For the two-dimensional Poiseuille flow of Prob. 4-100, calculate the linear strain rates in the x- and y-directions, and calculate the shear strain rate ɛxy.
PROBLEM: Consider fully developed two-dimensional Poiseuille flow—flow between two infinite parallel plates separated by distance h,with both the top plate and bottom plate stationary, and a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–48. (dP/dxis constant and negative.)
The flow is steady, incompressible, and two-dimensional in the xy-plane. The velocity components are given by
where μ is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the z-direction. Do fluid particles in this flow rotate clockwise or counterclockwise?
Solution to 97P
We need to find the linear strain rates in x and y directions and the shear strain rate εxy for the velocity field equation given below.
u is the component of velocity in x direction, v is the component of velocity in y direction, is the pressure gradient, μ is the fluid viscosity, h is the height.
The fluid is assumed to be incompressible, steady and two dimensional.
Linear strain rates:
We know that the linear strain rates are given by the following equations,
Where is the linear strain rate for a given coordinates, u and v are velocity components in the given direction.
Linear strain rate in x direction:
Linear strain rate in y direction:
Thus linear strain rates in both the directions is zero.