3D Stokes flow in a bumpy curved channel

Okechi, Nnamdi Fidelis

doi:10.1063/5.0232562

Three-dimensional (3D) Stokes flow of an incompressible fluid in a curved channel is analyzed. The flow field and characteristics under the influence of 3D surface bumps are studied. A mathematical model describing the 3D complex flow in the curvilinear geometry is formulated and validated. The solution of the coupled system of quasilinear partial differential equations associated with the model is obtained analytically. The behavior of the velocity distribution as the fluid moves through the undulating passage of the bumpy curved channel is examined and explained for each velocity component. The analysis of this flow problem indicates that the velocity vector of the flow is significantly altered by the presence of the structures on the bounding bumpy surfaces, depending on the alignment of the bumps. Furthermore, it is found that the difference between the flow rate in the bumpy curved channel and that in a smooth curved channel is essentially a function of the geometric characteristics of the surface bumps and the curved channel. Notably, the flow in a bumpy straight channel is a limiting case of that in the bumpy curved channel, but the overall effect of the bumps is more significant for a bumpy curved channel.

I. INTRODUCTION

Channel boundaries can be perturbed in the form of corrugations or bumps due to manufacturing defects, corrosion, abrasion, or deposition, among other causes. Flow in perturbed channels has been studied extensively to predict modifications in the flow characteristics and develop potential applications, especially in the context of optimizing the transport of fluids.^1–4

Among the early reports on the flow in perturbed conduits, Wang^5,6 investigated pressure-driven flow in corrugated channels. Phan-Thien^7,8 examined flow in corrugated tubes. Both of these authors^6–8 reported that the flow rates in channels and tubes were appreciably modified by the corrugations. Pozrikidis⁹ studied Stokes flow in a twisted tube, formulated in non-orthogonal curvilinear helical coordinates. Song et al.¹⁰ modeled surface roughness effects on Stokes flow in pipes for different roughness morphologies. Other related works include those of Wang¹¹ and of Duan and Muzychka,¹² who discussed the effects of corrugations on slip flows. Rahmani and Taghari¹³ recently analyzed the flow of a Bingham fluid in a channel with superhydrophobic grooves, for both creeping and inertial flow regimes. To explore the flow features in a curved tube with protrusions, Peterson¹⁴ studied steady flow through a curved tube with circumferential waviness. Further to these works, Okechi and Asghar^15–17 have conducted a number of studies on two-dimensional (2D) flows in corrugated curved channels, reporting the combined impact of the channel radius of curvature and the parameters of the corrugations. The problems considered in Refs. 5–17 are typically 2D in nature, owing to the 2D corrugations. However, there have been further developments concerning 3D flows in channels and tubes with perturbed boundaries. In particular, Wang studied 3D Stokes flows in a bumpy straight channel¹⁸ and tube,¹⁹ investigating the effects of the bumps for both of these flow geometries. The work of Wang^18,19 has been extended to pressure-driven flow in a porous bumpy tube²⁰ and channel,^21,22 and to electroosmotic flow through a bumpy tube²³ and channel,²⁴ among other configurations. In addition, pressure- and boundary-driven 3D flows in a bumpy shaft within a smooth cylinder have been discussed in detail by Song et al.²⁵ These studies have indicated that owing to the tortuosity of the path traveled by fluid in a bumpy channel or tube, the flow modification encountered in such bumpy conduits is more significant than that in a corrugated conduit, where the undulation of the flow passage is unidirectional.

Artificially or naturally curved channels arise in many physical and biological systems, and therefore it is important to model and analyze flow through curved geometries, with the aim of providing a better understanding of the flow mechanism and improve predictions of flow properties in practical situations where models of straight channel flow are rather unrealistic. In particular, bumpy curved channels occur in microfluidic devices with rough surface topography and in biological systems such as blood and lymphatic vessels with endothelial cell linings that exhibit micro protrusions. Several investigations reported in the literature have been devoted to analyzing viscous flows through bumpy straight channels, but to the best of the present author’s knowledge, there have been no analyses of viscous flow through a bumpy curved channel. Therefore, the primary purpose of the present work is to study the effects of channel geometry coupled with the structure of the surface bumps on the velocity distribution and the flow rate associated with bumpy curved channel flow. The study attempts to provide insight into the dynamics of flow in a bumpy curved channel, which could have potential applications in understanding the trends of the flow modifications that arise in curvilinear domains bounded by bumpy surfaces.

The present problem is essentially a generalization of bumpy straight channel flow.¹⁸ The flow will be investigated under the Stokes assumption, which describes the creeping flow under consideration. The solution of the 3D inertia-free model in curvilinear coordinates will be derived analytically for small amplitude of the bumps relative to the average height of the channel. The model and results are validated through comparison with previous results in the literature.^18–21 Owing to the complexity of the models governing flow problems in perturbed channels and tubes, perturbative approaches have generally had to be used to find analytical solutions of these models.^15–25 Keeping this in mind, a boundary perturbation method is used here to derive analytical solutions of the present flow problem. The remainder of this paper is organized as follows: Sec. II describes the problem formulation and analysis, Sec. III presents the results and discussion, and finally Sec. IV gives concluding remarks.

II. PROBLEM MODEL AND ANALYSIS

Consider flow through a gently curved channel of radius of curvature k, as shown in Fig. 1. Let (x, y, z) denote curvilinear coordinates in $R^{3}$ ⁠. The flow is driven by a mean pressure gradient G in the x direction and is bounded by two walls with bumps, described by $y_{T} = d + a \sin (2 π x / L_{1}) \sin (2 π z / L_{3})$ at the top and $y_{1 B} = - d + a \sin (2 π x / L_{2}) \sin (2 π z / L_{3})$ or $y_{2 B} = - d - a \sin (2 π x / L_{2}) \sin (2 π z / L_{3})$ at the bottom. When the bumpy walls are aligned in-phase with zero phase shift, we get the combination y_T and y_1B, while for the anti-phase alignment with maximum phase shift, we get y_T and y_2B. The boundaries are separated by a mean height 2d, and the wavelengths of the bumps of amplitude a in the x direction are taken to be L₁ and L₂, while the wavelength in the z direction is taken to be L₃.

FIG. 1.

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Flow schematic: (a) channel axial section; (b) channel cross section. The flow is driven along the x direction.

The vector equations governing the steady 3D incompressible flow are

\begin{gathered} \nabla \cdot v = 0, \\ μ \nabla^{2} v - \nabla p = 0, \end{gathered}

(1)

where v = (u, v, w) is the velocity vector in the directions (x, y, z), μ is the dynamic viscosity of the fluid, and p is the pressure. We normalize (x, y, z), (u, v, w), and the pressure gradient by d, Gd²/μ, and G, respectively, to obtain the dimensionless forms of the governing equations. In component form, the dimensionless forms of Eq. in curvilinear coordinates are written as

\partial_{y} [(y + k) v] + k \partial_{x} u + (y + k) \partial_{z} w = 0,

(2)

\begin{align} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) u]\} + 2 k \partial_{x} v \\ + k^{2} \partial_{x x} u + {(y + k)}^{2} \partial_{z z} u = k (y + k) \partial_{x} p, \end{align}

(3)

\begin{align} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) v]\} - 2 k \partial_{x} u \\ + k^{2} \partial_{x x} v + {(y + k)}^{2} \partial_{z z} v = {(y + k)}^{2} \partial_{y} p, \end{align}

(4)

(y + k) \partial_{y} [(y + k) \partial_{y} w] + k^{2} \partial_{x x} w + {(y + k)}^{2} \partial_{z z} w = {(y + k)}^{2} \partial_{z} p,

(5)

where ∂_ij( · ) is the partial derivative with respect to i and j. Equation is the equation of continuity and Eqs. – are the components of the equation of momentum in the x, y, and z directions, respectively. Taking the limit as the radius of curvature approaches infinity, the model of Wang¹⁸ (for a bumpy straight channel flow) can easily be recovered. The no-slip boundary conditions on the bumpy walls are

\begin{gathered} u = v = w = 0 on y = y_{T} = 1 + ε \sin (α_{T} x) \sin (ϑ z), \\ y = y_{B} = - 1 \pm ε \sin (α_{B} x) \sin (ϑ z), \end{gathered}

(6)

where ɛ = a/d is the relative amplitude, α_T = 2πd/L₁ and α_B = 2πd/L₂ are the wavenumbers along the x direction, and ϑ = 2πd/L₃ is the wavenumber in the z direction. For the present analysis, we assume that α_T ∼ α_B = α. The ± signs denote the cases of in-phase (zero phase shift) and antiphase (maximum phase shift) alignment, respectively. The bottom boundary conditions incorporate y_1B and y_2B. To obtain the solution, the velocities and pressure can be written in the following perturbative expansions for small ɛ (i.e., the case with very small amplitude of the bumps, a << d):

u = \sum_{β = 0}^{N} ε^{β} u_{β},

(7)

v = \sum_{β = 0}^{N} ε^{β} v_{β},

(8)

w = \sum_{β = 0}^{N} ε^{β} w_{β},

(9)

p = \sum_{β = 0}^{N} ε^{β} p_{β} .

(10)

The boundary conditions can be expanded as

{u|}_{y = y_{T}} = \sum_{β = 0}^{N} ε^{β} {si n^{β} (α x) si n^{β} (ϑ z) \partial_{y^{β}} u / β!|}_{y = 1} = 0,

(11)

{v|}_{y = y_{T}} = \sum_{β = 0}^{N} ε^{β} {si n^{β} (α x) si n^{β} (ϑ z) \partial_{y^{β}} v / β!|}_{y = 1} = 0,

(12)

{w|}_{y = y_{T}} = \sum_{β = 0}^{N} ε^{β} {si n^{β} (α x) si n^{β} (ϑ z) \partial_{y^{β}} w / β!|}_{y = 1} = 0,

(13)

{u|}_{y = y_{B}} = \sum_{β = 0}^{N} {(\pm 1)}^{β} ε^{β} {si n^{β} (α x) si n^{β} (ϑ z) \partial_{y^{β}} u / β!|}_{y = - 1} = 0,

(14)

{v|}_{y = y_{B}} = \sum_{β = 0}^{N} {(\pm 1)}^{β} ε^{β} {si n^{β} (α x) si n^{β} (ϑ z) \partial_{y^{β}} v / β!|}_{y = - 1} = 0,

(15)

{w|}_{y = y_{B}} = \sum_{β = 0}^{N} {(\pm 1)}^{β} ε^{β} {si n^{β} (α x) si n^{β} (ϑ z) \partial_{y^{β}} w / β!|}_{y = - 1} = 0,

(16)

where

\partial_{i^{β}} (\cdot)

is the βth derivative with respect to i and N∊

N

⁠. From Eqs. – we get differential equations and boundary conditions corresponding to each perturbation order. The terms with N > 2 are neglected because their contributions to the perturbation expansion are negligible for ɛ ≪ 1. That is to say, with increasing N (⁠

\geq 3

⁠), ɛ^N → 0 for ɛ ≪ 1. Thus, the impacts of the corresponding terms in the perturbation series become infinitesimal and can be ignored.^{5–8,14–24}

A. Zeroth-order problem

The velocity vector for the zeroth-order equation is (u₀, 0, 0). The velocities v₀ and w₀ in the y and z directions, respectively, are zero when ɛ = 0. The flow is purely 1D, and p₀ is a function of x alone. Therefore, Eqs. – with – reduce to

\begin{gathered} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) u_{0}]\} = k (y + k), \\ {u_{0}|}_{y = 1} = {u_{0}|}_{y = - 1} = 0 \cdot \end{gathered}

(17)

The solution of Eq. can be expressed as

u_{0} = a_{0} (y + k) + b_{0} {(y + k)}^{- 1} + k (y + k) [1 - 2 \ln (y + k)] / 4,

(18)

where

a_{0} = {(k + 1)}^{2} \ln (k + 1) / 8 - {(k - 1)}^{2} \ln (k - 1) / 8 - k / 4,

(19)

b_{0} = {(k + 1)}^{2} {(k - 1)}^{2} \ln (\frac{k - 1}{k + 1}) / 8 .

(20)

B. First-order problem

The velocity vector for the first-order equation is (u₁, v₁, w₁). The flow is now 3D, with p₁ as a function of (x, y, z) owing to the influence of bumps at this order. Thus, Eqs. – become

\partial_{y} [(y + k) v_{1}] + k \partial_{x} u_{1} + (y + k) \partial_{z} w_{1} = 0,

(21)

\begin{align} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) u_{1}]\} + 2 k \partial_{x} v_{1} \\ + k^{2} \partial_{x x} u_{1} + {(y + k)}^{2} \partial_{z z} u_{1} = k (y + k) \partial_{x} p_{1}, \end{align}

(22)

\begin{align} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) v_{1}]\} - 2 k \partial_{x} u_{1} \\ + k^{2} \partial_{x x} v_{1} + {(y + k)}^{2} \partial_{z z} v_{1} = {(y + k)}^{2} \partial_{y} p_{1}, \end{align}

(23)

(y + k) \partial_{y} [(y + k) \partial_{y} w_{1}] + k^{2} \partial_{x x} w_{1} + {(y + k)}^{2} \partial_{z z} w_{1} = {(y + k)}^{2} \partial_{z} p_{1},

(24)

and the boundary conditions read

{u_{1}|}_{y = 1} = - \sin (α x) \sin (ϑ z) {\partial_{y} u_{0}|}_{y = 1},

(25)

{u_{1}^{\pm}|}_{y = - 1} = \mp \sin (α x) \sin (ϑ z) {\partial_{y} u_{0}|}_{y = - 1},

(26)

{v_{1}|}_{y = 1} = {v_{1}^{\pm}|}_{y = - 1} = {w_{1}|}_{y = 1} = {w_{1}^{\pm}|}_{y = - 1} = 0 .

(27)

The coupled system – requires solutions of the form

u_{1} = U_{1} \sin (α x) \sin (ϑ z),

(28)

v_{1} = V_{1} \cos (α x) \sin (ϑ z),

(29)

w_{1} = W_{1} \cos (α x) \cos (ϑ z),

(30)

p_{1} = P_{1} \cos (α x) \sin (ϑ z) .

(31)

Using Eqs. –, we have

\partial_{y} [(y + k) V_{1}] + α k U_{1} - ϑ (y + k) W_{1} = 0,

(32)

\begin{align} (y + k) \partial_{y} [(y + k) \partial_{y} U_{1}] - 2 α k V_{1} - [ϑ^{2} {(y + k)}^{2} + α^{2} k^{2} + 1] U_{1} \\ = - α k (y + k) P_{1}, \end{align}

(33)

\begin{align} (y + k) \partial_{y} [(y + k) \partial_{y} V_{1}] - 2 α k U_{1} - [ϑ^{2} {(y + k)}^{2} + α^{2} k^{2} + 1] V_{1} \\ = {(y + k)}^{2} \partial_{y} P_{1}, \end{align}

(34)

(y + k) \partial_{y} [(y + k) \partial_{y} W_{1}] - [α^{2} k^{2} + ϑ^{2} {(y + k)}^{2}] W_{1} = ϑ {(y + k)}^{2} P_{1},

(35)

with boundary conditions

{U_{1}|}_{y = 1} = - {\partial_{y} u_{0}|}_{y = 1},

(36)

{U_{1}^{\pm}|}_{y = - 1} = \mp {\partial_{y} u_{0}|}_{y = - 1},

(37)

{V_{1}|}_{y = 1} = {V_{1}^{\pm}}_{y = - 1} = 0,

(38)

{W_{1}|}_{y = 1} = {W_{1}^{\pm}|}_{y = - 1} = 0 .

(39)

The solutions of the coupled system – subject to Eqs. – are obtained as

\begin{align} U_{1}^{\pm} & = \frac{1}{2} [a_{1}^{\pm} I_{α k + 1} (ϑ (y + k)) + b_{1}^{\pm} K_{α k + 1} (ϑ (y + k)) \\ + c_{1}^{\pm} I_{α k - 1} (ϑ (y + k)) + d_{1}^{\pm} K_{α k - 1} (ϑ (y + k))], \end{align}

(40)

\begin{align} V_{1}^{\pm} & = \frac{1}{2} [a_{1}^{\pm} I_{α k + 1} (ϑ (y + k)) + b_{1}^{\pm} K_{α k + 1} (ϑ (y + k)) - c_{1}^{\pm} I_{α k - 1} \\ \times (ϑ (y + k)) - d_{1}^{\pm} K_{α k - 1} (ϑ (y + k)) + e_{1}^{\pm} (y + k) I_{α k} (ϑ (y + k)) \\ + f_{1}^{\pm} (y + k) K_{α k} (ϑ (y + k))], \end{align}

(41)

\begin{aligned} W_{1}^{\pm} & = \frac{1}{2 ϑ} \{e_{1}^{\pm} ϑ (y + k) I_{α k + 1} (ϑ (y + k)) - f_{1}^{\pm} ϑ (y + k) K_{α k + 1} (ϑ (y + k)) \\ + [ϑ (a_{1}^{\pm} - c_{1}^{\pm}) + e_{1}^{\pm} (α k + 2)] I_{α k} (ϑ (y + k)) + [ϑ (d_{1}^{\pm} - b_{1}^{\pm}) \\ + f_{1}^{\pm} (α k + 2)] K_{α k} (ϑ (y + k))\}, \\ P_{1}^{\pm} & = e_{1}^{\pm} I_{α k} (ϑ (y + k)) + f_{1}^{\pm} K_{α k} (ϑ (y + k)) . \end{aligned}

(42)

Here, I_A and K_A are order-A modified Bessel functions of the first and second kinds, respectively. The coefficients $a_{1}^{\pm}, b_{1}^{\pm}, c_{1}^{\pm}, d_{1}^{\pm}, e_{1}^{\pm}$ ⁠, and $f_{1}^{\pm}$ follow directly from the boundary conditions (36)–(39). The details can be found in the Appendix.

C. Second-order problem

Here, the velocity vector is expressed as (u₂, v₂, w₂), and p₂ is also function of (x, y, z). Hence, Eqs. – now become

\partial_{y} [(y + k) v_{2}] + k \partial_{x} u_{2} + (y + k) \partial_{z} w_{2} = 0,

(43)

\begin{align} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) u_{2}]\} + 2 k \partial_{x} v_{2} + k^{2} \partial_{x x} u_{2} \\ + {(y + k)}^{2} \partial_{z z} u_{2} = k (y + k) \partial_{x} p_{2}, \end{align}

(44)

\begin{align} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) v_{2}]\} - 2 k \partial_{x} u_{2} + k^{2} \partial_{x x} v_{2} \\ + {(y + k)}^{2} \partial_{z z} v_{2} = {(y + k)}^{2} \partial_{y} p_{2}, \end{align}

(45)

(y + k) \partial_{y} [(y + k) \partial_{y} w_{2}] + k^{2} \partial_{x x} w_{2} + {(y + k)}^{2} \partial_{z z} w_{2} = {(y + k)}^{2} \partial_{z} p_{2},

(46)

and the boundary conditions are

{u_{2}|}_{y = 1} = - \sin (α x) \sin (ϑ z) {\partial_{y} u_{1}|}_{y = 1} - \frac{1}{2} \sin^{2} (α x) \sin^{2} (ϑ z) {\partial_{y y} u_{0}|}_{y = 1},

(47)

\begin{align} {u_{2}^{\pm}|}_{y = - 1} & = \mp \sin (α x) \sin (ϑ z) {\partial_{y} u_{1}^{\pm}|}_{y = - 1} \\ - \frac{1}{2} \sin^{2} (α x) \sin^{2} (ϑ z) {\partial_{y y} u_{0}|}_{y = - 1}, \end{align}

(48)

{v_{2}|}_{y = 1} = - \sin (α x) \sin (ϑ z) {\partial_{y} v_{1}|}_{y = 1},

(49)

{v_{2}^{\pm}|}_{y = - 1} = \mp \sin (α x) \sin (ϑ z) {\partial_{y} v_{1}^{\pm}|}_{y = - 1},

(50)

{w_{2}|}_{y = 1} = - \sin (α x) \sin (ϑ z) {\partial_{y} w_{1}|}_{y = 1},

(51)

{w_{2}^{\pm}|}_{y = - 1} = \mp \sin (α x) \sin (ϑ z) {\partial_{y} w_{1}^{\pm}|}_{y = - 1} .

(52)

The coupled system – with boundary conditions – requires solutions of the type

u_{2} = U_{2} + U_{3} \cos (2 ϑ z) + U_{4} \cos (2 α x) + U_{5} \cos (2 α x) \cos (2 ϑ z),

(53)

v_{2} = V_{2} \sin (2 α x) + V_{3} \sin (2 α x) \cos (2 ϑ z),

(54)

w_{2} = W_{2} \sin (2 α x) \sin (2 ϑ z),

(55)

p_{2} = P_{2} \sin (2 α x) + P_{3} \sin (2 α x) \cos (2 ϑ z) .

(56)

Using Eqs. –, we have

(y + k) \partial_{y} [(y + k) \partial_{y} U_{2}] - U_{2} = 0,

(57)

{(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) U_{3}]\} - 4 ϑ^{2} {(y + k)}^{2} U_{3} = 0,

(58)

\partial_{y} [(y + k) V_{2}] - 2 α k U_{4} = 0,

(59)

\begin{align} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) U_{4}]\} + 4 α k (V_{2} - α k U_{4}) \\ = 2 α k (y + k) P_{2}, \end{align}

(60)

\begin{align} {(y + k)}^{2} \partial_{y} \{{(y + k)}^{- 1} \partial_{y} [(y + k) V_{2}]\} + 4 α k (U_{4} - α k V_{2}) \\ = {(y + k)}^{2} \partial_{y} P_{2}, \end{align}

(61)

\partial_{y} [(y + k) V_{3}] - 2 α k U_{5} + 2 ϑ (y + k) W_{2} = 0,

(62)

\begin{align} (y + k) \partial_{y} [(y + k) \partial_{y} U_{5}] + 4 α k V_{3} - [4 ϑ^{2} {(y + k)}^{2} + 4 α^{2} k^{2} + 1] U_{5} \\ = 2 α k (y + k) P_{3}, \end{align}

(63)

\begin{align} (y + k) \partial_{y} [(y + k) \partial_{y} V_{3}] + 4 α k U_{5} - [4 ϑ^{2} {(y + k)}^{2} + 4 α^{2} k^{2} + 1] V_{3} \\ = {(y + k)}^{2} \partial_{y} P_{3}, \end{align}

(64)

(y + k) \partial_{y} [(y + k) \partial_{y} W_{2}] - [4 α^{2} k^{2} + 4 ϑ^{2} {(y + k)}^{2}] W_{2} = - 2 ϑ {(y + k)}^{2} P_{3},

(65)

with the set of boundary conditions

{U_{2}|}_{y = 1} = - {U_{3}|}_{y = 1} - {U_{4}|}_{y = 1} = {U_{5}|}_{y = 1} = - \frac{1}{4} {(\partial_{y} U_{1}^{\pm} + \frac{1}{2} \partial_{y y} u_{0})|}_{y = 1},

(66)

\begin{align} {U_{2}^{\pm}|}_{y = - 1} & = - {U_{3}^{\pm}|}_{y = 1} = - {U_{4}^{\pm}|}_{y = 1} = {U_{5}^{\pm}}_{y = 1} \\ = \frac{1}{4} {(\mp \partial_{y} U_{1}^{\pm} - \frac{1}{2} \partial_{y y} u_{0})|}_{y = - 1}, \end{align}

(67)

{V_{2}|}_{y = 1} = - {V_{3}|}_{y = 1} = - \frac{1}{4} {\partial_{y} V_{1}^{\pm}|}_{y = 1},

(68)

{V_{2}^{\pm}|}_{y = - 1} = {V_{3}^{\pm}|}_{y = - 1} = \mp \frac{1}{4} {\partial_{y} V_{1}^{\pm}|}_{y = - 1},

(69)

{W_{2}|}_{y = 1} = - \frac{1}{4} {\partial_{y} W_{1}^{\pm}|}_{y = 1},

(70)

{W_{2}^{\pm}|}_{y = - 1} = \mp \frac{1}{4} {\partial_{y} W_{1}^{\pm}|}_{y = - 1} .

(71)

After some algebra, the solutions of the coupled systems – subject to Eqs. – are obtained as

U_{2}^{\pm} = a_{2}^{\pm} (y + k) + b_{2}^{\pm} / (y + k),

(72)

U_{3}^{\pm} = a_{3}^{\pm} I_{1} (2 ϑ (y + k)) + b_{3}^{\pm} K_{1} (2 ϑ (y + k)),

(73)

\begin{align} U_{4}^{\pm} & = c_{4}^{\pm} {(y + k)}^{2 α k - 1} - d_{4}^{\pm} {(y + k)}^{- 2 α k - 1} + \{e_{4}^{\pm} [{(2 α k - 1)}^{- 1} - 1] \\ \times {(y + k)}^{1 - 2 α k} + f_{4}^{\pm} [{(2 α k + 1)}^{- 1} + 1] {(y + k)}^{1 + 2 α k}\} / 4, \end{align}

(74)

\begin{align} U_{5}^{\pm} & = \frac{1}{2} [a_{5}^{\pm} I_{2 α k - 1} (2 ϑ (y + k)) + b_{5}^{\pm} K_{2 α k - 1} (2 ϑ (y + k)) \\ + c_{5}^{\pm} I_{2 α k + 1} (2 ϑ (y + k)) + d_{5}^{\pm} K_{2 α k + 1} (2 ϑ (y + k))], \end{align}

(75)

\begin{align} \begin{gathered} \begin{matrix} V_{2}^{\pm} & = c_{4}^{\pm} {(y + k)}^{2 α k - 1} + d_{4}^{\pm} {(y + k)}^{- 2 α k - 1} + α k [e_{4}^{\pm} {(2 α k - 1)}^{- 1} \\ \times {(y + k)}^{1 - 2 α k} + f_{4}^{\pm} {(2 α k + 1)}^{- 1} {(y + k)}^{1 + 2 α k}] / 2, \end{matrix} \\ P_{2}^{\pm} = e_{4}^{\pm} {(y + k)}^{- 2 α k} + f_{4}^{\pm} {(y + k)}^{2 α k}, \end{gathered} \end{align}

(76)

\begin{align} V_{3}^{\pm} & = \frac{1}{2} [a_{5}^{\pm} I_{2 α k - 1} (2 ϑ (y + k)) + b_{5}^{\pm} K_{2 α k - 1} (2 ϑ (y + k)) - c_{5}^{\pm} I_{2 α k + 1} \\ \times (2 ϑ (y + k)) - d_{5}^{\pm} K_{2 α k + 1} (2 ϑ (y + k)) + e_{4}^{\pm} (y + k) I_{2 α k} \\ \times (2 ϑ (y + k)) + f_{4}^{\pm} (y + k) K_{2 α k} (2 ϑ (y + k))], \end{align}

(77)

\begin{align} \begin{gathered} \begin{matrix} W_{2}^{\pm} & = \frac{1}{2 ϑ} \{f_{5}^{\pm} ϑ (y + k) K_{2 α k + 1} (2 ϑ (y + k)) - e_{5}^{\pm} ϑ (y + k) I_{2 α k + 1} \\ \times (2 ϑ (y + k)) - [ϑ (a_{5}^{\pm} - c_{5}^{\pm}) + e_{1}^{\pm} (α k + 2)] I_{2 α k} (2 ϑ (y + k)) \\ - [ϑ (d_{5}^{\pm} - b_{5}^{\pm}) + f_{1}^{\pm} (α k + 2)] K_{2 α k} (2 ϑ (y + k))\}, \end{matrix} \\ P_{1}^{\pm} = e_{5}^{\pm} I_{2 α k} (2 ϑ (y + k)) + f_{5}^{\pm} K_{2 α k} (2 ϑ (y + k)) . \end{gathered} \end{align}

(78)

The coefficients $a_{2 - 5}^{\pm}, b_{2 - 5}^{\pm}, c_{2}^{\pm}, d_{2 - 5}^{\pm}$ ⁠, $e_{2}^{\pm}$ ⁠, and $f_{2}^{\pm}$ are obtained by using the boundary conditions (66)–(71). See the Appendix for details.

D. Flow rate evaluation

Considering the complexity of the flow domain, we proceed to obtain the normalized flow rate per unit area, which is evaluated at x = 0, where the cross section is simpler.^21,22 Thus, we have

Q = \frac{1}{2} \int_{- 1}^{1} {\bar{u}|}_{x = 0} d y = \frac{1}{2} \int_{- 1}^{1} {({\bar{u}}_{0} + ε^{2} {\bar{u}}_{2} + \dots)|}_{x = 0} d y,

(79)

where

\bar{u}

represents the average with respect to z and is given by

\bar{u} = \frac{ϑ}{2 π} \int_{- π / ϑ}^{π / ϑ} u d z .

(80)

Clearly, in Eq. , the first-order solution is periodic in z and will have no effect on Q, but the effect of the bumps on the flow rate rather begins from the second-order solution. On evaluation, Eq. can be written equivalently as

Q = \overset{⌣}{q} [1 - 3 ε^{2} χ^{\pm} / 4 + O (ε^{4})] / 2 .

(81)

Here,

\begin{align} q & = \overset{⌣}{q} / 2 = 2 k a_{0} + b_{0} \ln (\frac{k + 1}{k - 1}) - k \\ \times [{(k + 1)}^{2} \ln (k + 1) - {(k - 1)}^{2} \ln (k - 1)] / 4 . \end{align}

(82)

The function q is the volumetric flow rate of the smooth curved channel without bumps, and ɛ²χ^± is the fractional change in the flow rate due to the surface bumps. The function χ^± is expressed as

\begin{align} χ^{\pm} & = 8 \frac{a_{2}^{\pm} k}{3 \overset{⌣}{q}} + 4 b_{2}^{\pm} \ln [\frac{k + 1}{3 \overset{⌣}{q} (k - 1)}] \\ + \frac{2 c_{2}^{\pm} [{(k + 1)}^{2 α k} - {(k - 1)}^{2 α k}] + d_{2}^{\pm} [{(k + 1)}^{- 2 α k} - {(k - 1)}^{- 2 α k}]}{3 α k \overset{⌣}{q}} \\ + \frac{e_{2}^{\pm} [{(2 α k - 1)}^{- 1} - 1] [{(k + 1)}^{- 2 α k + 2} - {(k - 1)}^{- 2 α k + 2}]}{12 α k \overset{⌣}{q} (1 - α k)} \\ + \frac{f_{2}^{\pm} [{(2 α k + 1)}^{- 1} - 1] [{(k + 1)}^{2 α k + 2} - {(k - 1)}^{2 α k + 2}]}{12 α k \overset{⌣}{q} (1 + α k)} . \end{align}

(83)

III. RESULTS AND DISCUSSION

A. Velocity distribution

The validity of the model and analytical result is confirmed by comparing plots of the amplitudes of the first-order velocity components (in the limit k → ∞) with those of a bumpy straight channel studied by Yu and Wang²¹ (a special case of the present problem; for further verification, see the analytical results of Wang¹⁸). The profiles of the amplitudes of the velocity components in Fig. 2 coincide with the analytical results obtained by Wang¹⁸ and Yu and Wang.²¹

FIG. 2.

View large Download slide

Variation of first-order velocity amplitudes in (a) x direction, (b) y direction and (c) z direction, for a large value of k (=20), when α = 1 and ϑ = 1. The + and − signs indicate in-phase and antiphase alignments, respectively, of the bumpy walls.

Figures 3–5 illustrate the behavior of the 3D flow by showing each component of the velocity vector as a function of the spatial variables describing the dimensions of the flow and the parameters that define the geometry of the flow domain. In Fig. 3, the variation of the x component of the velocity, u, with y is shown for given values of x and z. We can observe that the peaks of the velocity remain approximately at certain positions, namely, z = 0, π, and 2π, irrespective of the phase shift between the bumpy walls. This is because the numerical values of u at these points remain unchanged. Notably, the profiles of the velocity when the bumps are aligned without phase shift have the same distance along y. Physically, this is because when the bumps are in phase, the fluid moves through a path with constant height between the bumpy walls. However, in the antiphase situation, the corresponding velocity profiles reveal a variation in the distance along y. This can be explained by the periodic change in the height between the bumpy walls as the fluid moves through the channel, alternating between wide and narrow paths as it passes along the passage, when the bumpy walls are in an antiphase configuration.

FIG. 3.

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Variation of x component of velocity, u, along y for different values of z (as indicated in the key) at x = π/3, when k = 2.5, ɛ = 0.1, α = 1, and ϑ = 1: (a) antiphase bumpy walls; (b) in-phase bumpy walls.

FIG. 4.

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Variation of y component of velocity, v, along z for different values of y (as indicated in the key) at x = π/3, when k = 2.5, ɛ = 0.1, α = 1 and ϑ = 1: (a) antiphase bumpy walls; (b) in-phase bumpy walls.

FIG. 5.

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Variation of z component of velocity, w, along z for different values of y (as shown in the key) at x = π/3, when k = 2.5, ɛ = 0.1, α = 1, and ϑ = 1: (a) antiphase bumpy wall; (b) in-phase bumpy walls.

The y component of the velocity, v, is depicted in Fig. 4. It can be seen that the magnitudes of the peaks of v in the lower section (y < 0) of the channel are higher than those in the upper section (y > 0). In each section, the magnitude of the peak increases away from the center of the channel at y = 0 when the bumpy walls are not in phase [see Fig. 4(a)], which is contrary to the observation in the case when the bumpy walls are in phase [Fig. 4(b)]. Overall, the velocity exhibits fluctuations between negative and positive values as the fluid moves through the undulations, owing to the bumpy surfaces of the channel. For the z component of the velocity, w, Fig. 5(a) shows that the profiles corresponding to different values of y exhibit similar trends along z, with slight variation in the peaks, for the antiphase bumpy walls. However, for the in-phase bumpy walls, the profiles depicted in Fig. 5(b) are distinct and significantly different in trend when compared with those in Fig. 5(a), at the same nominal values of y. Furthermore, it can also be seen that the magnitude of each peak of w is greater for the antiphase bumpy walls, which is not the case for v, as can be seen in Fig. 4.

Interestingly, the geometry of a bumpy straight channel is such that when the phase shift between the bumpy walls of the channel is maximum (i.e., when the walls are in antiphase), the channel is symmetric about the center plane; otherwise the channel is asymmetrical in nature. This geometric phenomenon is discussed by Wang.¹⁸ On the contrary, the results here indicate that for a bumpy curved channel, the channel is rather asymmetrical, regardless of the phase shift between the bumpy walls: This feature is demonstrated by the velocity profiles in Figs. 3–5. The dashed lines in Figs. 4 and 5 indicate the channel center at y = 0.

Figures 6–8 illustrate the effects of k on the velocity distributions. Regardless of the alignment between the bumpy walls, the x component of the velocity, u, is an increasing function of k, as can be seen in Fig. 6. We can also observe that for sufficiently large nominal values of k, only the profile in the case of antiphase bumpy walls approaches symmetry, as expected. On the other hand, the y and z components v and w are decreasing functions of k, when the effects of k are distinguishable: in Fig. 7, the influence of k on v is prominent only for the antiphase bumpy walls, but in Fig. 8, the effects of k on w are pronounced only when the bumpy walls are in phase.

FIG. 6.

View large Download slide

Variation of x component of velocity, u, along y for different values of k (shown in the key) when x = π/3, z = π/3, ɛ = 0.1, α = 1, and ϑ = 1: (a) antiphase bumpy walls; (b) in-phase bumpy walls.

FIG. 7.

View large Download slide

Variation of y component of velocity, v, along z for different values of k (shown in the key) when x = π/3, y = 0, ɛ = 0.1, α = 1, and ϑ = 1: (a) antiphase bumpy walls; (b) in-phase bumpy walls.

FIG. 8.

View large Download slide

Variation of z component of velocity, w, along z for different values of k (shown in the key) when x = π/3, y = 0, ɛ = 0.1, α = 1, and ϑ = 1: (a) antiphase bumpy walls; (b) in-phase bumpy walls.

B. Flow rate

Graphical depictions of the analytical results for the modified flow rate and confirmation of their validity are presented in this subsection. Figure 9 shows the variation of the normalized flow rate q with the radius of curvature k. The function q increases as k increases (owing to the decreasing effects of curvature). We can observe that q tends toward a constant as large values of k are approached, giving the flow rate (1/3) corresponding to Poiseuille flow in a smooth straight channel.

FIG. 9.

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Variation of q as a function of k when ɛ = 0.

From Eq. (81), it is evident that the function Q representing the flow rate in the bumpy channel depends significantly on the function χ resulting from the presence of surface bumps. Naturally, for a smooth channel, Q reduces to q; in other cases, Q may decrease below q subject to the variation of χ with the parameters defining the channel geometry and configuration. To further validate the present analytical result, graphical illustrations of χ as a function of α, ϑ, and k → ∞ are presented in Fig. 10. The observed trends for both zero and maximum phase shifts agree with those for the 3D bumpy straight channel flow studied by Wang¹⁸ and by Yu and Wang.²¹ In comparison with Fig. 10, the profiles in Fig. 11 reveal that χ increases when the value of k is reduced to 2.5. This can be explained by the effects of the curvature of the channel, which increase the flow resistance, leading to a reduction in the flow rate. The influence of α on χ is clearly greater than that of ϑ, which implies that the flow resistance is higher in the axial direction, where the pressure gradient generating the flow is applied.

FIG. 10.

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Variation of χ with wavenumbers α and ϑ when k = 20: (a) in-phase bumpy walls; (b) antiphase bumpy walls.

FIG. 11.

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Variation of χ with wavenumbers α and ϑ when k = 2.5: (a) in-phase bumpy walls; (b) antiphase bumpy walls.

Figure 12 further illustrates the effects of the variation of k on the flow rate. We can observe that χ decreases monotonically when k increases. The magnitude of χ for antiphase bumpy walls [Fig. 12(b)] is greater than that for in-phase bumpy walls [Fig. 12(a)], except at large values of ϑ(≥5), where the magnitudes are the same. These trends indicate that as the wavenumbers of the bumps increase (in addition to the increase in the flow resistance), the alignment of the bumps on the top wall of the channel with those on the bottom wall will gradually lose its significance for the flow. However, the effect of bump alignment is only substantial for small wavenumbers. It can be deduced that the surface bumps will invariably decrease the total flow rate, owing to the flow resistance that they imposed, with the decrease being greater when the wavenumbers are large (especially the wavenumber in the direction of the pressure gradient).

FIG. 12.

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Variation of χ with k for different values of ϑ when α = 2: (a) in-phase bumpy walls; (b) antiphase bumpy walls.

IV. CONCLUDING REMARKS

Steady Stokes flow of a Newtonian fluid in 3D under the influence of 3D surface bumps on the walls of a curved channel has been theoretically investigated. An analytical solution has been obtained via a boundary perturbation procedure appropriate for very small amplitudes of the surface bumps relative to the distance between the boundaries of the flow. The solution is only valid for small relative amplitude and for 1 < k < ∞. In terms of the flow field, the associated 3D velocity vector has been obtained and studied, and it has been found that the distributions of the x, y, and z components of the velocity are significantly influenced in different ways by the microperiodic structure on the boundaries. In addition, the characteristics of the normalized modified flow rate $Q \sim q (1 - 3 ε^{2} χ^{\pm} / 4)$ have been discussed through considering the factor χ^± as a function of the wavenumbers α and ϑ and the radius of curvature k. It has been found that χ^± > 0 always, for α > 0 and ϑ > 0, and consequently that Q < q. This implies that the flow rate in a bumpy curved channel decreases below that in a smooth curved channel. Furthermore, the following results should be noted:

The flow geometry of the bumpy curved channel is asymmetrical, irrespective of the phase shift between the bumps, unlike a bumpy straight channel, which can be symmetrical under the condition of a total phase shift between the bumpy walls.
The structural impact of the flow passage on the velocity distribution changes significantly with the phase shift between the bumpy walls.
χ^± decreases monotonically with k. Thus, the magnitude of χ^±(k → ∞) for a bumpy straight channel is less than that for a bumpy curved channel.
An increase in the wavenumbers will lead to an increase in χ^± for any given k, indicating that small wavenumbers impose less flow resistance, resulting in less reduction in the total flow rate. However, the effect of the wavenumber α in the direction parallel to the pressure gradient is much greater than that of the wavenumber ϑ in the perpendicular direction.
The magnitude of the impact of the bumps increases with increasing phase shift between the bumpy walls, except for large wavenumbers.

Overall, the results obtained here are of value in identifying and understanding the modifications of the flow field due to the existence of minute surface bumps on the boundaries of a curved channel, as well as the parameters relevant for control of flow reduction. Future work will be directed toward studying the flow behavior subject to the effects of surface bumps characterized by large relative amplitudes. In the latter context, it may not be possible to obtain an analytical solution of the flow model, and it will be necessary to resort to numerical methods.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

Author Contributions

Nnamdi Fidelis Okechi: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

NOMENCLATURE

a: Bump amplitude
d: Channel average height
G: Pressure gradient
k: Radius of curvature
L₁, L₂: Wavelengths in x direction
L₃: Wavelength in z direction
p: Pressure
q: Volumetric flow rate of a smooth curved channel
Q: Volumetric flow rate of the bumpy curved channel
u: velocity in x direction
v: Fluid velocity vector
v: Velocity in y direction
w: Velocity in z direction
x, y, z: Spatial variables
y_1B,2B: Bottom boundary functions
y_T: Top boundary function

Greek

ɛ: Dimensionless amplitude
α: Wavenumber in x direction
α_B: Wavenumber of bottom boundary
α_T: Wavenumber of top boundary
ϑ: Wavenumber in z direction
μ: Fluid viscosity
χ: Fractional change function

NOMENCLATURE

a: Bump amplitude
d: Channel average height
G: Pressure gradient
k: Radius of curvature
L₁, L₂: Wavelengths in x direction
L₃: Wavelength in z direction
p: Pressure
q: Volumetric flow rate of a smooth curved channel
Q: Volumetric flow rate of the bumpy curved channel
u: velocity in x direction
v: Fluid velocity vector
v: Velocity in y direction
w: Velocity in z direction
x, y, z: Spatial variables
y_1B,2B: Bottom boundary functions
y_T: Top boundary function

Greek

ɛ: Dimensionless amplitude
α: Wavenumber in x direction
α_B: Wavenumber of bottom boundary
α_T: Wavenumber of top boundary
ϑ: Wavenumber in z direction
μ: Fluid viscosity
χ: Fractional change function

APPENDIX: COEFFICIENTS

For the first-order coefficients, we can write Eqs. – applying the boundary conditions – in the form

a_{11} a_{1}^{\pm} + a_{12} b_{1}^{\pm} + a_{13} c_{1}^{\pm} + a_{14} d_{1}^{\pm} = g_{1}^{\pm},

(A1)

a_{21} a_{1}^{\pm} + a_{22} b_{1}^{\pm} + a_{23} c_{1}^{\pm} + a_{24} d_{1}^{\pm} = g_{2}^{\pm},

(A2)

a_{31} a_{1}^{\pm} + a_{32} b_{1}^{\pm} + a_{33} c_{1}^{\pm} + a_{34} d_{1}^{\pm} + a_{35} e_{1}^{\pm} + a_{36} f_{1}^{\pm} = 0,

(A3)

a_{41} a_{1}^{\pm} + a_{42} b_{1}^{\pm} + a_{43} c_{1}^{\pm} + a_{44} d_{1}^{\pm} + a_{45} e_{1}^{\pm} + a_{46} f_{1}^{\pm} = 0,

(A4)

a_{51} a_{1}^{\pm} + a_{52} b_{1}^{\pm} + a_{53} c_{1}^{\pm} + a_{54} d_{1}^{\pm} + a_{55} e_{1}^{\pm} + a_{56} f_{1}^{\pm} = 0,

(A5)

a_{61} a_{1}^{\pm} + a_{62} b_{1}^{\pm} + a_{63} c_{1}^{\pm} + a_{64} d_{1}^{\pm} + a_{65} e_{1}^{\pm} + a_{66} f_{1}^{\pm} = 0,

(A6)

where

\begin{align} \begin{gathered} a_{11} = a_{31} = I_{α k + 1} (ϑ (k + 1)) / 2, a_{12} = a_{32} = K_{α k + 1} (ϑ (k + 1)) / 2, \\ a_{13} = - a_{33} = I_{α k - 1} (ϑ (k + 1)) / 2, a_{14} = - a_{44} = K_{α k + 1} (ϑ (k + 1)) / 2, \\ a_{15} = (k + 1) I_{α k - 1} (ϑ (k + 1)) / 2, a_{16} = (k + 1) K_{α k + 1} (ϑ (k + 1)) / 2, \\ a_{21} = a_{41} = I_{α k + 1} (ϑ (k - 1)) / 2, a_{22} = a_{42} = K_{α k + 1} (ϑ (k - 1)) / 2, \\ a_{23} = - a_{43} = I_{α k - 1} (ϑ (k - 1)) / 2, a_{24} = - a_{44} = K_{α k + 1} (ϑ (k - 1)) / 2, \\ a_{51} = - a_{53} = I_{α k} (ϑ (k + 1)) / 2, a_{52} = - a_{54} = - K_{α k} (ϑ (k + 1)) / 2, \\ a_{55} = (k + 1) I_{α k + 1} (ϑ (k + 1)) / 2 + (α k + 2) I_{α k} (ϑ (k + 1)) / 2 ϑ, \\ a_{56} = - (k + 1) K_{α k + 1} (ϑ (k + 1)) / 2 + (α k + 2) K_{α k} (ϑ (k + 1)) / 2 ϑ, \\ a_{61} = - a_{53} = I_{α k} (ϑ (k - 1)) / 2, a_{62} = - a_{54} = - K_{α k} (ϑ (k - 1)) / 2, \\ \begin{matrix} a_{65} & = (k - 1) I_{α k + 1} (ϑ (k - 1)) / 2 + (α k + 2) I_{α k} (ϑ (k - 1)) / 2 ϑ, \\ g_{1}^{\pm} = - {\partial_{y} u_{0}|}_{y = 1}, \end{matrix} \\ \begin{matrix} a_{66} & = - (k - 1) K_{α k + 1} (ϑ (k - 1)) / 2 + (α k + 2) K_{α k} (ϑ (k - 1)) / 2 ϑ, \\ g_{2}^{\pm} = \mp {\partial_{y} u_{0}|}_{y = - 1} . \end{matrix} \end{gathered} \end{align}

(A7)

Equations – can be rewritten as

A X = G,

(A8)

where

A = [\begin{matrix} \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \begin{matrix} a_{14} & 0 & 0 \\ a_{24} & 0 & 0 \\ a_{34} & a_{35} & a_{36} \end{matrix} \\ \begin{matrix} a_{41} & a_{42} & a_{43} \\ a_{51} & a_{52} & a_{53} \\ a_{61} & a_{62} & a_{63} \end{matrix} \begin{matrix} a_{44} & a_{45} & a_{46} \\ a_{54} & a_{55} & a_{56} \\ a_{64} & a_{65} & a_{66} \end{matrix} \end{matrix}], X = [\begin{matrix} \begin{matrix} a_{1}^{\pm} \\ b_{1}^{\pm} \\ c_{1}^{\pm} \end{matrix} \\ d_{1}^{\pm} \\ e_{1}^{\pm} \\ f_{1}^{\pm} \end{matrix}], G = [\begin{matrix} \begin{matrix} g_{1}^{\pm} \\ g_{2}^{\pm} \\ 0 \end{matrix} \\ 0 \\ 0 \\ 0 \end{matrix}] .

(A9)

The coefficients in Eq. can be readily obtained algebraically, using appropriate software. For the second-order coefficients, those appearing in the expressions for U₂ subject to the corresponding boundary conditions are given by

[\begin{matrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{matrix}] [\begin{matrix} a_{2}^{\pm} \\ b_{2}^{\pm} \end{matrix}] = [\begin{matrix} h_{1}^{\pm} \\ h_{2}^{\pm} \end{matrix}],

(A10)

and similarly for U₄ and V₂ we have

[\begin{matrix} \begin{matrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix} \begin{matrix} c_{13} & c_{14} \\ c_{23} & c_{24} \end{matrix} \\ \begin{matrix} c_{31} & c_{32} \\ c_{41} & c_{42} \end{matrix} \begin{matrix} c_{33} & c_{34} \\ c_{43} & c_{44} \end{matrix} \end{matrix}] [\begin{matrix} c_{2}^{\pm} \\ d_{2}^{\pm} \\ e_{2}^{\pm} \\ f_{2}^{\pm} \end{matrix}] = [\begin{matrix} - h_{1}^{\pm} \\ - h_{2}^{\pm} \\ m_{1}^{\pm} \\ m_{2}^{\pm} \end{matrix}],

(A11)

where

\begin{gathered} b_{11} = k + 1, b_{12} = \frac{1}{k + 1}, b_{21} = k - 1, b_{22} = \frac{1}{k - 1}, \\ c_{11} = c_{31} = {(k + 1)}^{2 α k - 1}, c_{12} = - c_{32} - {(k + 1)}^{- 2 α k - 1}, \\ c_{13} = [{(2 α k - 1)}^{- 1} - 1] {(k + 1)}^{1 - 2 α k} / 4, \\ c_{14} = [{(2 α k + 1)}^{- 1} - 1] {(k + 1)}^{1 + 2 α k} / 4, \\ c_{21} = c_{41} = {(k - 1)}^{2 α k - 1}, c_{22} = - c_{42} = {(k - 1)}^{- 2 α k - 1}, \\ c_{23} = [{(2 α k - 1)}^{- 1} - 1] {(k - 1)}^{1 - 2 α k} / 4, \\ c_{24} = [{(2 α k + 1)}^{- 1} - 1] {(k - 1)}^{1 + 2 α k} / 4, \\ c_{33} = α k {(2 α k - 1)}^{- 1} {(k + 1)}^{1 - 2 α k} / 2, \\ c_{34} = α k {(2 α k + 1)}^{- 1} {(k + 1)}^{1 + 2 α k} / 2, \\ c_{43} = α k {(2 α k - 1)}^{- 1} {(k - 1)}^{1 - 2 α k} / 2, \\ c_{34} = α k {(2 α k - 1)}^{- 1} {(k + 1)}^{1 + 2 α k} / 2, \\ h_{1}^{\pm} = - {(\partial_{y} U_{1}^{\pm} + \partial_{y y} u_{0} / 2) / 4|}_{y = 1}, \\ h_{2}^{\pm} = {(\mp \partial_{y} U_{1}^{\pm} - \partial_{y y} u_{0} / 2) / 4|}_{y = - 1}, \\ m_{1}^{\pm} = - {\partial_{y} V_{1}^{\pm} / 4|}_{y = 1}, m_{2}^{\pm} = \mp {\partial_{y} V_{1}^{\pm} / 4|}_{y = - 1} . \end{gathered}

(A12)

The same procedure applies to the coefficients in the expressions for the remaining amplitude functions, and therefore they are not repeated here.

REFERENCES

1.

Z. G.

Mills

,

A.

Warey

, and

A.

Alexeev

, “

Heat transfer enhancement and thermal–hydraulic performance in laminar flows through asymmetric wavy walled channels

,”

Int. J. Heat Mass Transfer

97

,

450

–

460

(

2016

).

https://doi.org/10.1016/j.ijheatmasstransfer.2016.02.013

Google Scholar

Crossref

2.

A.

Mohammadi

and

J. M.

Floryan

, “

Numerical analysis of laminar-drag-reducing grooves

,”

J. Fluids Eng.

137

,

041201

(

2015

).

https://doi.org/10.1115/1.4028842

Google Scholar

Crossref

3.

J.

Hærvig

,

K.

Sørensen

, and

T. J.

Condra

, “

On the fully-developed heat transfer enhancing flow field in sinusoidally, spirally corrugated tubes using computational fluid dynamics

,”

Int. J. Heat Mass Transfer

106

,

1051

–

1062

(

2017

).

https://doi.org/10.1016/j.ijheatmasstransfer.2016.10.080

Google Scholar

Crossref

4.

W.

Wang

,

Y.

Zhang

,

K. S.

Lee

, and

B.

Li

, “

Optimal design of a double pipe heat exchanger based on the outward helically corrugated tube

,”

Int. J. Heat Mass Transfer

135

,

706

–

716

(

2019

).

https://doi.org/10.1016/j.ijheatmasstransfer.2019.01.115

Google Scholar

Crossref

5.

C. Y.

Wang

, “

Parallel flow between corrugated plates

,”

J. Eng. Mech. Div.

102

,

1088

–

1090

(

1976

).

https://doi.org/10.1061/jmcea3.0002181

Google Scholar

Crossref

6.

C. Y.

Wang

, “

On Stokes flow between corrugated plates

,”

J. Appl. Mech.

46

,

462

–

464

(

1979

).

https://doi.org/10.1115/1.3424575

Google Scholar

Crossref

7.

N.

Phan-Thien

, “

On the Stokes flow of viscous fluids through corrugated pipes

,”

J. Appl. Mech.

47

,

961

–

963

(

1980

).

https://doi.org/10.1115/1.3153825

Google Scholar

Crossref

8.

N.

Phan-Thien

, “

On Stokes flow of a Newtonian fluid through a pipe with stationary random roughness

,”

Phys. Fluids

47

,

579

–

582

(

1981

).

https://doi.org/10.1063/1.863423

.

Google Scholar

Crossref

9.

C.

Pozrikidis

, “

Stokes flow through a twisted tube

,”

J. Fluid Mech.

567

,

261

–

280

(

2006

).

https://doi.org/10.1017/s0022112006002242

Google Scholar

Crossref

10.

S. Y.

Song

,

X. H.

Yang

,

F. X.

Xin

, and

T. J.

Lu

, “

Modeling of surface roughness effects on Stokes flow in circular pipes

,”

Phys. Fluids

30

,

023604

(

2018

).

https://doi.org/10.1063/1.5017876

Google Scholar

Crossref

11.

C. Y.

Wang

, “

On Stokes slip flow through a transversely wavy channel

,”

Mech. Res. Commun.

38

,

249

–

254

(

2011

).

https://doi.org/10.1016/j.mechrescom.2011.02.006

Google Scholar

Crossref

12.

Z.

Duan

and

Y. S.

Muzychka

, “

Effects of corrugated roughness on developed laminar flow in microtubes

,”

J. Fluids Eng.

130

,

031102

(

2008

).

https://doi.org/10.1115/1.2842148

Google Scholar

Crossref

13.

H.

Rahmani

and

S.

Taghavi

, “

Poiseuille flow of a Bingham fluid in a channel with a superhydrophobic groovy wall

,”

J. Fluid Mech.

948

,

A34

(

2022

).

https://doi.org/10.1017/jfm.2022.700

Google Scholar

Crossref

14.

S. D.

Peterson

, “

Steady flow through a curved tube with wavy walls

,”

Phys. Fluids

22

,

023602

(

2010

).

https://doi.org/10.1063/1.3327239

Google Scholar

Crossref

15.

N. F.

Okechi

and

S.

Asghar

, “

Fluid motion in a corrugated curved channel

,”

Eur. Phys. J. Plus.

134

,

165

(

2019

).

https://doi.org/10.1140/epjp/i2019-12517-2

Google Scholar

Crossref

16.

N. F.

Okechi

and

S.

Asghar

, “

Oscillatory flow in a corrugated curved channel

,”

Eur. J. Mech.-B: Fluids

84

,

81

–

92

(

2020

).

https://doi.org/10.1016/j.euromechflu.2020.05.005

Google Scholar

Crossref

17.

N. F.

Okechi

and

S.

Asghar

, “

Two-phase flow in a groovy curved channel

,”

Eur. J. Mech.-B: Fluids

88

,

191

–

198

(

2021

).

https://doi.org/10.1016/j.euromechflu.2021.03.004

Google Scholar

Crossref

18.

C. Y.

Wang

, “

Stokes flow through a channel with three-dimensional bumpy walls

,”

Phys. Fluids

16

,

2136

–

2139

(

2004

).

https://doi.org/10.1063/1.1707023

Google Scholar

Crossref

19.

C. Y.

Wang

, “

Stokes flow through a tube with bumpy wall

,”

Phys. Fluids

18

,

078101

(

2006

).

https://doi.org/10.1063/1.2214883

Google Scholar

Crossref

20.

M. S.

Faltas

and

E. I.

Saad

, “

Three-dimensional Darcy–Brinkman flow in sinusoidal bumpy tubes

,”

Transp. Porous Media

118

,

435

–

448

(

2017

).

https://doi.org/10.1007/s11242-017-0865-5

Google Scholar

Crossref

21.

L. H.

Yu

and

C. Y.

Wang

, “

Darcy–Brinkman flow through a bumpy channel

,”

Transp. Porous Media

97

,

281

–

294

(

2013

).

https://doi.org/10.1007/s11242-013-0124-3

Google Scholar

Crossref

22.

M. S.

Faltas

and

K. E.

Ragab

, “

Darcy-Brinkman flow through a three-dimensional bumpy channel with stationary random model in the slip regime

,”

Colloid J.

84

,

485

–

497

(

2022

).

https://doi.org/10.1134/s1061933x22040044

Google Scholar

Crossref

23.

J. C.

Lei

,

C. C.

Chang

, and

C. Y.

Wang

, “

Electro-osmotic pumping through a bumpy microtube: Boundary perturbation and detection of roughness

,”

Phys. Fluids

31

,

012001

(

2019

).

https://doi.org/10.1063/1.5063869

Google Scholar

Crossref

24.

L.

Chang

,

Y.

Jian

,

M.

Buren

, and

Y.

Sun

, “

Electroosmotic flow through a microparallel channel with 3D wall roughness

,”

Electrophoresis

37

,

482

–

492

(

2016

).

https://doi.org/10.1002/elps.201500228

Google Scholar

Crossref

PubMed

25.

M. T.

Song

,

J. C.

Lei

,

C. C.

Chang

, and

C. Y.

Wang

, “

Stokes’s flow of a bumpy shaft inside a cylinder and a model for predicting the roughness of the shaft

,”

Phys. Fluids

32

,

032002

(

2020

).

https://doi.org/10.1063/1.5142050

Google Scholar

Crossref

This work is licensed under a Creative Commons Attribution 4.0 International License .

3D Stokes flow in a bumpy curved channel

I. INTRODUCTION

II. PROBLEM MODEL AND ANALYSIS

A. Zeroth-order problem

B. First-order problem

C. Second-order problem

D. Flow rate evaluation

III. RESULTS AND DISCUSSION

A. Velocity distribution

B. Flow rate

IV. CONCLUDING REMARKS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

NOMENCLATURE

Greek

NOMENCLATURE

Greek

APPENDIX: COEFFICIENTS

REFERENCES

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3D Stokes flow in a bumpy curved channel

I. INTRODUCTION

II. PROBLEM MODEL AND ANALYSIS

A. Zeroth-order problem

B. First-order problem

C. Second-order problem

D. Flow rate evaluation

III. RESULTS AND DISCUSSION

A. Velocity distribution

B. Flow rate

IV. CONCLUDING REMARKS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

NOMENCLATURE

Greek

NOMENCLATURE

Greek

APPENDIX: COEFFICIENTS

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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