found out that that the rate of heat transfer is significantly enhanced with increase in heat source while the contrast is true in presence of heat sink. Jha, Oni, and Aina (2018) presented the exact solution responsible for the mixed convection in an annular geometry when the fluid is either heat generating or heat absorbing. In another work, Jha and Ajibade (2009) presented the solution representing the time-dependent Couette flow of heat-generating/absorbing fluid in a permeable channel. Saba et al. (2018) Presented a study on thermal analysis of nanofluid in presence of internal heat generation. Dogonchi, Chamkha, Seyyedi, Hashemi-Tilehnoee, and Ganji (2019) investigated the impact of viscous dissipation on free convection flow of nanofluid in presence of internal heat source. Other-related articles include: Dogonchi, Chamkha, and Ganji (2019), Seyyedi, Dogonchi, Ganji, and Hashemi-Tilehnoee (2019), Seyyedi et al. (2019), Dogonchi, Waqas, Seyyedi, Hashemi-Tilehnoee, and Ganji (2019), Dogonchi et al. (2019), Dogonchi, Tayebi, Chamkha, and Ganji (2019), Dogonchi, Chamkha, Seyyedi, Hashemi-Tilehnoee, and Ganji (2019), Saba et al. (2018), Ahmed, Khan, and Mohyud-Din (2017), Khan, Ahmed, and Mohyud-Din (2017), Khan, Adnan, Ahmed, and Mohyud-Din (2017), Khan, Ahmed, Bin-Mohsen, and Mohyud-Din (2017), Adnan, Asadullah, Khan, Ahmed, and Mohyud-Din (2016) and Adnan, Khan, Ahmed, and Mohyud-Din (2016).

In the present work, we investigate combine effects of rotation and heat generation/absorption on flow formation of viscous incompressible fluid in a horizontal channel in presence of viscous dissipation. To the author’s knowledge, no studies have been conducted to study these effects. Analytic solution is obtained for governing problem and presented in dimensionless form while the effect of various non-dimensional parameters is highlighted and discussed in the course of the study. The current work is an extension of Jana and Datta (1977) by incorporating heat generating/absorbing fluid . The novelty of the present work is that an exact solution is obtained representing the flow of viscous fluid in a horizontal rotating channel when the fluid is either heat generating or heat absorbing in the existence of viscous dissipation. Also, the expressions for the critical Eckert number for the flow are obtained under both considered cases. Result obtained from this analysis is relevant in many heat transfer problem. Also, result obtained from the exact solution can serve as a benchmark for other numerical computation.

2. Mathematical analysis

Schematic diagram of the flow problem is displayed in Figure 1. Consider the steady fully developed Couette flow of viscous incompressible fluid in a horizontal rotating channel formed by parallel walls having width “$b$”. The channel walls are taken to be infinitely long so that the entry effect is neglected and all flow quantities depend on $y\prime$ alone. It is assumed that the upper wall with temperature $T_1$ moves with constant velocity $U_0$ along the $x\prime$ direction while the lower wall is fixed with temperature $T_0.$ The flow system rotates along the $y\prime$ axis with angular velocity $\Omega ,$ which is normal to the channel walls giving rise to another flow along the secondary direction ($z\prime$). . Under these assumptions, governing equations describing the flow problem are given in dimensional form under the Boussinessq’s approximation (1) $$\mu \frac{d^{2}u\prime }{dy{\prime }^2}-2\rho \Omega w\prime =0,$$(1) (2) $$-\frac{\partial \rho }{\partial y\prime }=0,$$(2) (3) $$\mu \frac{d^{2}w\prime }{dy{\prime }^2}+2\rho \Omega u\prime =0,$$(3) (4) $$\alpha \frac{d^{2}T\prime }{dy{\prime }^2}\pm Q+\frac{\mu }{\rho C_p}\left[{\left(\frac{d^{2}u\prime }{dy\prime }\right)}^2+{\left(\frac{d^{2}w\prime }{dy\prime }\right)}^2\right]=0,$$(4) where $Q=\frac{Q_0(T\prime -T_0)}{k}.$

Figure 1. Diagram of the flow problem.

With momentum and thermal boundary conditions as: (5) $$u\prime =0,w\prime =0,T\prime =T_0,\text{at}y\prime =0,$$(5) (6) $$u\prime =U_0,w\prime =0,T\prime =T_1,\text{at}y\prime =b.$$(6)

Introducing the dimensionless parameters, (7) $$y=\frac{y\prime }{b},u=\frac{u\prime }{U_0},w\prime =\frac{w\prime }{U_0},a^2=\frac{\Omega b^2}{\upsilon },T=\frac{T\prime -T_0}{T_1-T_0},E=\frac{{U_0}^2}{C_p(T_1-T_0)},S=\frac{Q_0{b}^2}{k},\mathrm{Pr}=\frac{\upsilon }{\alpha }.$$(7)

Equations (1)–(3) reduces to: (8) $$\frac{d^{2}u}{d{y}^2}-2a^{2}w=0,$$(8) (9) $$\frac{d^{2}w}{d{y}^2}+2a^{2}u=0,$$(9) (10) $$\frac{d^{2}T}{d{y}^2}\pm S^{2}T+\mathrm{Pr}E\left[{\left(\frac{d^{2}u}{dy}\right)}^2+{\left(\frac{d^{2}w}{dy}\right)}^2\right]=0.$$(10)

With momentum and thermal boundary conditions are: (11) $$u=0,w=0,T=0,\text{at}y=0,$$(11) (12) $$u=1,w=0,T=1,\text{at}y=1.$$(12)

By defining $F=u+iw$ and $\overline{F}=u-iw$ denoting the complex conjugate, Equations (8)(8) $$\frac{d^{2}u}{d{y}^2}-2a^{2}w=0,$$(8) and (9)(9) $$\frac{d^{2}w}{d{y}^2}+2a^{2}u=0,$$(9) can be rewritten as: (13) $$\frac{d^{2}F}{d{y}^2}+2i{a}^{2}F=0,$$(13) (14) $$\frac{d^{2}T}{d{y}^2}\pm S^{2}T+\mathrm{Pr}E\left[F\prime \overline{F}\prime \right]=0.$$(14)

Subject to the boundary condition: (15) $$F=0,T=0,\text{at}y=0,$$(15) (16) $$F=1,T=1,\text{at}y=1.$$(16)

Solution to Equation (13)(13) $$\frac{d^{2}F}{d{y}^2}+2i{a}^{2}F=0,$$(13) subject to boundary conditions (15)–(16) are given following Jana and Datta (1977) as: (17) $$F\left(y\right)=\frac{\sinh \left(a(1-i)y\right)}{\sinh \left(a(1-i)\right)},$$(17) (18) $$\overline{F}\left(y\right)=\frac{\sinh \left(a(1+i)y\right)}{\sinh \left(a(1+i)\right)}.$$(18)

Detailed analysis on the effect of rotation parameter “$a$” on primary velocity, secondary velocity and the shear stress on the channel boundaries are extensively discussed in Jana and Datta (1977). Hence, the main aim of this work is to investigate the effects of rotation and heat generation/absorption parameters on fluid temperature and rate of heat transfer when the fluid is either heat generating or heat absorbing in the existence of viscous dissipation.

Solution to the energy Equation (14)(14) $$\frac{d^{2}T}{d{y}^2}\pm S^{2}T+\mathrm{Pr}E\left[F\prime \overline{F}\prime \right]=0.$$(14) subject to boundary conditions (15)–(16) is obtained for two cases of interest:

Case I: Heat-generating fluid (heat source)

Using the method of undetermined coefficient, solution to the energy Equation (14)(14) $$\frac{d^{2}T}{d{y}^2}\pm S^{2}T+\mathrm{Pr}E\left[F\prime \overline{F}\prime \right]=0.$$(14) is obtained as (19) $$T(y)=\frac{\mathrm{Pr}E{a}^2}{\left[{\sinh }^2\left(a\right)+{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\left(a\right)\right]}\left[\begin{array}{l}\frac{1}{S^2+4a^2}\left(\frac{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(Sy\right)}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)}\cosh \left(2a\right)-\cosh \left(2ay\right)\right)\\ +\frac{1}{S^2-4a^2}\left(\frac{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(Sy\right)}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)}\text{cos}\hspace{0.17em}\left(2a\right)-\hspace{0.17em}\text{cos}\hspace{0.17em}\left(2ay\right)\right)\\ -\frac{2S^2}{S^4-16a^4}\frac{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\left(1-y\right)\right)}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)}\end{array}\right]+\frac{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(Sy\right)}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)},$$(19) where $\mathrm{Pr},S,E,a$ are, respectively, the Prandtl number, heat-generating parameter, Eckert number and the rotation parameter. For clarity, it is worth noting that the temperature distribution profile is induced due to the effect of viscous dissipation in presence of heat generation/absorption within the rotating system.

The rate of heat transfer represented by the Nusselt number at the wall $y=1$ is obtained as follows: (20) $$\frac{dT}{dy}|{}_{y=1}=\frac{\mathrm{Pr}E{a}^2}{\left[{\sinh }^2\left(a\right)+{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\left(a\right)\right]}\left[\begin{array}{l}\frac{1}{S^2+4a^2}\left(S\text{cot}\left(S\right)\cosh \left(2a\right)-2a\sinh \left(2a\right)\right)\\ +\frac{1}{S^2-4a^2}\left(S\text{cot}\left(S\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(2a\right)+2a\hspace{0.17em}\text{sin}\hspace{0.17em}\left(2a\right)\right)\\ +\frac{2S^3}{S^4-16a^4}\frac{1}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)}\end{array}\right]+S\text{cot}\left(S\right).$$(20)

From the above expression, it is obvious that there is transfer of heat from either the heated wall to the fluid or from the fluid to the heated wall . Table 1 presents the effect of $\mathrm{Pr},S\text{and}a$ on the rate of heat transfer.

Table 1. Effects of Prandtl number ($\mathrm{Pr}$), heat-generating parameter ($S$) and rotation parameter ($a$) on rate of heat transfer (heat-generating fluid) at wall $y=1.$

$\mathrm{Pr}$	$S$	$\frac{dT}{dy}|{}_{y=1}$ for $a=0.5\text{and}E=0.2$	$\mathrm{Pr}$	$a$	$\frac{dT}{dy}|{}_{y=1}$ for $S=0.5\text{and}E=0.2$
0.72	0.2	0.9782	0.72	0.5	0.8998
	0.4	0.9339		1.0	0.9062
	0.6	0.8569		2.0	0.8948
	0.8	0.7346		3.0	0.8787
1.0	0.2	0.9749	1.0	0.5	0.8938
	0.4	0.9291		1.0	0.9027
	0.6	0.8491		2.0	0.8869
	0.8	0.7181		3.0	0.8645
2.0	0.2	0.9631	2.0	0.5	0.8724
	0.4	0.9121		1.0	0.8901
	0.6	0.8211		2.0	0.8586
	0.8	0.6592		3.0	0.8138
3.0	0.2	0.9514	3.0	0.5	0.8510
	0.4	0.8952		1.0	0.8775
	0.6	0.7932		2.0	0.8302
	0.8	0.6003		3.0	0.7631

From the Table, it is observed that when the fluid is heat generating, increase in $\mathrm{Pr}$ or $S$ causes a reduction in the rate of heat transfer from the upper wall to the fluid leading to increase in fluid temperature. This is physically true since the thermal boundary layer thickness decreases with increase in $\mathrm{Pr},$ thus reducing the rate of heat transfer. Furthermore, the Table also reveals that for all values of $\mathrm{Pr},$ increase in $S$ causes a reduction in $\frac{dT}{dy}|{}_{y=1}$ while it displays a dual character when the rotation parameter raises. Hence, increase in $a$ leads to increase in the rate of heat transfer from $a=0.5\text{to}a=1.0$ while it decreases when $a$ is further increased from $a=2.0\text{to}a=3.0.$ The Table also suggests that the rate of heat transfer remains positive, i.e. $\frac{dT}{dy}|{}_{y=1}>0$ for all values of $\mathrm{Pr}$ and $S.$ This implies that there is significant transfer of heat from the heated wall to the fluid since the temperature of the wall is greater than the heat-generating fluid with viscous dissipation.

However, it is interesting to note that there exists a critical point where the there is no transfer of heat from either the fluid to the wall or from the wall to the fluid. This value is calculated from Equation (19)(19) $$T(y)=\frac{\mathrm{Pr}E{a}^2}{\left[{\sinh }^2\left(a\right)+{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\left(a\right)\right]}\left[\begin{array}{l}\frac{1}{S^2+4a^2}\left(\frac{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(Sy\right)}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)}\cosh \left(2a\right)-\cosh \left(2ay\right)\right)\\ +\frac{1}{S^2-4a^2}\left(\frac{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(Sy\right)}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)}\text{cos}\hspace{0.17em}\left(2a\right)-\hspace{0.17em}\text{cos}\hspace{0.17em}\left(2ay\right)\right)\\ -\frac{2S^2}{S^4-16a^4}\frac{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\left(1-y\right)\right)}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)}\end{array}\right]+\frac{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(Sy\right)}{\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)},$$(19) above and obtained when $E=E_{\text{crit}}$ (critical Eckert number), defined by (21) $$E_{\mathit{crit}}=\frac{S\hspace{0.17em}\text{cot}\left(S\right)\left[{\sinh }^2\left(a\right)+{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\left(a\right)\right]}{\mathrm{Pr}{a}^2\left[\begin{array}{l}\frac{1}{S^2+4a^2}\left(2a\hspace{0.17em}\sinh \left(2a\right)-S\hspace{0.17em}\coth \left(S\right)\cosh \left(2a\right)\right)\\ -\frac{1}{S^2-4a^2}\left(S\hspace{0.17em}\text{cot}\left(S\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(2a\right)+2a\hspace{0.17em}\text{sin}\hspace{0.17em}\left(2a\right)\right)\\ -\frac{2S^3}{\left(S^4-16a^4\right)\hspace{0.17em}\text{sin}\hspace{0.17em}\left(S\right)}\end{array}\right]}.$$(21)

Variations of the critical Eckert number with different flow parameters are presented in Table 2. From this Table, it is observed that $E_{\text{crit}}$ decreases with increase $\mathrm{Pr}.$ The Table also reveals that critical value of $E$ decreases with increase in heat generation ($S$), whereas it displays a dual character with increase in rotation parameter ($a$).

Table 2. Effects of Prandtl number ($\mathrm{Pr}$), heat-generating parameter ($S$) and rotation parameter ($a$) on critical Eckert number $E_{\text{crit}}$(heat-generating fluid).

$\mathrm{Pr}$	$S$	$E_{\text{crit}}$ (when $a=0.5$)	$\mathrm{Pr}$	$a$	$E_{\text{crit}}$ (when $S=0.5$)
0.72	0.2	2.4128	0.72	0.5	1.3128
	0.4	1.6890		1.0	2.2794
	0.6	0.9653		2.0	0.9512
	0.8	0.3932		3.0	0.5185
1.0	0.2	1.7372	1.0	0.5	0.9452
	0.4	1.2161		1.0	1.6412
	0.6	0.6950		2.0	0.6848
	0.8	0.2831		3.0	0.3733
2.0	0.2	0.8686	2.0	0.5	0.4726
	0.4	0.6080		1.0	0.8206
	0.6	0.3475		2.0	0.3424
	0.8	0.1415		3.0	0.1867
3.0	0.2	0.5791	3.0	0.5	0.3151
	0.4	0.4054		1.0	0.5471
	0.6	0.2317		2.0	0.2283
	0.8	0.0944		3.0	0.1244

Case II: Heat-absorbing fluid (heat sink)

Solution to the energy equation in Equation (14)(14) $$\frac{d^{2}T}{d{y}^2}\pm S^{2}T+\mathrm{Pr}E\left[F\prime \overline{F}\prime \right]=0.$$(14) when the fluid is assumed to be heat absorbing is obtained subject to the boundary conditions (15)–(16) as: (22) $$T(y)=\frac{\mathrm{Pr}E{a}^2}{\left[{\sinh }^2\left(a\right)+{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\left(a\right)\right]}\left[\begin{array}{l}\frac{1}{4a^2-S^2}\left(\frac{\sinh \left(Sy\right)}{\sinh \left(S\right)}\cosh \left(2a\right)-\cosh \left(2ay\right)\right)\\ +\frac{1}{S^2+4a^2}\left(\frac{\sinh \left(Sy\right)}{\sinh \left(S\right)}\text{cos}\hspace{0.17em}\left(2a\right)+\hspace{0.17em}\text{cos}\hspace{0.17em}\left(2ay\right)\right)\\ -\frac{2S^2}{16a^4-S^4}\frac{\sinh \left(S\left(1-y\right)\right)}{\sinh \left(S\right)}\end{array}\right]+\frac{\sinh \left(Sy\right)}{\sinh \left(S\right)},$$(22) where $\mathrm{Pr},S,E,a$ are, respectively, the Prandtl number, heat-absorbing parameter, Eckert number and the rotation parameter.

Rate of heat transfer presented by the Nusselt number at the wall $y=1$ is obtained as follows: (23) $$\frac{dT}{dy}|{}_{y=1}=\frac{\mathrm{Pr}E{a}^2}{\left[{\sinh }^2\left(a\right)+{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\left(a\right)\right]}\left[\begin{array}{l}\frac{1}{4a^2-S^2}\left(S\coth \left(S\right)\cosh \left(2a\right)-2a\sinh \left(2a\right)\right)\\ +\frac{1}{S^2+4a^2}\left(S\coth \left(S\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(2a\right)-2a\hspace{0.17em}\text{sin}\hspace{0.17em}\left(2a\right)\right)\\ -\frac{2S^3}{16a^4-S^4}\frac{1}{\sinh \left(S\right)}\end{array}\right]+S\coth \left(S\right).$$(23)

From Table 3, it is found out that for heat-absorbing fluid, simultaneously increasing $\mathrm{Pr}$ and $S$ leads to increase in the rate of heat transfer causing a decrease in fluid temperatures. Whereas it decreases when the rotation parameter raises leading to increase in fluid temperature.

Table 3. Effects of Prandtl number ($\mathrm{Pr}$), heat-generating parameter ($S$) and rotation parameter ($a$) on rate of heat transfer (heat-absorbing fluid) at wall $y=1.$

$\mathrm{Pr}$	$S$	$\frac{dT}{dy}|{}_{y=1}$ for $a=0.5\text{and}E=0.2$	$\mathrm{Pr}$	$a$	$\frac{dT}{dy}|{}_{y=1}$ for $S=0.5\text{and}E=0.2$
0.72	0.2	1.0136	0.72	0.5	1.0816
	0.4	1.0526		1.0	1.0722
	0.6	1.1165		2.0	1.0617
	0.8	1.2035		3.0	1.0462
1.0	0.2	1.0137	1.0	0.5	1.0814
	0.4	1.0526		1.0	1.0685
	0.6	1.1162		2.0	1.0539
	0.8	1.2031		3.0	1.0324
2.0	0.2	1.0141	2.0	0.5	1.0808
	0.4	1.0524		1.0	1.0549
	0.6	1.1153		2.0	1.0257
	0.8	1.2014		3.0	0.9827
3.0	0.2	1.0146	3.0	0.5	1.0802
	0.4	1.0522		1.0	1.0414
	0.6	1.1143		2.0	0.9976
	0.8	1.1997		3.0	0.9331

Similarly, the critical Eckert number is calculated when the fluid is heat absorbing and defined as $E=E_{\text{crit}}$ (24) $$E_{\text{crit}}=\frac{S\hspace{0.17em}\coth \left(S\right)\left[{\sinh }^2\left(a\right)+{\hspace{0.17em}\text{sin}\hspace{0.17em}}^2\left(a\right)\right]}{\mathrm{Pr}{a}^2\left[\begin{array}{l}\frac{1}{4a^2-S^2}\left(2a\hspace{0.17em}\sinh \left(2a\right)-S\hspace{0.17em}\coth \left(S\right)\cosh \left(2a\right)\right)\\ -\frac{1}{S^2+4a^2}\left(S\hspace{0.17em}\coth \left(S\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(2a\right)-2a\hspace{0.17em}\text{sin}\hspace{0.17em}\left(2a\right)\right)\\ -\frac{2S^3}{\left(16a^4-S^4\right)\sinh \left(S\right)}\end{array}\right]}.$$(24)

The effect of some governing parameter on critical Eckert number is depicted in Table 4.

Table 4. Effects of Prandtl number ($\mathrm{Pr}$), heat-generating parameter ($S$) and rotation parameter ($a$) on critical Eckert number $E_{\text{crit}}$ (heat-absorbing fluid).

$\mathrm{Pr}$	$S$	$E_{\text{crit}}$ (when $a=0.5$)	$\mathrm{Pr}$	$a$	$E_{\text{crit}}$ (when $S=0.5$)
0.72	0.2	−14.0075	0.72	0.5	−3.1218
	0.4	−4.7324		1.0	2.3211
	0.6	−2.1284		2.0	1.0697
	0.8	−0.8982		3.0	0.6056
1.0	0.2	−10.0854	1.0	0.5	−2.2477
	0.4	−3.4074		1.0	1.6412
	0.6	−1.5325		2.0	0.7702
	0.8	−0.6467		3.0	0.4361
2.0	0.2	−5.0427	2.0	0.5	−1.1239
	0.4	−1.7037		1.0	0.8356
	0.6	−0.7662		2.0	0.3851
	0.8	−0.3233		3.0	0.2180
3.0	0.2	−3.3618	3.0	0.5	−0.7492
	0.4	−1.1358		1.0	0.5571
	0.6	−0.5108		2.0	0.2567
	0.8	−0.2156		3.0	0.1454

From Table 4, it is evident that rate of heat transfer displays a negative value with increase in either $\mathrm{Pr}orS.$ It is also evident that the critical Eckert number is directly proportional to increase in Prandtl number ($\mathrm{Pr}$) and also the heat absorption parameter ($S$) resulting to increase in $E_{\text{crit}}.$

From the exact solution obtained for the energy equation and represented in Equations (19)–(24). Analysis on the effect of some governing parameters on fluid temperature, rate of heat transfer and critical Eckert number is highlighted and discussed with the aid of line graphs and Tables for selected range of values $0.72\le \mathrm{Pr}\le 4.0,$ $0.2\le S\le 2.0$ and $0.02\le E\le 0.2$ all chosen arbitrarily to investigate their effect on flow formation in the channel.

Numerical values of the present work in the absence of the heat-generating/absorbing parameter are validated with the work of Jana and Datta (1977) and presented in Table 5. From the Table, it is obvious that an excellent agreement is obtained when the heat generation/absorption parameter is negligible.

Table 5. Numerical comparison of the rate of heat transfer in the present work with those of Jana and Datta (1977) when $E=0.02.$

$\mathrm{Pr}$	$a$	$\frac{dT}{dy}|{}_{y=1}$(present work as $S\to 0$)	$\frac{dT}{dy}|{}_{y=1}$(Jana & Datta, 1977)
0.72	1.0	0.9916	0.9927
	2.0	0.9799	0.9799
	3.0	0.9639	0.9639
	4.0	0.9496	0.9495
1.0	1.0	0.9883	0.9883
	2.0	0.9720	0.9720
	3.0	0.9498	0.9498
	4.0	0.9300	0.9299
2.0	1.0	0.9766	0.9766
	2.0	0.9441	0.9441
	3.0	0.8996	0.8996
	4.0	0.8599	0.8599

Figure 2(a,b) displays the effect of heat generation/absorption parameter ($S$) on temperature distribution profile within the channel. From Figure 2(a), it is evident that for heat-generating situation, increase in heat source ($S$) reduces the thermal boundary layer causing an enhancement in fluid temperature all through the flow domain. The Figure shows that temperature attains its extremum at higher values of heat source parameter ($S$) as the fluid becomes warmer. This reduces the rate of heat transfer within the channel leading to increase in heat accumulation. The Figure also reveals that the increase in fluid temperature is more pronounced about the hot situated at $y=1$ as the heat source parameter ($S$) increases. For heat-absorbing fluid on the other hand, it is evident that increase in heat absorption ($S$) thickens the thermal boundary layer resulting to decrease in fluid temperature.

Figure 2. (a) Effect of heat generation parameter ($S$) on fluid temperature (when the fluid is heat generating) for $E=0.2,a=0.5\text{and}\mathrm{Pr}=0.72.$ (b) Effect of heat absorption parameter ($S$) on fluid temperature (when the fluid is heat absorbing) for $E=0.2,a=0.5\text{and}\mathrm{Pr}=0.72.$

Figure 3(a,b) illustrates the effect of Prandtl number ($\mathrm{Pr}$) on fluid temperature profiles. From the figures, it is interesting to observe that increase in $\mathrm{Pr}$ yields an observable decrease in rate of heat transfer causing an increase in fluid temperature. The figure also suggests that in presence of viscous dissipation and heat generation, fluid temperature increases. A similar phenomenon is observed from Figure 3(b) when the fluid is heat absorbing as the temperature profile is observed to be directly proportional to the Prandtl number ($\mathrm{Pr}$) of the fluid. .

Figure 3. (a) Effect of Prandtl number ($\mathrm{Pr}$) on fluid temperature (when the fluid is heat generating) for $E=0.2,a=0.5\text{and}S=0.5.$ (b) Effect of Prandtl number ($\mathrm{Pr}$) on fluid temperature (when the fluid is heat absorbing) for $E=0.2,a=0.5\text{and}S=0.5.$

Figure 4(a,b) reveals the effect of viscous dissipation parameter ($E$) on heat-generating/absorbing fluid. From the Figures, it is clear that with increase in $E,$ heat generated due to shear forces increases, causing an increase in fluid temperature for both situations (Figure 4(a,b)). In addition, it is also clear that for heat-absorbing fluid (Figure 4(b)), there exist a point within the channel in a region about the heated wall where the effect of $E$ becomes insignificant.

Figure 4. (a) Effect of Eckert number ($E$) on fluid temperature (when the fluid is heat generating) for $\mathrm{Pr}=0.72,a=0.5\mathit{and}S=0.5.$ (b) Effect of Eckert number ($E$) on fluid temperature (when the fluid is heat absorbing) for $\mathrm{Pr}=0.72,a=0.5\text{and}S=0.5.$

3. Conclusion

The present article investigates Couette flow formation and heat transfer of heat-generating/absorbing viscous fluid in a horizontal rotating channel with viscous dissipation. Governing flow equations are obtained and solved exactly using the method of undetermined coefficient to obtain expressions for fluid temperature, rate of heat transfer and also critical Eckert numbers when the fluid is either heat generating or when the fluid is heat absorbing. During the course of the analysis, it is observed that in the existence of viscous dissipation, fluid temperature could be enhanced with the increase in heat generation parameter while it decreases with heat absorption parameter. In addition, the rate of heat transfer at the hot wall can be increased by simultaneously increasing the Prandtl number and the rotation parameter in presence of heat source. Furthermore, it is interesting to mention that increase in heat absorption in the existence of viscous dissipation causes an increase in heat transfer from the fluid to the hot wall ($E>E_c$) while the contrast is true for heat absorption ($E<E_c$).

Nomenclature
Pr	=	Prandtl number
b	=	channel width
Cp	=	specific heat at constant pressure
u’	=	dimensional primary fluid velocity
W’	=	dimensional secondary fluid velocity
u′	=	dimensionless velocity component in $x\prime$ direction
$w$	=	dimensionless velocity component in $z\prime$ direction
$F$	=	dimensionless complex velocity
$\overline{F}$	=	dimensionless conjugate complex velocity
$T$	=	dimensionless temperature
$a^2$	=	rotation parameter
$E$	=	Eckert number
$S$	=	heat generation/absorption parameter
$T\prime$	=	dimensional temperature of the fluid ($K$)
$T_0$	=	temperature ($K$)
$T_1$	=	temperature at $y=0\left(K\right)$
$T_2$	=	temperature at $y=b\left(K\right)$

Greek letters
$\beta$	=	coefficient of thermal expansion
${\gamma }_s$	=	ratio of specific heat
$\mu$	=	dynamic viscosity
$\rho$	=	density ($kg/m^3$)
$\Omega$	=	angular velocity ($\mathit{radians}/s$)
$\alpha$	=	thermal diffusivity
$\upsilon$	=	momentum diffusivity (m2/s)

Disclosure statement

No potential conflict of interest was reported by the authors.