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In this work, the peristaltic motion of a nano non-Newtonian fluid which obeys Carreau model through a porous medium inside an asymmetric channel is investigated. The hall current effects with Joule heating and viscous dissipation are considered. The problem is modulated mathematically by a set of nonlinear partial differential equations which describe the conservation of mass, momentum, energy and concentration of nanoparticles. The non-dimensional form of these equations is simplified under the assumption of long wavelength and low Reynolds number, and then resulting equations of coupled nonlinear differential equations are tackled numerically with appropriate boundary conditions. Graphical results are presented for dimensionless velocity, temperature, concentration and pressure gradient in order to illustrate the variations of various parameters of this problem on these obtained solutions.

Nowadays, the study of nanofluids flow has the interest of researches because of its applications in medicine, biochemistry and industrial engineering. Nanofluids are moderately new category of fluids which consist of a base-fluid with nano-sized particles (1 - 100 nm) suspended within them. Choi [

Peristaltic motion in a channel or tube is considered as a type of flow that has great value in several physiological processes and industries. Peristalsis is a mechanism for mixing and transporting fluids through expansion and contraction of the wave propagation along the channel walls. This mechanism is seen in many biological system such as urine transport from kidney to bladder through the ureter, transport of lymph in the lymphatic vessels, swallowing food through the esophagus, the movement of chime in the gastrointestinal track, ovum movement in the fallopian tube, transport of spermatozoa vasomotion of small blood vessels such as venules and capillaries and blood flow in arteries, transport of corrosive fluids, sanitary fluid transport and blood pumps in heart lung machine etc. This analysis was first investigated by Latham [

Recently, Eldabe et al. [

Few attempts have been devoted to peristaltic flows in presence of heat and mass transfer; such investigations are of great importance, which is due to their extensive applications in medical and bio-engineering sciences, as it may be relevant in many processes in human body like oxygenation in lungs, hemodialysis and nutrients diffuse out of blood. Sohail and nadeem [

The main aim of this work is to study the peristaltic motion of a Carreau nanofluid with heat and mass transfer through a porous medium in an asymmetric channel under the effects of Hall current, viscous dissipation and Joule heating. The analysis is performed under the well-established long wavelength and low Reynolds number approximations. A detailed mathematical formulation is presented and numerical solution graphically for velocity, temperature, nanoparticle phenomena and pressure gradient have been presented.

Consider the peristaltic flow of an incompressible viscous electrically conducting nanofluid which obeys Carreau model inside a two dimensional vertical asymmetric channel of width d 1 + d 2 through a porous medium. Asymmetry in the channel is produced by propagation of waves along the channel walls traveling with different amplitudes, phases but with constant speed. In the Cartesian coordinates system (X, Y), the right-hand side wall Y = H 1 and the left-hand side wall Y = H 2 are given by

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

Y = H 2 ( X , t ) = − d 2 − b 1 cos ( 2 π λ ( X − c t ) + ϕ ) (2)

where, a 1 and b 1 are the amplitudes of right and left walls respectively, λ is the wavelength, t is the time, and ϕ is the phase which varies in the range 0 ≤ ϕ ≤ π . When ϕ = 0 then symmetric channel with waves out of the phase can be described (i.e. both walls move outward or inward simultaneously), and for ϕ = π , the waves are in phase. Further d 1 , d 2 , a 1 , b 1 and ϕ satisfy the condition:

a 1 2 + b 1 2 + 2 a 1 b 1 cos ϕ ≤ ( d 1 + d 2 ) 2 , (3)

so that the walls will not intersect with each other.

A strong uniform magnetic field with magnetic flux density B = ( 0 , 0 , B 0 ) is applied and the Hall effects are taken into account. The induced magnetic field is neglected by assuming a very small magnetic Reynolds number ( R e m < < < 1 ) , also it is assumed that there is no applied polarization voltage so that the total electric field E = 0 . The expression for the current density J including the Hall effect and neglecting ion-slip and thermoelectric effects [The generalized Ohm’s law] is given by [

J = σ [ E + V × B − 1 e n e ( J × B ) ] (4)

where σ is the electrical conductivity of the fluid, V is the velocity vector, e is the electric charge of electrons, n e is the number density of electrons. Equation (1) can be solved in J to yield the Lorentz force vector in the form:

J × B = − σ B 0 2 1 + m 2 [ ( U − m V ) i ^ + ( m U + V ) j ^ ] (5)

where U and V are the X and Y components of the velocity vector, m = σ B 0 e n e is

the hall parameter. The heat transfer and nanopatricle processes are maintained by considering temperature T 0 , T 1 and nanoparticle phenomena C 0 , C 1 to the right and left sides wall, respectively.

The constitutive equation for a carreau fluid is given by [

η − η ∞ η 0 − η ∞ = [ 1 + ( Γ γ ˙ ) 2 ] n − 1 2 , (6)

τ i j = η 0 [ 1 + n − 1 2 ( Γ γ ˙ ) 2 ] γ ˙ i j , (7)

in which τ i j is the extra stress tensor, η ∞ is the infinite shear rate viscosity, η 0 is the zero shear rate viscosity, Γ is the time constant, n is the power law index, and γ ˙ is defined as:

γ ˙ = 1 2 ∑ i ∑ j γ ˙ i j γ ˙ j i = 1 2 Π (8)

where Π = t r a c ( g r a d V + ( g r a d V ) T ) 2 is the second invariant strain tensor.

The governing equations for the present problem are described as:

Continuity equation

∂ U ∂ X + ∂ V ∂ Y = 0 (9)

Equations of motion

ρ f ( ∂ U ∂ t + U ∂ U ∂ X + V ∂ U ∂ Y ) = − ∂ P ∂ X + ∂ τ X X ∂ X + ∂ τ X Y ∂ Y − η 0 k 1 U + σ B 0 2 1 + m 2 ( m V − U ) + ρ f g α ( T − T 0 ) + + ρ f g α ( C − C 0 ) (10)

ρ f ( ∂ V ∂ t + U ∂ V ∂ X + V ∂ V ∂ Y ) = − ∂ P ∂ Y + ∂ τ Y X ∂ X + ∂ τ Y Y ∂ Y − η 0 k 1 V + σ B 0 2 1 + m 2 ( m U + V ) (11)

Energy equation

( ρ c ) f ( ∂ T ∂ t + U ∂ T ∂ X + V ∂ T ∂ Y ) = K ( ∂ 2 T ∂ X 2 + ∂ 2 T ∂ Y 2 ) + [ τ X X ∂ U ∂ X + τ X Y ∂ U ∂ Y + τ Y X ∂ V ∂ X + τ Y Y ∂ V ∂ Y ] + σ B 0 2 1 + m 2 ( U 2 + V 2 ) + ( ρ c ) p { D B [ ∂ C ∂ X ∂ T ∂ X + ∂ C ∂ Y ∂ T ∂ Y ] + D T T 0 [ ( ∂ T ∂ X ) 2 + ( ∂ T ∂ Y ) 2 ] } (12)

Concentration equation

∂ C ∂ t + U ∂ C ∂ X + V ∂ C ∂ Y = D B ( ∂ 2 C ∂ X 2 + ∂ 2 C ∂ Y 2 ) + D T T 0 ( ∂ 2 T ∂ X 2 + ∂ 2 T ∂ Y 2 ) (13)

where ρ f is the density of the fluid, P is the pressure, k 1 is the permeability of the porous medium, g is the acceleration due to gravity, α is the volume expansion coefficient, T is the temperature of the fluid, C is the nanoprticle concentration, ( ρ c ) f is the heat capacity of the fluid, K is the thermal conductivity, ( ρ c ) p is the effective heat capacity of the nanoparticle material, D B is the Brownian diffusion coefficient and D T is the thermophoretic diffusion.

Introducing a wave frame ( x , y ) moving with the velocity c away from the fixed frame ( X , Y ) by the transformation

x = X − c t , y = Y , u = U − c , p ( x ) = P ( X , t ) (14)

in which ( x , y ) , ( u , v ) and p are the coordinates, velocity components and pressure in the wave frame.

Defining the following non-dimensional quantities

x ¯ = x λ , y ¯ = y d 1 , u ¯ = u c , v ¯ = v c , t ¯ = t λ , δ = d 1 λ , d = d 2 d 1 , a = a 1 d 1 , b = b 1 d 1 , h 1 = H 1 d 1 , h 2 = H 2 d 2 , p ¯ = d 1 2 p η 0 c λ , Re = ρ f c d 1 η 0 , ψ ¯ = ψ c d 1 , τ ¯ x ¯ x ¯ = λ η 0 c τ x x , τ ¯ x ¯ y ¯ = d 1 η 0 c τ x y , τ ¯ y ¯ y ¯ = d 1 η 0 c τ y y , W e = Γ c d 1 , γ ˙ ¯ = d 1 γ ˙ c , s = k 1 d 1 2 , M = σ η 0 B 0 d 1 , θ ¯ = T − T 0 T 1 − T 0 , Ω ¯ = C − C 0 C 1 − C 0 , Pr = μ ρ f α , G r = ρ f g α d 1 2 ( T 1 − T 0 ) c η 0 , B r = ρ f g α d 1 2 ( C 1 − C 0 ) c η 0 , E c = c 1 2 c f ( T 1 − T 0 ) , S c = μ ρ f D B , N b = τ D B ( C 1 − C 0 ) α , N t = τ D B ( T 1 − T 0 ) α , α = K ( ρ c ) f , τ = ( ρ c ) p ( ρ c ) f . (15)

where δ is the dimensionless wave number, R e is the Reynolds number, W e is the Weissenberg number, s is the porosity parameter, M is the magnetic parameter, G r is the local temperature Grashof number, B r is the local nanoparticle Grashof number, P r is the Prandtl number, E c is the Eckert number, S c is the Schmidt number, N b is the Brownain motion parameter and N t is the thermophoresis parameter.

Using Equation (14) and the above set of non-dimensional quantities (15) into Equations (10)-(13), the resulting equations in terms of stream function

ψ ( u = ∂ ψ ∂ x , v = − δ ∂ ψ ∂ y ) can be written after dropping bars in the following

Non-dimensionless form as:

Re δ [ ( ∂ ψ ∂ y ∂ ∂ x − ∂ ψ ∂ x ∂ ∂ y ) ∂ ψ ∂ y ] = − ∂ p ∂ x + δ 2 ∂ ∂ x ( τ x x ) + ∂ ∂ y ( τ x y ) − δ m M 2 1 + m 2 ( ∂ ψ ∂ x ) − M 2 1 + m 2 ( ∂ ψ ∂ y + 1 ) − 1 s ( ∂ ψ ∂ y + 1 ) + G r θ + B r Ω (16)

Re δ 3 [ ( − ∂ ψ ∂ y ∂ ∂ x + ∂ ψ ∂ x ∂ ∂ y ) ∂ ψ ∂ x ] = − ∂ p ∂ y + δ 2 ∂ ∂ x ( τ y x ) + δ ∂ ∂ y ( τ y y ) − δ m M 2 1 + m 2 ( ∂ ψ ∂ y − 1 ) − δ 2 M 2 1 + m 2 ( ∂ ψ ∂ y + 1 ) − δ 1 s ( ∂ ψ ∂ x ) (17)

Re δ [ ∂ ψ ∂ y ∂ θ ∂ x − ∂ ψ ∂ x ∂ θ ∂ y ] = 1 P r ( ∂ 2 θ ∂ y 2 + δ 2 ∂ 2 θ ∂ x 2 ) + E c [ ( δ 2 τ x x ∂ 2 ψ ∂ x ∂ y + τ x y ∂ 2 ψ ∂ y 2 − δ 2 τ y x ∂ 2 ψ ∂ x 2 − δ τ y y ∂ 2 ψ ∂ y ∂ x ) ] + E c M 2 1 + m 2 [ δ 2 ( ∂ ψ ∂ x ) 2 + ( ∂ ψ ∂ y + 1 ) 2 ] E c M 2 1 + m 2 [ δ 2 ( ∂ ψ ∂ x ) 2 + ( ∂ ψ ∂ y + 1 ) 2 ] + N t Pr [ δ 2 ( ∂ θ ∂ x ) 2 + ( ∂ θ ∂ y ) 2 ] (18)

Re δ S c [ ∂ ψ ∂ y ∂ Ω ∂ x − ∂ ψ ∂ x ∂ Ω ∂ y ] = ( ∂ 2 Ω ∂ y 2 + δ 2 ∂ 2 Ω ∂ x 2 ) + N t N b ( ∂ 2 θ ∂ y 2 + δ 2 ∂ 2 θ ∂ x 2 ) (19)

where

τ x x = 2 [ 1 + n − 1 2 W e 2 γ ˙ 2 ] ∂ 2 ψ ∂ x ∂ y , (20)

τ x y = [ 1 + n − 1 2 W e 2 γ ˙ 2 ] ( ∂ 2 ψ ∂ y 2 − δ 2 ∂ 2 ψ ∂ x 2 ) = τ y x , (21)

τ y y = 2 δ [ 1 + n − 1 2 W e 2 γ ˙ 2 ] ∂ 2 ψ ∂ x ∂ y , (22)

and,

γ ˙ = [ 2 δ 2 ( ∂ 2 ψ ∂ x ∂ y ) 2 + ( ∂ 2 ψ ∂ y 2 − δ 2 ∂ 2 ψ ∂ x 2 ) 2 + 2 δ 2 ( ∂ 2 ψ ∂ y ∂ x ) 2 ] 1 2 (23)

Under the assumption of long wavelength δ ≪ 1 and low Reynolds number, Equations (16)-(19) become

0 = − ∂ p ∂ x + ∂ ∂ y [ 1 + n − 1 2 W e 2 ( ∂ 2 ψ ∂ y 2 ) 2 ] ∂ 2 ψ ∂ y 2 − ( M 2 1 + m 2 + 1 s ) [ ∂ ψ ∂ y + 1 ] + G r θ + B r Ω (24)

0 = − ∂ p ∂ y (25)

0 = 1 Pr ∂ 2 θ ∂ y 2 + E c [ ( ∂ 2 ψ ∂ y 2 ) 2 + n − 1 2 W e 2 ( ∂ 2 ψ ∂ y 2 ) 4 ] + E c M 2 1 + m 2 ( ∂ ψ ∂ y + 1 ) 2 + N b Pr ( ∂ Ω ∂ y ∂ θ ∂ y ) + N t Pr ( ∂ θ ∂ y ) 2 (26)

0 = ∂ 2 Ω ∂ y 2 + N t N b ∂ 2 θ ∂ y 2 (27)

Eliminating the pressure from Equations (24) and (25) yields

∂ 2 ∂ y 2 [ ( 1 + n − 1 2 W e 2 ( ∂ 2 ψ ∂ y 2 ) 2 ) ∂ 2 ψ ∂ y 2 − ( M 2 1 + m 2 + 1 s ) ψ ] + G r ∂ θ ∂ y + B r ∂ Ω ∂ y = 0 (28)

The non-dimensional boundaries will take the form

h 1 = 1 + a cos ( 2 π x ) , h 2 = − d − b cos ( 2 π x + ϕ ) (29)

where a , b , ϕ , d satisfies the relation

a 2 + b 2 + 2 a b cos ϕ ≤ ( 1 + d ) 2 (30)

The corresponding boundary conditions are

ψ = F 2 , ∂ ψ ∂ y = − 1 , θ = 0 , Ω = 0 , at y = h 1 ψ = − F 2 , ∂ ψ ∂ y = − 1 , θ = 1 , Ω = 1 , at y = h 2 (31)

where F is the dimensionless average flux in the wave frame defined by

F = ∫ h 2 ( x ) h 1 ( x ) ∂ ψ ∂ y d y (32)

The time mean Q in the wave frame is defined by

Q = F + 1 + d (33)

In this work, a program was designed by Mathematica Software (version 10) simulate the application of parametric ND Solve package to find the numerical behaviors of our dimensionless system. Graphical results to the velocity u, the temperature θ , the nano particle concentration Ω and the pressure gradient

d p d x are obtained under the impact of emerging parameters at moving boundaries

of the fluid.

Figures 1-5 represent the impact of the Weissenberg number we, the power law index n, the magnetic parameter M, the thermophoresis parameter n_{t} and flow rate Q on the velocity profile. _{t} on the velocity profile is the same and opposite to that of we and n, as by increasing M and n_{t} the velocity decreases near the channel walls and increases at the center of the channel, but a reduction in u occurs and variation in u becomes narrow in some sections of the

fluid motion see (

The variation of temperature profile for different values of the magnetic parameter M, the Brownain motion parameter Nb, the Prandtl number Pr the local nanoparticle Grashof number br and the thermophoresis parameter n_{t} are plotted in Figures 6-10. It is clear that the temperature profile increases when there is an increase in M, Nb and Pr, however it decreases with the increase in br and n_{t}.

Figures 11-15 describe the variation of the concentration profile for several values of the magnetic parameter M, the prandtl number Pr, the thermophoresis parameter n_{t}, the Brownain motion parameter Nb and the local nanoparticle Grashof number br. From Figures 11-13 it is clear that by increasing M, Pr, and n_{t}, the concentration profile desreases, while from _{t}.

Figures 16-21 examine the influence of the magnetic parameter M, the power law index n, the Weissenberg number we, the local nanoparticle Grashof

number br, the local temperature Grashof number g_{r} and the Brownian motion parameter Nb on the pressure gradient. It has been observed that because of successive contraction and relaxation of peristaltic walls the pressure gradient shows oscillatory behavior. It is clear from Figures 16-19, that the magnitude of the pressure gradient decreases in view of an increase in M, n, we and br. The situation is reversed in

The peristaltic flow of a Carreau nanofluid through a porous medium with heat and mass transfer in the presence of Hall current, Joule heating and viscous dissipation is studied under assumption of long wavelength and low Reynolds number. The main results are summarized as follow:

・ The velocity of Carraeu nanofluid increases at the neighborhood of the channel walls and decreases near the center of the channel by increasing the Weissenber number We and the power law index n.

・ It is observed that the magnetic parameter M and the thermophoresis parameter n_{t} have opposite effects on the velocity to that of we and n.

・ The temperature profile increases with an increase in the Brownian motion parameter Nb and decreases with an increase in the thermophoresis parameter n_{t}.

・ The concetration profile decreases with an increase in the Brownian motion parameter Nb and increases with an increase in the thermophoresis parameter n_{t}.

・ The pressure gradient decreases with the increase of the magnetic parameter M, the power law index n, the Weissenber number we and the local nanoparticle Grashof number br, while it increases with the increase of the local temperature Grashof number g_{r} and the thermophoresis parameter n_{t}.

Eldabe, N.T.M., Abo-Seida, O.M., Abo-Seliem, A.A.S., El- Shakhipy, A.A. and Hegazy, N. (2017) Peristaltic Transport of Magnetohydrodynamic Carreau Nanofluid with Heat and Mass Transfer inside Asymmetric Channel. American Journal of Computational Mathematics, 7, 1-20. https://doi.org/10.4236/acjm.2017.71001