COMBINED INFLUENCE OF HALL CURRENTS AND JOULE HEATING ON HEMODYNAMIC PERISTALTIC FLOW WITH POROUS MEDIUM THROUGH A VERTICAL TAPERED ASYMMETRIC CHANNEL WITH RADIATION

The aim of the present attempt is hall currents and joule heating on peristaltic blood flow in porous medium through a vertical tapered asymmetric channel under the influence of radiation. The Mathematical modeling is investigated by utilizing long wavelength and low Reynolds number assumptions. The indicates an appreciable increase in the axial velocity distribution with increase in hall current parameter and porosity parameter whereas the result in axial velocity distribution diminished by increase in magnetic field parameter. The result in pressure gradient reduces by rise in hall current parameter, porosity parameter and volumetric flow rate. The temperature of the fluid increases by increase in M, N, Pr and Br and decreases by increase in m and Da. Keywords: Hall current, porosity parameter, joule heating, radiation and tapered channel Frontiers in Heat and Mass Transfer (FHMT), 9, 19 (2017) DOI: 10.5098/hmt.9.19 Global Digital Central ISSN: 2151-8629 2 variable viscosity. Shehzad et al. (2014) investigated on hydromagnetic peristaltic transportation of nanofluid with Joule Heating and thermophoresis. The peristaltic flow of fourth-grade fluid with Dufour and Sore effects was studied by Mustafa et al. (2014). Recently, Ravi Kumar (2016) investigated on hydromagnetic peristaltic transportation through a perpendicular tapered uneven channel with radiation. Some pertinent studies on this topic will be found from the list of Refs. Such as Tasawar Hayat et al. (2017), Kothandapani et al. (2015) Abzal et al. (2016) , Jitendra Kumar Singh et al. (2016) and Veeresh et al. (2017). 2. FORMULATION OF THE PROBLEM We consider the MHD peristaltic transport of an incompressible viscous fluid in a two-dimensional uneven inclined perpendicular tapered channel under the influence of porous medium. The joule heating, hall currents and radiation is taken into the account. The left wall of the channel is maintained at temperature T0, whereas the right wall has temperature T1. We tend to assume that the fluid is subject to a relentless transverse magnetic field B0. The fluid is induced by sinusoidal wave trains propagating with constant speed c along the channel walls. The geometry of the wall deformations are drawn by the subsequent expressions =  = + + sin  − ̅ (1) =  = − − − sin  − ̅ + (2) Where b is the half-width of the channel, d is the wave amplitude, is the phase speed of the wave and m′ ( ) 1 << ′ m is the non-uniform parameter, is the wavelength, t is the time and X are the direction of wave propagation. The phase difference varies in the range 0 ≤ φ ≤ π, φ = 0 corresponds to the symmetric channel with waves out of phase and further b, d and φ satisfy the following conditions for the divergent channel at the inlet cos ! ≤ . It is assumed that the left wall of the channel is maintained at temperature T0 while the right wall has temperature T1. Fig. 1 Schematic diagram of the physical model The equations governing the motion for the present problem prescribed by The continuity equation is #$ #% + #& #' = 0 (3) The momentum equations are ) * #$ #% + + #$ #' = − #, #% + * #-$ #%+ #-$ #'+ . /0123 + − * + 4 − 5 67 * + (4) ) * #& #% + + #& #' = − #, #' + * #-& #%+ #-& #'− . /012* − 5 67 + + (5) The energy equation is ) 8, * # #% + + # #' 9 = : ##%+ ##'9 + ;< + =>< * − #? #% (6) u and v are the velocity components in the corresponding coordinates, k1 is the permeability of the porous medium,) is the density of the fluid, p is the fluid pressure, k is the thermal conductivity, @ is the coefficient of the viscosity, Q0 is the constant heat addition/absorption, Cp is the specific heat at constant pressure, σ is the electrical conductivity, g is the acceleration due to gravity and T is the temperature of the fluid. The relative boundary conditions are AB = 0,9 = 9< at = B AB = 0,9 = 9 at = B The radioactive heat flux (Cogley et al (1968)) is given by #? #% = 4D 9< − 9 (7) here α is the mean radiation absorption coefficient. Introducing a wave frame (x, y) moving with velocity c away from the fixed frame (X, Y) by the transformation x = X-ct, y = Y, u = U-c, v = V and p(x) = P(X, t) (8) Introducing the following non-dimensional quantities: λ x x = b y y = λ ct t = c u u = δ c v v = b H h 1 1 = b H h 2 2 = μ λ c p b p 2 = 0 1 0 T T T T − − = θ λ δ b = μ ρ b c = Re c g a μ λ ρ η 3 0 1 = μ σ b B M 0 = k C p μ = Pr ( ) 0 1 2 T T C c E p c − = k d N 2 1 2 2 4α = ( ) 0 1 2 0 T T C b Q p − = μ γ b d = ε c g a μ ρ η 2 0 = (9) where E = FG is the non-dimensional amplitude of channel, H = G is the wave number,: = 2I G is the non uniform parameter, Re is the Reynolds number, M is the Hartman number, J = 6 GPermeability parameter, Pr is the Prandtl number, Ec is the Eckert number, γ is the heat source/sink parameter, Br (= EcPr) is the Brinkman number, η andη1 are gravitational parameters and K is the radiation parameter. 3. SOLUTION OF THE PROBLEM In view of the above transformations (8) and non-dimensional variables (9), equations (3-6) are reduced to the following non-dimensional form after dropping the bars, LMH N* O* OP + + O* OQR = − OS OP + H O * OP + O * OQ + T123 H+ − * + 1 4 − V * − V (10) LMHW N* O+ OP + + O+ OQR = − OS OQ + H XH O + OP + O + OQ Y − ZT12* + 1 + H+ − H V + − H V (11) LM H* #[ #% + + #[ #' = \] H #-[ #%+ #-[ #'+ ^ + _ `a* + b-[ \] (12) Frontiers in Heat and Mass Transfer (FHMT), 9, 19 (2017) DOI: 10.5098/hmt.9.19 Global Digital Central ISSN: 2151-8629 3 Applying long wave length approximation and neglecting the wave number along with low-Reynolds numbers. Equations (10-12) become #-$ #'− T12+ V! * = #, #% + T12+ V! (13) #, #' = 0 (14) \] #-[ #'+ ^ + _ `a* + b-[ \] = 0 (15) The relative boundary conditions in dimensionless form are given by u = -1, θ = 0 at ( ) [ ] φ π ε + − − − = = t x x k h y 2 sin 1 1 1 (16) u = -1, θ = 1 at ( ) [ ] t x x k h y − + + = = π ε 2 sin 1 1 2 (17) The solutions of velocity and temperature with subject to boundary conditions (16) and (17) are given by * = c sinheD Qf + cosheD Qf + c (18) Where c = X 1 + c sinheD h fY + h cosheD h f 1 + c sinheD h f icosheD h f − cosheD h f sinheD h f − sinheD h f j sinheD h f − cosheD h fk c = h 1 + c icosheD h f − cosheD h f sinheD h f − sinheD h f j sinheD h f − cosheD h fk c = − l1 + X , m7no-1 7 pqYr, where S = #, #% s = ct coseK Qf + cu sineKQf − v\] b− T-/]wx yz7-1bM z7' − T-/]w{ yz7-1bM| z7' − T-/]w} z7-1bMz7' − T-/]w~ z7-1bM|z7' − T-/]w b(19) Where cW = lc 4 + c 4 + c c 2 r , cy = lc 4 + c 4 − c c 2 r c‚ = ec c + c f, cƒ = ec c − c f c„ = l− c 2 + c 2 + c r cu = l− coseKh f − c < − c − c − c W − c y − c ‚ c ƒ r ct = …†† †‡ −cu sineKh f + ^ˆ‰ K + l _ >‰cW 4D + K r M z7Š7 coseKh f ‹ŒŒ Œ + ŽN m-]q{ {7-n‘-R’“-7”71Nm-]q} 7-n‘-R’7”71Nm-]q~ 7-n‘-R’“7”71m-]q ‘•–—ebŠ7f ˜ c < = Ni^ˆ‰ K j cosheKh f − cosheKh f R c = lX _ >‰cW 4D + K Y M z7ŠcosheKh f − M z7ŠcosheKh f r c = lX _ >‰cy 4D + K Y M| z7ŠcosheKh f − M| z7ŠcosheKh f r c W = lX _ >‰c‚ D + K Y Mz7ŠcosheKh f − Mz7ŠcosheKh f r c y = lX _ >‰cƒ D + K Y M|z7ŠcosheKh f − M|z7ŠcosheKh f r c ‚ = lX_ >‰c„ K Y cosheKh f − cosheKh f r c ƒ = esinheKh f coseKh f − sinheKh f coseKh ff The coefficients of the heat transfer Zh1 and Zh2 at the walls y = h1 and y = h2 respectively, are given by x y h Zh 1 1 θ = (20) x y h Zh 2 2 θ = (21) The solutions of the coefficient of heat transfer at y = h1 and y = h2 are given by ™h = s' h % = X−ctK sineKQf + cuK coseKQf − l2D _ >‰cW 4D + K r M z7' + l2D _ >‰cy 4D + K r M| z7' − lD _ >‰c‚ D + K r Mz7' + lD _ >‰cƒ D + K r M|z7'Y −2š ›œe2š P − + ∅f − : (22) ™h = s' h % = X−ctK sineKQf + cuK coseKQf − l2D _ >‰cW 4D + K r M z7' + l2D _ >‰cy 4D + K r M| z7' − lD _ >‰c‚ D + K r Mz7' + lD _ >‰cƒ D + K r M|z7'Y 2šž ›œe2š P − f + : (23) The volumetric flow rate in the wave frame is defined by Ÿ = c sinheD Qf + cosheD Qf + c ŠŠ7 Q w7 z7 ecosheD h f − cosheD h ff + wz7 esinheD h f − sinheD h ff + ec h − h f (24) The pressure gradient obtained from equation (24) can be expressed as F, F% = T12+ V − ?1/¡|V¢ Š-|Š7 1V¢|/¡ T12+ V (25) The instantaneous flux Q (x, t) in the laboratory frame is ; = £ * + 1 Š7 ŠQ = Ÿ − h (26) The average volume flow rate over one wave period (T = λ/c) of the peristaltic wave is defined as ; = ¤ £ ; ¤ < = Ÿ + 1 + (27) From the equations (25) and (27), the pressure gradient can be expressed as F, F% = T12+ V − N3¥| |F41/¡|V¢ Š-|Š7 1V¢|/¡ R T12+ V (28) Where > = …†† †‡1 + h cosheD h f icosheD h f − cosheD h f sinheD h f − sinheD h f j sinheD h f − cosheD h f‹ŒŒ Œ ¦ = h 1 icosheD h f − cosheD h f sinheD h f − sinheD h f j sinheD h f − cosheD h fk ` = lcosheD h f − cosheD h f D sinheD h f r § = lsinheD h f − sinheD h f D r 4. DISCUSSION OF THE PROBLEM The objective of this research is to study hall currents and joule heating on peristaltic blood flow in the porous medium through a vertical tapered asymmetric channel under influence of radiation. In order to find out numerical solutions, MATHEMATICA software is used. Fig. 2 indicate the behaviour of axial velocity with y for different values of hall current parameter m with fixed Da = 0.1, M = 2, k1= 0.1, dp/dx = -1, t = 0.4,∅ = π/6, x = 0.6, ε = 0.2. It is clear from the figure that the velocity gradually enhances by increase in Hall current parameter m. Frontiers in Heat and Mass Transfer (FHMT), 9, 19 (2017) DOI: 10.5098/hmt.9.19 Global Digital Central ISSN: 2151-8629 4 The impact of porosity parameter Da on axial velocity distribution is shown in figure 3. We perceive from this graph that the axial velocity distribution gradually increases by increase in porosity parameter Da (Da = 0.1, 0.2, 0.3). Fig.4 depicts the axial velocity distribution (u) with dissimilar values of M (M = 2, 4, 6) with fixed m = 0.5, Da = 0.1, k1= 0.1, dp/dx = -1, t = 0.4,∅ = π/6, x = 0.6, ε = 0.2. It can be notice from this graph that with an increase in magnetic field parameter M, the results in velocity profile diminished. Fig. 2 Impact of m on axial velocity distribution with Da = 0.1, M = 2, k1= 0.1, dp/dx = -1