# Analysis of Heating Effects and Different Wave Forms on Peristaltic Flow of Carreau Fluid in Rectangular Duct

Authors: Safia Akram; Najma Saleem
Journal: Advances in Mathematical Physics
 (2020)

---

## Abstract

The existing analysis deals with heat transfer occurrence on peristaltic transport of a Carreau fluid in a rectangular duct. Flow is scrutinized in a wave frame of reference moving with velocityc away from a fixed frame. A peristaltic wave propagating on the horizontal side walls of a rectangular duct is discussed under lubrication approximation. In order to carry out the analytical solution of velocity, temperature, and pressure gradient, the homotopy perturbation method is employed. Graphical results are displayed to see the impact of various emerging parameters of the Carreau fluid and power law index. Trapping effects of peristaltic transport is also discussed and observed that number of trapping bolus decreases with an increase in aspect ratio β.


---

## Body

## 1. Introduction

The applications of peristaltic flows in medical and engineering sciences have attracted the attention of a number of researchers. Applications of peristalsis occur in swallowing food through the esophagus, urine transport from the kidney to the bladder through the ureter, transport of the spermatozoa in the efferent ducts of the male reproductive tract, movement of the ovum in the fallopian tube, movement of the chyme in the gastrointestinal tract, the transport of lymph in the lymphatic vessels, and vasomotion in the small blood vessels such as the arterioles, veins, and capillaries. The peristaltic phenomenon was first discussed by Latham [1]. Based on his experimental theory, numerous researchers have inspected the phenomenon of peristaltic transport under many conjectures [2–10].Another fascinating area in connection with peristaltic motion is the heat transfer which has industrial applications like sanitary fluid transport, blood pumps in the heart-lungs machine and transport of corrosive fluids where the contact of fluid with the machinery parts are prohibited. Only a limited attention has been focused to the study of peristaltic flows with heat transfer [11–15].An immense amount of literature is presented on two-dimensional peristaltic flow problems. The study of peristaltic phenomenon in a rectangular channel was first examined by [16]. Based on the theory of [16], several researchers have studied the phenomenon of peristaltic transport in a rectangular duct under various approximations [17–23]. In the papers cited above, the phenomena of heat transfer are not taken into account. Keeping in mind the present information, the heat transfer phenomena on the peristaltic flow of non-Newtonian fluid have not been discussed in a three-dimensional channel. So, the aim of the present problem is to discuss the effects of heat transfer on peristaltic flow of a non-Newtonian fluid in a rectangular duct with different wave forms. The governing equations for the three-dimensional rectangular channel are first modeled for Carreau fluid and then simplified under the long wavelength and low Reynolds number approximation. Homotopy perturbation technique is carried out to calculate the analytical solution of the highly nonlinear partial differential equations. The expressions for velocity, pressure rise, pressure gradient, and temperature have been computed and discussed through graphs. Three-dimensional graphical representations for the velocity field are also discussed through graphs. In the end, different wave forms are also used.


## 2. Mathematical Formulation of the Problem

Let us consider the peristaltic flow of an incompressible Carreau fluid in a duct of rectangular cross section having the channel width 2d and height 2a. We are considering the Cartesian coordinates system in such a way that theX‐axis is taken along the axial direction, Y‐axis is taken along the lateral direction, and Z‐axis is along the vertical direction of a rectangular duct.The peristaltic waves on the walls are represented as(1)Z=HX,t=±a±bcos2πλX−ct,where a and b are the amplitudes of the waves, λ is the wave length, c is the velocity of propagation, t is the time, and X is the direction of wave propagation. The walls parallel to the XZ plane remain undisturbed and are not subjected to any peristaltic wave motion. We assume that the lateral velocity is zero as there is no change in the lateral direction of the duct cross section. Let U,0,W be the velocity for a rectangular duct. The governing equations for the flow problem are
(2)∂U∂X+∂W∂Z=0,(3)ρ∂U∂t+U∂U∂X+W∂U∂Z=−∂P∂X+∂∂XSXX+∂∂YSXY+∂∂ZSXZ,(4)0=−∂P∂Y+∂∂XSYX+∂∂YSYY+∂∂ZSYZ,(5)ρ∂W∂t+U∂W∂X+W∂W∂Z=−∂P∂Z+∂∂XSZX+∂∂YSZY+∂∂ZSZZ,(6)ρC′∂T∂t+U∂T∂X+W∂T∂Z=K′∂2T∂X2+∂2T∂Y2+∂2T∂Z2+SXX∂U∂X++SYX∂U∂Y+SXZ∂W∂X+∂U∂Z+SYZ∂W∂Y+SZZ∂W∂Z,in which ρ is the density, P is the pressure, t is the time, andS,s is the stress tensor for the Carreau fluid; C′ is the specific heat; and T is the temperature. The stress tensor for the Carreau fluid is defined by [17].
(7)S=μ1+Γγ̇2n−1/2γ̇.Let us define a wave framex,y moving with the velocity caway from the fixed frame X,Y by the transformation
(8)x=X−ct,y=Y,z=Z,u=U−c,w=W,px,z=PX,Z,t.Defining the following nondimensional quantities,(9)x¯=xλ,y¯=yd,z¯=za,u¯=uc,w¯=wcδ,t¯=ctλ,h=Ha,p¯=a2pμcλ,Re=ρacδμ,β=ad,δ=aλ,S¯x¯x¯=aμcSxx,S¯x¯y¯=dμcSxy,S¯x¯z¯=aμcSxz,S¯y¯z¯=dμcSyz,S¯z¯z¯=λμcSzz,S¯y¯y¯=λμcSyy⋅Using the above nondimensional quantities in Equations (2), (3), (4), (5) and (7), the resulting equations after dropping the bars can be written as
(10)∂u∂x+∂w∂z=0,(11)Reu∂u∂x+w∂u∂z=−∂p∂x+δ∂∂xτxx+β2∂∂ySxy+∂∂zSxz,(12)0=−∂p∂y+δ2∂∂xSyx+δ2∂∂ySyy+δ∂∂zSyz,(13)Reδ2u∂w∂x+w∂w∂z=−∂p∂z+δ2∂∂xSzx+δβ2∂∂ySzy+δ2∂∂zSzz,(14)Reδu∂θ∂x+w∂θ∂z=1Prδ2∂2θ∂x2+β2∂2θ∂y2+∂2θ∂z2+EcδSxx∂u∂x+β2Sxy∂u∂y+β2δSyz∂w∂y+δ2Szz∂w∂z+Szxδ2∂w∂x+∂u∂z,where
(15)Sxx=−2δ1+n−12We2γ̇2∂u∂x,(16)Sxy=−1+n−12We2γ̇2∂u∂y,(17)Sxz=−1+n−12We2γ̇2∂u∂z+δ2∂w∂x,(18)Syy=0,(19)Syz=−δ1+n−12We2γ̇2∂w∂y,(20)Szz=−21+n−12We2γ̇2∂w∂z,(21)γ̇2=2δ2∂u∂x2+β2∂u∂y2+δ2β2∂w∂y2+δ2∂w∂z2+δ2∂w∂x+∂u∂z2⋅Under the assumption of long wavelengthδ≤1 and low Reynolds number Re⟶0, Equations (10), (11), (12), (13), (14) and (21) take the form
(22)dpdx=β2∂2u∂y2+∂2u∂z2+n−12We2β4∂∂y∂u∂y3+n−12We2∂∂z∂u∂z3+n−12We2β2∂∂y∂u∂y∂u∂z2+n−12We2β2∂∂z∂u∂z∂u∂y2⋅(23)∂2θ∂z2+β2∂2θ∂y2+EcPrβ2∂u∂y2+n−12β4We2∂u∂y4+n−12β2We2∂u∂y2∂u∂z2+∂u∂z2+n−12We2∂u∂z4+n−12β2We2∂u∂y2∂u∂z2=0.The corresponding boundary conditions are(24)u=−1aty=±1,u=−1atz=±hx=±1±φcos2πx,θ=0aty=1,θ=0atz=hx,θ=1aty=−1,θ=1atz=−hx.where 0≤φ≤1, φ=0 for straight duct and φ=1 corresponds to total occlusion.


## 3. Solution of the Problem

### 3.1. Homotopy Perturbation Method

The solution of the nonlinear partial differential Equations (22) and (23) has been calculated using homotopy perturbation technique. The homotopy perturbation method for Equations (22) and (23) can be defined as [24–30].
(25)Hv,q=1−q£v−£u0+q£v+β2∂2u∂y2+n−12We2β4∂∂y∂v∂y3+n−12We2∂∂z∂v∂z3+n−12We2β2∂∂y∂v∂y∂v∂z2+n−12We2β2∂∂z∂v∂z∂v∂y2−dpdx=0,HΘ,q=1−q£Θ−£θ~0+q£Θ+β2∂2Θ∂y2+EcPrβ2∂v∂y2+n−12β4We2∂v∂y4+n−12β2We2∂v∂y2∂v∂z2+∂v∂z2+n−12We2∂v∂z4+n−12β2We2∂v∂y2∂v∂z2=0,or
(26)Hv,q=£v−£u0+q£u0+qβ2∂2u∂y2+n−12We2β4∂∂y∂v∂y3+n−12We2∂∂z∂v∂z3+n−12We2β2∂∂y∂v∂y∂v∂z2+n−12We2β2∂∂z∂v∂z∂v∂y2−dpdx=0,(27)HΘ,q=£Θ−£θ~0+q£θ~0+qβ2∂2Θ∂y2+EcPrβ2∂v∂y2+n−12β4We2∂v∂y4+n−12β2We2∂v∂y2∂v∂z2+∂v∂z2+n−12We2∂v∂z4+n−12β2We2∂v∂y2∂v∂z2=0.Here,q is an embedding parameter which has the range 0≤q≤1, under the condition that for q=0, we get the initial solution, and for q=1, we seek the final solution. Here, £ is the linear operator which is taken here as £=∂2/∂z2. We define the initial guess as
(28)u0=−1+h2−z2+1−y2−h−z2h,θ~0=z2−h2+h−z2h+1β21−y2.Let us define(29)vx,y,z,q=v0+qv1+q2v2+⋯,(30)Θx,y,z,q=Θ0+qΘ1+q2Θ2+⋯,substituting Equation (30) into Equations (26) and (27), and then comparing the like powers of q, one obtains the following problems with the corresponding boundary conditions.


### 3.2. Zeroth Order System

(31)£v0−£u0=0,£Θ0−£θ~0=0,v0=−1aty=±1,v0=−1atz=±hx,Θ0=0aty=1,Θ0=0atz=hx,Θ0=1aty=−1,Θ0=1atz=−hx.


### 3.3. First-Order System

(32)£v1+£u0+β2∂2v0∂y2+n−12We2β4∂∂y∂v0∂y3+n−12We2∂∂z∂v0∂z3+n−12We2β2∂∂y∂v0∂y∂v0∂z2+n−12We2β2∂∂z∂v0∂z∂v0∂y2−dpdx=0,£Θ1+£θ~0+β2∂2Θ0∂y2+EcPrβ2∂v0∂y2+n−12β4We2∂v0∂y4+n−12β2We2∂v0∂y2∂v0∂z2+∂v0∂z2+n−12We2∂v0∂z4+n−12β2We2∂v0∂y2∂v0∂z2=0,v1=0aty=±1,v1=0atz=±hx,Θ1=0aty=1,Θ1=0atz=hx,Θ1=0aty=−1,Θ1=0atz=−hx.


### 3.4. Second-Order System

(33)£v2+β2∂2v1∂y2+n−12We2β4∂∂y3∂v0∂y2∂v1∂y+n−12We2∂∂z3∂v0∂z2∂v1∂z+n−12We2β2∂∂y2∂v0∂y∂v0∂z∂v1∂z+∂v1∂y∂v0∂z2+n−12We2β2∂∂z2∂v0∂y∂v0∂z∂v1∂y+∂v1∂z∂v0∂y2=0,£Θ2+β2∂2Θ1∂y2+EcPr2β2∂v0∂y∂v1∂y+n−12β4We24∂v0∂y3∂v1∂y+2n−12β2We22∂v0∂y2∂v0∂z∂v1∂z+2∂v0∂y∂v1∂y∂v0∂z2+2∂v0∂z∂v1∂z+n−12We24∂v0∂z3∂v1∂z=0,v2=0aty=±1,v2=0atz=±hx,Θ2=0aty=1,Θ2=0atz=hx,Θ2=0aty=−1,Θ2=0atz=−hx.The resulting series solutions after three iterations are determined using Equation (30) as (when q⟶1) and are evaluated as
(34)ux,y,z=−1+h2−z2+1−y2−h−z2h+112−12h4n−1We2+12n−1We2z4−6h22+dpdx+14n−1We2β2+6z22+dpdx+14n−1We2β2+196n−1We2h−zh+z384n−1We2h4+h2z2+z4+12n−1β2We2h2+z2+624h2+z2+β22+dpdx+14n−1We2β2,θx,y,z=−h2+h−z2h+z2+1−y2β2+160PrEc16n−1We2h6−z6+5PrEch4−z44+n−1We2β2+1516PrEch−zh+zβ28+n−1We2β2−11260PrEc212+dpdx+14n−1We2β232h6n−1We2+5h44+n−1We2β2+z4−20−n−1We232z2+5β2+24n−1We260h8n−1We2+7h64+n−1We2β2+z6−28−n−1We260z2+7β2.The volumetric flow rate is given by
(35)q=0hx01ux,y,zdydz=−hx+85h5n−1We2+247h7n−12We4+25h5n−12β2We4+196h3n−12We4β4+dpdx65h5n−1We2+124h3−8+n−1We2β2.The instantaneous flux is defined as(36)Q¯=0hx01u+1dydz=q+hx.The average volume flow rate over one periodT=λ/c of the peristaltic wave is defined as
(37)Q=1T0TQ¯dt=q+1⋅The pressure gradient is obtained from Equations (35) and (37) as
(38)dpdx=Q−1+hx−8/5h5n−1We2−24/7h7n−12We4−2/5h5n−12β2We4−1/96h3n−12β4We46/5h5n−1We2+1/24h3−8+n−1We2β2,Integration of Equation (38) over one wavelength yields
(39)Δp=01dpdxdx⋅It is noticed here that the limitβ⟶0 (keeping it fixed and d⟶∞), the rectangular duct reduces to a two-dimensional channel. It is also noticed that when β=1 the rectangular duct becomes a square duct.


## 3.1. Homotopy Perturbation Method

The solution of the nonlinear partial differential Equations (22) and (23) has been calculated using homotopy perturbation technique. The homotopy perturbation method for Equations (22) and (23) can be defined as [24–30].
(25)Hv,q=1−q£v−£u0+q£v+β2∂2u∂y2+n−12We2β4∂∂y∂v∂y3+n−12We2∂∂z∂v∂z3+n−12We2β2∂∂y∂v∂y∂v∂z2+n−12We2β2∂∂z∂v∂z∂v∂y2−dpdx=0,HΘ,q=1−q£Θ−£θ~0+q£Θ+β2∂2Θ∂y2+EcPrβ2∂v∂y2+n−12β4We2∂v∂y4+n−12β2We2∂v∂y2∂v∂z2+∂v∂z2+n−12We2∂v∂z4+n−12β2We2∂v∂y2∂v∂z2=0,or
(26)Hv,q=£v−£u0+q£u0+qβ2∂2u∂y2+n−12We2β4∂∂y∂v∂y3+n−12We2∂∂z∂v∂z3+n−12We2β2∂∂y∂v∂y∂v∂z2+n−12We2β2∂∂z∂v∂z∂v∂y2−dpdx=0,(27)HΘ,q=£Θ−£θ~0+q£θ~0+qβ2∂2Θ∂y2+EcPrβ2∂v∂y2+n−12β4We2∂v∂y4+n−12β2We2∂v∂y2∂v∂z2+∂v∂z2+n−12We2∂v∂z4+n−12β2We2∂v∂y2∂v∂z2=0.Here,q is an embedding parameter which has the range 0≤q≤1, under the condition that for q=0, we get the initial solution, and for q=1, we seek the final solution. Here, £ is the linear operator which is taken here as £=∂2/∂z2. We define the initial guess as
(28)u0=−1+h2−z2+1−y2−h−z2h,θ~0=z2−h2+h−z2h+1β21−y2.Let us define(29)vx,y,z,q=v0+qv1+q2v2+⋯,(30)Θx,y,z,q=Θ0+qΘ1+q2Θ2+⋯,substituting Equation (30) into Equations (26) and (27), and then comparing the like powers of q, one obtains the following problems with the corresponding boundary conditions.


## 3.2. Zeroth Order System

(31)£v0−£u0=0,£Θ0−£θ~0=0,v0=−1aty=±1,v0=−1atz=±hx,Θ0=0aty=1,Θ0=0atz=hx,Θ0=1aty=−1,Θ0=1atz=−hx.


## 3.3. First-Order System

(32)£v1+£u0+β2∂2v0∂y2+n−12We2β4∂∂y∂v0∂y3+n−12We2∂∂z∂v0∂z3+n−12We2β2∂∂y∂v0∂y∂v0∂z2+n−12We2β2∂∂z∂v0∂z∂v0∂y2−dpdx=0,£Θ1+£θ~0+β2∂2Θ0∂y2+EcPrβ2∂v0∂y2+n−12β4We2∂v0∂y4+n−12β2We2∂v0∂y2∂v0∂z2+∂v0∂z2+n−12We2∂v0∂z4+n−12β2We2∂v0∂y2∂v0∂z2=0,v1=0aty=±1,v1=0atz=±hx,Θ1=0aty=1,Θ1=0atz=hx,Θ1=0aty=−1,Θ1=0atz=−hx.


## 3.4. Second-Order System

(33)£v2+β2∂2v1∂y2+n−12We2β4∂∂y3∂v0∂y2∂v1∂y+n−12We2∂∂z3∂v0∂z2∂v1∂z+n−12We2β2∂∂y2∂v0∂y∂v0∂z∂v1∂z+∂v1∂y∂v0∂z2+n−12We2β2∂∂z2∂v0∂y∂v0∂z∂v1∂y+∂v1∂z∂v0∂y2=0,£Θ2+β2∂2Θ1∂y2+EcPr2β2∂v0∂y∂v1∂y+n−12β4We24∂v0∂y3∂v1∂y+2n−12β2We22∂v0∂y2∂v0∂z∂v1∂z+2∂v0∂y∂v1∂y∂v0∂z2+2∂v0∂z∂v1∂z+n−12We24∂v0∂z3∂v1∂z=0,v2=0aty=±1,v2=0atz=±hx,Θ2=0aty=1,Θ2=0atz=hx,Θ2=0aty=−1,Θ2=0atz=−hx.The resulting series solutions after three iterations are determined using Equation (30) as (when q⟶1) and are evaluated as
(34)ux,y,z=−1+h2−z2+1−y2−h−z2h+112−12h4n−1We2+12n−1We2z4−6h22+dpdx+14n−1We2β2+6z22+dpdx+14n−1We2β2+196n−1We2h−zh+z384n−1We2h4+h2z2+z4+12n−1β2We2h2+z2+624h2+z2+β22+dpdx+14n−1We2β2,θx,y,z=−h2+h−z2h+z2+1−y2β2+160PrEc16n−1We2h6−z6+5PrEch4−z44+n−1We2β2+1516PrEch−zh+zβ28+n−1We2β2−11260PrEc212+dpdx+14n−1We2β232h6n−1We2+5h44+n−1We2β2+z4−20−n−1We232z2+5β2+24n−1We260h8n−1We2+7h64+n−1We2β2+z6−28−n−1We260z2+7β2.The volumetric flow rate is given by
(35)q=0hx01ux,y,zdydz=−hx+85h5n−1We2+247h7n−12We4+25h5n−12β2We4+196h3n−12We4β4+dpdx65h5n−1We2+124h3−8+n−1We2β2.The instantaneous flux is defined as(36)Q¯=0hx01u+1dydz=q+hx.The average volume flow rate over one periodT=λ/c of the peristaltic wave is defined as
(37)Q=1T0TQ¯dt=q+1⋅The pressure gradient is obtained from Equations (35) and (37) as
(38)dpdx=Q−1+hx−8/5h5n−1We2−24/7h7n−12We4−2/5h5n−12β2We4−1/96h3n−12β4We46/5h5n−1We2+1/24h3−8+n−1We2β2,Integration of Equation (38) over one wavelength yields
(39)Δp=01dpdxdx⋅It is noticed here that the limitβ⟶0 (keeping it fixed and d⟶∞), the rectangular duct reduces to a two-dimensional channel. It is also noticed that when β=1 the rectangular duct becomes a square duct.


## 4. Expressions for Different Wave Shapes

The nondimensional expressions for five considered wave forms are given by [11]. The expression for the triangular, square, and trapezoidal waves are derived from the Fourier series.
(1)
Sinusoidal wave(40)hx=±1±ϕsin2πx(2)
Multisinusoidal wave(41)hx=±1±ϕsin2mπx(3)
Triangular wave(42)hx=±1±ϕ8π3∑m=1∞−1m+12m−12sin2π2m−1x(4)
Trapezoidal wave(43)hx=±1±φ32π2∑m=1∞sinπ/82m−12m−12sin2π2m−1x(5)
Square wave(44)hx=±1±φ4π∑m=1∞−1m+12m−1cos22m−1πx


## 5. Numerical Results and Discussion

In this segment, the graphical representation of the proposed problem is discussed. With the help of mathematics software Mathematica, the expression for pressure rise and pressure gradient is calculated. In order to see the behavior of pressure rise with volume flow rateQ for different values of aspect ratio β, power law index n, amplitude ratio ϕ, and Weissenberg number We. It is observed from Figure 1 that in the peristaltic pumping Δp>0,Q>0 and retrograde pumping Δp>0,Q<0 regions, the pumping rate decreases with an increase in aspect ratio β, while in the copumping region Δp<0,Q>0, the behavior is quite opposite; here, the pumping rate increases with an increase in aspect ratio β. Figures 2 and 3 show the variation of pressure rise with volume flow rate Q for different values of n and ϕ. It is observed from Figure 2 that in the retrograde pumping Δp>0,Q>0 and peristaltic pumping Δp>0,Q<0 regions, the pumping rate increases with an increase in n and ϕ, while in the copumping region Δp<0,Q>0, the pressure rise decreases with an increase in values of n and φ. In order to see the behavior of pressure rise with volume flow rate Q for different values of We, Figure 4 is plotted. It is depicted from Figure 4 that in the peristaltic pumping Δp>0,Q>0 and retrograde pumping Δp>0,Q<0 regions, the pumping rate decreases with an increase in We, while in the copumping region Δp<0,Q>0, the behavior is quite opposite; here, the pumping rate increases with an increase in We. The pressure gradient for different values of aspect ratio β, n, amplitude ratio ϕ, and We against space variable x is plotted in Figures 5–8. It is depicted from Figures 5–8 that for x∈0,0.2 and x∈0.8,1, the pressure gradient is small, i.e., the flow can easily pass without the imposition of a large pressure gradient, while in the region x∈0.2,0.8 that the pressure gradient decreases with an increase in aspect ratio β and We and increases with an increase in amplitude ratio ϕ and n, so much pressure gradient is required to maintain the flux to pass. Figures 9–12 show the stream lines for different wave forms. It is observed from Figures 9–12 that the pressure gradient retains the same shape of the waves as we consider.Figure 1
Variation of pressure rise with volume flow rateQ for different values of β for fixed values of φ=0.7, We=0.5, and n=0.6.Figure 2
Variation of pressure rise with volume flow rateQ for different values of n for fixed values of φ=0.8, We=0.6, and β=4.Figure 3
Variation of pressure rise with volume flow rateQ for different values of φ for fixed values of n=0.5, We=0.6, and β=4.Figure 4
Variation of pressure rise with volume flow rateQ for different values of We for fixed values of φ=0.7, n=0.3, and β=6.Figure 5
Variation of pressure gradient withx for different values of β for fixed values of ϕ=0.5, n=0.5, We=0.4, and Q=−2.Figure 6
Variation of pressure gradient withx for different values of n for fixed values of φ=0.5, β=4, We=0.6, and Q=−2.Figure 7
Variation of pressure gradient withx for different values of φ for fixed values of n=0.5, β=4, We=0.6, and Q=−1.Figure 8
Variation of pressure gradient withx for different values of We for fixed values of φ=0.5, β=4, n=0.5, and Q=−2.Figure 9
Variation of pressure gradient withx for different values of n for fixed values of φ=0.5, β=4, We=0.6, and Q=−2.Figure 10
Variation of pressure gradient withx for different values of n for fixed values of φ=0.5, β=4, We=0.6, and Q=−2.Figure 11
Variation of pressure gradient withx for different values of n for fixed values of φ=0.5, β=6, We=0.6, and Q=−3.5.Figure 12
Variation of pressure gradient withx for different values of n for fixed values of φ=0.5, β=4, We=0.6, and Q=−2.Figures13–16 show the temperature profile for different values of We, β, Pr, and Ec. It is depicted from Figure 13 that the magnitude value of the temperature profile increases with an increase in We. It is observed from Figure 14 that the magnitude value of the temperature profile decreases with an increase in β Figures 15 and 16 show the temperature profile for different values of Pr and Ec. It is observed from Figures 15 and 16 that temperature profile increases with an increase in Pr and Ec. Velocity profile for different values of β, Q, and We is plotted in Figures 17–19. It is observed from Figures 17 and 18 that the magnitude value of the velocity profile decreases with an increase in β volume flow rate Q. It is depicted from Figure 19 that the magnitude value of the velocity profile increases with an increase in We. Stream lines for different values of aspect ratio β amplitude ratio ϕ, power law index n, and Weissenberg number We is plotted in Figures 20–23. It is observed from Figure 20 that the number of the trapping bolus decreases with an increase in aspect ratio β. It is observed from Figures 21 and 22 that the size of the trapping bolus increases with an increase in power law index n and amplitude ratio ϕ Figure 23 shows the stream lines for different values of the Weissenberg number We. It is depicted from Figure 23 that the number of the trapping bolus increases in the lower half of the channel with an increase in We, while in the upper half of the channel the size of the trapping bolus increases. Figure 24 shows the stream lines for different wave forms.Figure 13
Temperature profile for different values ofWe for fixed values of φ=0.6, x=0, β=0.5, n=2, Q=−1, y=1, Pr=0.5, and Ec=0.2.Figure 14
Temperature profile for different values ofβ for fixed values of φ=0.6, x=0We=0.03, n=2, Q=−1, y=1, Pr=0.5, and Ec=0.2.Figure 15
Temperature profile for different values ofPr for fixed values of φ=0.6, x=0, We=0.03, n=2, Q=−1, y=1, β=0.2, and Ec=0.5.Figure 16
Temperature profile for different values ofEc for fixed values of φ=0.6, x=0, We=0.03, n=2, Q=−1, y=1, β=0.2, and Pr=0.5.Figure 17
Velocity profile for different values ofβ. The other parameters are φ=0.6, x=0, Q=−0.5, We=0.3, and n=2.Figure 18
Velocity profile for different values ofQ. The other parameters are φ=0.6, x=0, β=0.5, We=0.03, and n=1.Figure 19
Velocity profile for different values of We. The other parameters areφ=0.6, x=0, β=0.9, Q=4, and n=2.Figure 20
Stream lines for different values ofβ. The other parameters are φ=0.7, We=0.51, y=1, n=−0.8, and Q=2.0.
(a)(b)Figure 21
Stream lines for different values ofn. The other parameters are φ=0.7, β=4, We=0.51, y=1, and Q=2.0.
(a)(b)Figure 22
Stream lines for different values ofφ. The other parameters are n=−0.8, β=7, We=0.6, y=1, and Q=2.0.
(a)(b)Figure 23
Stream lines for different values of We. The other parameters aren=−0.8, β=4, φ=0.7, y=1, and Q=2.0.
(a)(b)Figure 24
Stream lines for different wave forms (a) for the sinusoidal wave, (b) for the multisinsoidal wave, (c) for the square wave, (d) for the trapezoidal wave, and (e) for the triangular wave. The other parameters aren=−0.8, β=6, φ=0.7, y=1, We=0.53, and Q=3.3.
(a)(b)(c)(d)(e)


## 6. Conclusion

We have discussed the impact of heat transfer on peristaltic flow of a Carreau fluid in a rectangular duct with different wave forms. The governing equations for a three-dimensional rectangular channel are the first model for Carreau fluid and then simplified under long wave length and low Reynolds number approximation. The analytical solutions of highly nonlinear equations are calculated. The results are discussed through graphs. It is observed that pumping rate decreases with an increase in aspect ratioβ and We in the peristaltic Δp>0,Q>0 and retrograde pumping Δp>0,Q<0 regions, while in the copumping region Δp<0,Q>0, the behavior is quite opposite. It is also noted that in the retrograde pumping Δp>0,Q<0 and peristaltic pumping Δp>0,Q>0 regions, the pumping rate increases with an increase in n and φ. It is observed that the profile of temperature increases with an increase in We, Pr, and Ec, while temperature profile shows the opposite behavior in the case of β. It is also observed that number of trapping bolus increases in the lower half of the channel with an increase in We, while in the upper half of the channel, size of the trapping bolus increases.



---

*Source: 8294318-2020-05-20.xml*