Heating-induced drag reduction in relative movement of parallel plates

An external force is required to maintain the relative movement of horizontal plates. It is shown that this force is reduced when the plates are subject to a spatially distributed heating. The largest reduction occurs for heating wavelengths of the order of distance between the plates with its magnitude increasing proportionally to the second power of the relevant Rayleigh number. It is shown that a sufficiently strong heating eliminates the need for the driving force altogether. The drag-reducing effect is active only in small Reynolds number flows and is stronger in fluids with smaller Prandtl numbers.

Figure 1

Schematic diagram of the flow system.

Figure 2

The flow and temperature fields for $\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\alpha =2$ and (a) $\mathrm{Re}=0$, (b) $\mathrm{Re}=1$, (c) $\mathrm{Re}=5$, (d) $\mathrm{Re}=50$. Thick lines identify streamlines, thin lines identify isotherms, gray shadings identify zones of cold fluid. Thick streamlines mark borders of bubbles trapping the fluid. Enlargement of box shown in panel (b) is displayed in Fig. 3. Flow conditions used in these plots are marked in Fig. 10 using squares.

Figure 3

Enlargement of the box shown in Fig. 2. The streamline emanating from the in-flow stagnation points corresponds to $\psi =0.286322$.

Figure 4

Variations of the local $|{\psi }_{\max }|$ associated with the upper plate (solid line), the clockwise rolls (dashed line), and the counterclockwise rolls (dashed-dotted line) as functions of Re for $\alpha =2,\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0$.

Figure 5

Distribution of the shear stress ${\tau }_U$ acting on the upper plate for $\alpha =2,\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0$ at $\mathrm{Re}=1$ (solid line) and $\mathrm{Re}=10$ (dashed line). Enlargement of the box shown in panel (a) is displayed in panel (b) including shear mean values.

Figure 6

Variations of Δ$F$/Re as a function of α for $\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines). Thin dotted lines identify asymptotes. The shaded area identifies conditions where the driving force must change direction and becomes a braking force.

Figure 7

The flow and temperature fields for $\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\mathrm{Re}=1$ and $\alpha =1$ (a), $\alpha =5$ (b), $\alpha =8$ (c), $\alpha =10$ (d). Thick lines identify streamlines, thin lines identify isotherms, gray shadings identify zones of cold fluid. Thick streamlines mark borders of various bubbles trapping the fluid.

Figure 8

Variations of change of the flow rate driven by movement of the upper plate Δ$Q$/Re as a function of α for $\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines). Thin dotted lines identify asymptotes. Dashed-dotted line identifies the negative values of Δ$Q$ for $\mathrm{R}{\mathrm{a}}_p=2000,\mathrm{Re}=1$.

Figure 9

The flow and temperature fields for $\mathrm{R}{\mathrm{a}}_p=2000,\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\mathrm{Re}=1,\alpha =0.25$. Enlargement of box in Fig. 6 is displayed in panel (b). Thick lines identify streamlines, thin lines identify isotherms, gray shadings identify zones of cold fluid. Thick streamlines mark borders of various bubbles trapping the fluid.

Figure 10

Pressure fields for $\alpha =2,\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Pr}=0.71,\mathrm{Re}=1$ (a) and $\mathrm{Re}=10$ (b). Dotted and dashed-dotted lines identify the positive and negative values, respectively. Thick solid lines illustrate streamlines. Shaded areas correspond to positive pressure.

Figure 11

Distributions of the pressure at the upper (black lines) and lower (gray lines) plates for $\alpha =2,\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Pr}=0.71,\mathrm{Re}=1$(solid lines) and $\mathrm{Re}=10$ (dashed lines). The average values of pressure at the lower wall overlap within the plotting accuracy as they are −32.2, −31.7 for $\mathrm{Re}=1$, 10, respectively. The characteristic points used in the evaluation of moments at the upper plate are marked with thin dotted lines using $\mathrm{Re}=1$ as an example.

Figure 12

Variations of the lifting force $G$ (a) and the moment $M$ (b) as functions of α for $\mathrm{Pr}=0.71,R{a}_{\mathrm{uni}}=0,\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines).

Figure 13

Variations of Δ$F$ (a) and Δ$Q$ (b) as functions of Re for $\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\alpha =2$ (solid lines) and $\alpha =1$ (dashed lines). Thin dotted lines identify asymptotes. Plots of flow and temperature field for conditions identified using squares are displayed in Fig. 2. See text for other details. The shaded area in panel (a) identifies conditions where the driving force must change direction and becomes a braking force.

Figure 14

Variations of Δ$F$ (a) and Δ$Q$ (b) as functions of $\mathrm{R}{\mathrm{a}}_p$ for $\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines). Plots of flow and temperature fields for conditions identified using squares are displayed in Fig. 15. See text for other details. The shaded area identifies conditions where the driving force must change direction and becomes a braking force when $\mathrm{Re}=1$ and the double shaded area identifies such conditions for $\mathrm{Re}=10$.

Figure 15

The flow and temperature fields for $\mathrm{Re}=1,\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\alpha =2$ at (a) $\mathrm{R}{\mathrm{a}}_p=500$ and (b) $\mathrm{R}{\mathrm{a}}_p=3000$. Thick lines identify streamlines, thin lines identify isotherms, gray shadings identify zones of cold fluid. Thick streamlines mark borders of various bubbles trapping the fluid. Flow conditions used in these plots are marked in Fig. 14 using squares.

Figure 16

Variations of ΔF/Re as a function of α (a) and as a function of $\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}$ (b) for $\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines), and $\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Pr}=0.71$. The shaded areas identify conditions where the driving force changes direction and becomes a braking force.

Figure 17

Variations of ΔQ/Re (a) as a function of α and as a function of $\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}$ (b) for $\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines), and $\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Pr}=0.71$.

Figure 18

Variations of (a) ΔF/Re and (b) ΔQ/Re as functions of Pr at $\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines) for $\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0$.

Figure 19

Variations of (a) ΔF/Re and (b) ΔQ/Re as functions of α at $\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines) for $\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0$.

Figure 20

Variations of $\mathrm{N}{\mathrm{u}}_{av}$ in panel (a) as a function of α for $\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines), in panel (b) as a function of Re for $\alpha =1$ (dashed lines) and $\alpha =2$ (solid lines), and in panel (c) as a function of $\mathrm{R}{\mathrm{a}}_p$ for $\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines), for $\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0$. Thin dotted lines identify asymptotes.

Figure 21

Variations of $\mathrm{N}{\mathrm{u}}_{av}$ in panel (a) as a function of α and in panel (b) as a function of $\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}$ for $\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines) for $\mathrm{R}{\mathrm{a}}_p=1000,\mathrm{Pr}=0.71$.

Figure 22

Variations of $\mathrm{N}{\mathrm{u}}_{av}$ in panel (a) as a function of α and in panel (b) as a function of Pr for $\mathrm{Re}=1$ (solid lines) and $\mathrm{Re}=10$ (dashed lines) for $\mathrm{R}{\mathrm{a}}_p=1000,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0$.

Figure 23

The flow and temperature fields for the same conditions as in Fig. 7 but with the upper plate heated and the lower plate moving. Thick lines identify streamlines, thin lines identify isotherms, gray shadings identify zones of cold fluid. Thick streamlines mark borders of bubbles trapping the fluid.

Figure 24

Variations of the difference between the actual and the isothermal $x$-velocity components, $\mathrm{\Delta }u(=u-u_{\mathrm{iso}})$, as functions of $y$ for $\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Re}=1,\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\alpha =30$, at $x/\lambda =0$ (solid line) and $x/\lambda =0.5$ (dashed line). Enlargement of the box shown in panel (a) is displayed in panel (b).

Figure 25

Variations of the temperature θ as functions of $y$ for $\mathrm{R}{\mathrm{a}}_P=1000,\mathrm{Re}=1,\mathrm{Pr}=0.71,\mathrm{R}{\mathrm{a}}_{\mathrm{uni}}=0,\alpha =30$ at $x/\lambda =0$ (solid line) and $x/\lambda =0.5$ (dashed line). Enlargement of the box shown in panel (a) is displayed in panel (b).