Determine also for each case which (viscous) normal and tangential stresses work on the sides of the fluid element.

PROBLEM 1.E – Deformation and vorticity

Each of the expressions given below represents a two–dimensional incompressible flow field, with
velocity components u and v in resp. the x– and y–direction of a Cartesian coordinate frame:

(1) u = Ax v = –Ay

(2) u = Ay v = Ax

Here A is a constant, with dimension of [velocity/length]. Assume that A has a positive value.

Show that both velocity fields are irrotational and satisfy the (incompressible) continuity equation.

Determine for each flow field the shape of the streamlines en sketch the flow pattern (indicate the flow direction).

Determine for each of these flow fields how a square fluid element is transported and deformed by the flow over a small time interval Δt. Make a clear sketch for each case. Determine also for each case which (viscous) normal and tangential stresses work on the sides of the fluid element. Indicate these stresses in the sketch, noting the proper direction of the stresses and their relative magnitude.