Analysis of entropy generation with heat generation in time dependent hydromagnetic flow of nanofluid in an oscillatory semi- porous curved channel

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Introduction
The analysis of various fluid dynamics in tubes or semi-porous/ porous channels has obtained a lot of attention from many scientists and researchers over the last few decades owing to its 1 Corresponding Author. Tel.: +92 62 9255480 e-mail address: rana.m.naveed@gmail.com (M. Naveed) broad range of practical applications in mechanical and biomedical engineering. Such applications includes permeable or semi-permeable pipes processing, movement of blood in oxygenators, capillary blood flow, filters design and blood dialysis in synthetic kidney. Berman [1] initiated the first research on steady flow phenomenon of viscous fluid in porous channel. He provided an exact solution to the acquired Navier stokes equations. After Berman [1], both for viscous and non-Newtonian liquids, several researchers extended his concept in different directions. White [2] examined incompressible viscous fluid flow via an uniformly permeable channel. Abbas et al. [3] conducted a analytical study for the hydromagnetic boundary layer Maxwell fluid flow inside a porous channel. Raftari and Vajravelu [4] has computed analytical results for hydromagnetic viscoelastic liquid flow and heat transfer in a stretchable wall channel by means of homotopy analysis method. Ali et al. [5] performed an analysis of the hydromagnetic Oldroyd-B liquid flow and heat transport inside a channel. Abbas et al. [6] performed an analytical examination of Maxwell liquid motion in an axis-symmetric semi-permeable channel by incorporating perturbation approach. Heat transfer research for channel flow of MHD Jeffery liquid with generalized boundary conditions was reported by Aleem et al. [7].
The examination of flow through some kind of narrow, curved type channel has acquired high significance as a result of its multiple physical applications in many biomedical and industrial processes. Khuri [8] conducted a study for the Stokes motion in a curved channel. The impacts of forced convection and porosity on curved reciprocating channel flow were tested by Fu et al. [9]. Abbas et al. [10] proposed a numerical study for nonlinear transfer of thermal energy with Hall impacts in flow of viscous liquid inside a curved semi-permeable channel. Naveed et al. [11] examine the impact of thermal radiation and permeability material on motion of flowing liquid via a curved semi-porous channel. Sajid et al. [12] has computed joule heating impacts on magnetic nanoparticles through a semi-porous curved channel. The impacts of the applied constant magnetic field on the thermally heated flow of Carreau liquid within a curved channel was evaluated by Abbas et al. [13]. Numerical outcomes for heat transfer process in Powell-Eyring liquid flow through a curved channel with Cattaneo-Christov heat flux model was computed by Abbas et al. [14].
The research of oscillatory flows is a fundamental theory in the field of biological and engineering processes such as oil drilling, blood flow control during surgical treatment, oil exploration, lungs respiratory functions processing, manufacturing and processing of foods and papers, cosmetic products, chemical /blood dispensing modeling in biochemistry /clinical laboratories etc. Misra et al. [15] evaluated the heat transfer in viscoelastic hydromagnetic fluid flow in a stretchable wall oscillating channel. Ali et al. [16] examined time dependent oscillatory flow of viscoelastic fluid in a permeable channel with heat and mass transfer. Ali and Asghar [17] conducted analytical solution for Jeffery fluid flow in an oscillatory channel. Khan et al. [18] investigated the heat transfer characteristics in hydromagnetic Maxwell fluid flow within an oscillatory channel with Cattaneo-Christov heat model. Abbas et al. [19] detected the influences of linear thermal radiation in time dependent motion of flowing liquid across a convectively heated curved oscillating stretchable surface. Very recently, Imran et al. [20] evaluated the impacts of applied magnetic field and heat production in flow of ferrofluid past over a curved stretching oscillatory sheet.
The analysis of heat transfer in flow of nanofluids has attained considerable attention by the researchers due to its numerous utilizations in the fields, like space cooling, microelectronic cooling and modern generation of cooling technology. Fluids such as ethylene glycol, oil and water are examples of base fluids having low thermal conductivity. With the addition of certain nano-sized particles (less than 1%), the thermal conductivity of such base fluids can be improved and they form nanofluids. Choi [21] initially presented the idea of nanofluids. After that, by considering the impacts of thermophoresis and the Brownian motion of the nanoparticles, Buongiorno [22] introduced a new definition of nanofluids. Sheikoleslami et al. [23] addressed the analytical outcomes for MHD flow of nanofluid in a semi-porous channel. Naveed et al. [24] deliberated the consequences of Brownian motion and thermophoretic in the existence of thermal radiation for the Blasius motion of nanoliquid across a curved stretchable sheet. Alblawi et al. [25] has detected Buongiorno's nanoliquid model across a curved exponentially stretchable wall. Rashed and Ahmed [26] has investigated peristaltic flow of dusty nanofluids inside curved channel. Riaz et al. [27] investigated the heat transfer mechanism in peristaltic flow of nanoparticles through a curved channel with second order slip condition.
The main purpose of the present study is to examine the entropy production rate in presence of applied magnetic field on time dependent flow of nanofluid in a semi-porous oscillatory curved channel. The implications of heat production are also included in heat equation. The governing partial differential equations describing the flow phenomenon are highly complex and nonlinear in nature which is solved analytically by utilizing an efficient analytical technique called homotopy analysis method. The description of this article is as follows: Section 2 gives the mathematical development of the flow problem with appropriate boundary conditions; Section 3 gives the rate of entropy generation on the flow; Section 4 is all about analytical simulation in series form; Section 5 comprises of discussion of obtained results and Section 6 summarizes some concluding remarks.

Mathematical development
Consider hydromagnetic time dependent and boundary layer two dimensional motion of an incompressible nanoliquid inside the walls of curved semi-permeable channel that are separated by distance H coiled in a semi-circle of radius .
A The viscous fluid is injected through upper wall of the channel which is considered porous and lower wall is considered oscillatory. The lower oscillatory wall having temperature w T moves continually to and fro about origin with periodic velocity sin .
  In overhead equations, wand v are considered to be the velocity parts along s and x  directions,  the density, p the pressure, Where 0 w U  denotes injection and 0 w U  denote the suction velocity respectively.
We will now initiate the following non-dimensional variables as Via the application of Eq. (7) Eq. (1) is verified on identical basis, and remaining Eqs.
represents the parameter of Brownian motion respectively.
After removing the term of pressure from the Eqs. (8) and (9), the liquid velocity can be accomplished as With following boundary conditions The drag surface force, the transmission heat and mass rate along the curved wall is characterized as here, , r s  w q and w j are depicted by Making apply of Eq. (7) in Eqs. (14) and (15), we attain here, 1 2 Re s w U s   represents the Reynolds number.

Entropy Generation
The velocity of the liquid, concentration and temperature fields once accomplished can be used for the calculation of the entropy production rate in an oscillatory curved porous channel. Entropy production depicts the irreversible action of the mechanism induced by heat flow, electric conduction of nanofluid and fluid friction. Entropy production rate in dimensional form for magnetohydrodynamic flow of nanofluid inside an oscillatory curved semi-porous channel can expressed as   Applying Eq. (7), the non-dimensional form of Eq. (19) is Be

Solution methodology
The prime focus of this section is to briefly explain the method of homotopy analysis that we used to calculate the series solution of governing nonlinear PDEs (10-12) with final boundary conditions (13). For this depiction, as initial assumption and linear auxiliary operators for concentration, temperature and velocity field we take the following expressions as The General solution of the problem is of the form   (29)

Results and discussion
In this section, we concentrate on briefly explaining the outcomes of various involved parameters including non-dimensional radius of curvature       Fig.3. (a-c) and Fig.4 (a-c)

Conclusions
In this current study, we have investigated entropy production and the effects of heat production on viscous nanoliquid motion by considering Buongiorno's model in a semiporous curved oscillating channel. The analytical outcomes of the governed nonlinear partial differential flow equations are accomplished by implementing HAM. Consequences of unalike variables on concentration, velocity, temperature, Bejan number, skin friction coefficient, entropy production, local Nusselt number and on Sherwood number are established through tables and graphs and deliberated in details. The following specific conclusions are commented from present examination which are stated as