Biological flow of thermally intense cilia generated motion of non-Newtonian fluid in a curved channel

Abstract

The energy loss during the beating cilia phenomenon in the human stomach causing acidity in the blood flow under certain conditions has been a serious topic in the modern medical field. Therefore, the current study intends to exhibit a theoretical analysis of mixed convective transport of non-Newtonian Casson fluid observed by ciliary motion walls in the curved channel. The flow of constitutive equations is used to modify in curvilinear coordinates into a wave frame for two-dimensional flow due to the complication of the flow regime. The attributes of biological ciliary approximation are revealed through the control of viscous and inertial impacts utilizing the long-wavelength assumption and obtained the analytical closed form solutions for the normalized equations. The impacts of physical parameters on the velocity profile and heat flow phenomena are discussed. It is observed that the flow velocity, the momentum bolus and the trapped bolus are reduced in the cilia transport channel by enhancing the channel curvature. A validity of admirable comparison is also noticed with previously results.

Introduction

Numerous microscopic organisms have hair-like projections called cilia. They are designed to generate fluid flow for cellular or transport purposes in the surrounding environment. Cilia perform important function motility in the living things by producing air flow that are transported out of the lungs. Additionally, there are known as investigational facts that cilia and flagella induced a flow during development and the placement of organs in body.^1,2 The possibility of motile and non-motile cilia of using in different biological phenomena has drawn in extensive research interest of many researchers in the special fields of medical, mathematics and engineering in recent years. The two-dimensional hydromagnetic transportation of viscous liquid through curled conduit reported by Siddiqui et al. ³ and attained analytical outcomes by utilizing lubrication theory. The heat transfer mechanism of the wavelike cilia motion of electrically conducting viscous liquid in a symmetric conduit was conveyed by Sher Akbar et al. ⁴ By taking into account the assumptions under that the corresponding creeping phenomena and long wavelength conditions Reynolds number is approach to small and the diameter ratio is large for the pressure force to be consideration, Maiti and Pandey ⁵ deliberated the ciliary approach of rheological fluid under variable Reynolds number in symmetry less tube. Nadeem and Sadaf ⁶ analyzed the liquid velocity with rising role of curvature close to the cilia-oriented surface in a two-dimensional curved channel due to cilia however, it is diminishing in the inner section of the conduit. Shaheen et al. ⁷ studied a case of heat transmission in a Williamson fluid flow through a ciliated porous channel with semi numerical approach. More significant applications of fluid flow based on ciliary motion can be seen in Sadaf and Nadeem, ⁸ Krishna et al., ⁹ Javid et al., ¹⁰ and Al-Zubaidi et al. ¹¹

The study of non-Newtonian fluids has gained great importance mainly due to their huge range of practical application in engineering and industry. Numerous research workers have studied diverse flow problem related to several non-Newtonian fluids. Among the non-Newtonian fluids, Casson fluid has attracted more attention of researchers due to its applications in the fields of food processing, drilling operation, bioengineering operations, and metallurgy. At zero shear rate, Casson fluid is a unmeasurable viscosity with a shear-thinning liquid. When shear rates are low, shear thinning fluids are more viscous compared to Newtonian fluid but when shear rates are high, they become less viscous. The influential models used for revealing characteristics of yield stress are the liquid flow model, which Casson ¹² first introduced in 1959. When yield strains exceed shear stresses, Casson fluid freezes. When the yield stresses are lower than the shear stresses, a movement begins. Some examples include sauce, soup, jam, fruit juices, and honey. The ciliary oriented in a rotating tube served as an inspiration for the transportation system of a Casson liquid flow was studied by Siddiqui et al. ¹³ Thermally radiative transportation of Casson liquid in an inclined conduit having sinusoidal walls was explored by Abbas et al., ¹⁴ wherein they attained exact solutions of normalized equations. The consequence of temperature-dependent variations electrically conducting cilia movement of Casson fluid in a conduit was evaluated by Divya et al. ¹⁵ Several notable works on the subject are available in the Ramesh and Devakar, ¹⁶ Givi and Sangeetha George, ¹⁷ Saleem et al., ¹⁸ Ali et al., ¹⁹ and Abbas and Rafiq. ²⁰

Cilia flow with heat transfer is of enormous significance in oxygenation and hemodialysis. The process of heat transfer is used in vasodilation, paper making, food processing, etc. A heat source/sink is used to transfer heat in flow fields. A heat sink is a passive heat exchanger that extracts extra heat created by a system and dissipates it away from the system allowing it to maintain an optimal temperature. A heat source is a device from which is used to add heat in the fluid medium. The process of heat generation/absorption is very helpful in controlling the thermal performance of various fluids. It occurs in the fabrication of plastic and rubber sheets, storage of victual stuff, disposal of radioactive waste materials, etc. Akram et al. ²¹ examined the effects of heat transfer and chemical reaction on Prandtl-Eyring fluid in an inclined channel. Qureshi et al. ²² investigated the impact of radially varying magnetic field during peristaltic flow with inner heat generation. Al Nuwairan et al. ²³ explained the solutions of magnetic filed hybrid nanofluid flow past a surface by heat generation/absorption. The influence of heat generation/absorption mechanism on stagnation point flow by rotating disk influenced by activation energy was explored by Al Nuwairan et al. ²⁴

Mixed convection is a common phenomenon in both engineering systems and environmental processes. Examples of such systems are reactor cooling, heat exchangers, cooling of electrical components, float glass processing, and indoor ventilation with radiators etc. Thus significance of mixed convective peristaltic transport of nanofluid in presence of Soret and Dufour effects is studied by Hayat et al. ²⁵ Mustafa et al. ²⁶ investigated the numerical solution for mixed convective peristaltic flow of fourth grade fluid. Influence of heat and mass transfer in terms of mixed convection, initial stress and radially varying magnetic field on the peristaltic flow in an annulus has been discussed by Abd-Alla et al. ²⁷ The effect of mixed convection flow of nanoparticles on the peristaltic motion of tangent hyperbolic fluid model in an annulus has been studied by Nadeem et al. ²⁸ Peristaltic transport of magneto-nanoparticles submerged in water for drug delivery system is discussed by Abbasi et al. ²⁹ Analysis of mixed convection and hall current for MHD peristaltic transport of nanofluid with compliant wall was reported by Alsaedi et al. ³⁰ Ahmed et al. ³¹ discussed the mixed convection and thermal radiation effect on MHD peristaltic motion of Powell-Eyring nanofluid using lubrication approximation theory. Some other inquiries dealing with the heat transfer can be seen in Riaz et al., ³² and Alolaiyan et al. ³³

The above literature survey clearly indicates that no study is available to cover the cilia flow analysis inside a curved channel for Casson fluid with heat transfer. The novelty of the problem is given in Table 1. The non-Newtonian fluids have received a great attention during the recent years. The flow of non-Newtonian fluids is widely observed in industry and biology, for example, enhanced oil recovery, chemical processes such as in distillation towers and fixed-bed reactors and in the applications of pumping fluids such as synthetic lubricants, colloidal fluids, liquid crystals, and biofluids (e.g. animal and human blood).Therefore, the current study is concerned with the mixed convective flow of non-Newtonian Casson fluid by metachronal waves of bio-mimetic cilia in the curved channel. The problem is formulated by considering the assumptions that the Reynolds number is small enough for inertial effects to be negligible and wavelength to diameter ratio is large enough for the pressure to be supposed uniform over the cross-section. The analytical solution is attained for the velocity and temperature distributions. The influence of different physical parameters on the fluid flow is observed. The computed results for the ciliated walls are discussed and shown in graphical forms. A motivation of the present analysis is the hope that such a problem will be applicable in many clinical applications such as the stomach problem.

Table 1.

Novelty of the present problem.

Ref.no.	Nadeem and Sadaf 6	Sadaf and Nadeem 8	Ijaz et al. 34	Present
Curved channel	Yes	Yes	Yes	Yes
Cilia induced flow	Yes	Yes	Yes	Yes
Casson fluid	No	No	No	Yes
Heat source/sink	No	Yes	Yes	Yes
Analytical solution	Yes	Yes	Yes	Yes
Channel symmetry	Yes	No	No	Yes

Mathematical formulation

We assume a curved two dimensional flow problem channel of an laminar incompressible non-Newtonian Casson fluid under the influence of heat source having flexible walls to be investigated as shown in (Figure 1) which curled in a circle of radius $R*$ having a center at $O.$ A ciliary oriented wave train flowing at wave speed $c$ along the walls of a curved regime controlled the fluid motion. These walls have different amplitudes. Flow has been examined in term of curvilinear $(\overline{U},\overline{V})$ coordinate system with $\overline{V}$ is representing radial and $\overline{U}$ is representing axial directions. The curved conduit upper wall has a temperature of $T_1$ while the lower wall is maintained at $T_0.$

Figure 1.

Problem’s geometry.

The fundamental equations governing the flow are given by:

$$\mathit{\nabla }.\mathit{V}=0,$$

(1)

$$\rho \frac{d\mathit{V}}{\mathrm{dt}}=\nabla .\mathit{\tau }-\rho \overrightarrow{g}\beta (T-T_1),$$

(2)

$$\rho c_p\frac{\mathrm{dT}}{\mathrm{dt}}=k^{*}{\nabla }^{2}T+Q_0,$$

(3)

where, $T$ is temperature, $\mathit{\tau }$ is Cauchy stress tensor, $c_p$ represents the specific heat, $k^{*}$ is the thermal conductivity, $\rho$ is fluid density, $Q_0$ is heat absorption parameter, and $\beta$ is the angular velocity. The relation for $\mathit{\tau }$ is

$$\mathit{\tau }=-P\mathit{I}+\mathit{S},$$

(4)

where, $\mathit{I}$ is identity tensor, $P$ is pressure, and $\mathit{S}$ is extra stress tensor of the non-Newtonian fluid presently taken as Casson fluid model ³⁵ :

$$\begin{array}{c}\multicolumn{1}{c}{{\tau }_{\mathrm{ij}}=\{\begin{array}{c}(p_y/\sqrt{2\pi c}+{\mu }_B)2e_{\mathrm{ij}},\pi <{\pi }_c,\hfill \\ \multicolumn{1}{c}{(p_y/\sqrt{2\pi }+{\mu }_B)2e_{\mathrm{ij}},\pi >{\pi }_c.}\end{array}}\end{array}$$

(5)

where ${\mu }_B$ is the viscosity of the Casson liquid, $\pi$ is the deformation rate product component, $p_y$ is the liquid yield stress, and ${\mu }_c$ is the non-Newtonian model’s product value, which is used to represent the non-Newtonian fluid’s component of the deformation rate and is written as $e_{\mathrm{ij}}.$

The velocity field is given by

$$\overline{\mathit{V}}=(\overline{V}(\overline{X},\overline{R},\overline{t}),\overline{U}(\overline{X},\overline{R},\overline{t}),0).$$

(6)

Using equations (1) to (6), the governing equations in the fixed frame for the non-Newtonian Casson fluid in the curvilinear coordinates are [6,8]

$$\frac{\partial }{\partial \overline{R}}\left(\overline{V}(R^{*}+\overline{R})\right)+\frac{\partial \overline{U}}{\partial \overline{X}}{R}^{*}=0,$$

(7)

$$\begin{array}{c}\frac{\partial \overline{V}}{\partial \overline{X}}\frac{R^{*}\overline{U}}{R^{*}+\overline{R}}+\frac{\partial \overline{V}}{\partial \overline{t}}-\frac{{\overline{U}}^2}{R^{*}+\overline{R}}+\overline{V}\frac{\partial \overline{V}}{\partial \overline{R}}=\frac{\mu }{\rho }(1+\frac{1}{\zeta })\hfill \\ \frac{\partial }{\partial \overline{R}}\left((R^{*}+\overline{R})\frac{\partial \overline{V}}{\partial \overline{R}}\right)\frac{1}{R^{*}+\overline{R}}-\frac{\overline{V}}{{(R^{*}+\overline{R})}^2}-\frac{\partial \overline{P}}{\partial \overline{R}}\hfill \\ +{\left(\frac{R^{*}}{R^{*}+\overline{R}}\right)}^2\frac{{\partial }^2\overline{V}}{\partial {\overline{X}}^2}-\frac{2R^{*}}{{(R^{*}+\overline{R})}^2}\frac{\partial \overline{U}}{\partial \overline{X}},\hfill \end{array}$$

(8)

$$\begin{array}{c}\rho (\overline{V}\frac{\partial \overline{U}}{\partial \overline{R}}+\frac{R^{*}\overline{U}}{R^{*}+\overline{R}}\frac{\partial \overline{U}}{\partial \overline{X}}-\frac{\overline{U}\overline{V}}{R^{*}+\overline{R}}+\frac{\partial \overline{U}}{\partial \overline{t}})\hfill \\ =-\frac{R^{*}}{R^{*}+\overline{R}}\frac{\partial \overline{P}}{\partial \overline{X}}+\frac{{\partial }^2\overline{U}}{\partial {\overline{X}}^2}{\left(\frac{R^{*}}{R^{*}+\overline{R}}\right)}^2+(\overline{T}-T_1)\hfill \\ \left(\rho \beta \right)g+\mu (1+\frac{1}{\zeta })\left(\frac{1}{R^{*}+\overline{R}}\frac{\partial }{\partial \overline{R}}\left(\frac{\partial \overline{U}}{\partial \overline{R}}(R^{*}+\overline{R})\right)\right)\hfill \\ +\frac{2R^{*}}{{(R^{*}+\overline{R})}^2}\frac{\partial \overline{V}}{\partial \overline{X}}-\frac{\overline{U}}{{(R^{*}+\overline{R})}^2},\hfill \end{array}$$

(9)

$$\begin{array}{c}(\frac{\partial \overline{T}}{\partial \overline{X}}\frac{R^{*}\overline{U}}{R^{*}+\overline{R}}+\frac{\partial \overline{T}}{\partial \overline{R}}\overline{V}+\frac{\partial \overline{T}}{\partial \overline{t}})=\frac{k}{\rho c_p}\hfill \\ (\frac{\partial \overline{T}}{\partial \overline{R}}\frac{1}{R^{*}+\overline{R}}+\frac{{\partial }^2\overline{T}}{\partial {\overline{R}}^2}+\frac{{\partial }^2\overline{T}}{\partial {\overline{X}}^2}\frac{2R^{*}}{{(R^{*}+\overline{R})}^2})+\frac{1}{\rho c_p}{Q}_0,\hfill \end{array}$$

(10)

where $\overline{U},$ $\overline{V}$ symbolize the mechanism of an axial and radial direction, $\zeta$ signify the Casson liquid parameter, and $\rho$ is the liquid density.

Envelope of ciliary tips can be mathematical form expressed below, taking into consideration the metachronal geometry of the wave shape:

$$\overline{R}=f(\overline{X},\overline{t})=[a+a\varphi \cos \left((\overline{X}-\overline{t}c)\frac{2\pi }{\lambda }\right)]=\overline{H}.$$

(11)

Here $a$ describe the channel radius, $\varphi$ is a dimensionless parameter of ciliary length, and $\lambda$ is the wavelength of the metachronal wave. Mathematically form of ciliary oriented tips in the axial path can be assumed to transport in an elliptical track given below

$$\overline{X}=\overline{g}(\overline{X},{\overline{X}}_0,\overline{t})={\overline{X}}_0+a\alpha \varphi \sin \left((\overline{X}-\overline{t}c)\frac{2\pi }{\lambda }\right),$$

(12)

where $\alpha$ and ${\overline{X}}_0$ display a measure of the elliptical motion eccentricity and the particle reference position, respectively. Axial and radial velocities of the ciliated walls are

$$\begin{array}{c}\overline{U}={\frac{\partial \overline{X}}{\partial \overline{t}}|}_{{\overline{X}}_0}=\frac{\partial \overline{g}}{\partial \overline{t}}+\frac{\partial \overline{g}}{\partial \overline{X}}\frac{\partial \overline{X}}{\partial \overline{t}}=\frac{\partial \overline{g}}{\partial \overline{t}}+\frac{\partial \overline{g}}{\partial \overline{X}}\overline{U},\hfill \\ \overline{V}={\frac{\partial \overline{R}}{\partial \overline{t}}|}_{{\overline{X}}_0}=\frac{\partial \overline{f}}{\partial \overline{t}}+\frac{\partial \overline{f}}{\partial \overline{X}}\frac{\partial \overline{X}}{\partial \overline{t}}=\frac{\partial \overline{f}}{\partial \overline{t}}+\frac{\partial \overline{f}}{\partial \overline{X}}\overline{U}.\hfill \end{array}$$

(13)

Invoking (11) and (12) into (13), we achieve

$$\begin{array}{c}\overline{U}=\frac{-\frac{2\pi }{\lambda }\mathrm{ca}\alpha \varphi \cos \left(2\pi (\overline{X}-\overline{t}c)\frac{1}{\lambda }\right)}{1-\frac{2\pi }{\lambda }\alpha a\varphi \cos \left(2\pi (\overline{X}-\overline{t}c)\frac{1}{\lambda }\right)}=\chi (\overline{H},\overline{t})\hfill \\ \overline{V}=\frac{\frac{2\pi }{\lambda }\mathrm{ca}\alpha \varphi \sin \left(2\pi (\overline{X}-\overline{t}c)\frac{1}{\lambda }\right)}{1-\frac{2\pi }{\lambda }\alpha a\varphi \cos \left(2\pi (\overline{X}-\overline{t}c)\frac{1}{\lambda }\right)}\mathrm{at}\overline{R}=\overline{H}.\hfill \end{array}$$

(14)

The corresponding boundary conditions for the current analysis are ⁶ :

$$\begin{array}{c}\frac{\partial \overline{U}}{\partial R}=0,\overline{T}=T_1\mathrm{at}\overline{R}=0,\hfill \\ \overline{U}=\chi \left(\overline{H}\right),\overline{T}=T_0\mathrm{at}\overline{R}=\overline{H}.\hfill \end{array}$$

(15)

Translate our quantities in dimensionless form between the fixed and moving frame via Galilean transformation

$$\overline{p}=\overline{P},\overline{v}=\overline{V},\overline{x}=\overline{X}-c\overline{t},\overline{u}=\overline{U}-c,\overline{r}=\overline{R}.$$

(16)

Defining the following dimensionless quantities are given by

$$\begin{array}{c}v=\frac{\overline{v}}{c\delta },\mathrm{Re}=\frac{\mathrm{ac}\rho }{\mu },k=\frac{R^{*}}{a},r=\frac{\overline{r}}{a},\delta =\frac{a}{\lambda },\hfill \\ u=\frac{\overline{u}}{c},B=\frac{Q_0{a}^2}{(T_0-T_1)k},x=\frac{\overline{x}}{\lambda },p=\frac{a^2\overline{p}}{c\lambda \mu },\theta =\frac{\overline{T}-T_1}{T_0-T_1},\hfill \\ h=\frac{\overline{H}}{a}=1+\varphi \cos \left(2\pi x\right),\zeta =\mu \beta \sqrt{2{\pi }_c}/p_y,\hfill \\ G_r=\frac{(T_0-T_1)g\beta a^2\rho }{c\mu }.\hfill \end{array}$$

(17)

Now invoking equations (16) and (17) in equations (7) to (10), and take the assumptions of negligible Reynolds number and large wavelength yield, we acquire the subsequent equations

$$\frac{\partial p}{\partial r}=0,$$

(18)

$$\frac{k}{r+k}\frac{\partial p}{\partial x}=(1+\frac{1}{\zeta })[\frac{1}{r+k}\frac{\partial }{\partial r}(\frac{\partial u}{\partial r}(k+r))-\frac{u}{{(r+k)}^2}]+G_r\theta ,$$

(19)

$$(r+k)\frac{{\partial }^2\theta }{\partial r^2}+(r+k)B+\frac{\partial \theta }{\partial r}=0.$$

(20)

To get the equations (19) and (20) of solutions while the equation (18) shows that $p\ne p\left(r\right)$ , and the boundary conditions are reduced to

$$u=-1-2\pi \varphi \alpha \delta \cos 2\pi x=\chi \left(h\right),\theta =1,\mathrm{at}r=h.$$

(21)

The channel symmetry suggests that

$$\frac{\partial u}{\partial r}=0,\theta =0,\mathrm{at}r=0.$$

(22)

Solution methodology

The exact solution to the boundary value problem (19) to (22) can be simply inscribed as

$$\theta =-1/4\mathrm{Br}(2k+r)+C_2+(\left(B{k}^2\right)/2+C_1)\log (k+r),$$

(23)

$$\begin{array}{l}u=-(1/\left(180A\left(kh2+2k^2\left(k+r\right)+h^2\right)\right)\left(A1+B{G}_{r}A2-45k\right)\left(h+2k\right)D)+2k\\ \left(G_{r}A3+90k^{3}D\right)r+\left(G_{r}A4+90k^{3}D\right)r^2-5G_r\left(h^2+2hk+2k^2\right)(16C_1-12C_2+11\\ B{k}^2)r^3-15B{G}_{r}k\left(h^2+2hk+2k^2\right)r^4-3B{G}_r\left(2hk+h^2+k^{2}2\right)r^5+180A\left(-r+h\right)\\ \left(\left(rk+\left(r+k\right)h\right)-\left(k+h\right)\left(r2k+r^2+2k^2\right)w\right)+30(4C_1{G}_r+2B{G}_r{k}^2-3D)\left(-r+h\right)\\ \left(k^3\left(2k+r+h\right)logk-{\left(k+h\right)}^2\left(G_r\left(h+k\right)\left(2C_1+B{k}^2\right)-3kD\right)\right)log\left(h+k\right)+\\ \left(2hk+2k^2+h^2\right){\left(k+r\right)}^2\left(2C_1{G}_r\right)\left(k+r\right)+\left(-3D+B{G}_{r}k\left(k+r\right)\right)klog\left(k+r\right)\end{array}$$

(24)

where

$$\begin{array}{c}A=1+\frac{1}{\zeta },A_1=-30C_2{G}_{r}h(2h+3k)+10C_1{G}_r(8h^2+15\mathrm{hk}+6k^2),\hfill \\ A_2=3h^4+15h^{3}k+55h^2{k}^2+75h{k}^3+30k^4,\hfill \\ A_3=h^2(80C_1-60C_2+3B{h}^2)+15h^2(8C_1-6C_2+B{h}^2)k+55B{h}^3{k}^2-60(C_1-B{h}^2)k^2-30B{k}^5,\hfill \\ A_4=h^3(80C_1-60C_2+3B{h}^2)+15B{h}^{2}k+5h(-48C_1+36C_2+11B{h}^2)k^2+60(-5C_1+3C_2)k^3\hfill \\ -120\mathrm{Bh}{k}^2-150B{k}^2,\hfill \\ D=-42f{h}^2+h^3+4\mathrm{fhk}+h^{2}k+4f{k}^2+2h{k}^2-h^{2}w-3h^2\mathrm{kw}-2h{k}^{2}w+2k(h+k)(1+w)\mathrm{logk}1/\hfill \\ h(h^2+4h^{2}k+8h{k}^2+8k^3)+4k^2(h+k)\mathrm{logk}-8k^2(h+k)\log (k+h)+4k^2{(k+h)}^2\log {(h+k)}^2,\hfill \\ C_1=-4+B{h}^2+2\mathrm{Bhk}+2B{k}^2\mathrm{logk}-2B{k}^2\log (h+k)/4\mathrm{logk},\hfill \\ C_2=4\mathrm{logk}-B{h}^2\mathrm{logk}-2\mathrm{Bhklogk}/4\log (h+k)-\mathrm{logk}.\hfill \end{array}$$

The term for the pressure gradient is achieved as

$$F=\underset{0}{\overset{h}{\int }}\left(u\right)\mathrm{dr}.$$

(25)

Pressure gradient, flow rate, and pressure rise can be calculated as

$$\frac{F-a_1}{a_2}=\frac{\mathrm{dp}}{\mathrm{dx}},Q=F+1,\mathrm{\Delta }p=\underset{0}{\overset{1}{\int }}\frac{\mathrm{dp}}{\mathrm{dx}}\mathrm{dx}.$$

(26)

In physical terms, a stream function is used to describe the dynamic information about the trapping phenomenon that allows observers to see streamlines close to boundary walls of channel in the form of trapped bolus. The channel walls and the confined bolus travel at the same speed. It has a substantial mathematical effect on the reduction of the number of dependent variables in governing equations and, ultimately, the transformation of the multiple dimensions into a one dimensional system of ordinary differential equations. Mathematically, the stream functions are given by^36,37:

$$u=-\frac{\partial \psi }{\partial r},v=\delta \frac{\partial \psi }{\partial x}\frac{k}{r+k}.$$

(27)

Results and discussion

This segment of manuscript, we interpret the physical response of different parameters on fluid velocity, temperature, streamlines, pressure gradient, and pressure rise. Table 2 and Figures 2 to 8 are offered for this purpose. In this analysis, the following default parameter values are adopted for computations: $B=0.2,k=2,\varphi =0.02,\delta =0.11,\zeta =0.2,a=0.32,x=0.23,G_r=1,F=0.1$ and $Q=0.03.$ We have designated the parameter values according to the previous literature^6,8,20 and these are beneficial for experimental purposes. We can see that Table 2 shows that by rising the values of the Casson parameter liquid velocity near the boundary profile in the section $(-1.0\le x\le -0.19$ , and $0.38\le x\le 0.99)$ enhances while opposed manners detected in the profile of boundary area $(0.01\le x\le 0.19).$ Furthermore, the outcomes of the present study are in decent agreement with the outcomes available in the literature, ⁶ which suggests the validity of the present model.

Table 2.

Momentum profile variation for various values of Casson parameter $\zeta$ with other values $\alpha =0.02,$ $\delta =0.11,$ $\varphi =0.02,$ $G_r=2.5,$ $k=0.2,$ and $x=0.2$ .

	$u$
$r$	$\zeta =0.4$	$\zeta =0.6$	$\zeta =0.8$
−1	−1.380383	−1.380378	−1.380355
−0.8	−0.787574	−0.7875731	−0.787523
−0.6	−0.354825	−0.354860	−0.354858
−0.4	−0.062462	−0.062453	−0.062401
−0.2	−0.275605	−0.275615	−0.275656
0.0	−0.158002	−0.158087	−0.158029
0.2	−0.107409	−0.107474	−0.107473
0.4	−0.039359	−0.039311	−0.039354
0.6	−0.275605	−0.275631	−0.275624
0.8	−0.595486	−0.595442	−0.595408
1.0	−0.993747	−0.993709	−0.993782

Figure 2.

Temperature profile for different values of (a) heat generation/absorption parameter $B$ and (b) curvature parameter $k$ with $\varphi =0.02,$ $\alpha =0.32,G_r=1,$ $\delta =0.11$ , and $x=0.23.$

Figure 3.

Pressure gradient for different values of (a) Grashof number $G_r,$ (b) curvature parameter $k$ and (c) cilia length parameter $\varphi$ with $\alpha =0.32,$ $\delta =0.11,$ and $\zeta =0.2.$

Figure 4.

Pressure rise for different values of (a) cilia length parameter $\varphi ,$ (b) curvature parameter $k$ , and (c) Casson parameter $\zeta$ with $G_r=1,$ $\alpha =0.32,$ $\delta =0.11$ , and $x=0.23.$

Figure 5.

Streamlines for Casson parameter $(a)$ $\zeta =0.2$ and $(b)$ $\zeta =0.24$ with the other parameters are $G_r=1,k=2,\alpha =0.32,\varphi =0.02,\delta =0.11$ , and $x=0.23.$

Figure 6.

Streamlines for curvature parameter $(a)$ $k=2$ and $(b)$ $k=2.2$ with the other parameters are $\delta =0.11,G_r=1,\alpha =0.32,\varphi =0.02,\zeta =0.2$ , and $x=0.23.$

Figure 7.

Streamlines for Grashof number $(a)$ $G_r=1$ and $(b)$ $G_r=1.2$ with the other parameters are $\delta =0.11,k=2,\alpha =0.32,\varphi =0.02,\zeta =0.2$ , and $x=0.23.$

Figure 8.

Comparison of limiting case of the present study with the results of Nadeem and Sadaf. ⁶

The impacts of sundry variables on thermal profile are accessible in Figure 2(a) and (b). Figure 2(a) explores the impacts of heat generation/absorption parameter $B$ on temperature distribution $\theta .$ It is depicted that liquid temperature rises as $B$ rising and the extreme amount of temperature in the middle of the curved channel occurs. This impression is physically valid since the improvement in heat generation/absorption parameter conveys the growing strength of the heat sink/source parameter which tends to enhance the liquid temperature. From Figure 2(b), special effects of curvature parameter $k$ on temperature profile $\theta$  are disclosed. Temperature is decreasing by enhancing curvature parameter values.

The impacts of various important parameters on pressure gradient $\mathrm{dp}/\mathrm{dx}$ are presented in Figure 3(a)–(c). Figure 3(a) is plotted to observe the deviations in pressure gradient for different values of Grashof number $G_r.$ It is revealed from Figure 3(a) that pressure gradient $\mathrm{dp}/\mathrm{dx}$ decreases in the sections $(-0.2\le x\le 0.2)$ and enhances in the area $(0.21\le x\le 0.76)$ and opposite behavior occurs in the area $(0.76\le x\le 1.2)$ . It is observed that pressure gradient increase and decrease in the narrow part of the curved channel. The outcome of the curvature parameter on the pressure gradient $\mathrm{dp}/\mathrm{dx}$ is analyzed in Figure 3(b) From this graph, it can be noticed that the pressure gradient is increased in the areas $(-0.2\le x\le 0.2\mathrm{and}0.75\le x\le 1.2)$ and diminishes in region $(0.25\le x\le 0.77)$ . Figure 3(c) is graphed to see the manners of $\mathrm{dp}/\mathrm{dx}$ for various values of the ciliated length parameter. It can be scrutinized that $\mathrm{dp}/\mathrm{dx}$ decreases in the sections $(-0.2\le x\le 0.2)$ and enhances in the area $(0.21\le x\le 0.76)$ and opposite behavior occur in the area $(0.76\le x\le 1.2)$ .

Figure 4(a)–(c) represent the effects of the Casson fluid parameter, curvature parameter, and cilia length parameter on the pressure rise $\mathrm{\Delta }p.$ Figure 4(a) show the outcome of the ciliated length parameter on the boundary of velocity profile. The pressure rise increase in the retrograde pumping region by enhancing the cilia length parameter. The pressure rise profile is addressed in Figure 4(b) due to the effect of the curvature parameter. This graph tells that pressure enhances in the region of retrograde pumping region and decreases in the region of enhanced pumping region. The outcome inspected in Figure 4(c) signified the change in pressure rise due to Casson liquid parameter. The pressure rise increase in the pumping area of retrograde by enhancing the cilia length parameter.

Trapping is one of the most important mechanisms in peristaltic movement. Trapping refers to the creation of a circulating bolus by the splitting of streamlines under specific circumstances. The trapped boluses move at fixed speed as the wave because it is contained by the peristaltic waves. With the help of this mechanism, blood thrombus can form and food can move across the gastrointestinal tract. Through Figures 5 to 7, for both uniform and non-uniform channels, we have illustrated the phenomenon of entrapment, or the variations in the size of the bolus under the changing of effective flow parameters. Stream function for special values of Casson parameter such as $\zeta =0.2$ and $\zeta =0.24$ are displayed in Figure 5(a) and (b). Figure 5 means that as the values of the Casson fluid parameter $B$ are increased, the size of the trapped bolus steadily diminishes. Streamlines for curvature parameter $k$ is shown in Figure 6(a) and (b). It can be distinguished that number of trapped bolus enhances with enhanced values of Casson fluid parameter. Figure 7(a) and (b) is graphed to see the alteration of a trapped bolus by enhancing values of $G_r$ . By rising the values of $G_r$ it can be seen that the number of trapped boluses diminishes.

Validation

The intention of this part is to verify the accuracy of our findings. To validate obtained results, a comparison of limiting case of present investigation for the velocity profile in the absence of Casson parameter, Grashof number and heat generation/absorption parameter with the results reported by Nadeem and Sadaf ⁶ (see Figure 8). This graph indicates that both results are in good agreement.

Conclusions

The effect of a heat transfer on the cilia flow is very important for medical diagnosis and it has many clinical applications. It is a very important tool for determining real reasons responsible for many problems in human organs in which the fluid is transported by cilia walls. Therefore, the current study is concerned with the thermal analysis of mixed convective transport of non-Newtonian Casson fluid observed by bio-mimetic cilia wall in the curved channel. The problem is formulated by considering the requirements that the wavelength is going to increase sufficient for the pressure to be regarded consistent over the cross-section and the associated Reynolds number is small adequate for inertial property to be insignificant. Analytical closed form solution is attained for the axial velocity, pressure gradient, stream functions, pressure rise for each metachronal wavelength and temperature distributions. The key arguments of the problem are as monitors:

• The temperature profile is directly proportional to heat generation/absorption and inversely proportional to curvature parameter.

• Momentum velocity profile enhance for the Grashof number, curvature parameter, and cilia length parameter.

• Pressure gradient decreases with the Grashof number and cilia length parameter in the sections $(-0.2\le x\le 0.2)$ and enhances in the area $(0.21\le x\le 0.76)$ and opposite behavior occurs in the area $(0.76\le x\le 1.2)$ .

• By increasing the Grashof number and cilia length parameter, pressure rise increases in the retrograde pumping region and diminishes in the enhanced pumping region.

• Increasing values of the Casson liquid parameter and Grashof number raise the number of trapped boluses while, enhancing values of curvature parameter reduce the number of trapped boluses.