Question

An incompressible fluid is flowing between two infinitely large parallel plates separated by 5 mm distance. The bottom plate is stationary and the top plate is moving at a constant velocity of 5 mm/s in the direction parallel to the bottom plate. The flow of the fluid between the plates is steady, two-dimensional, laminar, and the variation of fluid velocity is linear between the plates. A square fluid element of 1 mm side is considered at equal distance from both the plates in the flow field such that one of its sides is parallel to the plates. The magnitude of circulation in mm\(^2\)/s (in integer) along the edges of the square fluid element is ...........

Hint:

For flow between parallel plates with linear velocity variation, the circulation around a square fluid element is proportional to the size of the element and the velocity difference between the plates.

Answer 1

In the scenario where fluid flows between two parallel plates, the velocity distribution is linear due to the steady, incompressible, and laminar nature of the flow. The velocity of the fluid varies from zero at the stationary plate (bottom plate) to the maximum velocity at the moving plate (top plate). Since the velocity gradient is linear, the velocity at any point between the plates can be expressed as: \[ V = \left( \frac{V_{{top}}}{h} \right) y \] Where:
- \( V_{{top}} = 5 \, {mm/s} \) is the velocity of the top plate,
- \( h = 5 \, {mm} \) is the distance between the plates,
- \( y \) is the distance from the bottom plate.
Given this, the velocity at any point between the plates will be proportional to the distance from the bottom plate.
Now, to calculate the circulation along the edges of the square fluid element, we use the definition of circulation, which is the line integral of velocity around the boundary of the square fluid element. In this case, the fluid element is placed such that one of its sides is parallel to the plates. The magnitude of the circulation can be calculated by integrating the velocity along the path of the fluid element.
Since the velocity variation is linear, the circulation is directly related to the fluid velocity difference between the top and bottom plates and the size of the square fluid element. The fluid element has a side length of 1 mm, and the velocity difference between the plates is \( 5 \, {mm/s} \). The magnitude of circulation for this fluid element can be determined to be 1 mm\(^2\)/s.
Thus, the magnitude of circulation along the edges of the square fluid element is \( 1 \, {mm}^2/{s} \).