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In this paper, the study of non-Newtonian fluid flow with heat transfer in a porous asymmetric channel due to peristaltic wave was taken out. Hall current and Ohmic heating effects were introduced in the present study. A Casson non-Newtonian constitutive model was employed for the transport of fluid. Analytical solutions were obtained for stream function, temperature and heat transfer coefficient. The coupled nonlinear equations have also been solved numerically using MATLAB software (by bvp4c function). The influences of many evolving parameters on the flow characteristic have been explained with the help of 2D and 3D plots. Again the obtained results were compared with the results available in the literature and were found in good agreement.

At present, investigation regarding the peristaltic flow of non-Newtonian fluid cannot be ignored due to their vast applications in physiology and industry. This mechanism is associated with a spontaneous relaxing and compressing movement along the length of the fluid filled channel or tube. Digestive track, urine transport form kidney to bladder, blood flow in small vessels, egg movement in fallopian tube are few examples that can be seen in our body [

It is well known the industrial and physiological fluids are non-Newtonian in nature. Due to various rheological properties of non-Newtonian fluids, several constitutive relations have been suggested. Casson fluid is one of the non-Newtonian fluids which was introduced by Casson [

Consider the peristaltic flow of non-Newtonian fluid in a two dimensional porous asymmetric channel. We choose a stationary frame of reference ( X , Y ) such that X measured along the axis of the channel and Y perpendicular to it. The channel walls H 1 and H 2 are maintained at constant temperature T 0 and T 1 respectively. Let ( U , V ) be the velocity components in the frame. A strong magnetic field with magnitude B ¯ = ( 0 , 0 , B 0 ) is applied and the Hall and Joule heating effects are taken into consideration. The induced magnetic field is neglected for small magnetic Reynolds number and also the externally applied electric field is assumed to be zero. The geometry (in

Y = H 1 = d 1 + a 1 cos { 2 π λ ( X − c t ) } Y = H 2 = − d 2 − a 2 cos { 2 π λ ( X − c t ) + ϕ } } (1)

where a 1 , a 2 denote the waves amplitudes, d 1 + d 2 is the channel width, λ is the wave length, t is the time, c is the velocity of propagation and ϕ is the phase difference ( 0 ≤ ϕ ≤ π ) , in which ϕ = 0 corresponds to symmetric channel with waves out of phase and ϕ = π corresponds to waves in phase. Here a 1 , a 2 , d 1 , d 2 and ϕ satisfies the condition

a 1 2 + a 2 2 + 2 a 1 a 2 cos ϕ ≤ ( d 1 + d 2 ) 2 (2)

The rheological relation for non-Newtonian Casson [

τ i j = 2 ( μ b + P y 2 π ) e i j (3)

where e i j is the ( i , j ) th component of deformation rate, τ i j is the ( i , j ) th component of the stress tensor, π is the product of the component of deformation rate with itself and μ b is the plastic dynamic viscosity. The yield stress P y is expressed as P y = μ b 2 π / β , where β Casson fluid parameter. For non-Newtonian Casson fluid flow μ = μ b + P y / 2 π which gives υ ′ = υ ( 1 + 1 / β ) , where υ = μ b / ρ is the kinematic viscosity for Casson fluid. Again the yield stress P y = 0 for Newtonian case.

The equations governing the fluid motion can be written as follows

∂ U ∂ X + ∂ V ∂ Y = 0 . (4)

∂ U ∂ t + U ∂ U ∂ X + V ∂ U ∂ Y = − 1 ρ ∂ P ∂ X + υ ( 1 + 1 β ) ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) + σ B ∘ 2 ρ ( 1 + m 2 ) ( m V − U ) − υ ( 1 + 1 β ) U K ′ (5)

∂ V ∂ t + U ∂ V ∂ X + V ∂ V ∂ Y = − 1 ρ ∂ P ∂ Y + υ ( 1 + 1 β ) ( ∂ 2 V ∂ X 2 + ∂ 2 V ∂ Y 2 ) − σ B ∘ 2 ρ ( 1 + m 2 ) ( m U + V ) − υ ( 1 + 1 β ) V K ′ (6)

∂ T ∂ t + U ∂ T ∂ X + V ∂ T ∂ Y = k ρ c p ( ∂ 2 T ∂ X 2 + ∂ 2 T ∂ Y 2 ) + υ C p ( 1 + 1 β ) [ 2 { ( ∂ U ∂ X ) 2 + ( ∂ V ∂ Y ) 2 } + ( ∂ U ∂ Y + ∂ V ∂ X ) 2 ] + 1 ρ c p σ B ∘ 2 ρ ( 1 + m 2 ) ( V 2 + U 2 ) (7)

Generalized Ohm’s law is

J ¯ = σ [ E ¯ + q ¯ × B ¯ − γ ( J ¯ × B ¯ ) ] (8)

Lorentz force is

J ¯ × B ¯ = σ B ∘ 2 1 + m 2 [ i ¯ ( m V − U ) − j ¯ ( m U + V ) ] (9)

Ohmic heating (Joule heating) is

J ¯ ⋅ J ¯ σ = σ B 0 2 1 + m 2 ( V 2 + U 2 ) (10)

where J ¯ is the current density, q ¯ is the fluid velocity, P is the pressure, γ = 1 / e n e is the Hall factor/Hall Current, e is the charge electron, n e is the mass of the electron, E ¯ is the electric field.

The corresponding boundary conditions are

U = 0 , T = T 0 when Y = H 1 U = 0 , T = T 1 when Y = H 2 } (11)

The coordinates, velocity, pressure and temperature in the fixed frame ( X , Y ) and wave frame ( x , y ) are related by the following expression

x = X − c t , y = Y , u = U − c , v = V , p ( x , y ) = P ( X , Y , t ) , T ¯ ( x , y ) = T ( X , Y , t ) (12)

where u , v , p , T ¯ are the velocity components, pressure and temperature in the wave frame respectively.

Now we introduce the following dimensionless quantities

x ′ = x λ , y ′ = y d 1 , u ′ = u c , v ′ = v c δ , t ′ = c t λ , p ′ = p d 1 2 λ c μ b δ = d 1 λ , h 1 = H 1 d 1 , h 2 = H 2 d 1 , d = d 2 d 1 , a = a 1 d 1 , b = a 2 d 1 , θ = T ¯ − T ¯ 0 T ¯ 1 − T ¯ 0 } (13)

The governing Equations (4)-(7) under the assumptions of long wave length and low Reynolds number in terms of stream function ψ (dropping the das symbols) become

∂ p ∂ x = ( 1 + 1 β ) [ ∂ 3 ψ ∂ y 3 − α 2 ( ∂ ψ ∂ y + 1 ) ] (14)

∂ p ∂ y = 0 (15)

θ ' ' + ( 1 + 1 β ) B r ( ∂ 2 ψ ∂ y 2 ) 2 + B r M 2 1 + m 2 ( ∂ ψ ∂ y + 1 ) 2 = 0 (16)

The dimensionless boundary conditions become

ψ = F 2 , ∂ ψ ∂ y = − 1 , θ = 0 when y = h 1 ψ = − F 2 , ∂ ψ ∂ y = − 1 , θ = 1 when y = h 2 } (17)

where α 2 = M 2 ( 1 + m 2 ) ( 1 + 1 / β ) + 1 K , M = σ μ b B 0 d 1 is the magnetic field parameter, K = K ′ d 1 2 is the permeability parameter, F is the volume flow rate in the wave frame, β = μ b 2 π / P y is the Casson fluid parameter, m = σ γ B 0 is the Hall parameter, P r = ρ υ C p k is Prandtl number, E c = c 2 C p ( T ¯ 1 − T ¯ 0 ) is the Eckert number and B r = P r ⋅ E c is the Brinkman number.

Equation (15) gives that p ≠ p ( y ) . Eliminating the pressure terms from (14) we get

∂ 4 ψ ∂ y 4 − α 2 ∂ 2 ψ ∂ y 2 = 0 (18)

Solving Equations (18) and (16) with the boundary conditions (17), the stream function and temperature are obtained as

ψ = B 1 + B 2 y + B 3 cosh ( α y ) + B 4 sinh ( α y ) (19)

θ = − F 1 y 2 − F 2 cosh ( 2 α y ) − F 3 sinh ( 2 α y ) − F 4 sinh ( α y ) − F 5 cosh ( α y ) − B 5 y − B 6 (20)

where the constants involved in the solutions are given in the appendix.

The dimensionless mean flow rate Q ′ in the laboratory frame is related to the dimensionless mean flow rate F in the wave frame by

Q ′ = F + 1 + d (21)

in which

F = ∫ h 2 h 1 u d y (22)

Also note that h 1 and h 2 represent the dimensionless forms of the peristaltic walls

h 1 = 1 + a cos ( 2 π x ) h 2 = − d − b cos ( 2 π x + ϕ ) } (23)

Again the heat transfer coefficient at the upper wall ( y = h 1 ) is

Z 1 = h 1 x θ ′ (24)

The dimensionless equations have again been solved numerically for different values of model parameters with another code written in MATLAB (bvp4c function). It is a finite difference code that implements a collocation formula. Numerical computations have been carried out for magnetic field parameter (M), Hall parameter (m), Casson fluid parameter ( β ), permeability parameter (K) and Brinkman number (Br) at fixed values of a = 0.3 , b = 0.5 , d = 1.3 , ϕ = π / 3 , F = − 1.5 and x = 0 . The software ORIGIN has been used to show the numerical results graphically. Plots 2D and 3D are sketched to understand and explain the varying activities of model parameters in a better way.

The effect of magnetic field parameter M on velocity field is sketched in

Figures 7-11 show the temperature profile θ for different values of M , K , β , m and Br. Here we see that temperature profiles are almost parabolic in character and higher temperature is seen near the centre of the channel.

In order to see the job of Ohmic (Joule) heating effect we have resolved the heat equation after neglecting the Ohmic heating term and then investigative the magnetic field parameter M and Hall parameter m effects on temperature field through graphs.

Trapping, the procedure of contours of streamlines, is an attractive characteristic for peristaltic flow. To investigate the effects of M , m and ϕ on streamlines, we have plotted Figures 14-16. Here we notice that the size of trapped bolus (trapped bolus is a bolus of fluid which is enclosed by streamlines) decreases for large values of magnetic field parameter M as seen in

To verify the accuracy of numerical results, the present study is compared with the previous study of Hayat et al. [

a = 0.5 , b = 0.6 , d = 1.5 , M = 2 , ϕ = π / 4 , F = − 2 , B r = 3 | ||||
---|---|---|---|---|

x | Present | Hayat et al. [ | Present | Hayat et al. [ |

K = 0.1 | K = 1 | |||

0.1 | 2.31567 | 2.31697 | 1.98257 | 1.98138 |

0.2 | 1.96850 | 1.96953 | 1.86659 | 1.86796 |

0.3 | 2.01043 | 2.01085 | 1.98089 | 1.98128 |

This study is presented on the peristaltic motion of non-Newtonian fluid in a porous asymmetric channel with Hall current and Ohmic heating effects. A Casson non-Newtonian constitutive model was employed for this study. Both analytic and numerical solutions have been obtained. The effects of different parameters on flow characteristics are shown by using 2D and 3D plots. The main outcomes of the study are:

1) Magnetic field parameter M has decreasing effect but Hall parameter m has an increasing impact on velocity profile.

2) The fluid becomes less viscous when we increase β .

3) Temperature decreases when m increases and increases when M and Br increase.

4) The activities of M and m on temperature field are quite reverse when we neglect the Ohmic heating term.

5) The size of trapped bolus decreases for large values of phase difference ϕ and trapping vanishes when ϕ = π .

The authors are thankful to the referees for their useful suggestions.

The authors declare no conflicts of interest regarding the publication of this paper.

Hasan, M.M., Samad, M.A. and Hossain, M.M. (2020) Effects of Hall Current and Ohmic Heating on Non-Newtonian Fluid Flow in a Channel due to Peristaltic Wave. Applied Mathematics, 11, 292-306. https://doi.org/10.4236/am.2020.114022

The values involved in the solution expressions are presented below.