Numerical Treatment for the Second Law Analysis in Hydromagnetic Peristaltic Nanomaterial Rheology: Endoscopy Applications

: In current study, analysis is presented for peristaltic motion of applied magnetic field and entropy generation within couple stress (Cu/H 2 O) nanofluid through an endoscope. An endoscope contains two coaxial cylindrical tubes in which the internal tube is nonflexible while the external tube has sinusoidal wave passing through the boundary. Influences of mixed convection along with applied magnetic field are encountered as well. Formulated governing model is fabricated introducing long wavelength and creeping Stokesian flow approximation which are then analyzed numerically by utilizing Adams Bashforth method. For a physical insight, results are demonstrated to examine the behaviors of flow profiles and entropy generation number for emerging flow parameters with the help of graphs, bar-charts and tables.


1: Introduction
Researchers have gained much attention in the study of non-Newtonian fluid behaviors owing to its novel applications in physiology, industry and technological processes. Non Newtonian fluids possess nonlinear relationship among the rate of strain and shear stresses. Among the theories of non-Newtonian fluids, couple stress fluid theory is important one which is further a subclass of polar fluid theories, introduced by Stokes. Constitutive relation that describes the behavior of couple stress fluids encounters couple stresses along with classical Cauchy stress. Moreover, it is oversimplification of the conventional theory of Newtonian fluids which validates polar effects.
Such fluids include biological fluids, cosmetics, slurries and dairy wastes etc. Characteristically, Devakar and Iyengar [1] has been investigated flow dynamics of couple stress fluid configured inside two parallel plates. Geometries of the cylindrical pipes with slip wall conditions and analysis of couple stress fluid transport between the parallel surfaces have been obtained by Devakar et al. [2] and [3]. Srivastava [4] analyzed consequences of axially symmetric mild stenosis for blood transport presuming blood as couple stress fluid. In order to inspect the performance of rheological complex fluids, investigations pertaining couple stress fluid are incredibly constructive [5][6][7][8][9][10].
Furthermore, since several realistic fluids serve as couple stress fluids and owing to their remarkable applications in heat transfer fields, thermal characteristics can be amplified by suspending particles having nanometer size called nanoparticles pioneered by Choi [11]. For instance, Khan et al. [12] have been investigated couple stress nanofluid flow through an oscillatory stretching sheet presuming the impacts of mixed convection with heat generation/ absorption. Some remarkable applications regarding couple stress nanofluids are [13][14][15][16].
Peristaltic motion has extensive applications in engineering processes, physiology and industry. In many biological systems, peristalsis has become one of the major apparatus for the fluid transport, initially investigated by Engelman [17]. Recently, Hayat et al. [18] have been investigated impacts of convective conditions and nanoparticles on the peristaltic transport, simultaneously. Moreover, endoscope has many clinical applications. For medical recognition, the endoscope/annulus has important effects on the peristaltic flow. In cancer therapy, for desirable tissues removal; heat transfer is very extensively applicable. For instance, heat transfer in peristaltic flow through a vertical porous annulus has been presented by Vajravela et al. [19]. The closed form solution of a nanofluid for the peristalsis in an annular section has been presented by Shahzadi and Nadeem [20]. Entropy generation analysis in peristalsis of nanofluids due to complex flow structures has motivated the researchers. Entropy production in peristaltically occurred nanofluid flow has been analyzed by Hayat et al. [21]. The generation of entropy for couple stress fluid has been studied by Jangli et al [22]. Further studies for fluid flows with entropy generation analysis can be seen in references [23][24][25].
Magneto hydrodynamic explains the magnetic aspects of electrically conducting fluid and have numerous important usages in controlling the velocity of fluids by implementing magnetic field effects. Recently, Awan et al. [26] inspected numerically an unsteady hydro-magnetic nanofluid flow and heat transfer through channel. Simulation of computational fluid dynamics for suspension of nanoparticles in MHD liquid has been analyzed by Nawaz et al. [27]. Some ongoing researches can be seen through the references [28][29][30]. By utilizing the knowledge of pre-mentioned literature, the aim of current research is to solve numerically the flow and heat exchange for couple stress nanofluid with entropy generation in existence of applied magnetic field and viscous dissipation.
The influences of emerging flow parameters are studied and results are demonstrated through graphs.

2: Problem development and governing model
Assume the peristaltic motion of incompressible couple stress (Cu/water) nanofluids conducted in an endoscope. Along the tube wall, sinusoidal wave is transmitting with a uniform speed c.
Cylindrical coordinate structure (R, Z) is preferred where Z-axis is passing through the central line and R indicates the radial direction as depicted in Fig. 1a In which  is the wavelength, b is the amplitude of wave and t represents time of travelling wave.
Moreover, T1 and T0 denote temperature of internal and external cylinders, accordingly.
In which, m denotes the 1 3 trace of M ,  and 1  express coefficients of viscosity, C indicates vector of body couple while  and   stand for coefficients of couple stress viscosity. The inequalities constraints for these material constants are given as: In view of the above relations (3) and (4), the simplified equations in fixed frame of reference are [36]: In aforementioned model, P and T represent temperature and pressure of fluid whereas U and W express the R and Z components of velocity, respectively. Further, g expresses gravitational acceleration, 0 B indicates intensity of external applied magnetic field,  denotes dissipation function and 0 Q is heat generation parameter.
In the laboratory frame (R, Z), flow is unsteady. In order to obtain a steady flow, transformations of quantities from the laboratory structure (R, Z) to the wave structure (r, z) are [7,32]: Where, u and v denote the velocity components in the wave frame (r, z). Equations (5)-(8) yields: Where, constants involved in the above model are: With the help of long wavelength and creeping flow approximations equation (11) is identically fulfilled while equations (12)- (14) reduces to the following expressions: By differentiating equation (18) with respect to r, we get the following expression:

Result and Analysis
Results are demonstrated to examine the behaviors of flow profiles and entropy generation number for emerging flow parameters with the help of graphs, bar-charts and tables Variation in w(r) against M

Analysis of Entropy Generation
Entropy is the measure of molecular disorder. The volumetric entropy generation rate can be expressed as [31]: Moreover, characteristic entropy generation rate obtained by using boundary conditions in equation (23) is: By using equation (24) and dimensionless variables in equation (23), dimensionless entropy generation rate is: Where, represents temperature difference parameter.

Discussion of Results:
In this section, effects of important parameters on the velocity and temperature with dimensionless entropy generation number are portrayed graphically. Numerical computations have been taken out by employing Adams Bashforth method.   Magnitude of entropy generation number towards M is increasing near the walls and in the central region including points of intersection at which entropy remains constant. This trend is due to enhancing frictional effects of Lorentz force (Fig. 8).

Concluding Remarks:
In present research article, analysis for the impacts of endoscope going on peristaltic flow of couple stress nanofluid in existence of magnetic field and viscous dissipation is carried out. Major conclusions drawn from present investigation are: ➢ Velocity profile increases close to the endoscope and decreases close to the peristaltic vertical tube with increment of Gr and y while an opposite behavior is noticed for M.
➢ Temperature increases against higher values of  and Ec but an opposite behavior is depicted for M.
➢ Entropy is directly affected by buoyancy and viscous forces which are dominant near the endoscope and tube walls.
➢ The consequences of Newtonian fluid model can be obtained by taking the couple stress parameter y = 0 within the current model.