Shear Flow Around a Circular Cylinder

Fixed cylinder. Let us consider a fixed circular cylinder in an arbitrary steady-state linear shear flow of viscous incompressible fluid in the plane normal to the cylinder axis. The velocity field of such a flow remote from the cylinder in the Cartesian coordinates X, X% can be represented in the general case as follows  

The shear tensor in (2.7.8) can be represented as the sum of the symmetric and antisymmetric tensors G = E + fl that correspond to the straining and rotational components of the fluid motion at infinity, 

In the Stokes approximation (as Re — 0), the solution of the corresponding hydrodynamic problem with the boundary conditions (2.7.8) at infinity and noslip conditions at the boundary of the cylinder (V = 0 for 1Z = a) leads to the stream function [93,166] 

These expressions are written in the coordinates TZ, 0 obtained from the initial point by revolution by the angle Ad and related to the principal axes of the symmetric tensor E (in the principal axes, the tensor E can be reduced to the diagonal form with diagonal entries E and -E). Purely straining shear corresponds to the value fi = 0, and simple shear is determined by the parameters E = 0 and E2 = — fi. 

The fluid velocity field can be obtained by substituting (2.7.9) into (1.1.8). 

---