Two incompressible, viscous fluids having the some densities but different visco

Question

Two incompressible, viscous fluids having the some densities but different viscous are contained between two infinite, horizontal parallel plates. The bottom plate is fixed and the upper plate moves with a constant velocity U. Determine the velocity at the Interface. Express your answer in terms of U and fluid viscosities. The motion of the fluid is caused entirely by the movement of the upper plate, that is. there is no pressure gradient in the x-direction. The fluid velocity and shear stress are continuous across the interface between the two fluids.

Explanation / Answer

Since there is no pressure gradient in the x direction,for any fluid element dxdydz

Fx =0: y dx dz - y+dy dx dz =0    =>    y = y+dy

i.e., the shear stress is constant along the y direction.

From = du/dy, we have

u1 = /1 y +C1      and   u2 = /2 y +C2

Use boundary condition, u1(2h) = U,   u2(0) = 0 and the continuity condition at h: u1(h) = u2(h),

U = /1 (2h) +C1                   (1)

0 =/2 (0) +C2                      (2)

/1 h +C1   = /2 h +C2         (3)

From (2), we have C2=0, and then from (1) and (3), we have

=U 12/[h(1 + 2) ]                 C1=U(1- 2)/(1 + 2)

Therefore,

u1 =U2/[h(1 + 2)] y +U(1- 2)/(1 + 2)

u2 = 1/[h(1 + 2) ] y

The velocity at interface is when y = h, that is

uinter = u1(h) = u2(h) = U1/(1 + 2)

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.