Entropy optimized dissipative flow of effective Prandtl number with melting heat transport and Joule heating
Introduction
The boundary layer flows for both viscous and non-Newtonian fluids over a stretching sheet have remarkable involvement in many fields of interests. Examples of such processes include of polymer sheet industries, cooling processes in many industries, glass sheets, pharmacology, biotechnology, nuclear research, glass sheet industries, drawing of plastic sheets, polymers extrusion, cooling processes of melting materials etc. In all such cases the flow situation is of vital importance. The quality of end product mainly depends upon the rate of heat transfer, skin friction and wall shear stresses etc. Such processes mainly depend on the boundary layer phenomenon along stretching surfaces as well as the mass transfer rate. Rajagopal et al. [1] pioneered the work by studying viscoelastic fluid flow phenomenon caused by a stretching surface. MHD flow by vertical plate is examined by Riley [2] Fairbanks and Wike [3] analyzed effect of chemical reaction in a uniform fluid flow over a horizontal sheet. Andersson et al. [4] studied chemically reactive stretched flow. Chemically reactive flow of MHD Jeffrey fluid in the presence of heat source/sink is examined by Mohanty et al. [5]. Magyari and Keller [6] examined the exact solutions of boundary layer flows induced by stretching walls. Magyari and Keller [7] are also discussed the heat transfer analysis of steady flow over a moving sheet. Squeezed flow of revised Fourier heat conduction in second grade fluid with heat source/sink is analyzed by Hayat et al. [8] Mixed convective flow of nanofluid towards a stretching sheet is discussd by Tian et al. [9] Mahanthesh et al. [10] considered the numerical study for MHD flow of nanoliquid over a bidirectional extended surface. Hayat et al. [11] inspected stretched flow of Cattaneo-Christov heat flux with variable thicked sheet. Khan et al. [12,13] considered entropy generation, homogenous-heterogeneous reactions, Joule heating, slip conditions and nonlinear mixed convection in stretched flow of liquids. Heat transport in nonlinear stretched flow with thermal slip and radiation is scrutinized by Khan et al. [14].
Nanomaterial has innovative characteristics that make them more beneficial in numerous applications in heat transmission like medicinal procedures, fuel cells, hybrid-powered engines, domestic refrigerator, heat exchanger and automobile thermal management.Study of nanofluids with melting heat transfer past a stretching surface is examined by many researchers. It is extensive use in many industrial processes. The concept of nanofluid was put forwarded by Choi [15]. Turkyilmazoglu [16] addressed the analytical solution for heat and mass transfer of MHD slip flow in nanoliquid. Entropy optimized dissipative flow of cubic autocatalytic chemical reaction near stagnation point of Sisko nanofluid is considered by Khan et al. [17] Gul et al. [18] investigate the effective Prandtl number model influences on the Al2O3 − H2O and Al2O3 − C2H6O2 nanofluids spray along a stretching cylinder. Nasir et al. [19] studied three-dimensional rotating flow of MHD single wall carbon nanotubes over a stretching sheet in presence of thermal radiation. Impact of the Marangoni and thermal radiation convection on the graphene-oxide-water-based and ethylene-glycol-based nanofluids is derived by Gul et al. [20] An analysis on the accuracy of statistical declaration and probable in two phase nanomaterial radiative flow is studied by Hayat et al. [21] Dissipative rotating flow with Ohmic heating is explored by Hayat et al. [22] Gireesha et al. [23] discussed melting heat transport in stagnation point flow with induced magnetic flied over a stretchable surface. Hayat et al. [24] analyzed magneto-Ferroliquid flow with melting phenomenon through a porous space.Appropriate transformations are utilized to tackle the problems of flows analysis in the aforementioned situations for solving the obtained differential system. Melting parameter and Prandtl number are the two,main factors used to study in present flow. The importance of γAl2O3 nanofluids in cooling processes is given through the following studies [[25], [26], [27], [28], [29]]. Vishnu Ganesh et al. [30] and Rashidi et al. [31] examined enhancement of thermal properties of γAl2O3 through similarity solutions of nanofluids flow past a stretching sheet. More increase in thermal properties was shown experimentally in the above studies. Hayat et al. [32] discussed entropy generation in comparative study of both the presence and absence of effective Prandtl number considering γAl2O3 − H2O and γAl2O2 − C2H6O2 nanomaterials. Sumaira et al. [33] worked on the entropy optimized nonlinear dissipative flow by a rotating disk with thermal radiation. Nanomaterial flow of third grade fluid with heat source between two rotating disks is scrutinized by Hayat et al. [34] Sumaira et al. [35] analyzed dissipative flow of Williamson liquid between two disks with entropy generation. Hayat et al. [36] studied Darcy-Forchheimer CNTs based bidirectional flow with irreversibility.
Motivation here is to study mixed convective flow of MHD nanofluids over a stretchable surface. Melting heat transfer, viscous dissipation and Joule heating conditions are studied. Effective Prandtl number has been discussed for temperature distribution. Entropy has been optimized through various parameters. Graphical results for rate of entropy and Bejan number have been presented. System of equations are tackled numerically through built-in-ND solve technique [37]. Main observations are concluded.
Section snippets
Modeling
Consider flow of (γAl2O3 − H2O and γAl2O3 − C2H6O2) nanofluids. Various assumptions considered in mathematical model are as fallows;
- (1)
Here we assume that uw = ax be the stretching sheet velocity (where a is a positive constant).
- (2)
Induced magnetic field for small magnetic Reynold number is omitted.
- (3)
We consider dissipation and Joule heating in energy equation.
- (4)
Temperature of melting surface (Tm) is less than free stream temperature (T∞).
- (5)
Entropy generation effects is considered.
- (6)
Mixed convection is
Thermophysical characteristics of nanomaterials
The relationships for effective dynamic density (ρnf), heat capacitance (ρcp)nf, thermal expansion ((ρβ)nf), effective electric conductivity , dynamic viscosity , Effective Prandtl number and thermal conductivity of nanofluid is are [[38], [39], [40], [41]]:
Dimensionless expressions
Considering the appropriate transformations
The momentum and energy equations for both (γAl2O3 − H2O and γAl2O3 − C2H6O2) nanofluids take the following forms
Skin friction
Mathematical expression for skin friction iswhere (τw) is defined as
Insertion of Eq. (32) into Eq. (31) leads to
Heat transfer rate
Mathematically we havewhere qw is expressed as
Solving Eqs. (34) and (35) we have
Entropy modeling
Entropy rate is the ratio of volumetric entropy rate (Eg) and characteristic entropy rate ((Eg)0) i.e.,
The dimensionless form of entropy generation for both (γAl2O3 − H2O and γAl2O3 − C2H6O2) nanofluids are
In
Discussion
This portion is designed for showing effects of various involved variables for velocity, temperature distribution, skin friction, entropy and Bejan number via Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12. Here presence and absence of effective Prandtl number are discussed in all Figs. It is clearly shown that presence of effective Prandtl number is more advantageous for both γAl2O3−water and γAl2O3−Ethylene glycol nanofluids. Effect of
Final remarks
Following main points are worthmentioning
- •
Increase in velocity field for γAl2O3 − H2O and γAl2O3 − C2H6O2 nanofluids is noted for larger nanoparticles volume fraction and melting parameter.
- •
Entropy increases for Re but opposite responce is observed for Bejan number.
- •
Heat transfer rate increases for larger magnetic and melting parameters.
- •
Nusselt number enhances for higher Eckert number.
- •
Skin friction for magnetic parameter is increased.
Declaration of Competing Interest
None.
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