Chapter 5, Problem 5.11P

(a)

Interpretation Introduction

Interpretation:

The value of branch breadths $a\text{ and }b$ and total force on the plate are to be determined.

Concept Introduction:

The general continuity equation to be used for a controlled volume is:

  ${\displaystyle {\iint }_{C.S}\rho \left(v\cdot n\right)dA}+\frac{\partial }{\partial t}{\displaystyle {\iiint }_{C.V}\rho dV}=0$   ...... (1)

The general equation written for the overall linear momentum balance for a controlled volume is:

  ${\displaystyle \sum F}={\displaystyle {\iint }_{C.S}v\rho \left(v\cdot n\right)dA}+\frac{\partial }{\partial t}{\displaystyle {\iiint }_{C.V}\rho vdV}$   ...... (2)

Here, C.S is the control surface over which the integral dA is taken, C.V is the control volume over which the integral dV is taken, $\rho$ is the density of the fluid, $v$ is the velocity vector, $n$ is the direction of the vector $v$ , the product $v\cdot n$ is scalar defined as:

  $v\cdot n=\left|v\right|\left|n\right|\cos \theta$

(b)

Interpretation Introduction

Interpretation:

The value of $l$ is are to be determined.

Concept Introduction:

The general equation written for the total moment of all forces acting on a controlled volume is:

  ${\displaystyle \sum M}={\displaystyle {\iint }_{C.S}\left(r\times v\right)\rho \left(v\cdot n\right)dA}+\frac{\partial }{\partial t}{\displaystyle {\iiint }_{C.V}\left(r\times v\right)\rho dV}$   ...... (5)

Here, C.S is the control surface over which the integral dA is taken, C.V is the control volume over which the integral dV is taken, $\rho$ is the density of the fluid, $v$ is the velocity vector, $n$ is the direction of the vector $v$ , $r$ is the position vector, $v\cdot n$ is a dot product and $\left(r\times v\right)$ is a cross product.