2. Problem formulation

Consider a long slot with one smooth bounding side while the other is corrugated. The slot is inclined at an angle $\beta$ to the horizontal, as illustrated in figure 1. We align x- and y-axes parallel and perpendicular to the slot and non-dimensionalise all lengths using the averaged half-width of the slot $h$ . The geometry of the slot is then supposed to be given by

(2.1a,b) \begin{equation}y_{R}(x)=-1+\frac{1}{2}A\cos \alpha x \qquad \text{and} \qquad y_{L}(x)=1,\end{equation}

in which we use the subscripts R and L to denote the right and left plates, respectively. We note that the left surface is smooth while the right side is corrugated; the grooves are characterised by a peak-to-trough amplitude $A$ and a wavenumber $\alpha$ or, equivalently, a wavelength

(2.2) \begin{equation}\lambda =\frac{2\pi }{\alpha }.\end{equation}

The grooved plate is also heated with the applied temperature profile being periodic of wavelength $\lambda$ . The smooth plate is kept isothermal so that the temperature distributions on the two sides can be written

(2.3a,b) \begin{equation}\theta _{R}(x)=\frac{1}{2}{\textit{Ra}}_{p,R}\cos (\alpha x+\varOmega )\quad \text{and}\quad \theta _{L}(x)=0\end{equation}

where the dimensionless temperature $\theta$ is defined to be $\theta =T{-}T_{L}$ ; here, $T_{L}$ is the temperature of the left plate and is used as the reference level. The scale of temperature is based on $\kappa \nu /(g\varGamma h^{3})$ , where $g$ is the acceleration due to gravity, $\varGamma$ and $\nu$ are the thermal expansion coefficient and the kinematic viscosity, respectively, while $\kappa$ denotes the thermal diffusivity. The angle $\varOmega$ in (2.3a ) measures the phase difference between the corrugation geometry given by (2.1a ) and the heating pattern. The periodic Rayleigh number ${\textit{Ra}}_{p,R}$ governs the intensity of the heating applied to the right plate. We emphasise that the same wavenumber $\alpha$ is used for both the heating and groove patterns, which ensures that the two patterns are perfectly tuned.

The relevant conservation equations which govern the two-dimensional convection of Boussinesq fluid in the slot are

(2.4a,b) \begin{align} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}&=0,\qquad u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{\partial p}{\partial x}+{\nabla} ^{2}u+{\textit{Pr}}^{-1}\theta \sin \beta , \end{align}

(2.4c,d) \begin{align} u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}&=-\frac{\partial p}{\partial y}+{\nabla} ^{2}v+{\textit{Pr}}^{-1}\theta \cos \beta ,\qquad u\frac{\partial \theta }{\partial x}+v\frac{\partial \theta }{\partial y}={\textit{Pr}}^{-1}{\nabla} ^{2}\theta . \end{align}

Here, u(x,y) and $v$ (x,y) denote the velocity components in the x and y directions, respectively. These dimensionless quantities have been scaled on $U_{v}=v /h$ while $p$ is the pressure relative to the hydrostatic component based on $\rho U_{v }^{2}$ . The Prandtl number $Pr=v/\kappa$ is defined in the usual way; in the subsequent computations, it is assumed that $Pr = 0.71$ (the value appropriate for air) unless otherwise specified. The system of (2.4) is derived using the Boussinesq approximation, and the text by Tritton (Reference Tritton1977) discusses the circumstances when this is appropriate. Some experiments conducted under conditions similar to those employed in our analysis here suggest that the Boussinesq approximation can effectively model the fluid response (Floryan & Inasawa Reference Floryan and Inasawa2021; Inasawa et al. Reference Inasawa, Taneda and Floryan2019, Reference Inasawa, Hara and Floryan2021).

The boundary conditions comprise the no-slip, no-penetration conditions combined with the requirement that the fluid temperature matches the specified plate temperatures at the edges. Consequently, we require that

(2.5) \begin{align}&u(x,y_{R})=u(x,y_{L})=0,\qquad v(x,y_{R})=v(x,y_{L})=0,\qquad \theta (x,y_{R})=\theta _{R}(x),\nonumber\\ &\theta (x,y_{L})=0,\end{align}

where $y_{R}$ and $y_{L}$ were defined in (2.1), while the temperature profile $\theta _{R}(x)$ is given by (2.3a ). As no externally imposed mean pressure gradient is present, the problem formulation must include a constraint of the form

(2.6) \begin{equation}\left.\frac{\partial p}{\partial x}\right| _{\textit{mean}}=0.\end{equation}

To solve the system (2.4)–(2.6), the velocities ( $u$ , $v$ ) were written in terms of a streamfunction $\psi$ so that $u=\partial \psi /\partial y$ and $v=-\partial \psi /\partial x$ . The pressure was eliminated between (2.4b,c ), and then $\psi$ and $\theta$ were represented using Fourier expansions in the x-direction combined with Chebyshev expansions in the y-direction.

A significant challenge arises from the irregularity of the flow domain, leading to a need for an algorithm capable of handling a broad range of slot geometries characterised by the groove wavenumber and amplitude. We used a fixed computational region (CR) with the actual flow domain placed well within it; this means that there is an area inside the CR but outside the flow zone. The discretised flow equations are solved throughout the CR, including at those points exterior to flow. Using a streamfunction enables the solution procedure to be split into two distinct steps. In the first of these, the flow field is determined, while in the second stage, the pressure field is fixed by solving the relevant Poisson equation. The value of the streamfunction is zero at one wall, while the streamfunction at the other follows from the imposition of the pressure gradient constraint. The boundaries of the flow lie inside the CR, and the requisite conditions (2.5) are enforced as constraints using the $\tau $ -method (Canuto et al. Reference Canuto, Hussaini, Quarteroni and Zang1992). Details as to how these constraints are imposed may be found in Szumbarski & Floryan (Reference Szumbarski and Floryan1999), Husain & Floryan (Reference Husain and Floryan2008, Reference Husain and Floryan2010), or Husain, Szumbarski & Floryan (Reference Husain, Szumbarski and Floryan2009). This computational strategy eliminates the need for any labour-intensive and error-prone grid construction to re-create the precise groove geometry and bypasses the requirement for delicate grid convergence studies. Spectral accuracy is maintained for all discretisation elements, which controls the global accuracy by adjusting the number of Fourier modes and Chebyshev polynomials. All the results reported in this paper maintain an accuracy of at least four digits. Finally, we note that the groove geometry is coded within the algorithm using Fourier expansions, which has the advantage that its shape can be changed by simply adjusting the requisite Fourier coefficients. Panday & Floryan (Reference Panday and Floryan2020) have given a detailed description of this process.

The net streamwise flow rate $Q$ can be determined during the post-processing phase using the definition

(2.7) \begin{equation}Q=\int _{y_{R}(x)}^{y_{L}}u(x,y)\mathrm{d}y;\end{equation}

this flow rate is equivalent to defining the Reynolds number of the flow based on the mean streamwise velocity $U_{m}$ .

The flow mechanics can be assessed by analysing the surface forces that act on the fluid. The stress vector $\vec {\boldsymbol{\sigma }}_{R}$ at the corrugated surface has the form

(2.8) \begin{equation}\vec {\boldsymbol{\sigma }}_{R}= [\!\!\begin{array}{cc} \sigma _{x,R} & \sigma _{y,R } \end{array}\!\!]=[\!\!\begin{array}{cc} n_{x,R} & n_{y,R} \end{array}\!\!] \left[\!\!\begin{array}{cc} 2\dfrac{\partial u}{\partial x}-p & \dfrac{\partial u}{\partial y}+\dfrac{\partial v}{\partial x}\\[12pt] \dfrac{\partial u}{\partial y}+\dfrac{\partial v}{\partial x} & 2\dfrac{\partial v}{\partial y}-p \end{array}\!\!\right]_{y={y_{R}}};\end{equation}

here, the normal unit vector $\vec {n}_{R}$ that points outwards is given by

(2.9) \begin{equation}\vec {n}_{R}=[\!\!\begin{array}{cc} n_{x,R} & n_{y,R} \end{array}\!\!]=N_{R}\left(\frac{\partial y_{R}}{\partial x}, -1\right)\!,\qquad N_{R}=\left[1+\left(\frac{\partial y_{R}}{\partial x}\right)^{2}\right]^{-{1}/{2}}.\end{equation}

The x-component of the stress vector can be decomposed using the representation

(2.10) \begin{equation}\sigma _{x,R}=\sigma _{xv,R}+\sigma _{xp,R}=N_{R}\left[2\frac{\partial y_{R}}{\partial x}\left.\frac{\partial u}{\partial x}\right| _{{y_{R}}}-\left.\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right| _{{y_{R}}}\right]-N_{R}\left.\frac{\partial y_{R}}{\partial x}p\right| _{{y_{R}}}.\end{equation}

Here, $\sigma _{xv,R}$ and $\sigma _{xp,R}$ denote the viscous and pressure stresses, respectively, and the pressure has been normalised by bringing its mean value to zero. The x-component of the total force $F_{x,R}$ (per unit length and unit width of the slot) can similarly be written as the sum of pressure $F_{xp,R}$ and viscous $F_{xv,R}$ contributions so that

(2.11) \begin{align}F_{x,R}&=F_{xv,R}+F_{xp,R}\nonumber\\&=\lambda ^{-1}\int _{x_{0}}^{x_{0}+\lambda }\left[2 \frac{\partial y_{R}}{\partial x}\left.\frac{\partial u}{\partial x}\right| _{{y_{R}}}-\left.\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right| _{{y_{R}}}\right]dx-\lambda ^{-1}\int _{x_{0}}^{x_{0}+\lambda }\frac{\partial y_{R}}{\partial x}\left.p\right| _{{y_{R}}}\mathrm{d}x,\end{align}

where $x_{0}$ is a convenient reference point. Analogous considerations for the other boundary show that the force $F_{x,L}$ on this plate is

(2.12) \begin{equation}F_{x,L}=F_{xv,L}=\lambda ^{-1}\int _{x_{0}}^{x_{0}+\lambda }\left.\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\right| _{{y_{L}}}\mathrm{d}x\qquad F_{xp,L}=0.\end{equation}

Last, we note that the total (buoyancy) body force per unit length has $x$ - and $y$ -components $F_{xb}$ and $F_{yb}$ given by

(2.13a,b) \begin{equation}F_{xb}=\lambda ^{-1}{\textit{Pr}}^{-1}\sin \beta \int _{y_{R}}^{y_{L}}\!\!\int _{0}^{\lambda }\theta \, \mathrm{d}x\,\mathrm{d}y\qquad \text{and}\qquad F_{yb}=\lambda ^{-1}{\textit{Pr}}^{-1}\cos \beta \int _{y_{R}}^{y_{L}}\!\!\int _{0}^{\lambda }\theta \, \mathrm{d}x\,\mathrm{d}y.\end{equation}

We aim to understand the interaction between the distributed heating and plate geometry and its implications for the induced flow rate and the heat transfer within the inclined slot. To this end, we note several inherent symmetries that can be exploited to simplify our task. First, if we transform $\beta \rightarrow 2\pi -\beta$ and $\varOmega \rightarrow 2\pi -\varOmega$ then the problem specification is unchanged if we switch the sign of $u$ and translate the coordinate $x\rightarrow 2\pi /\alpha -x$ . Therefore, we can reduce the problem to one in which the inclination of the slot $\beta \in [0,\pi ].$ Moreover, if one applies the transformations $(x,y,u,v,p,\theta ,\beta ,\varOmega )\mapsto (2\pi /\alpha -x,-y,-u,-v,p,-\theta ,\pi -\beta ,\pi -\varOmega )$ the governing equations are invariant while the boundary conditions are exchanged. The upshot is that we can further reduce the problem and suppose that $\beta \in [0,\pi /2]$ .

3. Natural convection in a smooth inclined slot

We shall begin our investigation by examining the problem when the grooving is absent (A = 0) so that both sides of the slot are flat. Some representative flow, temperature and pressure fields are displayed in figure 2. When the slot is either horizontal or vertical, the periodic heating generates pairs of counter-rotating convection rolls with no net movement along the slot. Although the flow patterns in these two cases are similar, the temperature and pressure fields are not, implying that the mechanisms responsible for the flow fields are probably different. In the case of a horizontal slot, the buoyancy forces make fluid directly above hot spots move upward while fluid above cold spots moves down. This generates low- and high-pressure regions over the hot and cold spots, which induces movement along the slot. Roll structures form with the borders between neighbouring rolls coinciding with the hot and cold spots. By contrast, in a vertical slot, the upward-moving hot fluid impacts the downward-moving cold fluid; the consequence is that both streams deviate toward the opposite wall of the slot. This results in the formation of counter-rotating rolls with a boundary midway between the hot and the cold spots. The streamwise components of the two viscous forces acting on the plates and the streamwise buoyancy force all vanish whenever the slot is horizontal or vertical. On the other hand, while the cross-slot buoyancy force is zero for a vertical slot, it is not when the channel is horizontal, see figure 3(a).

Figure 2.

The flow and temperature fields (top row) and the flow and pressure fields (bottom row) for a slot with both sides flat $(A=0)$ . Five cases are considered with the inclination angle $\beta =0$ (horizontal slot), $\pi /8, \pi /4, 3\pi /8$ and $\pi /2$ (vertical slot). Calculations performed with heating intensity ${\textit{Ra}}_{p,R}=300$ and wavenumber $\alpha =1$ . In all the cases, the temperature field has been normalised so that its maximum value is unity.

Figure 3.

(a) The x- and $y$ -components of the buoyancy force ( $F_{xb}$ and $F_{yb}$ ) together with the shear forces acting on the fluid at the right ( $F_{xv,R}$ ) and left ( $F_{xv,L}$ ) plates as a function of the inclination angle $\beta .$ Calculations performed for no grooves ( $A=0$ ), a heating intensity ${\textit{Ra}}_{p,R}=300$ and wavenumber $\alpha =1.$ (b) The flow rate $Q_{S}$ as a function of the inclination angle $\beta$ for the three wavenumbers $\alpha =0.1, 1$ and $10$ . Other parameters are the same as those used for (a).

In figure 3(b), we show the net flow along the slot for an inclination angle between $\beta =0$ and $\beta =\pi /2$ and a selection of wavenumbers $\alpha$ . When the slot is either horizontal or vertical, the flow is zero; in the former case, this is because the mean buoyancy force is across the slot while, when $\beta =\pi /2,$ the mean buoyancy force is zero. In other cases, the flow fields consist of counter-rotating rolls with a stream tube that meanders between them and carries the fluid in the positive $x$ -direction. The width of the stream tube is a function of the inclination angle; it grows with increasing $\beta$ up to a maximum when $\beta \approx \pi /4,$ but the tube narrows again with any further increase in $\beta$ . For the particular heating intensity ${\textit{Ra}}_{p,R}$ used in figures 2 and 3, the maximum flow rate is achieved at $\beta \approx \pi /4$ . It is worth pointing out that the inclination angle corresponding to the greatest flow rate depends on the heating intensity; the optimum value of $\beta$ decreases as the degree of heating is increased. Floryan et al. (Reference Floryan, Baayoun, Panday and Bassom2022a ) gave a detailed discussion of this effect.

4. Thermal drift in a horizontal slot

We now introduce the pattern interaction effect and, to simplify matters, we initially focus on a horizontal slot. Some typical flow, temperature and pressure fields are displayed in figure 4. We consider the four phase angles $\varOmega =i\pi /2 (i=0,1,2,3$ ), where it is recalled that this phase describes the offset between the groove geometry and the heating pattern. When the hot spots are located either at the groove peaks ( $\varOmega =0$ ) or troughs ( $\varOmega =\pi$ ) the flow field comprises symmetric counter-rotating rolls with a zero net flow. Both the temperature and the pressure fields are symmetric with respect to the groove peaks and troughs. The symmetry of the pressure field means that no average pressure force is exerted on the fluid at the lower plate.

Figure 4.

The flow and temperature fields (top row) and the flow and pressure fields (bottom row) for four choices of the phase angle $\varOmega =0$ , $\pi /2, \pi $ and $3\pi /2$ . Calculations performed for a heating intensity ${\textit{Ra}}_{p,R}=300$ , $\text{groove amplitude} A=0.05$ and a wavenumber $\alpha =1$ . In all cases the temperature field has been normalised to a maximum value equal to unity.

For phase angles $\varOmega \neq 0,\pi$ a non-zero net flow is obtained. The flow fields now consist of a pair of convection rolls separated by a stream tube; the fluid is transported to the right if $0\leq \varOmega \leq \pi$ and to the left when $\pi \leq \varOmega \leq 2\pi$ . Net flow in the horizontal direction is caused by the asymmetry of the pressure field with respect to groove position (see the pressure fields when $\varOmega =\pi /2$ or $3\pi /2$ in figure 5 a). The flow rate $Q$ is antisymmetric with respect to $\varOmega =0$ and $\pi$ . In essence, one can deduce that thermal drift is created by placing the heating pattern to break the symmetry between the grooves and the heating patterns. The form of the induced asymmetry dictates the direction of the net fluid movement (Abtahi & Floryan Reference Abtahi and Floryan2017a , Reference Abtahi and Floryan2018).

Figure 5.

(a) The x-component of the pressure stress $\sigma _{xp,R}$ acting on the fluid at the lower plate y $=y_{R}(x)$ for the four phase angles $\varOmega =0, \pi /2, \pi$ and $3\pi /2$ . The dashed lines indicate the non-zero mean values of $\sigma _{xp,R}$ . (b) The variation of the flow rate $Q$ as a function of the phase difference $\varOmega$ . Calculations performed for a heating intensity ${\textit{Ra}}_{p,R}=300$ , $\text{groove amplitude} A=0.05$ and wavenumber $\alpha =1$ .

5. Thermal drift in an inclined slot

The introduction of grooves on one edge of an inclined slot activates the combined effects of the patterned convection and thermal drift discussed in §§ 3 and 4. Our first results are summarised in figure 6(a–c), which illustrate the influence of the phase difference $\varOmega$ and the inclination angle $\beta$ on the flow rate $Q$ . It seems, in general, that as the imposed wavenumber increases, the inclination angle at which the maximum net flow changes is obtained increases from $\beta \approx \pi /4$ to $\pi /2$ . Concomitantly, the optimal phase shift difference reduces from $\varOmega \approx \pi /2$ to $0$ .

Figure 6.

The flow rate $Q$ as a function of the phase shift $\varOmega$ and inclination angle $\beta$ for three chosen wavenumbers: (a) $\alpha =0.1,$ (b) $\alpha =1$ and (c) $\alpha =10$ . The red circles identify the parameter choices for achieving the maximum flow rate. In the bottom row, we show the quantity $Q_{comp}=(Q-Q_{S})/Q_{S}$ where $ Q_{S}$ denotes the flow rate in a slot with both sides perfectly flat. The form $Q_{comp}$ is plotted for the same three wavenumbers: (d) $\alpha =0.1,$ (e) $\alpha =1$ and (f) $\alpha =10$ . The other flow parameters are taken to be ${\textit{Ra}}_{p,R}=300$ , $Pr=0.71$ and $A=0.05$ .

An alternative perspective in understanding the importance of corrugation on the flow rate is illustrated in figure 6(d–f). Here, we have introduced the quantity $Q_{comp}=(Q-Q_{S})/Q_{S}$ , where $Q_{S}$ is defined to be the flow rate in a smooth slot, which was plotted in figure 3(b); thus $Q_{comp}$ measures the increase in flow rate that the inclusion of grooves can achieve. Clearly, if $Q_{comp}\gt 0$ , the addition of corrugation increases the flow rate. Of course, corrugations have the most dramatic effect when the slot is horizontal or vertical because there is no net flow at all without grooving.

In figure 7, we depict the flow and temperature field topologies and show how they change with slot orientation $\beta$ and phase difference $\varOmega$ for the wavenumber choice $\alpha =1$ . For a horizontal slot, thermal drift (when $ \varOmega =\pi /2$ or $3\pi /2$ ) is created by imposing a heating pattern that breaks the symmetry between the grooves (Abtahi & Floryan Reference Abtahi and Floryan2017a , Reference Abtahi and Floryan2018). When the slot is vertical a pair of counter-rotating convection rolls appears, and there is no net movement along the channel when the phase angle is either $\varOmega =\pi /2$ or $\varOmega =3\pi /2$ . At any other phase angle, we observe a stream tube that snakes between pairs of rolls. A non-zero net flow is obtained when the slot is neither horizontal nor vertical, irrespective of the phase shift. The structure of the rolls does not change markedly as the phase angle is adjusted; typically, the fluid flows downward toward cold spots and upward from hot spots. For a prescribed phase difference we find that, as the inclination angle increases, the stream tube becomes wider, at least until a maximum width of the stream tube at $\beta \approx \pi /4$ . Any further increase in $\beta$ sees a gradual narrowing of the stream tubes again. As the geometry of the stream tubes evolves with $\beta$ , the rolls adjacent to the unheated plate shrink as $\beta$ grows from $0$ to approximately $\pi /4$ before expanding again as $\beta$ increases towards $\pi /2$ . We point out that a stream tube with the net flow directed in the negative $x$ -direction is apparent for a horizontal slot with a phase $\varOmega =3\pi /2$ , and also for a vertical slot with $\varOmega =\pi$ . These observations are entirely consistent with the results shown in figure 6(b).

Figure 7.

Details of the flow and temperature fields for five inclination angles $\beta =0$ (top row), $\pi /8$ (2nd row), $\pi /4$ (3rd row), $3\pi /8$ (4th row) and $\pi /2$ (bottom row) and four values of the phase angle $\varOmega =0$ (left column), $\pi /2$ (2nd column), $\pi $ (3rd column) and $3\pi /2$ (right column). Other parameters are ${\textit{Ra}}_{p,R}=300$ , $Pr=0.71$ , $\alpha =1$ and $A=0.05$ .

The influence of the projection of the surface pressure $\sigma _{xp,R}$ on the heated plate is illustrated in figure 8. The values of the x-component of the buoyancy force $F_{xb}$ are listed in the caption of this figure; we remark that for a smooth-walled slot, this buoyancy force $F_{xb}=$ 4.78 when $\beta =\pi /8$ ; 6.48 if $\beta =\pi /4$ and 4.86 if $\beta =3\pi /8$ . For hot spots that coincide with the groove peaks ( $\varOmega =0$ ), the incorporation of the grooves leads to a slight increase in $F_{xb}$ compared with its value for a smooth slot, and when the slot is inclined, we obtain a positive value for $F_{xp,R}$ (figure 8 a). We, therefore, find an increase in the flow rate compared with the smooth inclined slot. When hot spots are located to the left of peaks ( $\varOmega =\pi /2$ ), the presence of the grooves tends to reduce $F_{xb}$ . However, this reduction is more than adequately compensated by the positive value of $F_{xp,R}$ for a grooved slot, which increases the flow rate compared with a smooth slot (figure 8 b). We note that horizontal and vertical slots have zero values for $F_{xb}$ and $F_{xp,R}$ , which leads to a pair of rolls with no net movement along the slot (see figure 7). In contrast, when the hot spots either overlap with groove troughs ( $\varOmega =\pi$ ) or are located to the right of peaks ( $\varOmega =3\pi /2$ ), the introduction of corrugation and inclination for the slot has the opposite effects compared with the cases $\varOmega =0$ or $\pi /2$ (see figure 8 c,d); we note that $F_{xp,R}$ is negative, and a decrease in $F_{xb}$ when $\varOmega =\pi$ , and a negative $F_{xp,R}$ and a slight increase in $F_{xb}$ when $\varOmega =3\pi /2$ . Therefore, we see a decrease in the flow rate compared with the smooth slot shown in figure 6(e). We note in passing that the size of the buoyancy force $F_{xb}$ is typically an order of magnitude larger than that of the pressure force $F_{xp,R}$ .

Figure 8.

The streamwise component of the pressure force $\sigma _{xp,R}$ that acts on the fluid at the heated plate ( $\sigma _{xp,R}$ ) for four phase angles: (a) $\varOmega =0,$ (b) $\pi /2,$ (c) $\pi$ and (d) $3\pi /2$ . The mean values of $\sigma _{xp,R}$ are indicated by the dashed lines. The sizes of the associated buoyancy force $F_{xb}$ corresponding to $\beta =j\pi /8(j=0-4)$ are as follows: $\varOmega =0\colon F_{xb}=(0, 4.82, 6.56, 4.95, 0.17) ; \varOmega =\pi /2\colon F_{xb}=(0, 4.58, 6.16, 4.59, 0) ; \varOmega =\pi \colon F_{xb}= (0, 4.75, 6.43, 4.79, -0.17)$ and $\varOmega =3\pi /2\colon F_{xb}=(0, 4.96, 6.79, 5.11, 0)$ . The other parameters are ${\textit{Ra}}_{p,R}=300$ , $\alpha =1$ and $A=0.05$ .

Figure 9 explores the influence of the wavenumber $\alpha$ and inclination angle $\beta$ on the flow rate $Q$ for four values of the phase difference $\varOmega$ . We see that, at least over the range of wavenumbers considered, the net flow for a slot is in the positive $x$ -direction when $\varOmega =0$ or $\pi /2$ , while it is directed in the opposite sense when $\varOmega =\pi$ and $\alpha$ is large or when $\varOmega =3\pi /2$ and in the long-wavelength limit. Of particular interest is, for a given phase $\varOmega$ , the question: What is the slot inclination that leads to the greatest possible flow rate? This optimal angle varies with wavenumber; when $\varOmega =0$ it increases from $\beta \approx \pi /4$ to $\pi /2$ as $\alpha$ grows (figure 9 a), shifts from $\beta \approx 0$ to $\pi /4$ when $\varOmega =\pi /2$ (figure 9 b), decreases from $\beta \approx \pi /4$ to $0$ when $\varOmega =\pi$ (figure 9 c) and changes from $\beta \approx \pi /2$ to $\pi /4$ when $\varOmega =3\pi /2$ (figure 9 d). We re-emphasise that the net flow direction can switch in some circumstances.

Figure 9.

The flow rate $Q$ as a function of the wavenumber $\alpha$ and inclination angle $\beta$ for four phase angles: (a) $\varOmega =0$ , (b) $\varOmega =\pi /2,$ (c) $\varOmega =\pi$ and $(d) \varOmega =3\pi /2$ . The red curves indicate the optimal inclination $\beta (\alpha )$ corresponding to the maximum flow. Calculations performed with ${\textit{Ra}}_{p,R}=300$ and $A=0.05$ .

Additional insight as to how the flow rate $Q$ varies with wavenumber $\alpha$ can be derived by a study of suitable cuts taken through the contour plots shown in figure 9. Such information is presented in figure 10. Here, is shown how changes in the wavenumber $\alpha$ affect $Q$ , the mean pressure force $F_{xp,R}$ acting on the fluid at the right plate and the $x$ -component of the buoyancy force $F_{xb}$ at different inclination angles $\beta$ and phase angles $\varOmega$ .

Figure 10.

(a)–(d) The flow rate $Q$ as a function of the wavenumber $\alpha$ for the four phase angles, i.e. $\varOmega =0$ , $\pi /2$ , $\pi$ and $3\pi /2$ . In plots (e)–(h) are the forms of the pressure force $F_{xp,R}$ for the same phases while (i)–(l) illustrate the x-component of the buoyancy force $F_{xb}$ . In all cases, the dashed lines identify negative values. Calculations performed with ${\textit{Ra}}_{p,R}=300$ and $A=0.05$ .

Figures 10(a) and 10(c) suggest that an inclined slot with $\beta =\pi /4$ induces the largest net flow when $\varOmega =0$ or $\pi$ for relatively long-wavelength heating with $\alpha \lt 2$ . On the other hand, for a vertical slot, the greatest flow rate occurs as $\alpha \rightarrow 10$ (also see figure 6 c). A plausible explanation for the transition from an inclination angle $\beta =\pi /4$ to $\pi /2$ corresponding to the greatest flow rate can be ascribed to the form of the buoyancy force $F_{xb}$ . It is noted that, as $\alpha \rightarrow 10$ , the flow in a vertical slot possesses a stronger buoyancy force $F_{xb}$ than does a more gently inclined channel. The significance of the pressure $F_{xp,R}$ for the propulsion also diminishes; this hinders the net flow by acting as a surface roughness (see figures 10 e, g, i and k). Furthermore, figures 10(a) and 10(c) suggest that the magnitude of the flow rate rises in proportion to $\alpha ^{2}$ for small wavenumbers but seems to vary linearly with $\alpha$ when $\alpha$ is larger. However, this rise in $Q$ does not continue indefinitely but instead eventually plateaus with a finite upper bound approached as $\alpha \rightarrow \infty$ . This short-wavelength limit corresponds physically to neighbouring hot and cold spots approaching each other, and then it might be expected that the flow would move in a way similar to that seen should the heated wall be an isothermal smooth plate located at $y=-1+A/2$ held at a temperature ${\textit{Ra}}_{uni}={\textit{Ra}}_{p,R}/2 (=150$ for the values used in figure 10). Similarly, the change in the direction of the flow rate for a vertical slot but with phase difference $\varOmega =\pi$ as $\alpha \rightarrow \infty$ is because the right plate is now acting as a smooth cold plate. When $\alpha \rightarrow \infty$ the temperature, velocity fields and the flow rate approach

(5.1a–c) \begin{align}\theta &=\frac{{\textit{Ra}}_{p,R}}{4-A}(1-y),\quad u=\frac{{\textit{Ra}}_{p,R}\sin \beta }{24(A-4)Pr}(y-1)(A-6+2y)(A-2-2y),\nonumber\\&\qquad\qquad\qquad\qquad Q=-\frac{1}{384 \textit{Pr}}(A-4)^{3}{\textit{Ra}}_{p,R}\sin \beta .\end{align}

Figure 11.

(a) The flow field when $\varOmega =0.$ (b) Distribution of the x-velocity component u (black lines, left axis) and of the temperature $\theta$ (red lines, right axis) as functions of y. (c) As in (a) but with $\varOmega =\pi /2.$ (d) As in (b) with $\varOmega =\pi /2$ . Other parameters are ${\textit{Ra}}_{p,R}=300$ , $Pr=0.71$ , $\alpha =10$ , $A=0.05$ and $\beta =\pi /4$ and the temperature has been normalised with its maximum value.

When the phase difference is changed to either $\varOmega =\pi /2$ or $3\pi /2$ , the net flow rate in an inclined slot with $\beta =\pi /4$ is larger than the net flow rate seen at other angles, except in the small wavenumber limit, see figure 10(b,d). The flow rate grows roughly linearly with $\alpha$ for small wavenumbers but possesses a maximum at $\alpha \approx 1.5$ . Unless the slot is horizontal or vertical, the dependence of $Q$ on $\alpha$ is closely related to $F_{xb}$ (see figures 10 j and 10 l). The buoyancy force $F_{xb}$ is zero for a horizontal slot, and in this case, the maximum of the flow rate is related to the size of $F_{xp,R}$ (figures 10 f and 10 h). As $\alpha \rightarrow 10$ , although the magnitude of $F_{xp,R}$ remains large, the zones with pressure gradient (and temperature gradient) are localised between the peaks of the grooves, thereby ameliorating the relevance of $F_{xp,R}$ for the flow rate (figures 10 b and 10 d). As indicated in figure 6(a), when $\alpha =0.1$ the maximum flow rate occurs when $\beta \approx \pi /4$ and $\varOmega =\pi /2$ . The reason for that can be inferred from the results shown in figures 10(e, f,i, and j); both $F_{xp,R}$ and $F_{xb}$ tend to propel the fluid forward in the positive $x$ -direction when $\varOmega =\pi /2$ (figure 10 f, j), but only $F_{xp,R}$ does this if $\varOmega =0$ (figure 10 e,i).

Further details of the flow topology in the short-wavelength limit when $\beta =\pi /4$ together with $\varOmega =0$ or $\pi /2$ are illustrated in figure 11. The temperature fields in figures 11(a) and 11(c) demonstrate that a thermal boundary layer forms adjacent to the heated plate while the fluid outside this boundary layer becomes isothermal. These properties are underlined by the temperature distributions across the slot displayed in figure 11(b,d). The rolls evident in figure 7 have vanished for these parameter choices and the flow is uni-directional. However, it should be appreciated that the mechanisms responsible for this behaviour are different depending on whether $\varOmega =0$ or $\pi /2$ . In the former case, the important features of the flow field can be understood in terms of a strengthening of the stream tube owing to the absence of a velocity boundary layer (also see figure 11 b). The flow field is reminiscent of that in a slot with uniform heating at the right plate as the hot peaks of the right plate approach each other. In contrast, when $\varOmega =\pi /2,$ the points where $\theta =0$ overlap with the peaks of the grooved wall, so the right plate becomes akin to a plate without heating. This weakens the flow along the conduit. In this case, the temperature variations are confined to the trough zones, forming a velocity boundary layer near the right plate (see figure 11 d).

The influence of the corrugation amplitude $A$ on the flow rate $Q$ is shown in figure 12. This set of results reinforces some of the conclusions that were drawn after the analysis of data displayed in figure 6(e); the inclusion of a grooved wall at the right plate enhances the flow rate compared with a smooth slot only for a suitably chosen combinations of the inclination angle $\beta$ and the phase $\varOmega$ . As would be anticipated, as $A$ decreases, the results for a smooth slot are recovered and approaches a constant value for $\beta \neq 0$ and $\beta \neq \pi /2$ , while becoming zero linearly (in A) for $\beta =0$ and $\varOmega =\pi /2, 3\pi /2$ (figure 12 b,d) or for $\beta =\pi /2$ and $\varOmega =0,\pi$ (figure 12 a,c) (also see figure 3). We note, from figure 6, that there is no net flow for a horizontal slot ( $\beta =0$ ) when $\varOmega =0$ and $\pi$ , and for a vertical slot ( $\beta =\pi /2$ ) when $\varOmega =\pi /2$ and $3\pi /2$ . Moreover, if $\varOmega =0$ or $\pi$ , any change in the groove amplitude has the greatest impact on the flow rate when $\beta =3\pi /8$ ; it experiences a faster increase ( $\varOmega =0$ ) or decrease ( $\varOmega =\pi$ ) compared with a slot with $\beta =\pi /8$ or $ \pi /4$ (figure 12 a,b). In contrast, as the amplitude of the grooves increases, it is the slot with $\beta =\pi /8$ that suffers the most rapid increase in the flow rate for $\varOmega =\pi /2$ and decreases for $3\pi /2$ when compared with slots with $\beta =\pi /4$ or $3\pi /8$ , respectively (figure 12 b,d).

Figure 12.

Variations of flow rate $Q$ as a function of corrugation amplitude $A$ for selected slot inclinations $\beta .$ The plots show result for four values of the phase $\varOmega \colon$ (a) 0, (b) $\pi /2,$ (c) $\pi$ and (d) $3\pi /2$ . Other parameters are ${\textit{Ra}}_{p,R}=300$ , $Pr=0.71$ , $\alpha =1$ . Dashed lines identify negative values.

We further assess the effects of the heating intensity ${\textit{Ra}}_{p,R}$ on the flow rate $Q$ in figure 13. The quantity $Q$ grows linearly with ${\textit{Ra}}_{p,R}$ for the relatively small sizes of ${\textit{Ra}}_{p,R}$ ; a result that holds irrespective of the inclination of the slot. We see that the growth in $Q$ is somewhat arrested at a larger ${\textit{Ra}}_{p,R}$ and eventually saturates; the strength of the applied heating leading to saturation is a function of the precise flow configuration. Saturation occurs at the lowest ${\textit{Ra}}_{p,R}$ in a vertical slot and at the greatest ${\textit{Ra}}_{p,R}$ in the horizontal case. For example, the flow rate in a slot with $\beta =\pi /8$ becomes larger than that of a slot with $\beta =\pi /4$ when the heating intensity increases beyond approximately ${\textit{Ra}}_{p,R}=1000$ . It is either the horizontal or vertical slots that perform best when ${\textit{Ra}}_{p,R}$ is small. Moreover, we note that for weak heating in a slot with phase difference $\varOmega =\pi$ or $3\pi /2$ , the net flow actually reverses and is in the negative $x$ -direction. We have limited our calculations to relatively modest heating levels; although there is the temptation to consider greater values, these are probably of no practical relevance. If ${\textit{Ra}}_{p,R}$ is above critical, the flow becomes prone to the possible formation of secondary states (Hossain & Floryan Reference Hossain and Floryan2013), thereby invaliding any deductions predicated on a purely two-dimensional model.

Figure 13.

The flow rate $Q$ as a function of the heating intensity ${\textit{Ra}}_{p,R}$ for selected inclination angles and four choices of $\varOmega$ . (a) $\varOmega =0,$ (b) $\pi /2,$ (c) $\pi$ and (d)  $3\pi /2$ . Other parameters are $Pr=0.71$ , $\alpha =1,$ and $A=0.05$ . Dashed lines identify negative values.

Finally, figure 14 addresses the issue of how the flow rate varies with the Prandtl number. These results demonstrate that the flow rate diminishes with a growing Pr and eventually is almost proportional to ${\textit{Pr}}^{-1}$ when it is large. For a narrow range of relatively small $Pr$ , an inclined slot at a small inclination angle undergoes the largest decrease in the flow rate. The decrease in $Q$ with $Pr$ is to be expected since an enhanced $Pr$ tends to strengthen the convection, which smooths out horizontal temperature variations, which in turn weakens the gradients in the buoyancy force. We note that the slot with an inclination angle $\beta =\pi /4$ performs best in achieving the net flow rate compared with other configurations, except for a narrow range of relatively small $Pr$ .

Figure 14.

The flow rate $Q$ as a function of the Prandtl number $Pr$ for (a) $\varOmega =0$ , $(b) \pi /2,$ (c) $\pi$ and (d)  $3\pi /2$ . Other parameters are ${\textit{Ra}}_{p,R}=300$ , $\alpha =1$ and $A=0.05$ . Dashed lines identify negative values.

6. The long-wavelength limit $\boldsymbol{\alpha }\rightarrow \mathbf{0}$

It is possible to gain further insight into flow characteristics by examining certain appropriate limits that allow asymptotic progress. First, we examine long-wavelength modes with $\alpha \rightarrow 0$ . To facilitate progress, it is helpful to first transform the problem from the original $(x,y)$ -coordinate system to a more convenient set that enables a straightforward description of the edges of the slot. We therefore write

(6.1) \begin{equation}X=\alpha x,\qquad \eta =1+\frac{4(y-1)}{H(X)},\quad \text{where}\quad H(X)\equiv 4-A\cos X,\end{equation}

and then, in terms of these new coordinates, the edges of the slot are given by $\eta =\pm 1$ . The various differential operators expressed in terms of these new coordinates are given by

(6.2a,b) \begin{equation}\frac{\partial }{\partial x}\rightarrow \alpha \left(\frac{\partial }{\partial X}-\frac{(\eta -1)H'}{H}\frac{\partial }{\partial \eta }\right)\!,\frac{\partial }{\partial y}\rightarrow \frac{4}{H}\frac{\partial }{\partial \eta },\end{equation}

where a dash denotes differentiation with respect to $X$ . Furthermore

(6.2c) \begin{align}{\nabla} ^{2}&\equiv \alpha ^{2}\left[\frac{\partial ^{2}}{\partial X^{2}}-\frac{2(\eta -1)H'}{H}\frac{\partial ^{2}}{\partial X\partial \eta }+\left(\frac{(\eta -1)H'}{H}\right)^{2}\frac{\partial ^{2}}{\partial \eta ^{2}}\right .\nonumber\\&\quad \left .+\frac{(\eta -1)}{H^{2}}[2(H')^{2}-HH'']\frac{\partial }{\partial \eta }\right]+\frac{16}{H^{2}}\frac{\partial ^{2}}{\partial \eta ^{2}}.\end{align}

These transformations must be applied to the system (2.4), but we have suppressed the details in the interest of brevity. What we do note is that in these new coordinates, the boundary conditions become $u=v=0$ on $\eta =\pm 1$ together with $\theta =0$ on $\eta =1$ and $\theta =({1}/{2}){\textit{Ra}}_{p,R}\cos (X+\varOmega )$ on $\eta =-1$ .

When $\alpha \rightarrow 0,$ we seek solutions with the structure

(6.3) \begin{align}(u,v,p,\theta )&=\alpha ^{-1}(0,0,P_{-1},0)+(U_{0},0,P_{0},\theta _{0})+\alpha (U_{1},V_{0},P_{1},\theta _{1})\nonumber\\&\quad +\alpha ^{2}(U_{2},V_{1},P_{2},\theta _{2})+\cdots,\end{align}

in which all the unknowns are functions of $X$ and $\eta$ . The cross-stream momentum (2.4c ) shows that the leading-order pressure term $P_{-1}=P_{-1}(X)$ and the energy equation gives that

(6.4a) \begin{equation}\theta _{0}(X,\eta )=\frac{1}{4}{\textit{Ra}}_{p,R}(1-\eta )\cos (X+\varOmega ).\end{equation}

The continuity and streamwise momentum equations become

(6.4b) \begin{equation} 0=-P_{-1}^{\prime}+\frac{16}{H^{2}}\frac{\partial ^{2}U_{0}}{\partial \eta ^{2}}+{\textit{Pr}}^{-1}\theta _{0}\sin \beta,\qquad H\frac{\partial U_{0}}{\partial X}-A(\eta -1)\frac{\partial U_{0}}{\partial \eta }+ 4\frac{\partial V_{0}}{\partial \eta }=0 ,\end{equation}

where a dash denotes differentiation with respect to $X$ . These equations need to be solved subject to the fact that there should be no mean pressure gradient along the slot. Accordingly, we find that

(6.4c) \begin{equation}P_{-1}(X)=\frac{{\textit{Ra}}_{p,R}}{4Pr}\sin \beta \cos (X+\varOmega ),\end{equation}

and it follows that

(6.4d,e) \begin{align}U_{0}&=\frac{1}{96}H^{2}P_{-1}^{\prime}\eta (\eta ^{2}-1), \nonumber\\V_{0}&=\frac{1}{1536}H^{2}\big[(\eta -3)(\eta -1){H'P}_{-1}^{\prime}+(\eta ^{2}-1)HP_{-1}\big] (\eta ^{2}-1).\end{align}

We remark that $ U_{0}$ is an odd-valued function of $\eta$ so that the leading-order flux along the slot vanishes at $O(1)$ . This is in accord with the numerical results in figure 10(a–d), which shows that in cases $Q\rightarrow 0$ as $\alpha \rightarrow 0$ .

We need to proceed to at least one more order to deduce the flux through the slot. The energy equation at $O(\alpha )$ gives

(6.5) \begin{align}\theta _{1}(X,\eta )&=\frac{{\textit{Pr}}^{2}H^{3}P_{-1}^{\prime}}{184320 \sin \beta }(\eta ^{2}-1)[(3\eta ^{4}-6\eta ^{3}-2\eta ^{2}+14\eta -17)\textit{HP}_{-1}\nonumber\\&\quad+3(\eta ^{4}-4\eta ^{2}+11){H'P}_{-1}^{\prime}].\end{align}

The cross-slot equation may be integrated to conclude that the pressure term

(6.6) \begin{equation}P_{0}(X,\eta )=\frac{1}{8}\eta (2-\eta ) \textit{HP}_{-1}^{\prime}+\hat{P}_{00}(X),\end{equation}

for some function $\hat{P}_{00}(X)$ . The streamwise equation for $U_{1}$ may be cast as

(6.7a) \begin{align}U_{0}\left[\frac{\partial U_{0}}{\partial X}-\frac{H'(\eta -1)}{H}\frac{\partial U_{0}}{\partial \eta }\right]+\frac{4V_{0}}{H}\frac{\partial U_{0}}{\partial \eta }&=-\left[\frac{\partial P_{0}}{\partial X}-\frac{H'(\eta -1)}{H}\frac{\partial P_{0}}{\partial \eta }\right]\nonumber\\&\quad +\frac{16}{H^{2}}\frac{\partial ^{2}U_{1}}{\partial \eta ^{2}}+\frac{\theta _{1}}{\textit{Pr}}\sin \beta\end{align}

and then $V_{1}$ follows from the continuity equation. Fortunately, there is no need to find $V_{1}$ explicitly; instead, if we integrate the continuity balance across the slot and demand that $V_{1}(X,\pm 1)=0$ , it follows that

(6.7b) \begin{equation}H\left(\int _{-1}^{1}U_{1}(X,\eta ) {\rm d}\eta \right)=\text{constant}.\end{equation}

If we solve (6.7a ) with $P_{0}$ given by (6.6) and such that (6.7b ) holds, it can be shown that

(6.8) \begin{align}U_{1}(X,\eta )&=\frac{H^{5}P_{-1}^{\prime}(1-\eta ^{2})}{495452160}\big\{F_{1}(\eta ) {H'P}_{-1}^{\prime}+F_{2}(\eta ) \textit{HP}_{-1}+\textit{Pr}\big[F_{3}(\eta ) {H'P}_{-1}^{\prime}\nonumber\\&\quad +F_{4}(\eta ) \textit{HP}_{-1}\big]\big\}+\frac{H^{2}\cot \beta }{7680}F_{5}(\eta )\big[{H'P}_{-1}^{\prime}+\textit{HP}_{-1}\big]+\frac{K}{32H}(1-\eta ^{2});\end{align}

here, $K$ is a constant to be determined in due course, while the polynomials $F_{1}(\eta )-F_{5}(\eta )$ are given by

(6.9a) \begin{align}F_{1}(\eta )&\equiv 15\eta ^{6}-125\eta ^{4}+365\eta ^{2}-63, \end{align}

(6.9b) \begin{align}F_{2}(\eta )&\equiv 15\eta ^{6}-13\eta ^{4}-83\eta ^{2}+17, \end{align}

(6.9c) \begin{align}F_{3}(\eta )&\equiv 3(3\eta ^{6}-25\eta ^{4}-185\eta ^{2}-35), \end{align}

(6.9d) \begin{align}F_{4}(\eta )&\equiv 9\eta ^{6}-24\eta ^{5}-19\eta ^{4}+144\eta ^{3}-229\eta ^{2}-248\eta +47 \end{align}

and

(6.9e) \begin{equation}F_{5}(\eta )=(\eta ^{2}-1)(5\eta ^{2}-20\eta -1).\end{equation}

With this form of $U_{1}$ it then turns out that

(6.10) \begin{align}\frac{\mathrm{d}\hat{P}_{00}}{\mathrm{d}X}&=\frac{H^{3}P_{-1}^{\prime}}{241920}\big[(5+18 \textit{Pr} )\textit{HP}_{-1}-(13+33\textit{Pr}){H'P}_{-1}^{\prime}\big]\nonumber\\&\quad -\frac{1}{40}\cot \beta \big(11{H'P}_{-1}^{\prime}+\textit{HP}_{-1}\big)-\frac{K}{H^{3}}.\end{align}

It is now that we can isolate the constant $K$ . To ensure that there is a zero mean pressure along the slot, the integral of the right-hand side of (6.10) over a wavelength must vanish. The requisite calculations can be performed analytically and leads to the conclusion that

(6.11) \begin{equation}K=\frac{(16-A^{2})^{5/2}\!A {\textit{Ra}}_{p,R}\sin \varOmega }{40(32+A^{2}){\textit{Pr}}^{2}}\left[\!3\textit{Pr} \cos \beta +(\textit{Pr}-1)\frac{A(A^{2}+96){\textit{Ra}}_{p,R}}{129024}\cos \varOmega \sin ^{2} \beta \!\right]\!.\end{equation}

With the streamwise velocity defined by $u=U_{0}(X,\eta )+\alpha U_{1}(X,\eta )+\cdots$ , where $U_{0}$ and $U_{1}$ are given by (6.4c ) and (6.8), it is easily shown that the flux $Q$ is evaluated as

(6.12) \begin{equation}Q=\frac{\alpha }{4}H\int _{-1}^{1}U_{1}(X,\eta ) \mathrm{d}\eta +\cdots =\left(\frac{K}{96}\right)\alpha +\cdots ,\end{equation}

where $K$ is as defined in (6.11).

To assess the accuracy of our prediction, in figure 15, we show some sample comparisons between the numerical and theoretical results. These calculations, which relate to the case when ${\textit{Ra}}_{p,R}=300$ , $\varOmega =\pi /8$ and $A=0.05$ , show excellent agreement between the two sets of results and that the error of the analytical prediction appears to be $O(\alpha ^{2})$ as would be expected. (The one exception appears to be the instance when $\beta =0$ ; in this, the error is much smaller at $O(\alpha ^{3})$ .) The only example that seems somewhat anomalous relates to the vertical channel with $\beta =\pi /2$ . In this instance, the flux seems to fall roughly proportional to $\alpha ^{2}$ until $\alpha \approx 3\times 10^{-3}$ ; for smaller wavenumbers, the flux falls more gently. This behaviour can be explained by reference to the definition (6.11) for K. This expression for K consists of two parts, and for general values of the various parameters, the second is substantially smaller than the first, which, therefore, dominates. The only instance when this argument fails is when $\beta =\pi /2$ ; then the first term vanishes, and K is tiny. The expectation is that the $O(\alpha ^{2})$ contribution to $Q$ is much larger than the $O(\alpha )$ part unless ${\unicode[Arial]{x03B1}}$ is really small. This then explains the rather strange appearance of the $Q(\alpha )$ curve for a vertical channel.

Figure 15.

A comparison of the numerically and analytically determined flow rate $Q$ for long-wavelength grooves and heating when ${\textit{Ra}}_{p,R}=300$ , $\varOmega =\pi /8$ and $A=0.05$ . Three inclinations are considered: $\beta =0$ (red curves), $\pi /4$ (green) and $\pi /2$ (blue). The dash-dotted lines identify the analytical solution $Q_{a}$ given by (6.12), the solid lines identify numerical solutions $Q_{n}$ of the complete equations and dotted lines identify the differences ${\unicode[Arial]{x0394}} Q =| Q_{n}-Q_{a}|$ . The blue dashed line identifies the negative values of $Q_{n}$ for $\beta =\pi /2$ .

The flux prediction (6.12) also helps explain some other features of the results displayed in figure 10(a–d). In figure 10(b,d), we see that for small $\alpha ,$ the flux is proportional to $O(\alpha ),$ but in figure 10(a,c) the typical flux values are much smaller. In these instances, $\varOmega =0$ or $\pi ,$ and it is clear from (6.11) that the constant K then vanishes. In seems that in these circumstances $Q\sim \alpha ^{2}$ ; our long-wavelength analysis cannot confirm this, but the weak convection approximation discussed in § 7 can assist in resolving this.

We can use the above asymptotic results to deduce expressions for the pressure $F_{xp,R}$ and the buoyancy $F_{xb}$ forces in the long-wavelength limit. Routine manipulations lead to the results that

(6.13) \begin{equation}F_{xp,R}=- F_{xb}=\frac{A {\textit{Ra}}_{p,R}}{16 \textit{Pr}}\sin \beta \cos \varOmega +{O}({\unicode[Arial]{x03B1}} ),\end{equation}

but, if $\cos \varOmega =0,$ then

(61.4) \begin{equation}F_{xp,R}=\alpha \frac{A\sin \varOmega {R}a_{p,R}\cos \beta }{80 Pr}\frac{(224-11A^{2})}{32+A^{2}}+O(\alpha ^{2}), F_{xb}= O(\alpha ^{2}).\end{equation}

All these results are consistent with the calculations summarised in figure 10.

7. The weak convection limit

The other limit amenable to asymptotic study relates to the situation when the convection is relatively weak and the amplitude A of the grooving is small. To investigate this problem, we adopt a hybrid approach in which

(7.1) \begin{align}(u,v,p,\theta )&={\textit{Ra}}_{p,R} \big[\big(\hat{U}_{0},\hat{V}_{0},\hat{P}_{0},\hat{\Theta }_{0}\big)+A\big(\hat{U}_{1},\hat{V}_{1},\hat{P}_{1},\hat{\Theta }_{1}\big)+\cdots .\nonumber\\&\quad +{\textit{Ra}}_{p,R} \big(\hat{U}_{2},\hat{V}_{2},\hat{P}_{2},\hat{\Theta }_{2}\big)+\cdots \big]+\cdots ,\end{align}

where all the unknowns are functions of $X$ and $\eta$ as defined in (6.1). Before we proceed, it is important to explain our notation. We take the flow variables with a subscript 0 to denote the weak convection solution appropriate to a flat wall without grooves. Then, the unknowns with subscript 1 capture the variations induced by the inclusion of small amplitude corrugations, while those with subscript 2 describe the leading-order corrections arising from the heating being small but non-zero.

7.1. The basic solution

If we solve the energy equation, we deduce that

(7.2) \begin{equation}\hat{\Theta }_{0}(X,\eta )=\frac{\sinh \alpha (1-\eta )}{2\sinh 2\alpha }\cos (X+\varOmega ).\end{equation}

The leading-order flow variables are best written as

(7.3) \begin{align}\hat{U}_{0}&=U_{0C}\cos (X+\varOmega )+U_{0S}\sin (X+\varOmega ), \nonumber\\\big\{\hat{V}_{0},\hat{P}_{0}\big\}&=\{V_{0C},P_{0C}\}\sin (X+\varOmega )+\{V_{0S},P_{0S}\}\cos (X+\varOmega ).\end{align}

It is convenient to write these variables in terms of two further functions: one is the even-valued solution of

(7.4a) \begin{align}&V''''-2\alpha ^{2}V''+\alpha ^{4}V=\cosh \alpha \eta , V(1)=V'(1)=0\Rightarrow V_{even}=\frac{1}{8\alpha ^{2}}\eta ^{2}\cosh \alpha \eta \nonumber\\&\quad +\frac{[(\sinh 2\alpha -2\alpha )\cosh \alpha \eta -4{\eta }\cosh ^{2} \alpha \sinh \alpha \eta ]}{8\alpha ^{2}(2\alpha +\sinh 2\alpha )};\end{align}

here, a dash denotes differentiation with respect to $\eta$ . We also record the odd-valued solution of the same equation with the right-hand side replaced by $\sinh \alpha \eta$ . If we denote this function $V_{odd}$ it follows that

(7.4b) \begin{equation}V_{odd}=\frac{1}{8\alpha ^{2}}\eta ^{2}\sinh \alpha \eta +\frac{[(\sinh 2\alpha +2\alpha )\sinh \alpha \eta -4{\eta }\sinh ^{2} \alpha \cosh \alpha \eta ]}{8\alpha ^{2}(\sinh 2\alpha -2\alpha )}.\end{equation}

With these definitions, it follows that

(7.5a) \begin{align}& U_{0C}(\eta )=\frac{\alpha \sin \beta }{4 Pr}\left[\frac{V_{even}^{\prime}}{\sinh \alpha }-\frac{V_{odd}^{\prime}}{\cosh \alpha } \right], V_{0C}(\eta )=\frac{\alpha ^{2}\sin \beta }{4 Pr}\left[\frac{V_{even}}{\sinh \alpha }-\frac{V_{odd}}{\cosh \alpha } \right],\nonumber\\&\quad \alpha P_{0C}=U_{0C}^{\prime\prime}-\alpha ^{2}U_{0C}+\frac{\sinh a(1-\eta )}{2\textit{Pr } \sinh 2\alpha }\sin \beta , \end{align}

(7.5b) \begin{align}& U_{0S}(\eta )=\frac{\alpha \cos \beta }{4 Pr}\left[\frac{V_{odd}^{\prime}}{\sinh \alpha }-\frac{V_{even}^{\prime}}{\cosh \alpha } \right], V_{0S}(\eta )=\frac{\alpha ^{2}\cos \beta }{4 Pr}\left[\frac{V_{even}}{\cosh \alpha }-\frac{V_{odd}}{\sinh \alpha } \right],\nonumber\\&\quad \alpha P_{0S}=\alpha ^{2}U_{0S}-U_{0S}^{\prime\prime}. \end{align}

7.2. Effect of the grooving

Since the velocity component $\hat{U}_{0}$ has no mean component, it does not contribute to the flux Q and we need to proceed to the next order. The temperature equation gives that

(7.6) \begin{align}\hat{\Theta }_{1}(X,\eta )&=\frac{\alpha (1-\eta )}{16\sinh 2\alpha } [\cosh 2\alpha -\cosh \alpha (1-\eta )]\cos \varOmega +\Phi (\eta )\cos (2X+\varOmega )\nonumber\\&\equiv (\hat{\Theta }_{1})_{\textit{mean}}+\Phi (\eta )\cos (2X+\varOmega ),\end{align}

where $\Phi (\eta )$ is a function whose precise form is immaterial for what follows.

The mean part of the streamwise momentum equation shows that the X-independent part of the component $\hat{U}_{1}$ , say $\hat{U}_{1M}$ , satisfies

(7.7) \begin{align} \frac{d^{2}\hat{U}_{1M}}{{\rm d}\eta ^{2}}&=-\frac{\alpha }{8}(\eta -1)\big[P_{OC}^{\prime}+\alpha U_{OC}^{\prime}\big]\cos \varOmega +\frac{\alpha }{8}(\eta -1)\big[P_{OS}^{\prime}-\alpha U_{OS}^{\prime}\big]\sin \varOmega \nonumber\\&\quad -\frac{1}{4}\big(U_{0C}^{\prime\prime}\cos \varOmega +U_{0S}^{\prime\prime}\sin \varOmega \big)-\frac{\sin \beta }{\textit{Pr}}\big( \hat{\Theta }_{1}\big)_{\textit{mean}} . \end{align}

This equation is to be solved subject to the boundary conditions that $ \hat{U}_{1M}(\pm 1)=0$ . Then, taking into account the change of variables, it follows that the fluid flux is

(7.8) \begin{equation}Q=A {\textit{Ra}}_{p,R} \int _{-1}^{1}\hat{U}_{1M} \mathrm{d}\eta .\end{equation}

After some tedious algebra, it is found that the flux due to these geometrical effects $, Q_{G}$ , is

(7.9) \begin{equation}Q_{G}=A {\textit{Ra}}_{p,R} \int _{-1}^{1}\hat{U}_{1M} \mathrm{d}\eta =A\frac{{\textit{Ra}}_{p,R}}{16\textit{Pr } {\unicode[Arial]{x03B1}} N_{3}(\alpha )} [\cos \varOmega \sin \beta N_{1}(\alpha )+2\sin \varOmega \cos \beta N_{2}(\alpha )]\end{equation}

in which the functions are

(7.10) \begin{align}N_{1}(\alpha )&=-\,8\alpha (1+2\alpha ^{2})\sinh 2\alpha +\frac{2}{3}\!\left(\!16\alpha ^{4}\!+13\alpha ^{2}\!-\frac{3}{4}\right)\!\cosh 2\alpha -\frac{1}{6}(4\alpha ^{2}-3)\cosh 6\alpha,\nonumber\\ N_{2}(\alpha )&=32\alpha ^{3}\cosh 2\alpha +3\sinh 2\alpha -\sinh 6\alpha\quad{\rm and}\nonumber\\ N_{3}(\alpha )&=(3+16\alpha ^{2})\sinh 2\alpha -\sinh 6\alpha.\end{align}

7.3. Effect of the heating

If we now turn our attention to the effects of having finite (although small) heating, we begin by noting that what turns out to be important are the streamwise-independent corrections to the temperature profile, say $\hat{\Theta }_{2M}(\eta )$ , and the mean part of the along-slot velocity $\hat{U}_{2M}(\eta )$ . The former satisfies the equation

(7.11) \begin{equation}\frac{\mathrm{d}^{2}\hat{\Theta }_{2M}}{\mathrm{d}\eta ^{2}}=\frac{\alpha ^{2}\cos \beta }{16 \sinh 2\alpha }\times \frac{\mathrm{d}}{\mathrm{d}\eta }\left[\left(\frac{V_{even}}{\cosh \alpha }-\frac{V_{odd}}{\sinh \alpha }\right)\sinh \alpha (1-\eta )\right],\end{equation}

which needs to be integrated subject to the boundary conditions that $\hat{\Theta }_{2M}(\pm 1)$ = 0. The streamwise momentum equation shows that

(7.12) \begin{equation} \frac{\mathrm{d}^{2}\hat{U}_{2M}}{\mathrm{d}\eta ^{2}}=\frac{\alpha ^{2}\sin \beta }{16 {\textit{Pr}}^{2}}\left[\frac{V_{even}V_{odd}^{\prime\prime}-V_{odd}V_{even}^{\prime\prime}}{\cosh ^{2} \alpha \sinh ^{2} \alpha }\right]-\frac{\alpha ^{2}\sin 2\beta }{32 Pr\sinh 2\alpha }\hat{\Theta }_{2M} ; \hat{U}_{2M}(\pm 1)=0.\end{equation}

It is noted that our ultimate aim is to determine the flux through the slot given by the integral of $\hat{U}_{2M}$ across the domain. The first part of the right-hand side of (7.12) is an odd-valued function, so the corresponding solution is also odd valued and will not contribute to the flux calculation. When we solve the rest of the equation, we find that the flux due to the heating $ Q_{R}$ becomes

(7.13) \begin{equation}Q_{R}= {\textit{Ra}}_{p,R}^{2}\int _{-1}^{1}\hat{U}_{2M} \mathrm{d}\eta = {\textit{Ra}}_{p,R}^{2}\times \frac{N_{4}(\alpha )}{122\,880 \textit{Pr} \alpha ^{6}\sinh 2\alpha (\sinh ^{2} 2\alpha -4\alpha ^{2})}\times \sin 2\beta,\end{equation}

in which

(7.14) \begin{align}N_{4}(\alpha )&=10\alpha (2\alpha ^{2}-9)\cosh 6\alpha +75\sinh 6\alpha +(1536\alpha ^{6}-1760\alpha ^{4}-2160\alpha ^{2}-225)\nonumber\\&\quad\times\sinh 2\alpha-2\alpha (256\alpha ^{6}+320\alpha ^{4}-1670\alpha ^{2}-45)\cosh 2\alpha .\end{align}

Although it might seem unlikely, it turns out that the function $N_{4}(\alpha )=O(\alpha ^{13})$ as $\alpha \rightarrow 0$ so that overall $Q_{R}\propto \alpha ^{2}$ in the long-wavelength limit. When $\alpha$ is large, it is clear that $Q_{R}\propto \alpha ^{-3}$ .

7.4. Further comments

Here, we have derived some approximations of the effect on the flux in the case of small corrugations and relatively weak heating. First, we point out that the numerical value of ${\textit{Ra}}_{p,R}$ can be quite sizeable before this analysis fails; a simple-minded justification for this suggestion can be ascribed to the fact that the rather large denominator in (7.8) means that the correction to the basic state remains acceptably small until ${\textit{Ra}}_{p,R}$ is of quite an appreciable size. We also comment on the fact that it seems that the contribution $Q_{G}$ is more significant in the large- $\alpha$ limit.

We can also make some additional remarks relating to the results presented in figure 10. We focus on the results in figure 10(a,b); our comments can be easily adapted to apply to figure 10(c,d). If we study long-wavelength modes, it is seen that when $\varOmega =0,$ then $Q\propto \alpha ^{2}$ as $\alpha \rightarrow 0$ irrespective of the inclination $\beta ,$ although the flux seems to be significantly smaller for a vertical slot compared with other angles. When $\varOmega =\pi /2,$ it appears that $Q\propto \alpha$ . We can account for these behaviours by noting that for small $\alpha$ then

(7.15) \begin{equation}Q_{G}\sim \frac{A {\textit{Ra}}_{p,R}}{\textit{Pr}}\left[\frac{1}{36}\alpha ^{2}\cos \varOmega \sin \beta +\frac{1}{40}{\unicode[Arial]{x03B1}} \sin \varOmega \cos \beta \right], Q_{R}\sim \frac{ {\textit{Ra}}_{p,R}^{2}}{{\textit{Pr}}^{2}}\frac{\sin 2\beta }{18\,900} \alpha ^{2},\end{equation}

which, for the values used in the computations above, becomes

(7.16) \begin{align}&Q_{G }\sim (0.59\alpha ^{2}+\cdots) \cos \varOmega \sin \beta +(0.53{\unicode[Arial]{x03B1}} +\cdots) \sin \varOmega \cos \beta ,\nonumber\\& Q_{R}\sim (9.45\alpha ^{2}+\cdots )\sin 2\beta .\end{align}

Let us look at figure 10(a). With $\varOmega =0,$ the second term in $Q_{G }$ vanishes, and the contribution from $Q_{R}$ is an order of magnitude greater than that from $Q_{G }$ unless $\sin 2\beta =0$ . This explains why the flux for a vertical slot when $\alpha$ is small appears to be substantially less than that for other inclinations. In the case when $\varOmega =\pi /2$ it is the first term in $Q_{G }$ that is switched off; now the remaining part of $Q_{G }$ dominates $Q_{R}$ because it only decreases proportionally to $\alpha$ rather than $\alpha ^{2}$ . These findings align with our earlier long-wavelength studies conducted in § 6.

Now let us turn to the short-wavelength limit $\alpha \ll 1$ . When $\alpha$ is large, simple algebra shows that $N_{1}/N_{3}\sim ({2}/{3})\alpha ^{2}$ and $N_{1}/N_{3}\sim 1$ so, with the usual values of the parameters

(7.17) \begin{equation}Q_{G} \sim (0.88\alpha +\cdots )\cos \varOmega \sin \beta +(0.66\alpha ^{-1}+\cdots )\sin \varOmega \cos \beta ;\end{equation}

we note that, since $Q_{R}\propto \alpha ^{-3}$ , this part is not important. When $\varOmega =0$ we expect $Q$ to grow linearly with $\alpha$ as noted in figure 10(a). In contrast, when $\varOmega =\pi /2$ , $Q$ falls proportional to $\alpha ^{-1}$ ; see figure 10(b). The other feature of this plot that we ought to comment upon relates to the behaviour of $Q$ at intermediate values of $\alpha$ ; it appears that the flux in a horizontal slot is substantially less than within an inclined channel. This is associated with the fact that for ${\textit{Ra}}_{p,R}=300$ , $A=0.05$ and $\sin 2\beta \neq 0$ the contribution from $Q_{R}$ greatly exceeds that from $Q_{G}$ unless the wavenumber is large.

We make a few last remarks relating to the behaviour of the system. In this section we have focused on the properties of the flux through the slot. Of course, we could discuss other features of interest, such as forces or the details of the fluid flow, and the structures we have set out can easily be adapted to explain other properties of the flow. We have chosen not to do this, partly in the interest of brevity and partly because details are likely to be highly dependent on the particular parameter choices. With the case considered here, it is seen that the contribution to the flux attributable to the heating is significantly greater than that owing to the grooves; however, this behaviour could well change for other combinations. We also note that we have explored the flow structure under the assumption that the corrugations are, in some sense, small. Strictly speaking, we could have relaxed this restriction on the analysis, but the flow without any applied heating can then only be determined numerically. We could still formulate a weak heating theory, but its usefulness is somewhat arguable since the results could only be expressed in quantities that rely on an approximate determination.

8. Heat transfer effects

There is of course an energy that needs to be supplied in order to maintain the fluid motions described in the preceding sections. This quantity is of interest as it guides us as to the efficiency of generating the flow. Unfortunately, it is not totally straightforward to quantify the energy required as the heat flows in two directions, i.e. both across and along the slot. The cross-slot component can be measured in terms of the mean Nusselt number which is defined to be

(8.1) \begin{equation}{\textit{Nu}}_{\textit{av}}=\lambda ^{-1}\int _{x_{0}}^{x_{0}+\lambda }\left.\frac{\partial \theta }{\partial y}\right| _{y=1}\mathrm{d}x;\end{equation}

here, $x_{0}$ is an arbitrary location along the plate. With this definition, a positive value of ${\textit{Nu}}_{\textit{av}}$ implies that heat enters the fluid through the left wall. We point that there is no unambiguous way to estimate energy costs; some alternatives have been proposed by Maxworthy (Reference Maxworthy1997), Siggers et al. (Reference Siggers, Kerswell and Balmforth2004), Hughes & Griffiths (Reference Hughes and Griffiths2008) and Winters & Young (Reference Winters and Young2009).

It is known (Abtahi & Floryan Reference Abtahi and Floryan2017b ) that the intensity of convection in a horizontal slot rapidly decreases for $\alpha \rightarrow 0$ and $\alpha \rightarrow \infty$ with the heat flow in these two limits being driven by conduction. Once the slot is inclined, these properties are retained and some illustrative results are shown in figure 16. When $\varOmega =\pi /2$ or $3\pi /2$ the conductive effects disappear (Abtahi & Floryan Reference Abtahi and Floryan2017b ) and the heat flow is driven solely by convection. For $\pi /2\lt \varOmega \lt 3\pi /2$ the heat flow is positive for moderate wavelengths as it is dominated by convection. On the other hand it changes direction and becomes negative should $\alpha \rightarrow 0$ or $\alpha \rightarrow \infty$ because in these limits it is dominated by conduction (see figure 16 d–f). This only exception to this general behaviour occurs as the slot approaches the vertical in which case the heat flow is always negative. The heat flow is always positive for $0\lt \varOmega \lt \pi /2$ and $3\pi /2\lt \varOmega \lt 2\pi$ as conductive heat flow for $\alpha \rightarrow 0$ and $\alpha \rightarrow \infty$ is positive (see figure 16 a,b,h). For the parameter choices used for this study it seems to the largest heat flow always occurs for $\alpha \approx 1$ .

Figure 16.

The mean Nusselt number ${\textit{Nu}}_{\textit{av}}$ as a function of the wavenumber $\alpha$ for three inclination angles $\beta =0,\pi /4\text{ and } \pi /2$ and eight phase angles $\varOmega$ . Results shown for ${\textit{Ra}}_{p,R}=300$ and $A=0.05$ . Dashed lines identify negative values.

We can deduce some asymptotic expressions for the form of the mean Nusselt number in the case of gentle heating.

The use of the definitions of $\hat{\Theta }_{0}(X,\eta )$ combined with the mean parts $( \hat{\Theta }_{1})_{\textit{mean}}$ and $\hat{\Theta }_{2M}(\eta )$ enable us to conclude that

(8.2) \begin{align} &{\textit{Nu}}_{\textit{av}}\sim \frac{\alpha }{16}\coth 2\alpha \cos \varOmega A {\textit{Ra}}_{p,R}\nonumber\\&-\frac{3\sinh 8\alpha -8\alpha (3-\!16\alpha ^{2}) \cosh 4\alpha\! -6 (1\!+8\alpha ^{2}+32\alpha ^{4})\sinh 4\alpha +8\alpha (3+8\alpha ^{2}-\!32\alpha ^{4})}{12\,288 \alpha ^{3}\sinh ^{2} 2\alpha (\sinh ^{2} 2\alpha -4\alpha ^{2})}\nonumber\\&\times{\textit{Ra}}_{p,R}^{2}\cos \beta .\end{align}

It is straightforward to conclude that when $\varOmega =\pi /2$ or $3\pi /2$ then for small $\alpha$

(8.3) \begin{equation}{\textit{Nu}}_{\textit{av}}\sim -\frac{1}{1400}\alpha ^{2}{\textit{Ra}}_{p,R}^{2}\cos \beta +\cdots,\end{equation}

while, for large $\alpha$ , ${\textit{Nu}}_{\textit{av}}\sim -({1}/{512})\alpha ^{-3}{\textit{Ra}}_{p,R}^{2}\cos \beta$ . Conversely, for other phase values, we can conclude that ${\textit{Nu}}_{\textit{av}}\sim ({1}/{32})\cos \varOmega A {\textit{Ra}}_{p,R}$ for $\alpha \rightarrow 0$ and ${\textit{Nu}}_{\textit{av}}\sim ({\alpha }/{16})\cos \varOmega A {\textit{Ra}}_{p,R}$ when $\alpha \gg 1$ . These limiting values are all consistent with the computations summarised in figure 16.

We mention that the cost of moving energy within the right wall to create the local cold and hot spots is difficult to assess but it does contribute to the overall energy costs. The energy that has to be moved along the right wall can be determined by evaluating the longitudinal heat fluxes between the hot and cold wall segments. These fluxes can be quantified in terms of heat leaving each wall per half-heating wavelength. This can be expressed by the longitudinal Nusselt number which is defined to be

(8.4) \begin{equation}{\textit{Nu}}_{Lg,R}=2\lambda ^{-1}\int _{x_{1}}^{x_{1}+\lambda /2}\left.\left(-\frac{\partial \theta }{\partial x}\frac{\partial y_{R}}{\partial x}+\frac{\partial \theta }{\partial y}\right)\right| _{{y_{R}}}\mathrm{d}x-{\textit{Nu}}_{\textit{av}},\end{equation}

where $x_{1}$ is a point along the wall where the wall temperature changes sign. The evaluation of this integral using the expression (7.2) for $\hat{\Theta }_{0}(X,\eta )$ shows that at leading order

(8.5) \begin{equation} {\textit{Nu}}_{Lg,R}\sim \frac{{\textit{Ra}}_{p,R}}{\pi }\alpha \coth 2\alpha . \end{equation}

This simple expression shows that ${\textit{Nu}}_{Lg,R}$ is a monotonic function of $\alpha$ . Using the values taken in figure 16, this means that ${\textit{Nu}}_{Lg,R} \rightarrow 47.7$ as $\alpha \rightarrow 0$ and ${\textit{Nu}}_{Lg,R} \sim 94.5\alpha$ for $\alpha \gg 1$ . We note in particular that leading-order result is independent of the phase value $\varOmega$ .