The escalation process of heat and species transport in different flow phases with practical significance is an interesting and important subject in thermodynamics and fluid mechanics. Both the formation of fluid mixtures and changes in the functional thermophysical characteristics of the fluids fall within the category of escalation processes (Bergles and Webb, 1985; Jensen et al., 1997; Ligrani et al., 2003). Integrated heat exchangers that operate at low Reynolds numbers in case of laminar flows are treated specially for improved mixing (Webb and Bergles, 1981). It is often accepted that improved mixing may be achieved by driving a laminar-turbulent transformation under difficult conditions or by inserting vortex generators, which are efficient but have a substantial drag cost (Fiebig, 1995a; Fiebig, 1995b; Jacobi and Shah, 1995; Fiebig, 1998; Fiebig and Chen, 1999). In most cases, the hydrodynamic stabilities have been used for the transition of laminar flow to turbulent flow and, therefore, transverse grooves are used. However, a novel model has been demonstrated with the most accurate simulations, i.e., to shift laminar states without transiting to turbulent ones by using hydrodynamic instabilities. In this novel model, separating local flow is demonstrated well; the flow is driven by a slight oscillatory component and enters the resonant and stable separated shear layer (Patera and Mikic, 1986) Another type of bifurcation, i.e., centrifugal instability, has been taken into account and it produces simple vortices without suffering from the maximum drag cost associated with conventional vortex generators. Instability can improve vortex generators in limiting vortices decay, which decreases the minimum number of such generators even when the flow is only locally asymptotically stable. Both directly (by establishing a transverse transition) and indirectly (by creating a bypass transition), streamwise vortices can boost heat transfer. In most cases, a three-dimensional flow field has been developed at the end and transverse shear layers with inflection points and rapidly expanding secondary instability are formed (Floryan, 1991). Recent investigations have reported an alternate approach being taken into account of a fluid of higher thermal conductivity used with properly shaped channels while, at the same time, flow pressure being reduced immediately (Mohammadi and Floryan, 2013). This technique has provided an alternative solution to the most common development programs used on macro channels (Xu et al., 2016).

It has been known for nearly 100 years that the rotating shear layers are subjected to centrifugal instabilities. This instability has been taken into account for the situation of simple geometry and canonical flow, providing an example where it is possible to forecast the curvature of lines with ease. Using the flow state between rotating cylinders, Rayleigh introduced the inviscid technique in 1920 and determined the necessary stabilization condition through circulating distribution (Rayleigh, 1917). In 1923, Taylor included the full viscous problem and identified the crucial conditions that emerge as a result of the secondary flow (Taylor, 1923). Similar instability in curved channels was examined by Dean (Dean, 1928). The occurrence of centrifugal instability in the context of boundary layers on concave surfaces was proved by Görtler (Görtler, 1941). If the streamwise velocity distribution is not monotonic (Floryan, 1986), demonstrated that the instability is active in flows over concave as well as convex surfaces. A clear relationship between the streamline curvature and the wall curvature was provided by all of these investigations where the wall curvature was either constant or had been approximated as a constant. As a result, the critical stability condition might be expressed in terms of one parameter.

There are very few theoretical investigations that can pinpoint the starting conditions for flows in complex geometries where the wall curvature varies spatially. However, there are large number of numerical models and experimental studies that offer qualitative data (Gschwind et al., 1995). used a meandering amplitude of the same order of magnitude as the channel height, and (Nishimura et al., 1990; Tatsuo et al., 1990) used a large amplitude compared to the channel height to demonstrate the existence of streamwise vortices attributed to the centrifugal instability in sinusoidal channels. Rush et al. (Rush et al., 1999) has qualitatively identified the additional types of instability for corresponding geometries. Theoretical investigations which employed two-dimensional models were able to identify the effect of flow separation on heat transfer (Metwally and Manglik, 2004; Zhang et al., 2004) and identify the conditions that led to either single or double Hopf bifurcations or self-sustained oscillations (Guzmán et al., 2009). These models failed to identify the formation of the vortices. In their 2012 study, (Sui et al., 2012), took into account a three-dimensional rectangular channel with a very large meandering amplitude and applied numerical simulations to find a complex pattern of Dean’s vortices that changed over time and space. In the case of turbulent flow, consistent structures were found by (Pham et al., 2008). Other types of geometries, such as boundary layer flows over wavy surfaces (Saric and Ali, 1991) and Couette flows over wavy walls (Floryan, 2002), have also been found to demonstrate centrifugal instability. According to recent findings, streamwise vortices may be created by the placement of different triangular surface obstacles (Floryan and Asai, 2011) The transport of heat and fluid flow in various mediums and the studies considered the impact of magnetic fields, chemical reactions, porous media on fluid flow and thermal transport. The studies considered the impact of magnetic fields, chemical reactions, and porous media on fluid flow and thermal transport The results suggest that the addition of tri-hybrid nanoparticles and magnetic dipoles can enhance thermal transportation in Carreau Yasuda liquid, but may decrease the flow profile. This study also investigated the effects of different parameters on the peristaltic motion of hyperbolic tangent fluid in a curved compliant channel, which has potential applications in explaining blood transport dynamics. Numerical solutions and perturbation techniques were used to analyze and evaluate the results (Javed et al., 2021; Naseem et al., 2021; Wang et al., 2022). Different approaches related to the studies of fluid flow in microfluidic systems for biomedical engineering. This study analyzed the behavior of different types of fluids in different channel geometries by taking into account, convective conditions, thermal deposition effect, and chemical reactions. They investigated the impact of various parameters on flow quantities such as velocity, temperature, and concentration. This study also suggest the viability of electro-osmotic pumps for fluid flow in large osteoarticular implants (Hayat et al., 2015; Yasmin et al., 2020a; Yasmin et al., 2020b; Mehmood et al., 2020; Yasmin and Iqbal, 2021; Alyousef et al., 2023).

The main objective of the current analysis is to investigate the three-dimensional structure of the meandering geometries that cause the centrifugal force mechanism to produce streamwise vortices at the lowest possible cost as measured by pressure losses and without the interference of the traveling wave instability. This is the first comprehensive analysis of three-dimensional flow in a meandering tube that takes all potential instabilities into consideration. The new information would provide an accurate mathematical simulation for engineers and be a reasonable foundation upon which they may construct small heat exchangers that operate in the laminar flow domain. The study is divided into three main parts, specifically: i) mathematical modeling of flow in the meandering tube; ii) calculating the flow losses related to the meandering tube; and iii) identifying the geometric and flow characteristics that cause the centrifugal instability to predominate. The source in engineering procedures might be a mechanical pressure gradient. In this study, we look at the pressure gradient-driven (Poiseuille flow) flow of viscous fluid in a meandering tube with waves that is made up of fixed walls. A lot of interest has been shown in the flow of viscous fluid in a wavy meandering tube due to its applications in engineering and biological sciences, including regarding the development of muddy waves in river channels, the generation of wind waves on water and sandbanks in deserts, the movement of melting slides, rocket boosters, and the evaporation of film in burning chambers. Furthermore, physiologists and technicians have often attempted to create and explain blood and urine flow in terms of meandering channels (tubes).

The flow of a viscous fluid in a meandering tube, whose geometry is shown in Figure 1, is the main focus of this paper.

The tube of variable radius is taken, whose radius is determined by R * = r m * 1 + α   sin ⁡   δ . The polar coordinate system r , θ , z is connected to the Cartesian coordinate system x , y , z for this particular problem in such a way that x = η   1 + α   sin ⁡ δ cos ⁡ θ , y = η   1 + α   sin ⁡ δ sin ⁡ θ , z = z where δ = β θ + k z and = r R . Note that x , y , z , and η are dimensionless variables.

In order to express the current problem in proper coordinates, it is appropriate to choose a suitable coordinate system for the simulated problem. The well-established relationship between the Cartesian coordinates x * , y * , z * and cylindrical ones r * , θ , z * has been presented above. For the meandering, we took the following transformation to define a tube of non-uniform radius R * as: R * = r m * 1 + α   Sin   δ   w h e r e   δ = β θ + k * z * where r m * is the mean radius of the tube while “ α ” represents the dimensionless amplitude, “ β ” represents the number of helixes starts, and “ k * ” represents the wave number in the axial direction of the wall of the tube.

The governing equations are non-dimensionalized by introducing dimensionless variables and the length is non-dimensionalized by r m * (mean radius), the velocity components ( u * , v * , w * ) by υ * / r m * , the static pressure by ρ * υ * / r m * 2 , ∇ * 2 by 1 / r m * 2 , and V * ∙ ∇ * by υ * / r m * 2 .

Note that the asterisk “*” represents the dimensional quantities and υ * and ρ * are kinematic viscosity and density of the fluid, respectively. The governing equations in cylindrical coordinates r * , θ , z * are:

Continuity equation, r * , θ , and z * - components of Navier-Stokes equations in cylindrical coordinates r * , θ , z * are given as: ∂ ∂ r * r * u * + ∂ ∂ θ v * + r * ∂ ∂ z * w * = 0 (1) V * ∙ ∇ * u * − v * 2 r * = − 1 ρ * ∂ P * ∂ r * + υ * ∇ * 2 u * − u * r * 2 − 2 r * 2 ∂ v * ∂ θ (2) V * ∙ ∇ * v * − u * v * r * = − 1 ρ * r * ∂ P * ∂ θ + υ * ∇ * 2 v * − v * r * 2 − 2 r * 2 ∂ u * ∂ θ (3) V * ∙ ∇ * w * = − 1 ρ * ∂ P * ∂ z * + υ *   ∇ * 2 w * (4) V * ∙ ∇ * = u * ∂ ∂ r * + 1 r * v * ∂ ∂ θ + w * ∂ ∂ z * (5) ∇ ∗ 2 = 1 r * ∂ ∂ r * r * ∂ ∂ r * + 1 r * 2 ∂ 2 ∂ θ 2 + ∂ 2 ∂ z * 2 (6)

The no slip boundary conditions at the wall and the symmetry conditions at the center of the meandering tube are given as: u * = v * = w * = 0   a t   r * = R *   a n d   u * = v * = 0 , w * = U *   a t   r * = 0 (7) where U * represents stream velocity at the center of the meandering tube.

By using the dimensionless variables as defined above, the continuity Eq. 1 is transformed as: ∂ r m * ∂ r * r m * r m * r * r m * υ * r m * u * υ * r m * + ∂ ∂ θ υ * r m * v * υ * r m * + r m * r * r m *   ∂ r m * ∂ z * r m * υ * r m * w * υ * r m * = 0 (8)

From above definitions of dimensionless variables, we have: u * υ * r m * = v r , v * υ * r m * = v θ , w * υ * r m * = v z , r = r * r m * , z = z * r m * (9) where v r , v θ , and v z represent the dimensionless velocity components in r, θ , and z directions, respectively: ∂ ∂ r r   v r + ∂ ∂ θ v θ + r ∂ ∂ z v z = 0 (10) V ∙ ∇ v r − 1 r v θ 2 = − ∂ P ∂ r + ∇ 2 v r − v r r 2 − 2 r 2 ∂ v θ ∂ θ (11) V ∙ ∇ v θ + v r v θ r = − 1 r   ∂ P ∂ θ + ∇ 2 v θ − v θ r 2 + 2 r 2 ∂ v r ∂ θ (12) V ∙ ∇ v z = − ∂ P ∂ z + ∇ 2 v z (13) V ∙ ∇ = r m ∗ 2 υ * V * ∙ ∇ * (14) ∇ 2 = r m ∗ 2 ∇ ∗ 2 (15)

Where Eqs. 10–13 represent the dimensionless form of continuity and components of Navier-Stokes equations, respectively.

The dimensionless form of no slip boundary conditions at the wall and the symmetry conditions at the center of the meandering tube are obtained as: v r = v θ = v z = 0   a t   r = R   a n d   v r = v θ = 0 , v z = R e   a t   r = 0 where R e = ρ * U *   r m * υ * represents the Reynolds number.

The velocity vector V b for the modeled problem is decomposed as V b = V 0 + V 1 where the velocity V 0 , pressure P 0 , the total volume flow rate Q 0 , the stream function ψ 0 , and the vorticity function ξ 0 for the fully developed flow in a straight duct (circular pipe) becomes purely radial in a pipe. The solution is obtained from the continuity equation, η , θ , and z momentum equations. Note that the fluid flow is directed in the direction along positive z -axis. The velocity field and other field quantities for fluid motion in the meandering tube are approximated as: V b = V 0 + V 1 = 0 , 0 , w 0 η + u 1 η , θ , z , v 1 η , θ , z , w 1 η , θ , z   P b = P 0 η + P 1 η , θ , z   ψ b = ψ 0 η + ψ 1 η , θ , z   Q b = Q 0 η + Q 1 η , θ , z   ξ b = ξ 0 η + ξ 1 η , θ , z V 0 η , θ , z = u 0 η , θ , z , v 0 η , θ , z , w 0 η , θ , z = 0 , 0 , R e 1 − η 2 , P 0 = − 4 R e z + c 0 , Q 0 = π R e 2 , ψ 0 = − η 2 2   R e   1 − η 2 2 + c 1   a n d   ξ 0 = 2   η   R e

Further, the velocity vector V b = u b , v b , w b needs to be determined; therefore, an appropriate approximation technique is used to get this part of velocity V b .

Next, our aim is to determine the solution of dimensionless Eqs. (10 - 16). As these equations have a parameter α which may be a small quantity in many practical problems, we therefore consider the case of small amplitude waviness, i.e.; α → 0 and the flow domain has been regularized for the radial coordinate of the form when η = r R where R = 1 + α   sin δ where δ = β θ + k z (the tube or its wall is located at η = 1 for this new variable), and the dimensionless governing equations are transformed from r , θ , z to η , θ , z such that the dimensionless continuity and Navier-Stokes Eqs. (10-13) in cylindrical coordinates r , θ , z take the form below: 1 + α sin ⁡ δ u b + η 1 + α sin ⁡ δ 2 ∂ w b ∂ z + ∂ v b ∂ θ + η   ∂ u b ∂ η + α sin ⁡ δ ∂ v b ∂ θ + α   η sin ⁡ δ ∂ u b ∂ η − α   η cos ⁡ δ   β ∂ v b ∂ η + k η   α   η cos ⁡ δ 1 + α sin ⁡ δ ∂ w b ∂ z = 0 (18) − η 1 + α sin ⁡ δ 3 v b 2 − 1 + α sin ⁡ δ 2 ∂ 2 u b ∂ θ 2 + η 2 1 + α sin ⁡ δ 3 ∂ P b ∂ η − 2 α 2 β 2 η cos 2 δ ∂ u b ∂ η − α β 2 η sin ⁡ δ 1 + α sin ⁡ δ ∂ u b ∂ η + η 2 1 + α sin ⁡ δ 3 u b ∂ u b ∂ η + η 2 1 + α sin ⁡ δ 3 w b 1 + α sin ⁡ δ ∂ u b ∂ z − k α η cos ⁡ δ ∂ u b ∂ η + η 1 + α sin ⁡ δ 2 v b 1 + α sin ⁡ δ ∂ u b ∂ θ − α β η cos ⁡ δ   ∂ u b ∂ η + 2 1 + α sin ⁡ δ 1 + α sin ⁡ δ ∂ u b ∂ θ − α β η cos ⁡ δ ∂ u b ∂ η + α β η cos ⁡ δ 1 + α sin ⁡ δ ∂ 2 u b ∂ η ∂ θ − η 2 1 + α sin ⁡ δ 3 ∂ u b ∂ η + η ∂ 2 u b ∂ η 2 + α β η cos ⁡ δ 1 + α sin ⁡ δ ∂ 2 u b ∂ η ∂ θ − α β η cos ⁡ δ ∂ 2 u b ∂ η 2 − η + α η sin ⁡ δ 2 ( 1 + α sin ⁡ δ 2 ∂ 2 u b ∂ z 2 + 1 2 k α η ( k 3 α + α cos ⁡ δ + 2 ⁡ sin ⁡ δ ∂ u b ∂ η + 2 ⁡ cos ⁡ δ   − 2 1 + α sin ⁡ δ ∂ 2 u b ∂ η ∂ z + k α η   cos ⁡ δ ∂ 2 u b ∂ η 2 ) ) = 0 (19) − η 1 + α sin ⁡ δ 2 v b + η 1 + α sin ⁡ δ 3 u b   v b − 1 + α sin ⁡ δ 2 ∂ 2 v b ∂ θ 2 + η 1 + α sin ⁡ δ 2 1 + α sin ⁡ δ ∂ P b ∂ θ − α β η cos ⁡ δ ∂ P b ∂ η − 2 1 + α sin ⁡ δ 1 + α sin ⁡ δ ∂ u b ∂ θ − α β η cos ⁡ δ ∂ u b ∂ η − 2 α 2 β 2 η cos 2 ⁡ δ ∂ v b ∂ η − α β 2 η sin ⁡ δ 1 + α sin ⁡ δ ∂ v b ∂ η + η 2 1 + α sin ⁡ δ 3 u b ∂ v b ∂ η + η 2 1 + α sin ⁡ δ 3 w b 1 + α sin ⁡ δ ∂ v b ∂ z − k α η cos ⁡ δ ∂ v b ∂ η + η 1 + α sin ⁡ δ 2 v b 1 + α sin ⁡ δ ∂ v b ∂ θ − α β η cos ⁡ δ ∂ v b ∂ η + α β η cos ⁡ δ 1 + α sin ⁡ δ ∂ 2 v b ∂ η ∂ θ − η 2 1 + α sin ⁡ δ 3 ∂ v b ∂ η + η ∂ 2 v b ∂ η 2 + α β η cos ⁡ δ 1 + α sin ⁡ δ ∂ 2 v b ∂ η ∂ θ − α β η cos ⁡ δ ∂ 2 v b ∂ η 2 − η + α η sin ⁡ δ 2 ( 1 + α sin ⁡ δ 2 ∂ 2 v b ∂ η 2 + 1 2 k α η ( k 3 α + α cos ⁡ δ + 2 sin ⁡ δ ∂ v b ∂ η + 2 cos ⁡ δ k α η cos ⁡ δ ∂ 2 v b ∂ η 2 − 2 1 + α sin ⁡ δ ∂ 2 v b ∂ η ∂ z ) ) = 0 (20) − 1 + α sin ⁡ δ 2 ∂ 2 w b ∂ θ 2 + η 2 1 + α sin ⁡ δ 3 1 + α sin ⁡ δ ∂ P b ∂ z − k α η cos ⁡ δ ∂ p b ∂ η − 2 α 2 β 2 η cos 2 ⁡ δ ∂ w b ∂ η − α β 2 η sin ⁡ δ 1 + α sin ⁡ δ ∂ w b ∂ η + η 2 1 + α sin ⁡ δ 3 u b ∂ w b ∂ η + η 2 1 + α sin ⁡ δ 3 w b 1 + α sin ⁡ δ ∂ w b ∂ z − k α η cos ⁡ δ ∂ w b ∂ η + η 1 + α sin ⁡ δ 2 v b 1 + α sin ⁡ δ ∂ w b ∂ θ − α β η cos ⁡ δ ∂ w b ∂ η + α β η cos ⁡ δ 1 + α sin ⁡ δ ∂ 2 w b ∂ η ∂ θ − η 1 + α sin ⁡ δ 2 ∂ w b ∂ η + η ∂ 2 w b ∂ η 2 + α β η cos ⁡ δ 1 + α sin ⁡ δ ∂ 2 w b ∂ η ∂ θ − α β η cos ⁡ δ ∂ 2 w b ∂ η 2 − η + α η sin ⁡ δ 2 ( 1 + α sin ⁡ δ 2 ∂ 2 w b ∂ z 2 + 1 2 k α η ( k 3 α + α cos ⁡ 2 δ + 2 sin ⁡ δ ∂ w b ∂ η + 2 cos ⁡ δ k α η cos ⁡ δ ∂ 2 w b ∂ η 2 − 2 1 + α sin ⁡ δ ∂ 2 w b ∂ η ∂ z ) ) = 0 (21)

Similarly, the dimensionless Eq. 16 has transformed (by introducing η = r R ; R = 1 + α   sin ⁡ δ where δ = β θ + k z ) into the following form: u b = v b = w b = 0   a t   η = 1   a n d   u b = v b = 0 , w b = R e   a t   η = 0 (22)

In the case of small amplitude waviness, i.e., α → 0 , the velocity components, i.e., u b , v b , and w b in η ,   θ , and z-directions, respectively, and the pressure term P b are expanded in a series of α as: u b = u ^ 0 + α   u ^ 1 + O α 2 , v b = v ^ 0 + α   v ^ 1 + O ( α 2 ) , w b = w ^ 0 + α   w ^ 1 + O ( α 2 )   a n d P b = P ^ 0 + α P ^ 1 + O α 2 (23) where u 0 = u ^ 0 , v 0 = v ^ 0 , w 0 = w ^ 0 , a n d   P 0 = P ^ 0 and they are determined previously for the fully developed flow in straight duct.

The values of u b , v b , w b , and P b are substituted from Eq. 23 into the equations of continuity, motion, and the relevant boundary conditions, i.e., Eqs. 18–21 which are described by means of ( η , θ , z ), and terms of the same order of α are collected on each side of these equations.

The solution of the zeroth-order system, which is obtained by putting value from Eq. 23 into Eqs 18–21 and equating like powers of α 0 on both sides of them, is given below: u ^ 0 = v ^ 0 = 0 , w ^ 0 = R e 1 − η 2 , P ^ 0 = − 4 R e z + c 0 , Q ^ 1 = π R e 2 , ψ 0 = ψ ^ 0 = − η 2 2   R e   1 − η 2 2 + c 1   a n d   ξ ^ 0 = 2   η   R e (24)

Note that the solution in Eq. 24 for the fully developed flow in a straight duct has been reported in F.M. White (White and Majdalani, 2006) and Schlichting (Schlichting and Kestin, 1961).

Similarly, the first-order system is obtained by equating like powers of α 1 on both sides of Eqs 18–21 and then by substituting Eq. 24 into them. The unknowns in this system are further expressed by the following series: ψ ^ 1 = f a η sin ⁡ δ + f b η cos ⁡ δ , w ^ 1 = f c η sin ⁡ δ + f d η cos ⁡ δ , u ^ 1 = 1 η ∂ ψ ^ 1 ∂ θ − 1 2 k R e η 4 cos ⁡ δ , v ^ 1 = − ∂ ψ ^ 1 ∂ η − 1 β k η   w ^ 1 , f a η = ∑ p = 0 ∞ a p η p , f b η = ∑ p = 0 ∞ b p η p , f c η = ∑ p = 0 ∞ c p η p   a n d   f d η = ∑ p = 0 ∞ d p η p where the functions f a , f b , f c , and f d depend only on η and the coefficients a p , b p , c p , and d p are obtained by substituting the series into the first-order system. The coefficients of the above series are: a 0 = 0 , a 1 = 0 , a 2 = 0 , a 3 = 0 , a 4 = 12 k R e + 5 k R e β 2 2 β 12 + β 2 , a 5 = 54 k R e β 12 + β 2 15 + 2 β 2 , a 6 = − 2 R e β − 576 k + 15 k 3 + 2 k 3 β 2 6 + β 2 12 + β 2 15 + 2 β 2 , a 7 = − 2 k R e β − 72000 + k 2 2523 + 358 β 2 6 + β 2 12 + β 2 15 + 2 β 2 21 + 4 β 2 , a 8 = 1 6 + β 2 12 + β 2 15 + 2 β 2 21 + 4 β 2 24 + 5 β 2 2 k R e β 15552000 + 5 k 4 315 + 102 β 2 + 8 β 4 − k 2 72 8409 + 1234 β 2 + 5 R e 2 315 + 102 β 2 + 8 β 4

Moreover, the coefficients a I are recursively obtained as: a I + 4 = 1 I + 4 β 2 + 3 − 3 β 2   { I + 3 I + 1 I + 2 + 1 − 1 a I + 3 − k R e I − 1 b I − k 2 I + 1 a I + 2 + k R e I + 1 b I + 2 + k R e I + 1 b I + 1 − k β β 2 + 3 c I + 2 + k β I + 1 I + 3 + 1 c I + 1 − k 3 β c I + k 2 R e β d I − k 2 R e β d I − 2 }

For I = 5,6,7 , … …

Furthermore, the coefficients of the second series in the expression of ψ ^ 1 are obtained as: b 0 = 0 , b 1 = 0 , b 2 = 0 , b 3 = 0 , b 4 = 0 , b 5 = 0 , b 6 = − 2 k 2 R e 2 β β 2 + 6 β 2 + 12 , b 7 = − 2 k 2 R e 2 β 2523 + 358 β 2 β 2 + 6 β 2 + 12 2 β 2 + 15 4 β 2 + 21 , b 8 = 1 β 2 − 36 β 2 + 6 β 2 + 12 2 β 2 + 15 4 β 2 + 21 5 β 2 + 24 2 k 2   R e 2 β 21580668 + 2489292 β 2 − 105021 β 4 − 762 β 6 + 8 β 8 + 10 k 2 β − 6 β + 6 15 + 2 β 2 21 + 4 β 2 ,

The recursive formula for b I is obtained as: b I + 4 = 1 β 2 + 3 I + 4 − 3 β 2 { − k R e I + 1 a I + 2 + k R e I − 1 a I + I + 3 I + 1 I + 2 + 1 − 1 b I + 3 − k 2 I + 1 b I + 2 − k β β 2 + 3 d I + 2 + k β I + 1 I + 3 + 1 d I + 1 + k 3 β d I }

For I = 5,6,7 , … … c 0 = 0 , c 1 = 0 , c 2 = − 2 R e , c 3 = 0 , c 4 = 0 , c 5 = 0 , c 6 = 0 , c 7 = 0 ,

The recursive formula for c I is obtained as: c I + 2 = 1 I + 2 2 − β 2 2 R e β b I + k 2 c I − k R e d I + k R e d I − 2

For I = 6,7,8 , … … d 0 = 0 , d 1 = 0 , d 2 = − 2 R e , d 3 = 0 , d 4 = 0 , d 5 = 0 , d 6 = 4 k R e 2 β 2 β 2 − 36 β 2 + 12 , d 7 = 108 k R e 2 β 2 β 2 − 49 β 2 + 12 2 β 2 + 15 ,

And the recursive formula for d I is obtained as: d I + 2 = 1 I + 2 2 − β 2 − 2 R e β a I + k 2 c I + k R e c I + k R e c I − 2

For I = 6,7,8 , … …

Figure 2, Figure 3, Figure 4, Figure 5 describe the impact of meanders on the formation and behavior of vortices in the flow, producing complex patterns of stream contours. Vortices form near the wall of the tube and interact with the fluid flow. In Figure 6, fluid flow patterns are affected in several ways with variation in the number of helixes. As the number of helixes increases, the amplitude of the meanders increases, leading to more complex flow patterns with multiple recirculation zones near the wall of the tube. This also causes an increase in the pressure drop and the overall mixing in the flow.

In a meandering tube, streamwise vortices can improve the transportation of heat and species mass in a transverse direction. To create these vortices, a specific meandering wavelength is required, which can be measured using linear instability theory. Short wavelengths cause flow separation, while long wavelengths result in a rectilinear stream resembling a flat plate. Shear-driven instability also contributes to vortex formation. The effect of a steady laminar flow in a meandering tube with small amplitude wavy walls was studied. The governing equations were constructed using the continuity equation and Navier-Stokes equations with no slip boundary conditions. Suppositions were made to simplify the complex non-linear problem, including linear instabilities, uniform thermal characteristics, and laminar flow conditions. The equations were further simplified using available dimensionless variables and new transformations. The perturbation and power series approaches were used to solve the equations with the help of Mathematica, and the velocity profiles and stream contours were graphed using standard codes and definitions.

The flow in a meandering tube becomes unstable due to centrifugal impact, which results in the formation of streamwise vortices. This research has focused on identifying the lowest meandering amplitude that can create these vortices with minimal pressure loss. Two primary forms of vortex instability have been studied, with the most effective parameter being better for creating vortex instability than centrifugal instability. The meandering geometry that is most successful in producing vortices does not encourage traveling wave instability, which can delay the onset of laminar-turbulent transition. By using the reduced geometry model approach, the findings can be applied to various types of meandering tubes.

Further investigations can be conducted through experiments, simulations, and new applications. Experiments can provide valuable insights into the complex nature of the flow using advanced techniques like PIV, LDV, and HWA to measure velocities, turbulence, and vortices. Numerical simulations using CFD techniques can also be performed to investigate flow behavior, providing detailed information on velocity, pressure, and vortices, and help optimize tube design.