A fluid is constrained between two smooth surfaces a distance, \(d\), apart in a gravitational field with a temperature gradient across the fluid. If a Bénard instability develops, show that it is possible to have square cells. Find the horizontal width of the cells in terms of \(d\). Find an expression for the velocity, \(\boldsymbol{v}\), of the fluid and sketch the flow in each cell. Assume that, at the cell walls, the fluid flows upward and has its maximum speed.

Convection currents are fundamental in understanding how fluid moves when subjected to a temperature gradient. In our scenario, the fluid is heated from below and cooled from above. This heating makes the fluid at the bottom less dense, causing it to rise. Conversely, the cooler fluid at the top is denser and sinks. This continuous rise and fall within the fluid is what we call convection currents. These currents form specific patterns or 'cells' within the fluid.

Each convection current circulates in a loop: rising in the middle of the cell, moving to the sides at the top, sinking down along the cell walls, and then moving back to the center at the bottom. Understanding this basic concept helps us visualize the flow within the Bénard convection cells.

The temperature gradient is the key driver behind convection currents. It represents the difference in temperature between the top and bottom surfaces of the fluid. In the case of Bénard convection cells, the bottom surface is heated, while the top surface is kept cooler.

The strength of the temperature gradient determines how vigorous the convection currents will be. A larger temperature difference results in stronger currents and higher velocities within the fluid. Essentially, the temperature gradient acts as the engine for the entire convection process, dictating the movement and speed of the fluid.

Fluid dynamics is the study of how fluids behave when they're in motion. For Bénard convection cells, fluid dynamics principles help us understand and calculate the velocity and pattern of fluid flow.

The key equation governing fluid motion is the Navier-Stokes equation. It takes into account factors like the viscosity of the fluid, the pressure within the fluid, and external forces such as gravity. By applying these principles, we can determine that the maximum velocity of the fluid occurs at the vertical edges of each cell, where the fluid rises or sinks.

Fluid dynamics provides the mathematical framework to describe the velocity distribution and the shape of the convective cells.

In Bénard convection, the cells can take various shapes, including hexagonal and square forms. For simplicity, let's focus on square cells. These cells have the same width and height, leading to an easy-to-follow pattern.

The width of the cell, denoted as \(L\), equals the distance between the two surfaces, \(d\). Therefore, each square cell has a dimension of \(d \times d\). The formation of square cells means that convective currents generate a neatly ordered pattern, where the fluid rises at the center of the cell and descends along the walls.

This specific configuration helps simplify the visualization and calculations involved in understanding Bénard convection.

The velocity distribution within a Bénard convection cell varies depending on the location within the cell. In our scenario, the maximum vertical velocity, \(v\), occurs at the cell walls where the fluid rises and sinks.

To get an expression for the velocity, we assume that the fluid's speed is at its peak at these locations. This velocity depends on factors like the temperature gradient and the properties of the fluid itself. Using fluid dynamics principles, we can create a formula for \(v\), which will typically be proportional to \(d\), the distance between the surfaces, and the temperature difference across the fluid.

Visualize the velocity pattern as having a peak at the edges of the square cell, tapering off towards the center where the fluid's horizontal movement is more dominant.