Chapter 5, Problem 5.10P

Interpretation Introduction

Interpretation: The value of forces ${\text{F}}_{\text{x}}$ and ${\text{F}}_{\text{y}}$ required to maintain its position should be calculated for the given condition.

Concept introduction:

Incompressible flows are those in which density remains constant within the fluid material. The equation for steady state incompressible flow with respect to the conservation of momentum is given as,

  ${\displaystyle \sum {\text{F}}_{\text{n}}}={\displaystyle \underset{c.s}{\iint }{v}_x\text{ρ}\left(\text{v}\cdot \text{n}\right)}\text{dA}$   ..........…......(1)

Where,

$v_x$ is the velocity in x direction.

  $\text{ρ}$ is the density.

  $\text{dA}$ is the differential change in area.

  ${\text{F}}_{\text{n}}$ is the force normal to the plate.

Considering the momentum equation for no drag on plate and water as fluid,

  ${\text{F}}_{\text{n}}=-\stackrel{•}{v}\cdot \text{ρ}\left(v\cos \text{θ}\right)$   ....................(2)

Using the equation for density,

Density is defined as the ration of mass per unit volume,

  $\text{ρ}=\frac{\stackrel{\cdot }{m}}{v}$

  $\text{ρ}\times v=\stackrel{\cdot }{m}$

Replacing the term $\text{ρ}\times v=m$ in equation (2),

  ${\text{F}}_{\text{n}}=-\stackrel{\cdot }{m}\left(v\cos \text{θ}\right)$   ....................(3)

By using equation (1) and (3) the net force in n- direction is given as,

  ${\displaystyle \sum {\text{F}}_{\text{n}}}={\text{F}}_{\text{n}}$ .