This study investigates the inherent irreversibility associated with the Couette flow of a reacting variable viscosity combustible material under Arrhenius kinetics. The nonlinear equations of momentum and energy governing the flow system are solved both analytically using a perturbation method and numerically using the standard Newton Raphson shooting method along with a fourth-order Runge Kutta integration algorithm to obtain the velocity and temperature distributions which essentially expedite to obtain expressions for volumetric entropy generation numbers, irreversibility distribution ratio, and the Bejan number in the flow field.

In fluid dynamics, Couette flow refers to the laminar flow of a viscous fluid in the space between two parallel plates, one of which is moving relative to the other. This type of flow is named in honor of Maurice Marie Alfred Couette, a Professor of Physics at the French University of Angers in the late 19th century [

From the application point of view, the determination of thermal criticality in a flow system is extremely important. For example, special attention must be paid to the heating of lubricant by the frictional force and Arrhenius kinetics since viscosity is temperature dependent. Thermal criticality occurs when the rate of heat generation within the flow system exceeds the heat dissipation to the surroundings [

Meanwhile, efficiency calculation of heat exchange systems has been very much restricted to the first law of thermodynamics. Calculations using the second law of thermodynamics, which is related to entropy generation, are more reliable than first law-based calculations. Therefore, the second law of thermodynamics can be applied to investigate the entropy generation rate in the flow system due to fluid friction and heat transfer. The determination of entropy generation is also important in upgrading the system performance because the entropy generation is the measure of the destruction of the available work of the system [

In this study, the variable viscosity reactive Couette flow is considered and the inherent irreversibility together with thermal criticality in the flow system is investigated. The plan of this paper is as follows: in Section

The configuration of the problem studied in this paper is depicted in Figure

Schematic diagram of the problem.

Under these conditions the continuity, momentum, and energy equations governing the problem in dimensionless form may be written in Cartesian coordinate

Due to the nonlinear nature of the velocity and temperature field equations in (

and so on. The above equations for the coefficients of solution series are solved iteratively for the velocity and temperature fields, and we obtain^{-4}). However, using Hermite-Padé approximation technique, we have extended the usability of the solution series beyond small parameter values as illustrated in the following section.

From the application point of view, it is extremely important to determine the appearance of criticality or nonexistence of steady-state solution for certain parameter values. In order to achieve this, we first derived a special type of Hermite-Padé approximant. Let

The numerical technique chosen for the solution of the coupled ordinary differential (^{-7}) of the results obtained is achieved.

Flow and heat transfer processes between two parallel plates are irreversible. The nonequilibrium conditions arise due to the exchange of energy and momentum within the fluid and at solid boundaries, thus resulting in entropy generation. A part of the entropy production is due to the heat transfer in the direction of finite temperature gradients and the other part of entropy production arises due to the fluid friction. The general equation for the entropy generation per unit volume is given by [

We emphasize here that an increase in the parameter value of ^{-8}.

Comparison between analytical and numerical results (

0 | 0 | 0 | 0 |

0.1 | 0.02360044943 | 0.02360039520 | 5.423 ^{-8} |

0.2 | 0.04208380022 | 0.04208374384 | 5.638 ^{-8} |

0.3 | 0.05535532038 | 0.05535525228 | 6.810 ^{-8} |

0.4 | 0.06334613033 | 0.06334604947 | 8.086 ^{-8} |

0.5 | 0.06601440811 | 0.06601432697 | 8.114 ^{-8} |

0.6 | 0.06334613023 | 0.06334604947 | 8.076 ^{-8} |

0.7 | 0.05535532037 | 0.05535525228 | 6.809 ^{-8} |

0.8 | 0.04208380020 | 0.04208374384 | 5.636 ^{-8} |

0.9 | 0.02360044944 | 0.02360039520 | 5.424 ^{-8} |

1.0 | 0 | 0 | 0 |

The Hermite-Padé approximation procedure in Section

Computations showing the criticality procedure rapid convergence (

2 | 4 | 5.0854548671499 | 3.9528766995579 |

3 | 8 | 5.0849831249732 | 3.9528312115207 |

4 | 13 | 5.0849831815807 | 3.9528312148390 |

5 | 19 | 5.0849831815664 | 3.9528312148383 |

6 | 26 | 5.0849831815664 | 3.9528312148383 |

Computations showing thermal criticality for different parameter values.

0 | 0.1 | 0.15 | 0.2 | |
---|---|---|---|---|

Nu | 4.0000000000000 | 5.0849831815664 | 6.0215731738934 | 7.717815638192141 |

_{c} | 3.51383071912516 | 3.9528312148383 | 4.2506038264647 | 4.647918009128950 |

The results in Table

The velocity profiles are reported for increasing values of

Velocity profile:

Velocity profile:

Typical variations of the fluid temperature profiles in the normal direction are shown in Figures

Temperature profile:

Temperature profile:

A slice of the bifurcation diagram for

A slice of approximate bifurcation diagram in the (

Figures

Entropy generation rate:

Entropy generation rate:

Figures

Bejan number:

Bejan number:

The evaluation of the entropy production rates for variable viscosity reactive Couette flow was carried out using both analytical and numerical techniques. Solutions are obtained for fluid velocity and temperature profiles. Using a special type of Hermite-Padé approximation technique, we obtain accurately the thermal criticality conditions and the solution branches. The volumetric entropy generation rate and the Bejan number depend on fluid viscosity variation and activation energy parameter (

The second author would like to thank the National Research Foundation of South Africa Thuthuka programme for financial support.