Irreversibility analysis in Darcy-Forchheimer flow of nanofluid by a stretched surface

: The aim of this articles is to investigate the entropy optimization in unsteady MHD flow Darcy-Forchheimer nanofluids towards a stretchable sheet. The surface we tend to think about is porous and stretchy under acceleration. Flow occurs due to the stretching of the surface. Four distinct types of aqueous nanostructures are taken in this examination where copper oxide ( C u O ), copper ( C u ), titanium dioxide ( 2 T i O ) and aluminum oxide ( 2 3 A l O ) are the nanoparticles. Irreversibility analysis are discussed through second law of thermodynamics. The expression of energy is mathematically designed and discussed according to heat generation / absorption, dissipation, thermal radiation, and joule heating. The nonlinear PDE (partial differential conditions) is first changed to ODE (normal differential conditions) through the use of appropriate similarity variables. Here we used the numerically embedded solution technique to develop a numerical result for the obtained nonlinear flow expression. Influence of various flow parameter velocity temperature distribution and entropy generation are discussed. Reduction occurs in velocity profile for larger porosity and magnetic parameters. An enhancement in entropy generation and temperature distribution is seen for Brinkman number. An opposite effect is noticed in velocity and temperature through solid volume friction.

(EG) depends mainly on the following factors: heat transfer, chemical reactions, mass transfer, power transmission, viscosity losses and simultaneous effects of thermal and solutal. Bianco et al. [25] investigated the entropy generation of turbulent conductive flow water nanofluids in a tube in which constant wall heat flow conditions, in which they reported that for low concentrations of nanoparticles, the total entropy generation could be minimized. The stagnation point flow of MHD nanomaterials with minimal entropy generation on a stretched surface is explored by Rashidi and Bhatti [26]. Some modern irreversibility problem are mentioned in Refs. [27][28][29][30][31][32][33][34][35][36][37]. The theme of this article is to investigate the entropy generation in time dependent Darcy-Forchheimer flow towards a stretch surface. Energy expression are modeled through joule heating, heat source/sink and dissipation effects. Physical features of irreversibility analysis are examined thermodynamic second law. Ordinary system are obtained through similarity transformation. The obtained system are numerically solved through bvp4c Matlab software. Influence of various flow variables on velocity field temperature distribution and entropy generation rate. Thermo physical behaviors of nanolfluid and base fluid are highlighted in Tables 1 and 2.

2: Modeling
Consider magnetohydrodynamics time dependent Darcy-Forchheimer flow towards a stretchable surface. Here water is considered as base liquid and four different type of nano particles (i.e. copper oxide ( CuO ), copper ( Cu ), titanium dioxide ( 2 TiO ) and aluminum oxide ( 23 Al O )) are considered. Energy expression are developed through joule heating, heat generation/absorption and dissipation effect. Physical description of irreversibility analysis are discusses through second law of thermodynamics.

3: Engineering Quantities
Velocity and temperature gradient is given by

4: Entropy Modeling
Mathematically it is expressed as 22   Here our main focus is on copper ( Cu ) nanoparticles.

5.1: Velocity field
Performance of velocity versus volume friction is revealed in Fig.2 Large volume friction reduces the velocity profile. Large magnetic parameter correspond to improve Lorentz force which augments the resistance to the liquid flow. Therefore velocity is diminished (see Fig.3). Fig.4 elucidates the variation of Forchheimer number on velocity. Velocity is reduced against Forchheimer number. Porosity parameter impact on velocity is illustrated in Fig.5.An increment in porosity variable corresponds to decays velocity field.

5.2: Temperature
Physical description of temperature versus volume friction variable ( ) is illustrated in Fig 6. An augmentation in temperature is enhanced for volume friction variable. Unsteadiness variable impact on temperature is portrayed in Fig.7. A reduction is seen in temperature distribution with variation of unsteadiness parameter () A . Significant effect of ( M ) on temperature is displayed in Fig.8.An improvement in temperature is noted for magnetic parameter. Physically larger magnetic variable augments the disturbance in liquid particles which improve collision among the liquid particles. Thus temperature is boosted. Fig.9 sketch to see the Brinkman number effect on temperature. Larger Brinkman number enhance the kinetic energy and as a result temperature is augmented.     Fig.10 sketch to show the effect of porosity variable on entropy rate. An enhancement in entropy rate is seen for porosity variable (  ). Fig.11 exhibit outcome of magnetic variable on entropy optimization. An augmentation in magnetic variable improve Lorentz force, which augments the collision amongst the nano particles. Therefore entropy optimization is boosted. An increment in Brinkman number leads to augments heat generation which is created by fluid friction and as a result entropy rate is boosted (see Fig 12). Influence of Reynold number ( Re ) on entropy generation is revealed in Fig.13 Higher Reynold number ( Re ) improve the entropy rate.

5.4.1: Skin friction coefficient
Fig. 14 sketch to show the variations of skin friction coefficient against A and  .
Here clearly noted that velocity gradient has opposite effect for A and  . Skin friction coefficient reduces versus higher magnetic parameter.

5.4.2: Nusselt number
An increment in magnetic parameter reduces the heat transfer rate (see Fig. 16). Influence of stretching parameter on Nusselt number is revealed in Fig. 17. An enhancement in stretching parameter improve the heat transfer rate.

6: Conclusions
Key observation of the present flow are given below  Reduction in velocity field is noticed through magnetic and porosity parameters.  Higher Forchheimer number and unsteadiness parameters leads to reduce the velocity profile.  An augmentation in temperature distribution and entropy generation are observed with variation of Brinkman number.  Larger Magnetic variable corresponds to augments the entropy generation rate and temperature distribution.  Temperature distribution is reduced with variation of unsteadiness parameter.  Similar characteristic for entropy generation rate is seen through porosity parameter.  Magnetic parameter has reducing effect on both velocity gradient and heat transfer rate.  An opposite effect is observed in temperature gradient and surface drag force for stretching parameter.

Declaration of Competing Interest
None.