This article offers an analysis of entropy generated in mixed convection due to a vertical stretching cylinder placed in heat-generating fluid-saturated porous medium. The governing system of partial differential equations is subjected to a similarity transformation that results in boundary value problem (BVP) comprising ordinary differentials. The BVP solved numerically by Runge-Kutta integration scheme supplemented with shooting method provides momentum and thermal regimes which are readily exploited to enumerate entropy. The entropy distribution is traced and analysed for various parameters’ values.

#### Keywords

- mixed convection
- stretching cylinder
- porous media

#### MSC 2010

- 76S05
- 76E06

Theoretical/experimental studies on heat transfer in fluids due to stretching surfaces are available in the literature. These studies were prompted by a wide array of industrial applications such as fibre production, wire drawing, metallic plates cooling, extrusion processes, manufacture of plastic, glass fibre production, and so on. The thermo-fluidics involving a stretching surface has been much investigated for a variety of assumptions.

Flow in the boundary-layer on a continuous solid surface was first pioneered by Sakiadis [1] considering the two-dimensional, axisymmetric boundary-layer flow over a flat surface moving with a constant velocity. Due to the entrainment of ambient fluid, this phenomenon represented a boundary-layer problem differing from Blasius flow over a semi-infinite flat plate. Crane [2] considered flow over a moving strip with velocity proportional to the distance from the slit and obtained closed-form exponential solution. Afzal and Varshney [3] examined the cooling of a low heat resistance stretching sheet moving through a fluid. Chiam [4] examined micropolar fluid flow over a stretching sheet. Gupta and Gupta [5] examined heat and mass transfer over an isothermal stretching sheet with suction or blowing. Banks [6] devised a similarity solution of the boundary-layer equations for the stretching wall. Grubka and Bobba [7] extracted solutions in terms of Kummer's function for the phenomena constrained to prescribed wall temperature and heat flux. Chen and Chen [8] investigated the problem for viscoelastic fluid and reported solutions in terms of Kummer's function. Chauhan and Vyas [9] examined heat transfer in MHD viscous flow due to the stretching of boundary in the presence of a naturally porous bed. Chiam [10] studied hydromagnetic flow over a surface stretching with power-law velocity. Anderson and Valnes [11] investigated ferro-fluid flow over a stretching sheet in the presence of a magnetic dipole to explore the effects of the magneto-thermo-mechanical interaction on skin friction and heat transfer. Boutros et al. [12] considered a steady two-dimensional boundary-layer stagnation point flow towards a heated stretching sheet placed in a porous medium and exploited Lie group method for solving the problem. Mjankwi et al. [13] investigated variable properties effects on unsteady nanofluid flow over a stretching sheet. Vyas and Rai [14], Vyas and Ranjan [15], Vyas and Srivastava [16] and Vyas and Srivastava [17] examined various configurations involving the stretching surface. Grosan and Pop [18] reported an axisymmetric mixed convection boundary-layer flow of nanofluid past a vertical cylinder. Hayat et al. [19] investigated mixed convection flow of a Casson nanofluid over a stretching sheet with convective heated reaction and heat source/sink.

It is pertinent to record that these studies and references contained therein pivoted on the first law of thermodynamics only. That is, these systems got treated primarily for heat transfer aspects and the thermodynamic irreversibility involved was not looked into. However, from physical understanding we know that the quantification of entropy is instrumental in optimal design. The quantification of entropy is facilitated through a second law analysis that helps to understand the energy losses in the system.

Bejan [20,21], in his seminal works, created a breakthrough in entropy management in thermo-fluidics. This opened a new horizon in thermal systems where entropy generation analysis was well received, given its wide spectrum of utility. Butt and Ali [22] reported entropy generation due to a stretching cylinder. Das et al. [23] discussed entropy due to convective heating of the stretching cylinder. Vyas and Khan [24] reported entropy generation distribution for Casson fluid flow caused by a stretching cylinder. Vyas and Soni [25] analysed entropy generation for boundary-layer flow of fluid due to the melting stretching sheet. Vyas and Ranjan [26] examined the entropy analysis of radiative MHD forced convection flow in a porous medium channel with weakly temperature-dependent convection coefficient. Vyas and Srivastava [27] studied entropy analysis of generalized MHD Couette flow inside a composite duct with asymmetric convective cooling. Mukhopadhaya and Ishak [28] discussed mixed convection flow in a thermally stratified medium along with a stretching cylinder. Kumam et al. [29] presented entropy generation in the MHD flow of CNTs Casson nanofluid in rotating channels.

It is expected that the work presented here would be a basis for future explorations wherein the configuration may be a part of larger systems to be designed and simulated.

We consider an axisymmetric, steady 2-D heat-generating fluid flow over a stretching cylinder placed vertically in fluid-saturated porous medium. The fluid is uniformly heat generating. We assume that the surface of the cylinder remains at constant temperature _{W} and the temperature far away from the cylinder's surface is _{∞} where _{W} > _{∞}. The cylinder stretches with a velocity in a fashion that its surface velocity

The governing equations under boundary-layer assumptions read:
_{o} is the characteristic velocity, and

We prescribe the similarity transformation

The non-linear boundary value problem (BVP) described by Eqs (8)–(11) has been solved numerically using the fourth-order Runge-Kutta method along with the shooting method. To employ the shooting method, the BVP is reduced to following initial value problems:

The aforementioned system of IVP is solved with the proper guess values _{0} and _{0} for the unknown quantities ^{″}(0) and ^{′}(0) respectively, such that the end conditions ^{′}(∞) = 0 and _{1} and _{1} for ^{′}(∞) and ^{−6} magnitude together with grid space Δ

The local volumetric rate of entropy generation _{G} for the configuration under consideration is given as:

Consequently, entropy generation number _{S} using Eqs (19) and (20) reads as

The Bejan number

As evident from the expression for entropy number _{S} we need distributions for velocity and temperature and respective gradients to compute _{S}. The BVP governing momentum and thermal regimes have been solved numerically by the fourth-order R-K integration scheme supplemented with the shooting method. In this strategy, the BVP is reoriented to a system of initial problems where unknown gradients are guessed in an iterative procedure until the convergence is achieved.

The numerically computed values were then utilized to compute the entropy generation number. The plots for the quantities of interest were portrayed graphically to have an insight of the phenomenon.

Figure 1 displays that increasing Brinkmann number Br (which is a measure of dissipative effect) leads to a rise in entropy generation number. Here, it should be noted from the expression of entropy generation number _{S} that Br features in _{2} with curvature parameter

Before analysing Bejan number Be (0 ≤ Be ≤ 1), it is not out of place to recall that Be is the ratio of heat transfer irreversibility and irreversibility due to heat transfer and dissipation. When Bejan number vanishes, it means that there is no thermodynamic irreversibility due to heat transfer. Be being unity signifies the absence of irreversibility due to dissipation. However, Be ≥ 1/2 signifies the situation when heat transfer irreversibility contribution to total entropy is greater than or equal to that of fluid friction irreversibility. Figures 7–12 display that Bejan number has one thing common, that is, Bejan number Be attains maxima and or minima at various spatial distances depending upon the parameters values. These figures also reveal that Bejan number Be is larger at far away distances from the stretching surface compared to that of close to the region adjacent to the stretching surface. Furthermore, we also observe that the choice of selected parameters values Be ≤ 0.5.

Figure 7 displays that increasing values of Brinkman number reduce Bejan number at any spatial distance. However, Be becomes zero at

The entropy generation analysis for mixed convection in a vertical stretching cylinder embedded in a porous medium has been undertaken. Momentum and thermal regimes were obtained for velocity and temperature distributions and respective gradients. These quantities were utilised to compute the entropy generation number and Bejan number. The plots of quantities of interest show the qualitative and quantitative impact of parameters entering into the problem which has been discussed in the preceding section. The values of parameters chosen here are just representative and the analysis is very much an attempt to display that entropy management can be achieved by selecting parameters wisely. We expect that the work would be a formidable baby model to be extended for a much larger configuration wherein the findings could provide substantial insight.

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