Solution Found!
Combine your results from below to form the
Chapter 4, Problem 98P(choose chapter or problem)
Problem 98P
Combine your results from Problem given below to form the two-dimensional strain rate tensor in the ɛij in the xy-plane,
Are the x- and y-axes principal axes?
PROBLEM: 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 4 - 100: 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.)
FIGURE P11-48
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?
Questions & Answers
QUESTION:
Problem 98P
Combine your results from Problem given below to form the two-dimensional strain rate tensor in the ɛij in the xy-plane,
Are the x- and y-axes principal axes?
PROBLEM: 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 4 - 100: 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.)
FIGURE P11-48
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?
ANSWER:
Solution to 98P
Step 1
We need to find the two dimensional strain rate tensor for a two-dimensional Poiseuille flow with the velocity field equation as given below,
Where, 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.
The strain rate tensor εij in the x and y plane is given by,
Where, εxx and εyy are the linear strain rates in x and y directions respectively and εxy and εyx are the shear strain rates in x and y directions respectively.
By symmetry,
εxy=εyx