(1) VISCOUS FLOWS – AE 4120 1
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.

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 zero