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.
a. 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).
b. 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.
c. Show that the flow field of (2) is identical to that of (1) and can be obtained by a rotation of the coordinate frame over an angle of 45o. Compare the stress situations that have been derived for each case above, and use this to determine the consequence which the orientation of the fluid element with respect to the flow field has on the stress situation.
d. As the velocity field is irrotational (potential flow), it satisfies the flow equations for both inviscid and viscous flow, and the pressure can be obtained from Bernoulli’s relation (see White, section 2–10).
– Compute the pressure field p(x,y) and determine the shape of the isobars in the flow.
– Compute explicitly the components of the gradient of the viscous stress tensor ij j / x ; use this result to explain how the “inviscid potential flow” can also satisfy the viscous flow equations, even though the viscous stresses are not
