Figure P6.3 depicts flow in a horizontal parallel-plate channel in which v_x = U at the upper wall, the lower wall is fixed, and P is independent of x. This is plane Couette flow, as introduced. The wall spacing is H, the length under consideration is L, and the width (not shown) is W. Assuming that the flow is steady, laminar, and fully developed, the local velocity is found to be (Example 7.3-1)

(a) Calculate each component of τ for this flow.

(b) Use a force balance on the control volume in Fig. P6.3 to confirm that the average pressures at surfaces S3 and S4 (x = 0 and x = L, respectively) are equal. In using the results from part (a) to calculate the x component of the viscous force, be careful with the directions of the four unit normals and with the algebraic signs.

Transcribed Image Text:

vx(y) = y Uy H S3 V₂ = 0 = V₂. S S₂ H SA U (P6.3-1) g L Figure P6.3 Plane Couette flow in a horizontal channel. The control volume for an overall force balance is shown by the dashed lines.