The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 1.  Time-periodic flow reversal inside an oscillating square cylinder for η = 100. The animation, which corresponds to Figure 1 in the paper, visualises a sequence of instantaneous streamline patterns in a non-inertial frame of reference oscillating with the cylinder. Topological changes in the streamline pattern remain qualitatively the same throughout the low-η regime but the time-scale of the process decreases with decreasing η.

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 1.  Time-periodic flow reversal inside an oscillating square cylinder for η = 100. The animation, which corresponds to Figure 1 in the paper, visualises a sequence of instantaneous streamline patterns in a non-inertial frame of reference oscillating with the cylinder. Topological changes in the streamline pattern remain qualitatively the same throughout the low-η regime but the time-scale of the process decreases with decreasing η.

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 2.  Time-periodic flow reversal inside an oscillating square cylinder in the high-η regime; η = 5000 (compare this with Figure 14 in the paper). The animation is played four times slower than Movie 1. Note that in contrast to the flow reversal in the low-η regime (Movie 1) the corner eddies blend into Stokes layers through the emergence of wall eddies which develop near the boundary. The flow reversal in the interior is realised through a series of global bifurcations involving these wall eddies rather than the primary corner eddies.

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 2.  Time-periodic flow reversal inside an oscillating square cylinder in the high-η regime; η = 5000 (compare this with Figure 14 in the paper). The animation is played four times slower than Movie 1. Note that in contrast to the flow reversal in the low-η regime (Movie 1) the corner eddies blend into Stokes layers through the emergence of wall eddies which develop near the boundary. The flow reversal in the interior is realised through a series of global bifurcations involving these wall eddies rather than the primary corner eddies.

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 3.  Time-periodic flow reversal inside a square cavity driven by an oscillating lid (top) for η = 100. The animation visualises a sequence of instantaneous streamline patterns in an inertial frame fixed at the centre of the cavity. The time-scale of the reversal decreases with decreasing η but the sequence of topological changes in the streamline pattern remains the same.

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 3.  Time-periodic flow reversal inside a square cavity driven by an oscillating lid (top) for η = 100. The animation visualises a sequence of instantaneous streamline patterns in an inertial frame fixed at the centre of the cavity. The time-scale of the reversal decreases with decreasing η but the sequence of topological changes in the streamline pattern remains the same.

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 4.  Time-periodic flow reversal inside an oscillating cylinder with rhomboidal cross-section for η = 100. The animation, which corresponds to Figure 2 in the paper, visualises a sequence of instantaneous streamline patterns in a non-inertial frame of reference which oscillates with the cylinder. Note that the structure in the acute-angle corners is more complicated than in the case of the right-angle corners (applies only to the oscillating-cylinder problem). The local structure of the corresponding corner flow is shown in Movie 8.

The animations present some examples of the time-periodic evolution of oscillatory Stokes flows inside finite domains which have sharp corners on the boundary. For an oscillatory Stokes flow we have η >> Re where η = ω L^2/ ν ; ω is the frequency of oscillations, L is the characteristic length of the domain and ν is the viscosity. Such flows are in general not quasi-steady and flow reversals associated with the oscillations of the forcing are accomplished in a non-trivial manner. When following the flow evolution by tracing the changes in the instantaneous streamline pattern, as in the animations below, the flow reversals are represented by a sequence of global bifurcations in the streamline pattern. Movie 4.  Time-periodic flow reversal inside an oscillating cylinder with rhomboidal cross-section for η = 100. The animation, which corresponds to Figure 2 in the paper, visualises a sequence of instantaneous streamline patterns in a non-inertial frame of reference which oscillates with the cylinder. Note that the structure in the acute-angle corners is more complicated than in the case of the right-angle corners (applies only to the oscillating-cylinder problem). The local structure of the corresponding corner flow is shown in Movie 8.

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 5.  Evolution of the infinite eddy structure, characteristic of Regime I (oscillating-lid) and Regime III (oscillating-cylinder), near a 60 deg corner for η = 0.1 (compare with Figure 6 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign. (The full evolution is time-periodic but the eddy structure is almost stationary for most of the period.)

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 5.  Evolution of the infinite eddy structure, characteristic of Regime I (oscillating-lid) and Regime III (oscillating-cylinder), near a 60 deg corner for η = 0.1 (compare with Figure 6 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign. (The full evolution is time-periodic but the eddy structure is almost stationary for most of the period.)

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 6.  Emergence of a single eddy during the flow reversal in a 160 deg corner (Regime II in the oscillating-lid problem; Regime IV in the oscillating-cylinder problem) for η = 0.1 . There are no infinite eddy structures in the flow at any stage (compare with Figure 7 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign.

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 6.  Emergence of a single eddy during the flow reversal in a 160 deg corner (Regime II in the oscillating-lid problem; Regime IV in the oscillating-cylinder problem) for η = 0.1 . There are no infinite eddy structures in the flow at any stage (compare with Figure 7 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign.

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 7.  Flow evolution near a corner of reflex angle 270 deg (Regime II in the oscillating-lid problem; Regime IV in the oscillating-cylinder problem) for η = 0.1 ; this does not involve any eddy during the reversal process (compare with Figure 8 in the paper). As before, this animation shows the flow evolution within a very short time interval when the forcing changes sign.

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 7.  Flow evolution near a corner of reflex angle 270 deg (Regime II in the oscillating-lid problem; Regime IV in the oscillating-cylinder problem) for η = 0.1 ; this does not involve any eddy during the reversal process (compare with Figure 8 in the paper). As before, this animation shows the flow evolution within a very short time interval when the forcing changes sign.

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 8.  Flow reversal in an oscillating cylinder near a sharp corner representative of Regime II; 2α = 60 deg, η = 0.1, non-inertial frame of reference was adopted which oscillates with the cylinder (compare with Figure 11 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign. Note the emergence of pairs of side wall eddies and their subsequent merger followed by a reversed process closer to the corner. Such pairs of side wall eddies are also involved in the flow reversal in corner angles in Regime I but, in contrast to the situation in Regime II, they are only finite in number.

In the case of a corner flow in the oscillating-lid problem two distinct dynamical regimes were identified in the paper. Regime I (2α < 146.3 deg) is characterised by existence of infinite structure of time-dependent eddies in the corner. For corner angles that belong to Regime II (2α > 146.3 deg) there is either a single eddy created during the flow reversal or there are no eddies at all. In the case of a corner flow in the oscillating-cylinder problem four distinct dynamical regimes were identified in the paper. The structure in Regime IV (2α > 146.3 deg) is equivalent to that of Regime II in the oscillating-lid problem. The structure in Regime III (81.9 deg < 2α < 146.3 deg) is equivalent to that of Regime I in the oscillating-lid problem. In Regimes I and II (oscillating-cylinder problem) pairs of side wall eddies are responsible for the flow reversal. In Regime II (48.7 deg 2α < 81.9 deg) infinite number of such pairs is present at some point of the reversal. In Regime I (2α < 48.7 deg) there is only a finite number of such pairs. Movie 8.  Flow reversal in an oscillating cylinder near a sharp corner representative of Regime II; 2α = 60 deg, η = 0.1, non-inertial frame of reference was adopted which oscillates with the cylinder (compare with Figure 11 in the paper). This animation shows the flow evolution within a very short time interval when the forcing changes sign. Note the emergence of pairs of side wall eddies and their subsequent merger followed by a reversed process closer to the corner. Such pairs of side wall eddies are also involved in the flow reversal in corner angles in Regime I but, in contrast to the situation in Regime II, they are only finite in number.