A spectrally accurate and very efficient algorithm suitable for prediction of pressure losses in heated grooved channels has been developed. Heating and topography patterns are used to create spatial flow modulations resulting in a pattern interaction problem. Search for combinations of patterns resulting in the reduction of pressure losses requires development of a very accurate and efficient algorithm. The proposed algorithm uses a combination of the Fourier expansions in the horizontal directions and the Chebyshev expansions in the vertical direction to provide a very good resolution of the near wall regions. The immersed boundary conditions (IBC) method is used to enforce flow boundary conditions at the geometrically irregular boundaries. The resulting gridless discretization can be easily adapted to handle a wide range of topography patterns. Various tests demonstrate that the algorithm delivers spectral accuracy and can provide machine level accuracy. Comparisons with the standard open‐source codes based either on the finite volume or on the spectral element discretization demonstrate several orders of magnitude better efficiency of the proposed algorithm.

Peristaltic pumping in a two-dimensional conduit using vibrations in the form of traveling waves has been investigated. Two qualitatively different responses producing vastly different flow rates have been identified, with a transition occurring at wavelengths of the order of the conduit opening. The flow rate is always proportional to the wave phase speed and the second power of the amplitude. Long waves produce sloshing which extends across the whole conduit producing a small, nearly wave-number-independent flow rate. The use of such in-phase waves on both walls nearly eliminates this flow while the use of out-of-phase waves maximizes it. Short waves affect the near-wall regions, which appear to the bulk of the fluid as moving walls. Such waves produce an order of magnitude larger flow rate, with its magnitude increasing proportionally to the second power of the wavenumber. Each vibrating wall produces its own wall boundary layer with an unmodulated core flow in the central zone of the conduit. The core flow looks like a Couette flow and reduces to a plug flow when both waves have identical amplitudes. The phase difference between such waves does not affect the flow rate. Wave tilting increases the flow rate similarly to the increase in distance between these waves. The use of waves characterized by a combination of wavenumbers increases the flow rate but only when the commensurability index is greater than one. The best performance is achieved by concentrating all wave energy in a single and largest achievable wavenumber.

Laminar natural convection is investigated in an infinite vertical slot which has one wall with a corrugated profile, and which is subject to either a uniform or periodic heating profile. This configuration has the attractive feature that it enables a study of the effects that may be produced via the interaction of heating and topography patterns. It is found that the addition of the grooves to an isothermal plate leads to a reduction in the vertical fluid flow and an increase of the transverse heat flow. In contrast, imposing sinusoidal heating on a flat surface generates convection that appears as counter-rotating rolls but there is no net vertical flow. The combination of the two effects of corrugation together with periodic heating leads to a plethora of flow patterns involving a combination of rolls and stream tubes that carry the fluid along the slot. The details of this vertical flow are governed by a pattern interaction effect dictated by the relative positions of the heating and corrugation patterns; when hot spots of the imposed heating overlap the peaks in the grooves the net flow is upward; in contrast, when they lie over the troughs the resultant flow is downward. The interplay between the thermal and geometrical effects weakens as the wavelength of the structure is reduced. The inclusion of a sufficiently strong uniform heating also seems to wash away the pattern interaction effect.

An analysis of natural convection in horizontal slots has been carried out. It is demonstrated that a proper combination of heating and groove patterns can create a net horizontal fluid movement which we refer to as the horizontal chimney effect. Groove shapes that can be easily manufactured as well as heating patterns that can be easily created using heating wires were considered. It has been shown that both patterns must be properly tuned. The direction of the net horizontal flow can be changed by changing the relative positions of the patterns. Changes of groove geometry can change the flow rate by up to 100%. Simultaneous use of grooves and heating at both plates can nearly double the system effectiveness. The strength of the flow increases with reduction of the Prandtl number.

An analysis of laminar natural convection in inclined slots subjected to patterned heating has been performed. The imposed heating takes a simple form characterized by a single Fourier mode combined with uniform heating. It is shown that periodic heating applied at the lower plate produces no net flow when the slot is either horizontal or vertical, but a net upward flow is generated when the slot is tilted. Periodic heating applied at the upper plate produces net downward flow in the inclined situation. The addition of uniform heating promotes the upward flow while cooling has the opposite effect. There is a critical inclination angle at which the maximum net flow rate is greatest. Dynamic and thermal boundary layers are present when the wavenumber of the imposed heating is large. The use of heating at both plates, with the same wavenumber, leads to a flow dominated by the plate exposed to a more intense heating; when the two plates are heated equally no net flow is observed irrespective of the inclination angle. Changes of the relative positions of the two patterns can change the net flow rate by up to 50 %. The intensity of the flow increases with reduction of the Prandtl number. If the heating applied to the plates is of different wavelength, but of the same intensity, a wide range of behaviours of the flow system is possible. The details of this response are sensitive to the ratio of the two wavenumbers.

A highly accurate and fully implicit algorithm for analyses of transient effects in heated grooved channels, including chaotic responses and transition to secondary states, has been developed. The algorithm can handle pattern interaction problems arising from combinations of geometric and heating patterns which are expected to be beneficial in the development of energy efficient stirring systems. The algorithm uses spectral spatial discretization and up to sixth‐order temporal discretization, providing the means to deliver machine accuracy. The spatial discretization relies on a combination of Chebyshev and Fourier expansions, guaranteeing very good resolution in the vicinity of the grooved walls. The enforcement of boundary conditions along the irregular boundaries is carried out using the immersed boundary conditions. The overall discretization leads to a gridless algorithm and provides the geometric flexibility required for efficient analyses of multitude of topography patterns. Extensive testing demonstrates that the algorithm achieves the theoretically predicted accuracy.

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We demonstrate that the relative motion of horizontal parallel plates can be generated using patterned heating. This movement is driven by nonlinear thermal streaming associated with a pitchfork bifurcation. The propulsive effect is strongest when all the heating energy is concentrated in a single Fourier mode of the spatial heating pattern; it increases with a decrease in the Prandtl number and increases with the addition of a uniform heating component.