Thermocapillary and buoyant flows with low frequency jitter. II. Spanwise jitter

A temperature gradient is applied along a fluid filled slot. A basic state is considered where the slot is subject to thermocapillary forces and vertical mean gravity, each of which produces a parallel flow and a vertical advected temperature gradient, and is also subject to streamwise mean gravity, which will make the applied temperature stratification either stable or unstable. When this basic state is perturbed by jitter imposed in the spanwise direction, normal to the plane of the basic flow, the resulting fluid motion is three dimensional. The flow and temperature fields are found to have a simple functional dependence on streamwise and spanwise coordinates, but retain a complicated dependence on vertical coordinate. Perturbation equations describing the vertical variation of these fields are derived when the jitter is weak. At first order in the spanwise jitter, there is a time periodic spanwise-streamwise circulation around the slot. As this circulation also advects heat, it produces spanwise temperature gradients, enabling thermocapillarity and vertical gravity to generate subsidiary spanwise flows. At next order in the weak spanwise jitter, parallel streamwise flows are encountered, along with streamwise and vertical temperature gradients. In most parameter regimes these are opposed to the flow and temperature fields in the basic state. A thorough parametric investigation is performed where the weak spanwise jitter equations are solved, assuming for simplicity that streamwise gravity is absent. This leads to comparatively simple polynomial solutions in vertical coordinate for the various fields. A large number of parameters can still affect the solutions, however, and a detailed parametric investigation is performed. Interesting behavior is found at small Biot number, with trapping of heat producing large temperatures in the slot and large subsidiary flows. The spanwise to streamwise aspect ratio is another influential parameter, since geometric constraints encountered at extreme values of this ratio suppress certain velocity components of the flow and enhance others, thereby suppressing or enhancing temperature advection. These advected temperature fields themselves produce subsidiary velocities and subsidiary temperatures, which can exhibit a subtle and often counterintuitive dependence on the spanwise-streamwise aspect ratio.