Since the present analysis aims to find out the effect of viscous heating as produced by the rotation of the inner cylinder on the underlying thermal transport in an asymmetrical heating environment, we vary the temperature distribution, Nusselt number and the entropy-generation number under different conditions. In this study we have depicted the variation of all numbers and the temperature for different values
and
. We have clearly specified the different parameters considered in plotting a figure in the respective figure caption as well as in the text. However, we consider
and
, which are typically used in the literature [
7,
8,
15,
24,
28]. The Brinkman number is defined in this study as follows:
. If we assume water is used as a fluid in this analysis, and the change in temperature
, then the typical value of Brinkman number, for
,
,
and
, becomes
. Moreover, we have considered
in the range of 1.2 to 2. Thus, for
, the minimum gap of the annulus becomes ~2000 μm. However, in the present study, we have considered a higher value of
to investigate the effect of viscous dissipation on the underlying thermo-hydrodynamics.
3.1. Variation of Temperature in the Flow Field
Figure 2a–c shows the variation of temperature distribution inside the annulus for different degrees of asymmetrical wall heating paremeter
.
Figure 2a–c corresponds to
, −0.5 and 1, respectively. Note that
indicates a symmetrical heating condition in the present scenario. We have considered
in this analysis only to demarcate the effect of thermal asymmetry on the thermal transport characteristics of heat as manifested clearly by the variation of temperature distribution in the flow field.
For each case of wall heating, we have considered three different values of
to obtain the temperature profile as influenced by the shear-heating generated by the rotating cylinder. We should mention here that viscous dissipation always generates a distribution of heat source and stimulates the internal energy in the fluid, which in turn distorts the temperature profile as envisaged from the figures under present consideration. In fact, a strong influence of the viscous heating effect on the temperature distribution in the flow field is supported by an observation that, in cases with
, the profile of the dimensionless temperature is altered in comparison with the case having
, although the imposed boundary condition on the walls remains invariant. In the thermal entrance region, a linear trend of developing dimensionless temperature
is observed for all the cases of wall heating with
, which is a pure conduction profile. It can be observed from the figures that the nonexistence of viscous dissipation makes the temperature distribution independent, irrespective of the asymmetry of the wall heating considered in the analysis. Two points can be made about
Figure 2 as follows: first, positive values of
are compatible with the wall-heating case, which resembles the situation of heat transfer to the fluid across the wall. Therefore, for cases with positive values of
, the fluid temperature is increased, as seen from
Figure 2a–c. On the contrary, negative values of
represent the wall-cooling case and weaken the fluid temperature by transferring heat from the fluid to the wall. We can confirm the reduction in fluid temperature for negative
in the present figures as well. Second, the fluid temperature, for positive
, in the region closer to the inner cylinder shows an increasing trend for
and 1, respectively, which one may see from
Figure 2a,c. In this context, the rotation of the inner cylinder deforms the surrounding fluid layer, leading to the development of shear heating in that region. We mimic the rotation-induced development of shear heating by the positive values of
. Therefore, shear heating, stemming from the rotational effect of the inner cylinder, heats up the fluid in that region and results in an increase in the fluid temperature as clearly reflected in
Figure 2a,c. Having a closer look at
Figure 2a–c, we also observe a quantitative change in the fluid temperature with a change in
, albeit the parameter
remains unaltered. Also, it is worth noting from
Figure 2c that the variation of temperature with
becomes perfectly symmetrical. We attribute these two observations to the the effect of the degree of the asymmetrical-heating parameter on the thermal-transport characteristics of heat. However, the variations in the temperature profile with different values of Brinkman number, as observed in the above figures, are in tune with what it is expected and show similarity with reported results [
7,
28,
29].
The temperature inside the annulus is highly non-uniform, with this non-uniformity stemming from two important effects: one is the effect of thermal asymmetry, and the second is the effect of the continuous rotation of the inner cylinder. Taking these two important issues into account, we calculate the mean temperature for a fair assessment about the heat transfer rate at the walls of the annulus. Accordingly, we show in
Figure 3a,b the variation of bulk mean temperature
versus
(aspect ratio of the annulus) in the annulus for different values of
, obtained for two different degrees of asymmetrical heating
and
, respectively. However, one may note from the above figures that the profiles of the dimensionless bulk mean temperature with
are strongly dependent on the degree of asymmetry in wall heating
. Since at
, the solution becomes undefined (the effect of the point of singularity), we vary
in the range
, while plotting
Figure 3a,b as delineated above. We observe that the mean temperature for
becomes almost positive in the annulus, whereas it becomes negative for
. The behaviour of the mean temperature as seen in
Figure 3a,b is expected, since positive
mimics a situation of heating up the fluid, and so increases the mean temperature. By contrast, negative
weakens the fluid temperature, thus resulting in a negative mean temperature, as clearly reflected in
Figure 3b. One may also note from
Figure 3a,b that the variation of mean temperature with
, in the region closer to the inner cylinder, exhibits a step gradient. This observation once again signifies the effect of shear heating developed due to the rotational effect of the cylinder on the underlying thermal transport. It is interesting to observe from
Figure 3a,b that, for any non-zero values of
, the variation of
with
obtained at
are symmetrical about the profile of the same order obtained at
, while we observe an asymmetrical variation for
. Moreover, one can clearly see from
Figure 3a,b that the variation of
with
, for all degrees of thermal asymmetry considered, becomes independent of
near the outer cylinder. We attribute these observations to the inequality of the wall temperature of the annulus together with the effect of viscous heating induced by the rotation of the inner cylinder.
3.2. Variation of the Nusselt Number
Here we discuss variation of the Nusselt number, which gives the quantitative estimate of the heat transfer rate at the walls of the annulus, for the different physical parameters considered in this study. We depict in
Figure 4a–d the variation of the Nusselt number at the inner cylinder graphically for two different values of the Brinkman number
, as considered in this study. Note that
Figure 4a,b correspond to the cases of asymmetric wall heating
, while in
Figure 4c,d we consider
to obtain the desired variation. Equations (19) and (20) represent the expression of the Nusselt number on the inner cylinder and outer cylinder, respectively. One may clearly observe from Equations (19) and (20) that both Nusselt numbers are functions of two independent variables, e.g., the degree of asymmetry in the wall heating
and the Brinkman number
. However, both the Nusselt numbers will have parametric variation with
for
and with
for
.
Note that for
, the variation of
with
is not continuous for all cases of the degree of asymmetry considered in the study (see
Figure 4a,c); rather, for each case the variation of the Nusselt number exhibits a discontinuity in behaviour, leading to a point of singularity at different
. As mentioned, the positive
represents a wall-heating case, which mimics a situation of transferring heat from the wall to the fluid. It needs to be mentioned here that shear heating generated by the rotational effect of the inner cylinder increases the fluid temperature too. Since the fluid temperature, in the region closer to the inner cylinder increases owing to induced viscous heating, the variation of
in this region becomes flat, signifying no driving temperature difference for heat transfer. Thus, for a positive value of
, the system encounters a situation where the temperature within the fluid becomes equal to the wall temperature at that prevailing thermal-boundary condition, which in turn gives rise to an unbounded swing, as apparent from
Figure 4a,c. Careful observation of
Figure 4a,c reveals that the location of the onset of the point of singularity changes with the alteration in the degree of asymmetrical wall heating, largely attributed to the thermal condition (temperature) of the wall. We should mention here that the onset of the point of singularity is an outcome of the local equilibrium of energy wherein the internal heat generation due to viscous dissipation balances the heat supplied by the wall. A closer scrutiny of
Figure 2a confirms the appearance of inflections on the variation of the temperature profile specific to the case of
and
. To be precise, the temperature profile exhibits global maxima and global minima for
and
, respectively, and at this point the supplied heat from the wall and the dissipative heat generated due to fluid friction become equal, causing no heat to flow in either direction and leading to the appearance of an unbounded swing, as seen in
Figure 4a,c.
By contrast, we observe an increasing trend on the variation of
for
even for all the cases of asymmetrical heating (see
Figure 4b,d). A negative value of
resembles a situation of wall cooling, i.e., transferring heat from the fluid to the wall. So an increasing trend of
for a negative value of
, as seen from
Figure 4b,d, once more signifies the effect of viscous heating generated by the rotational effect of the inner cylinder.
Figure 5a–d show the influence of
on the variation of the Nusselt number at the outer wall of the annulus
with
for different degrees of asymmetrical wall heating
and −0.5, respectively. We still observe the point of singularity on the variation of
for positive
irrespective of
. A closer look at
Figure 5a,c reveals that the location of the onset of the point of singularity on the variation of
is different from those appearing on the variation of
, primarily attributed to the effect of viscous heating as induced by the inner cylinder rotation. For both cases of wall heating, one may observe an increasing trend of
with
for
, as apparent from
Figure 5b,d. We can also observe from
Figure 5b,d that the rate of heat transfer at the outer wall is more for
, signifying the effect of thermal asymmetries on the underlying transport of heat. From the point of singularity, both the Nusselt numbers decrease in both directions and asymptotically reach at constant value at both the inner and outer walls of the annulus. An important observation can be made from
Figure 4 and
Figure 5 that, for the cases with
, the value of the Nusselt number on both the walls of the annulus does not match as
. This is primarily attributable to the rotation of the inner cylinder, which enhances the temperature of the fluid layer in the region close to the inner wall following a significant contribution to the viscous dissipation effect. The physical explanation for the negative Nusselt number, as seen from the figures presented above, stems from the appearance of inflections on the temperature profile, where the cross-sectional averaged fluid temperature becomes larger than the temperature imposed at the wall and hence heat transfer takes place in the reverse direction.
It can be seen that the rotation of the inner cylinder imposes additional shear stress on the fluid, which aggravates the viscous heating resulting in reduction in the heat transfer at the inner wall of the annulus. Additionally, from the figures presented above, one may find that the dependence of Nusselt numbers on
,
and
is important for
. In an effort to bring out the effect of
on the heat-transfer rate, we depict in
Figure 6 the variation of both the Nusselt numbers for the case of symmetrical wall heating
. We consider
in plotting
Figure 6.
It is important to observe from
Figure 6 that, in the region closer to the outer wall of the annulus,
is significantly larger than
even in a symmetrical wall-heating scenario. Such behaviour of the Nusselt number can be explained from the induced viscous-heating effect stemming from the rotation of the inner cylinder. The induced viscous heat increases the surrounding fluid temperature in the region closer to the inner wall of the annulus. On the other hand, away from the inner wall, the rotational effect of the inner cylinder is not felt so severely, leading to a relatively lower fluid temperature. Since the fluid temperature in the zone closer to the outer wall of the annulus is relatively lower, we find an augmentation in the heat-transfer rate due to the increased driving potential of temperature difference, as clearly reflected in
Figure 6.
Here we discuss the appearance of point of singularity on the variation of the Nusselt number from the second law of thermodynamics. In the context of micro-scale convective heat transport analysis, there could be a situation where the dissipative heat arising even for a small
becomes important, since the viscous heat significantly alters the heat-transfer characteristics over small scales [
11,
12,
13,
14]. It is imperative to mention that all thermodynamic devices working with any system should ensure the maximum possible heat transfer from the processes involved in order to increase system efficiency. By contrast, an enhancement in the rate of heat transfer imposes a constraint on the same as far as the thermodynamic irreversibility of the system is concerned, since an increased heat-transfer rate increases the system irreversibility as well. This contradictory behavior of heat-transfer enhancement with the increased rate of entropy generation makes it obligatory for the thermal systems/devices to compromise on the viscous dissipation dominant irreversibility in the process. To be precise, the onset of the point of singularity on the variation of the Nusselt number is essentially a reflection of such a minimization process, as discussed in the next paragraph.
In this analysis two different degrees of asymmetrical wall heating have been taken into account to obtain the variation of both the Nusselt numbers evident from the figures portrayed above. The irreversibility associated with the heat-transfer rate for two different degrees of asymmetry parameter, however, compromises on the viscous heating dominant irreversibility in the system to obtain the maximum possible heat-transfer rate in the process. However, this process of the minimization of viscous heating dominant irreversibility of the system as considered in the present study leads to the onset of the point of singularity, as observed by the variation of the Nusselt number for both the cases of asymmetrical wall heating.
3.3. Thermodynamic Irreversibility Analysis: Variation of the Entropy-Generation Number and the Bejan Number
We show the variation of the volumetric entropy-generation number versus
in
Figure 7a,b for different values of
, as considered in this study.
Figure 7a,b correspond to the degree of asymmetrical parameter
and −0.5, respectively. We also consider the following parameter in plotting the figures as:
. One may see from
Figure 7a,b that, for both the cases of wall heating, entropy generation at the inner wall of the annulus is higher compared to that of the outer wall. We also find that the entropy generation at the inner wall of the annulus becomes maximum and minimum for
and −0.1, respectively. By contrast, a reverse phenomenon occurs at the outer wall of the annulus. In fact, these observations hold true for both wall-heating scenarios, as apparent from
Figure 7a,b.
A positive
represents a wall-heating case, which indicates that heat is being transferred from the wall to the fluid. Since the rotation of the inner cylinder induces viscous heating in the surrounding fluid layer, the heat-transfer rate is reduced in that region owing to the diminished strength of the driving temperature difference potential. A relatively lesser heat transfer rate, intrinsic to
, reduces the entropy-generation rate in the region of the inner cylinder for both cases of asymmetrical wall heating, as seen from
Figure 7a,b. On the other hand, since negative
represents the wall-cooling case, heat transfer is enhanced in the region closer to the inner cylinder owing to the increasing strength of the driving temperature difference, which in turn culminates in an enhancement of the entropy-generation rate. We confirm this phenomenon in
Figure 7a,b as well. By contrast, we can observe a reverse scenario at the outer wall of the annulus and its surrounding zone. At the outer wall of the annulus, no source contributes viscous heating to the fluid, except that which arises due to the frictional effect of fluid layers; thus, the transfer and so the entropy-generation rate become higher for positive
(since for positive
the driving temperature difference for heat transfer becomes higher) for both the cases of wall heating. It is worth mentioning here that such behaviour of the entropy generation rate
with
inside the annulus gives rise to a distinct crossover at some
, as clearly visible from the figures being considered. However, closer scrutiny of
Figure 7a,b reveals a quantitative distinction in the entropy-generation rate inside the annulus for varying degrees of asymmetrical heating
, largely attributed to the heat-transfer rate. From the above discussion, it may be inferred that the effect of viscous heating takes a lead role in contributing the total entropy-generation rate in the inner wall of the annulus, while heat transfer influences the entropy generation rate surrounding the outer wall of the annulus considerably. We will discuss the relative contribution of the two important factors viz., the viscous dissipation effect and heat transfer on the total entropy-generation of the system being considered in the next section.
Figure 8a,b depicts variation of the Bejan number
with
for two different cases of asymmetrical wall heating
and −0.5, respectively. While plotting the Bejan number, we consider three different values of
to see the effect of viscous dissipation on the irreversibility-generation behavior. Note that all the values of
chosen in depicting the present figures (
Figure 8a,b) are in compliance with those considered in
Figure 7a,b. One can see that in the region closer to the inner wall of the annulus, viscous heating plays a dominating role over the heat transfer in governing the irreversibility for
, while the effect of heat transfer become crucial in contributing the thermodynamic irreversibility for
. This observation is true for both cases of wall heating, as evident from
Figure 8a,b. The observations reflected in
Figure 8a,b are in clear support of the behaviour of the entropy-generation rate as delineated in
Figure 7a,b.
Figure 9a,b shows the variation of the entropy-generation rate inside the annulus as influenced by the group parameter
. In plotting
Figure 9a,b, we consider three different values of the group parameter
, respectively. The other parameters considered for plotting the figures are
and
. Note the group parameter is an important parameter, which signifies the ratio of the effect of viscous heating and thermal asymmetry. The values of the group parameter chosen in depicting the present figure are those typically used in the literature [
24,
30].
A closer look at
Figure 9a reveals that the entropy generation number shows a continual decreasing trend with
for
, while the variation becomes almost flat for
. We can also observe from above figure that, in the inner wall and its surrounding region, the entropy-generation number is higher for a relatively higher value of the group parameter. An increase in the value of
, for a given
(
), indicates a decrease in
. Therefore, with an increasing value of
the heat-transfer rate at both the walls of the annulus drops for a given degree of asymmetrical parameter, which in turn lessens the heat-transfer dominant irreversibility in the system as well. Thus, the increasing rate of entropy generation in the region closer to the inner wall as seen from
Figure 9a is largely attributed to the effect of the viscous heating developed due to the rotational effect of the cylinder. We confirm this in
Figure 9b as well, where the Bejan number
is seen to be almost zero for
, signifying the leading role of the viscous-heating effect in inviting total irreversibility into the system. However, with increasing
, the rotation-induced frictional effect decreases, which in turn decreases the entropy-generation rate, as confirmed in
Figure 9a as well. In precise terms, at the outer wall and its surrounding region heat transfer takes a lead role in inviting irreversibility, as largely confirmed in
Figure 9b, where the Bejan number becomes closer to one.