Chapter 5, Problem 5.1P

Interpretation Introduction

Interpretation: The equation for the laminar flow of a liquid between the 2 infinite parallel plates is to be proved.

Concept introduction: The flow in a liquid is said to be Laminar when the Reynold’s number is less than 2100. The equation for laminar flow can be derived by the force balance between the plates. The equation for the shear stress is a very useful measure here as it can help in achieving the final result. The shear stress equation is,

  $\text{τ}\text{=}\text{-μ}\frac{\text{du}}{\text{dy}}\mathrm{.......}\text{(1)}$

The notations used here are,

  $\begin{array}{l}\text{τ}\text{=}\text{Shear stress}\\ \text{μ}\text{=}\text{Fluid viscosity}\\ \frac{\text{du}}{\text{dy}}\text{=}\text{Strain}\text{rate}\end{array}$

Answer to Problem 5.1P

The equation for the flow of a laminar liquid between two infinite parallel plates is,

  ${\text{p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}\text{=}\frac{\text{12μ<mover accent="true"><mo>V</mo><mo>¯</mo></mover><mo>L</mo>}}{{\text{b}}^{\text{2}}}$ .

Explanation of Solution

The two infinite parallel plates can be given as,

  

In the given diagram,

b = Distance between the plates

L = Length of plate in direction of flow

The force balance between the two plates is given as,

  ${\text{2yp}}_{\text{a}}\text{-}{\text{2yp}}_{\text{b}}\text{=}{\text{2τ}}_{\text{v}}\text{L}\mathrm{.......}\text{(2)}$

In the above expression,

  $\begin{array}{l}{\text{p}}_{\text{a}}\text{=}\text{Pressure}\text{on}\text{one}\text{side}\text{of}\text{plate}\\ {\text{p}}_{\text{b}}\text{=}\text{Pressure}\text{on}\text{other}\text{side}\text{of}\text{plate}\\ \text{y}\text{=}\text{Distance}\text{on}\text{the}\text{plate}\end{array}$

Rearrange equation (2),

  $\frac{{\text{p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}}{\text{L}}\text{=}\frac{{\text{τ}}_{\text{v}}}{\text{y}}\mathrm{.......}\text{(3)}$

Substitute equation (1) in equation (3),

  $\frac{{\text{p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}}{\text{L}}\text{=}\frac{\text{-μdu}}{\text{ydy}}\mathrm{.......}(4)$

Integrate equation (4) with limits b/2 to y and 0 to u,

  $\begin{array}{l}\frac{{\text{(p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}\text{)}}{\text{2Lμ}}{\displaystyle \underset{\text{b/2}}{\overset{\text{y}}{\int }}\text{ydy}}\text{=}\text{-}{\displaystyle \underset{\text{0}}{\overset{\text{u}}{\int }}\text{du}}\\ \text{Solving}\text{above}\text{within}\text{given}\text{limits,}\\ \text{u}\text{=}\frac{{\text{(p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}\text{)}}{\text{2Lμ}}\left(\frac{{\text{b}}^{\text{2}}}{\text{4}}\text{-}{\text{y}}^{\text{2}}\right)\mathrm{.......}(5)\end{array}$

  $\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}$ , the average velocity between the plates is given by integrating equation (5),

  $\begin{array}{l}\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover><mo> = </mo>}\frac{\text{1}}{\text{b/2}}{\displaystyle \underset{\text{0}}{\overset{\text{b/2}}{\int }}\text{udy}}\\ \text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover><mo> = </mo>}\frac{\text{1}}{\text{b/2}}{\displaystyle \underset{\text{0}}{\overset{\text{b/2}}{\int }}\frac{{\text{(p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}\text{)}}{\text{2Lμ}}\left(\frac{{\text{b}}^{\text{2}}}{\text{4}}\text{-}{\text{y}}^{\text{2}}\right)\text{dy}}\end{array}$

Solving the above integrand within the given limits we get,

  $\begin{array}{l}\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}\text{=}\frac{{\text{(p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}{\text{)b}}^{\text{2}}}{\text{12Lμ}}\\ {\text{p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}\text{=}\frac{\text{12μ<mover accent="true"><mo>V</mo><mo>¯</mo></mover><mo>L</mo>}}{{\text{b}}^{\text{2}}}\mathrm{.......}(6)\end{array}$

Conclusion

The equation for the flow of a laminar liquid between two infinite parallel plates is,

  ${\text{p}}_{\text{a}}\text{-}{\text{p}}_{\text{b}}\text{=}\frac{\text{12μ<mover accent="true"><mo>V</mo><mo>¯</mo></mover><mo>L</mo>}}{{\text{b}}^{\text{2}}}$ .