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Entropy Analysis of Pressure Driven Flow in a Curved Duct V. K. Narla 1,a) and Vijayasekhar Jaliparthi 1, b) 1 Department of Mathematics, GITAM University, Hyderabad, India. a) Corresponding author: [email protected] b) [email protected] Abstract. This paper aims to present a theoretical model describing entropy generation analysis using second law of thermodynamics. A two-dimensional, incompressible, viscous MHD fluid flow in a curved duct undergoing peristalsis with prescribed wall motions in the presence of heat transfer is applied and demonstrated. In this problem, It is assumed that the inertial effect is very small and the wall wave length is comparatively large with duct width. The velocity and temperature fields are obtained analytically by solving momentum and energy equations. The entropy generation number is calculated by utilizing velocity and temperature profiles. The influence of various physical parameters on entropy generation are discussed numerically with the help of graphs. INTRODUCTION Peristaltic pumping is a kind of fluid transport that occurs when progressive wave of contraction and expansion passed along a conduit containing liquid. The fluid mechanics of this phenomenon has been studied by several researchers because of its application in biology and industry. Sato et al. [1] developed a peristaltic model in a twodimensional curved channel based on lubrication theory. In a curved channel with sinusoidally oscillating walls, they demonstrate that the peristaltic transport phenomena are strongly influenced by the curvature. This model was extended by Ali et al. [2] to capture heat transfer effects using the assumptions long wavelength and low Reynolds number. Hayat et al. [3] examined the effect of the compliant wall properties and heat transfer on the peristaltic flow in a curved channel. Ramanamurthy et al. [4] studied unsteady peristaltic transport in curved channels to capture spacial and temporal effects of flow phenomena. The impact of entropy generation due to heat transfer and fluid friction for the peristaltic flow in a tube was investigated Akbar [5]. Recently, Narla et al. [6] discussed entropy generation analysis for peristaltic flow in curved channels. FIGURE 1: Schematic diagram of peristaltic transport in a two-dimensional curved channel. MATHEMATICAL FORMULATION The flow configuration is shown in Figure 1. The cartesian system ( ′, ′, ′) is related to the intrinsic coordinates ( , , ) by the relations ′ = ( + ) ( ), ′ = ( + ) ( ), ′ = . (1) A two-dimensional peristaltic flow in a curved channel is filled with an incompressible viscous fluid under the influence of radial magnetic field. The unperturbed width of the channel is 2 coiled in a circle with center and the radius of curvature is . The -axis lies along the center line of the curved channel, -axis is normal to it and is measured from central line with scaling factors ℎ = , ℎ = 1 and ℎ = 1. There is no component in direction. The flow in the channel gives velocity vector in the form = !( , , ")#̂ + %( , , ")#̂ . The fluid motion within the channel is driven by two infinite trains of sinusoidal waves that are propagated along the channel. The temperatures of the lower and upper walls are maintained at constant temperatures ( &' and & respectively. An external magnetic field of strength ) is applied in radial direction as shown in Figure 1. The fluid-wall interface is time-dependent and is given as follows: = ±ℎ( , ") = ± ± + [ . ( − ")]. (2) Here, is the axial distance, the radius of the stationary curved channel, + is the wave amplitude, 1 the wave length, " the time, the velocity of the wave, and ℎ the radial displacement of the wave from the centerline. The mass conservation and the fluid momentum equations at the axial and radial directions, which are given respectively by: 23 2 23 27 + ( ⋅ ∇)! + 3: + ( ⋅ ∇)% − 27 3> 2 2 2< = − ;( 2: + )2 {( + )%} = 0, 3 + =[∇ ! − ( )> 2< = − ; 2 + =[∇ % − ( 2B C (3) 2: +( : )> )> 2 )> 2 23 : 2: 23 2: where ( ⋅ ∇) = ( 3 2 )2 2 − 2 + %2 ,∇ = ( 3 23 −( A< [ 27 + ( ⋅ ∇)&] = ; ∇ & + =[2{(2 ) + ( +( 2 + ] − ?@ ( 2 + () ) !, ], (5) ) } ) ], ) 2> 2 > (4) (6) 2 + 2 2> + 2 >. In order to describe the fluid flow pattern in dimensionless form, the variables are scaled as follows: 3 : G I 7 M ′ = . , ′ = D , !′ = E , %′ = FE , ℎ′ = D , H = D , J = D , "′ = B ∗ , L′ = ., N′ = D> < , #= OE. EDF P R T , Q′ = DE , S = DE , U = ;PVW C ,X = V E> W (BY ZB) ) BZB , [ = B ZB) , Y ) (7) where # is Reynolds number, S is volume flow rate, J is the curvature parameter, \ is D Hartman number, ] = . is wavelength ratio, H is the amplitude ratio or the occlusion parameter, L is the passage length of the channel, U is the Prandtl number and X is the Eckert number. The quantities ! and % in equations (3)-(6) are related to the stream function Q as follows: 2R ^ 2R ! = 2 ,% = − ^ 2 , (8) Using (7) and (8), the governing equations (3)-(6) can be written (after eliminating the pressure term) under the assumption of long wavelength (] ≪ 1) and small Reynolds number ( # → ∞) approximation as 2 2 2 2R 2 2R ( {( + J) } ) − ((1 + \ ) ) = 0, (9) 2 ^2 2 2> b 2 2 2> R 2b ^( ^) 2 2R + ^ 2 + c ( 2 > − ^ 2 ) = 0, (10) 2 where c = U X is the Brinkman number and primes are dropped for simplicity. The corresponding dimensionless boundary conditions are obtained by: 2R d = 0, = ±ℎ( , "), Q = ± , = ±ℎ( , ") (11) > 2 [ = 0, = −ℎ( , "), [ = 1, = ℎ( , ") (12) SOLUTION OF THE PROBLEM The obtained boundary value problem is solved for stream function with the boundary conditions consisting of the equation (11): > > Q( , ) = A + A ( + J) + A ( + J) Z√ f + Ag ( + J) √ f , (13) where, A = A = Here ZdhgG^dY √ gh(^ > dk> f> d> ( G> dl> )√ > m d(^ > ZG> ) Yno ( h(^ > dk> S = (J − ℎ) G> dl> )√ √ f> f> )(G> i > )j G^d> ( √ f> )(^dk Gdl ) f> f> )j G^d> ( + (ℎ + J) > f> )j ,A = f> √ f> > , Ag = gh(^ > dk> d> f> d > > G dl )√ f> d(Z h(^ > dk> , S = (J − ℎ) f> )[Z^dk Gdl ] √ G> dl> )√ f> G^d> ( f> − (ℎ + J) √ , f> )j G^d> ( > . f> )j √ f> > , S = (J − ℎ)√ f − (ℎ + J)√ f , Sg = (J − ℎ)√ f + (ℎ + J)√ f . The transformations between the wave and the laboratory frames, in dimensionless form, are given by : = − ", = , p = ! − 1, = %, q = S − 2ℎ, Ψ = Q − , (14) where the left hand side parameters are in the wave frame and the right hand side parameters are in the laboratory frame. Averaging the volumetric flow rate along time period & gives B B sB = B t' S( , ")u" = B t' (q + 2ℎ)u" = q + 2. (15) FIGURE 2:: Effect of curvature on temperature distribution. FIGURE 3:: Effect of magnetic parameter on temperature distribution. The solution of the energy equation (10) using equations (12) and (13) is obtained as: [ = v + v ln( + J) − Ag ( + J) where, √ v = f> f> ) g( [4(1 + \ )A Ag ln ( + J) + A ( + J) J Z ]. ( fl g( ( fl f> √ + (16) f> ) [4(1 + \ ))A Ag ln (J − ℎ) + A (J − ℎ)Z − z{(^ZG) |n} z{( ) }~| v2 = ( fl [1 − g( |n} z{( ) }~| [1 1− f> ) ( fl g( √ f> + Ag (J − ℎ) √ (4(1 + \ )A Ag • + A • + Ag S )],, f> ) (4(1 + \ )A Ag • + A • + Ag S )], ) • = ln (J − ℎ) − ln (ℎ + J), • = (J − ℎ)Z √ f> − (ℎ + J)Z FIGURE 44: Effect of curvature on entropy. √ f> . f> ] FIGURE 55: Effect of magnetic parameter on entropy. ENTROPY GENERATION Entropy generation is closely associated with thermodynamic irreversibility, which is encountered in all practical heat transfer processes. Different sources are responsible for entropy generation such as the heat transfer in the presence of temperature di difference fference and the viscous dissipation. The volumetric rate of entropy generation is defined as ^ O XŒ = B > (∇&) + B Φ + S “ 0. (17) •Ž•Ž• •Ž•Ž• ) ) ‘' ‘' The non-dimensional dimensional entropy generation equation under long wavelength approximation can be expressed as follows (Narla et al. [[6]): 2b X€ = X• + Xd = ( 2 ) + where, Eƒ = „… „† ‡> ˆ>) „… = ‰(ˆ ˆ) )> Y Zˆ ( ” 2> R [( 2 > − 2R i2 ) +\ ( 2R ^2 ) ], ] (18) is non dimensional entropy generation, BrΩ Br Z is viscous dissipation parameter, EŠ is entropy generation by heat transfer due to transverse heat conduction and E‹ is entropy generation due to the fluid friction. RESULTS AND DISCUSSION A graphical analysis of the solutions of temperature distribution and entropy generation is presented. Figure 2 shows the influence of channel curvature on the dimensionless temperature at fixed values of the parameters H = 0.4, c = 2, = 0.2, " = 0.5, sB = 1.5, \ = 1. The fluid temperature decreases with increasing curvature parameter J.. The temperature distribution distrib is parabolic nature. This is due to the presence of viscosity dissipation that increases the fluid’s temperature in the central part of the channel. This means the temperature distribution in the curved channel is very high than in the straight channel. The variation of magnetic parameter \ on temperature is presented in figure 3 at fixed parameters H = 0.4, J = 3, c = 2, = 0.4, " = 0.5, sB = 1.5. The temperature is a decreasing function of magnetic parameter. This means that the high magnetic strength decreases the temperature of the fluid in a curved channel. The distribution of entropy generation for different values of curvature parameter J is plotted in figure 4 at fixed parameters H = 0.6, c = 2, = 0.2, " = 0.5, sB = 1, \ = 1, c /Ω = 1.5. The increasing channel curvature decreases the entropy generation rate at upper wall and increases much at lower wall. For the decreasing channel curvature (as J → ∞) the entropy generation rate variation is not much at upper and lower walls. This shows that the entropy generation rate is strongly depends on curvature of the channel. Figure 5 illustrates the variation of X€ with magnetic parameter \ at fixed values c = 1, H = 0.6, J = 3, = 0.3, " = 0.5, sB = 1, c /Ω = 1. It is observed that the entropy generation rate increases with an increase in the magnitude of magnetic parameter. REFERENCES 1. 2. 3. 4. 5. 6. H. Sato, T. Kawai, T. Fujita, and M. Okabe, Trans. Jpn. Soc. Mech. Eng. Ser. B 66, 679–685 (2000). N. Ali, M. Sajid, T. Javed, and Z. Abbas, Int. J. Heat Mass Transfer 53, 3319–3325 (2010). T. Hayat, M. Javed, and A. A. Hendi, Int. J. Heat Mass Transfer 54, 1615–1621 (2011). J. V. Ramanamurthy, K. M. Prasad, and V. K. Narla, Phys. Fluids 25, 091903(1–20) (2013). N. S. Akbar, Energy 82, 23–30 (2015). V. K. Narla, K. M. Prasad, and J. V. R. Murthy, Journal of Engineering Physics and Thermophysics 89, 441–448 (2016).