Peristaltic transport is a fundamental mechanism of fluid motion generated by progressive wave-like contractions of flexible channel walls. This mechanism plays a crucial role in many physiological processes, including blood circulation, gastrointestinal motility, urine transport, and the operation of catheter- and endoscope-based medical devices. In addition to biomedical applications, peristaltic transport is widely utilized in industrial processes involving complex fluids, particularly in situations where direct mechanical pumping is undesirable or impractical.

Most biological and industrial fluids exhibit non-Newtonian behavior, making classical Newtonian fluid models inadequate for realistic flow prediction. Among the various rheological models, the Carreau fluid model has received considerable attention due to its ability to accurately represent shear-thinning behavior and viscosity variation with shear rate. This feature makes it especially suitable for modeling physiological fluids such as blood. Several studies have investigated peristaltic transport under the Carreau fluid assumption. Ali and Hayat ⁷ analyzed pumping characteristics, axial pressure gradients, and trapping phenomena, providing a comparative study between Newtonian and Carreau fluids. Nadeem et al. ¹¹ examined peristaltic wave propagation of Carreau fluid in a rectangular duct under the assumptions of long wavelength and low Reynolds number. Ullah et al. ¹⁶ studied Carreau fluid flow in an elastic tube, emphasizing the significant role of wall flexibility. In addition, peristaltic transport of other non-Newtonian fluids, such as Jeffrey fluid, in flexible channels has been reported by Al-Khalidi and Al-Khafajy, ⁶ highlighting the importance of rheological complexity in peristaltic flow modeling.

Viscosity is one of the most influential physical properties governing fluid motion, particularly in biomedical and food-processing applications. In realistic physiological environments, viscosity is not constant but depends strongly on temperature and concentration variations, which significantly affect flow behavior, heat transfer, and mass diffusion. Several researchers have reported that although fluid velocity exhibits relatively small variations with concentration and spatial location, it increases noticeably with increasing temperature. ^{1, 5, 10, 12} Nadeem et al. ⁹ investigated peristaltic flow of a reactive viscous fluid with temperature-dependent viscosity. Akram and Akbar ² analyzed biological flow of a Carreau nanofluid in an endoscopic system incorporating variable viscosity effects. The influence of temperature and concentration on oscillatory flow in an inclined porous channel was studied by Al-Khafajy and Labban, ⁴ while Al-Delfi and Al-Khafajy ³ examined peristaltic transport of Williamson fluid through a hollow flexible channel, accounting for coupled thermal and concentration effects.

Recent developments in peristaltic flow research have emphasized the necessity of incorporating multiple physical mechanisms to achieve realistic modeling of physiological and industrial transport processes. These mechanisms include heat transfer, mass diffusion, nanofluid effects, magnetohydrodynamics (MHD), electroosmosis, and complex wave motions in flexible biomedical geometries. In this context, Tanveer et al. ¹³ investigated flow and heat transfer characteristics in a fallopian tube with a metachronal wave of cilia, demonstrating the crucial role of coordinated wall motion in physiological transport. Tanveer et al. ¹⁴ further explored dynamic interactions in MHD Jeffrey fluid flow under peristalsis combined with electroosmotic effects and homogeneous–heterogeneous chemical reactions, highlighting the strong coupling between electromagnetic forces, chemical reactions, and non-Newtonian rheology. Moreover, Tanveer et al. ¹⁵ analyzed the enhancement of heat generation using a ternary hybrid nanofluid in a periodic channel, reflecting the growing interest in advanced nanofluid models for thermal management applications.

Furthermore, Vijayan and Sucharitha ¹⁷ examined electroosmotic effects on peristaltic transport of a Ree–Eyring nanofluid with double diffusive convection in a symmetric microchannel, demonstrating the strong interaction between electric fields, thermal gradients, and concentration gradients on flow behavior. Jagadesh et al. ⁸ studied convective peristaltic pumping of an MHD Ree–Eyring nanofluid in a chemically reacting flexible divergent channel, incorporating the effects of activation energy and thermal radiation. Their findings emphasize the importance of coupling electromagnetic, chemical, and thermal effects with non-Newtonian rheology in advanced peristaltic flow models.

These studies collectively indicate that realistic modeling of physiological and industrial transport processes requires the simultaneous consideration of thermal effects, concentration gradients, non-Newtonian fluid behavior, and geometrical features such as wall flexibility and internal devices. Despite these extensive efforts, the combined influence of temperature- and concentration-dependent viscosity on the peristaltic flow of a Carreau fluid through a flexible axisymmetric wavy channel containing an internal catheter has not yet been adequately investigated. In particular, the interaction between variable viscosity, channel flexibility, and internal catheter geometry remains insufficiently explored, despite its direct relevance to catheter-based and endoscopic medical applications.

Motivated by this research gap, the present study develops a mathematical model for the peristaltic transport of an incompressible, non-Newtonian Carreau fluid with variable viscosity flowing through a flexible axisymmetric wavy channel containing an internal catheter placed along the centerline. The effects of temperature and concentration variations at the channel walls are incorporated into the viscosity formulation. Under the assumptions of long wavelength and low Reynolds number, the governing equations are simplified and solved analytically. Unlike earlier investigations that considered either constant viscosity or rigid geometries, the present model simultaneously accounts for variable viscosity, wall flexibility, and internal catheter effects, thereby providing a more comprehensive and physiologically realistic representation of peristaltic transport phenomena.

We study the peristaltic flow of an incompressible Carreau fluid between two cylinders that are in a central location, with an endoscope in the middle of the main channel that has a flexible wall structured like a sine wave. A cylinder’s coordinates are specified by the radius of the channel (R) and the tube’s axis (Z).

The geometry wall of the flow channel form is r ¯ = r 1 ¯ ( z ¯ , t ¯ ) = a 1 , r ¯ = r 2 ¯ ( z ¯ , t ¯ ) = a 2 + b sin ( 2 π γ ( z ¯ − c t ¯ ) )

Here “the unobstructed radius of the pipe” is represented by a 1 , the radius of the disturbed tube is represented by a 2 , b is “amplitude of the peristaltic wave”, γ is “a wave length”, c is “a wave propagation speed”, and t ¯ is “a time”.

The basic governing equations of the problem system ∇ . U ¯ = 0 ( continuity equation ) (1) ρ ( U . ¯ ∇ ) U ¯ = ∇ σ ´ + ρg β 1 ( T − T 0 ) + ρg β 2 ( ∁ − ∁ 0 ) ( momentum equation ) (2) T p . ρ ( U . ¯ ∇ ) T = K . ∇ 2 T − ∇ . Q r − Q ( T − T 0 ) ( temperature equation ) (3) ( U . ¯ ∇ ¯ ) ∁ = D n ∇ 2 ∁ + D n K T T n ∇ 2 T ( concentration equation ) (4)

Where ∇ 2 = 1 r ∂ ∂ r ( r ∂ ∂ r ) “Laplace operator”, U ¯ = ( U ¯ 1 , 0 , U ¯ 3 ) is “the velocity field”, ρ is a “density”, σ ´ is “the Cauchy stress tensor”, T is “the temperature”, ∁ is a concentration of the fluid, T p is “the specific heat capacity at constant pressure”, Q r is “the radiation heat flux”, D n is “the coefficient of mass diffusivity”, T n is “the mean fluid temperature”, K T is “the thermal diffusion ratio”.

The equation of incompressible Carreau fluid with variable viscosity as the distance travelled is given by ⁷ σ ´ = − P ¯ I ¯ + S ¯ , (5) S ¯ = − μ ( T ) [ 1 + ( n − 1 2 ) Γ 2 α ̇ ¯ 2 ] α ̇ ¯ (6)

Where S ¯ is “extra stress tensor”, P ¯ "pressure", I ¯ “identity tensor”, μ “dynamic viscosity”, Γ “time constant”, n “dimensionless power law index” and α ̇ is defined as; α ̇ = 1 2 ∑ i = 1 3 ∑ j = 1 3 α ̇ ij α ̇ ji (7)

The model can be reduced to a Newtonian model n = 1 or Γ = 0 , so we investigate the case for Γ ≠ 0 . To understand how an elastic wall behaves, the equation L ∗ ∗ = P ¯ − p ¯ 0 , where L ∗ ∗ is “an operator”, which is used to represent the motion of stretched membrane with viscosity damping forces such that, see ³ L ∗ ∗ = B ∂ 4 ∂ Z ¯ 4 − ℂ ∂ 2 ∂ Z ¯ 2 + m ∂ 2 ∂ t ¯ 2 + D ∂ ∂ t ¯ + A L

Wall flexural rigidity is denoted by B, longitudinal tension per unit width by ℂ , mass per unit area by m, coefficient of viscous damping by D, and spring stiffness by A L .

This is the equation that controls the properties of a flexible wall canal at r ¯ = r 2 ¯ , is obtained as; ∂ P ¯ ∂ Z ¯ = ∂ ∂ Z ¯ ( B ∂ 4 ∂ Z ¯ 4 − ℂ ∂ 2 ∂ Z ¯ 2 + m ∂ 2 ∂ t ¯ 2 + D ∂ ∂ t ¯ + A L ) ( r 2 ¯ ) (8)

For the sake of accuracy in writing the continuity equation and the momentum equations, in addition to the temperature and concentration equations, we use the velocity components U 1 ¯ ( R ¯ , Z ¯ , t ¯ ) and U 3 ¯ ( R ¯ , Z ¯ , t ¯ ) , which represent the radial and axial velocity components, respectively, in an unsteady two-dimensional flow. The fluid temperature and concentration functions are expressed in terms of T = T ( R ¯ , Z ¯ , t ¯ ) and ∁ = ∁ ( R ¯ , Z ¯ , t ¯ ) , respectively. Now, by substituting the governing equations for the problem (1) - (4), we obtain the following system of nonlinear, nonhomogeneous partial differential equations; ∂ U 1 ¯ ∂ R ¯ + U 1 ¯ R ¯ + ∂ U 3 ¯ ∂ Z ¯ = 0 (9) ρ ( ∂ U 1 ¯ ∂ t ¯ + U 1 ¯ ∂ U 1 ¯ ∂ R ¯ + U 3 ¯ ∂ U 1 ¯ ∂ Z ¯ ) = − ∂ p ¯ ∂ R ¯ + 1 R ¯ ∂ ∂ R ¯ ( R ¯ S ¯ R ¯ R ¯ ) + ∂ S ¯ R ¯ Z ¯ ∂ Z ¯ (10) ρ ( ∂ U 3 ¯ ∂ t ¯ + U 1 ¯ ∂ U 3 ¯ ∂ R ¯ + U 3 ¯ ∂ U 3 ¯ ∂ Z ¯ ) = − ∂ p ¯ ∂ Z ¯ + 1 R ¯ ∂ ∂ R ¯ ( R ¯ S ¯ Z ¯ R ¯ ) + ∂ S ¯ Z ¯ Z ¯ ∂ Z ¯ + ρg β 1 ( T − T 0 ) + ρg β 2 ( ∁ − ∁ 0 ) (11) ∂ T ∂ t ¯ + U 1 ¯ ∂ T ∂ R ¯ + U 3 ¯ ∂ T ∂ Z ¯ = T n T p ρ ( 1 R ¯ ∂ T ∂ R ¯ + ∂ 2 T ∂ R ¯ 2 + ∂ 2 T ∂ Z ¯ 2 ) + 16 σ 0 T 2 E 3 k 0 T p ρ ( 1 R ¯ ∂ T ∂ R ¯ + ∂ 2 T ∂ R ¯ 2 ) − q T p ρ ( T − T 0 ) (12) ∂ ∁ ∂ t ¯ + U 1 ¯ ∂ ∁ ∂ R ¯ + U 3 ¯ ∂ ∁ ∂ Z ¯ = D n ( 1 R ¯ ∂ ∁ ∂ R ¯ + ∂ 2 ∁ ∂ R ¯ 2 + ∂ 2 ∁ ∂ Z ¯ 2 ) + D n K T T n ( 1 R ¯ ∂ T ∂ R ¯ + ∂ 2 T ∂ R ¯ 2 + ∂ 2 T ∂ Z ¯ 2 ) (13)

The component S ¯ R ¯ Z ¯ of the shear stress is S ¯ R ¯ Z ¯ = μ ( T ) { 1 + ( n − 1 2 ) Γ 2 ( 2 [ ( ∂ U 1 ¯ ∂ R ¯ ) 2 + ( U 1 ¯ R ¯ ) 2 + ( ∂ U 3 ¯ ∂ Z ¯ ) 2 ] + [ ( ∂ U 1 ¯ ∂ Z ¯ + ∂ U 3 ¯ ∂ R ¯ ) 2 ] ) } ( ∂ U 1 ¯ ∂ Z ¯ + ∂ U 3 ¯ ∂ R ¯ )

We use generic and specific frame coordinate transformations as shown below. U 1 ¯ = u 1 ¯ , U 3 ¯ = u 3 ¯ + c , R ¯ = r ¯ , and Z ¯ = z ¯ . Substituting these transformations into a system (9) - (13), we get: ∂ u 1 ¯ ∂ r ¯ + u 1 ¯ r ¯ + ∂ ( u 3 ¯ + c ) ∂ z ¯ = 0 (14) ρ ( ∂ u 1 ¯ ∂ t ¯ + u 1 ¯ ∂ u 1 ¯ ∂ r ¯ + ( u 3 ¯ + c ) ∂ u 1 ¯ ∂ z ¯ ) = − ∂ p ¯ ∂ r ¯ + 1 r ¯ ∂ ∂ r ¯ ( r ¯ S ¯ r ¯ r ¯ ) + ∂ S ¯ r ¯ z ¯ ∂ z ¯ (15) ρ ( ∂ ( u 3 ¯ + c ) ∂ t ¯ + u 1 ¯ ∂ ( u 3 ¯ + c ) ∂ r ¯ + ( u 3 ¯ + c ) ∂ ( u 3 ¯ + c ) ∂ z ¯ ) = − ∂ p ¯ ∂ z ¯ + 1 r ¯ ∂ ∂ r ¯ ( r ¯ S ¯ r ¯ z ¯ ) + ∂ S ¯ z ¯ z ¯ ∂ z ¯ + ρg β 1 ( T − T 0 ) + ρg β 2 ( ∁ − ∁ 0 ) (16) ∂ T ∂ t ¯ + u 1 ¯ ∂ T ∂ r ¯ + ( u 3 ¯ + c ) ∂ T ∂ z ¯ = T n T p ρ ( 1 r ¯ ∂ T ∂ r ¯ + ∂ 2 T ∂ r ¯ 2 + ∂ 2 T ∂ z ¯ 2 ) + 16 σ 0 T 2 E 3 k 0 T p ρ ( 1 r ¯ ∂ T ∂ r ¯ + ∂ 2 T ∂ r ¯ 2 ) − q T p ρ ( T − T 0 ) (17) ∂ ∁ ∂ t ¯ + u 1 ¯ ∂ ∁ ∂ r ¯ + ( u 3 ¯ + c ) ∂ ∁ ∂ z ¯ = D n ( 1 r ¯ ∂ ∁ ∂ r ¯ + ∂ 2 ∁ ∂ r ¯ 2 + ∂ 2 ∁ ∂ z ¯ 2 ) + D n K T T n ( 1 r ¯ ∂ T ∂ r ¯ + ∂ 2 T ∂ r ¯ 2 + ∂ 2 T ∂ z ¯ 2 ) (18)

The corresponding boundary conditions of the problem are: u 1 ¯ = 0 , u 3 ¯ + c = 0 , T = T 0 , ∁ = ∁ 1 at r ¯ = r 1 ¯ = a 1 u 1 ¯ = 0 , u 3 ¯ + c = 0 , T = T 1 , ∁ = ∁ 0 at r ¯ = r 2 ¯ = a 2 + b sin ( 2 π γ ( z ¯ − c t ¯ ) ) } (19)

Where the motion equation with condition of the elastic wall as follows: ∂ ∂ z ¯ ( B ∂ 4 ∂ Z ¯ 4 − C ∂ 2 ∂ Z ¯ 2 + m ∂ 2 ∂ t ¯ 2 + D ∂ ∂ t ¯ + A L ) ( r 2 ¯ ) = ∂ p ¯ ∂ z ¯ = − ρ ( ∂ ( u 3 ¯ + c ) ∂ t ¯ + u 1 ¯ ∂ ( u 3 ¯ + c ) ∂ r ¯ + ( u 3 ¯ + c ) ∂ ( u 3 ¯ + c ) ∂ z ¯ ) + 1 r ¯ ∂ ∂ r ¯ ( r ¯ S ¯ r ¯ z ¯ ) + ∂ S ¯ z ¯ z ¯ ∂ z ¯ + ρg β 1 ( T − T 0 ) + ρg β 2 ( ∁ − ∁ 0 ) (20)

To simplify the governing equations of the problem and to show the important parameters that affect the fluid flow, we introduce the following dimensionless transformations ^{2, 10} : u 1 = u 1 ¯ γ a 2 c , u 3 = u 3 ¯ c , r = r ¯ a 2 , z = z ¯ γ , S = a 2 S ¯ μ v c , p = a 2 2 p ¯ μ v cγ , φ = b a 2 , t = c t ¯ γ , r 2 = r 2 ¯ a 2 , r 1 = r 1 ¯ a 2 , a 1 a 2 = ε < 1 , Ω = q a 2 2 μ v T p , δ = a 2 γ , Re = ρc a 2 μ v , P r = μ v T p T n , M ( H ) = μ ( T ) μ v , W e = Γc a 2 , α ̇ = a 2 α ̇ ¯ c , R n = K 0 μ T p 4 T 2 E σ 0 , H = T − T 0 T 1 − T 0 , ξ = ∁ − ∁ 0 ∁ 1 − ∁ 0 , G 1 = ρg β 1 a 2 2 ( T 0 − T 1 ) μ v s , G 2 = ρg β 2 a 2 2 ( C 1 − C 0 ) μ v s , S 1 = ρ D n K T ( T 1 − T 0 ) μ v T n ( C 1 − C 0 ) , S 2 = μ v ρ D n } (21)

where φ “amplitude ratio”, Re “Reynolds number”, P r “Prandtl number”, R n “thermal radiation parameter”, S 2 “Schmidt number”, S 1 “Soret number”, G 1 “thermal Grashof number”, G 2 “Solutal Grashof number”, δ “dimensionless wave number”, Ω “heat source/sink parameter”, and W e is the Weissenberg number, μ v “viscosity constant”.

Substituting Equations (21) into Eqs. (14) - (20), we reformulate the governing equations and accompanying boundary conditions as follows: ( c γ ) ( ∂ u 1 ∂ r + u 1 r + ∂ u 3 ∂ z ) = 0 (22) Re δ 3 ( ∂ u 1 ∂ t + u 1 ∂ u 1 ∂ r + ( u 3 + 1 ) ∂ u 1 ∂ z ) = − ∂ p ∂ r + δ 1 r ∂ ∂ r ( r S rr ) + δ 2 ∂ S rz ∂ z (23) Re δ ( ∂ u 3 ∂ t + u 1 ∂ u 3 ∂ r + ( u 3 + 1 ) ∂ u 3 ∂ z ) = − ∂ p ∂ z + 1 r ∂ ∂ r ( r S rr ) + δ ∂ S zz ∂ z + G 2 ξ + G 1 H (24) Re δ ( ∂ H ∂ t + u 1 ∂ H ∂ r + ( u 3 + 1 ) ∂ H ∂ z ) = 1 P r ( 1 r ∂ H ∂ r + ∂ 2 H ∂ r 2 + δ 2 ∂ 2 H ∂ z 2 ) + 4 3 Rn ( 1 r ∂ H ∂ r + ∂ 2 H ∂ r 2 ) − ΩH (25) Re δ ( ∂ ξ ∂ t + u 1 ∂ ξ ∂ r + ( u 3 + 1 ) ∂ ξ ∂ z ) = 1 S 2 ( 1 r ∂ ξ ∂ r + ∂ 2 ξ ∂ r 2 + δ 2 ∂ 2 ξ ∂ z 2 ) + S 1 ( 1 r ∂ H ∂ r + ∂ 2 H ∂ r 2 + δ 2 ∂ 2 H ∂ z 2 ) (26)

The component S rz of the shear stress in dimensionless transformation form is S rz = M ( H ) { 1 + ( n − 1 2 ) We 2 ( 2 δ 2 [ ( ∂ u 1 ∂ r ) 2 + ( u 1 r ) 2 + ( ∂ u 3 ∂ z ) 2 ] + [ ( δ 2 ∂ u 1 ∂ z + ∂ u 3 ∂ r ) 2 ] ) } ( δ 2 a 2 ∂ u 1 ∂ z + ∂ u 3 ∂ r ) (27)

The corresponding dimensionless boundary conditions of the problem are u 1 = 0 , u 3 = − 1 , H = 0 , ξ = 1 at r = r 1 = ε u 1 = 0 , u 3 = − 1 , H = 1 , ξ = 0 at r = r 2 = 1 + φ sin ( 2 π ( z − t ) ) } (28) ( e 1 ∂ 5 ∂ z 5 − e 2 ∂ 3 ∂ z 3 + e 3 ∂ 3 ∂z ∂t 2 + e 4 ∂ 2 ∂z∂t + e 5 ∂ ∂z ) r 2 = 1 r ∂ ∂ r ( r S rz ) + δ ∂ S zz ∂ z − Re δ ( ∂ u 3 ∂ t + u 1 ∂ u 3 ∂ r + ( u 3 + 1 ) ∂ u 3 ∂ z ) + G 2 ξ + G 1 H (29) where e 1 = B a 2 3 μ v c γ 5 is the flexural stiffness of the wall, e 2 = C a 2 3 μ v c γ 3 is the longitudinal tension per unit width, e 3 = mc a 2 3 μ v γ 3 is the mass per unit area, e 4 = D a 2 3 μ v γ 2 is the coefficient of viscid damping, and e 4 = A L a 2 3 μ v cγ is spring stiffness, respectively.

These parameters are selected within physically realistic ranges commonly reported in the literature on peristaltic flow through elastic or compliant channels. ^{3, 5} They represent the resistance of the wall to bending and stretching, its inertia, viscous damping, and elastic support, respectively. The chosen values ensure numerical stability and allow a clear illustration of the essential physical behavior of the system. Moreover, it has been observed that moderate variations in these parameters do not significantly affect the qualitative trends of the results; therefore, a representative set of parameter values is employed throughout the analysis.

It is very difficult to solve the system of Equations (22) - (27) and (29), so we assume a very small wave number ( δ ≪ 1) concerning the width of the channel to its length. Thus, the system becomes in the following form after abbreviating its writing, taking into account the condition of the flexibility of the outer wall of the flow channel: ( e 1 ∂ 5 ∂ z 5 − e 2 ∂ 3 ∂ z 3 + e 3 ∂ 3 ∂z ∂t 2 + e 4 ∂ 2 ∂z∂t + e 5 ∂ ∂z ) r 2 = 1 r ∂ ∂ r ( r S rz ) + G 2 ξ + G 1 H (30) ( 1 Pr + 4 3 Rn ) ( ∂ 2 H ∂ r 2 + 1 r ∂ H ∂ r ) − ΩH = 0 (31) 1 S 2 ( ∂ 2 ξ ∂ r 2 + 1 r ∂ ξ ∂ r ) + S 1 ( ∂ 2 H ∂ r 2 + 1 r ∂ H ∂ r ) = 0 (32) with S rz = M ( μ ) { 1 + ( n − 1 2 ) We 2 ( ∂ u 3 ∂ r ) 2 } ( ∂ u 3 ∂ r ) (33)

This section involves solving the heat and concentration equations, then substituting the result into the velocity equation to solve it.