Problem Statement:
A smooth flat rectangular plate is placed edgewise in a stream of fluid. At what fraction of the length from the leading edge would the drag force on the front portion be equal to half of the total drag force? Assume the boundary layer to be laminar.

Solution:
To find the fraction of the length from the leading edge where the drag force on the front portion is equal to half of the total drag force, we need to analyze the flow around the flat rectangular plate.

1. Drag Force on a Flat Plate:
The drag force on a flat plate in a fluid flow is given by the equation:

F = 0.5 * ρ * V^2 * A * Cd

Where:
- F is the drag force
- ρ is the density of the fluid
- V is the velocity of the fluid
- A is the area of the plate perpendicular to the flow
- Cd is the drag coefficient

2. Flow Around a Flat Plate:
When a flat plate is placed edgewise in a stream of fluid, the flow pattern can be divided into two regions: the boundary layer and the wake.

2.1 Boundary Layer:
The boundary layer is the thin layer of fluid adjacent to the surface of the plate. It can be further divided into two regions: the laminar boundary layer and the turbulent boundary layer.

2.2 Laminar Boundary Layer:
In the laminar boundary layer, the fluid flows smoothly in parallel layers along the surface of the plate. The flow is characterized by smooth streamlines and low drag.

2.3 Turbulent Boundary Layer:
In the turbulent boundary layer, the flow becomes chaotic and irregular, resulting in increased drag.

3. Drag Force Distribution:
In the laminar boundary layer, the drag force is relatively low compared to the turbulent boundary layer. Therefore, to find the fraction of the length from the leading edge where the drag force on the front portion is equal to half of the total drag force, we need to consider the transition from laminar to turbulent flow.

4. Transition from Laminar to Turbulent Flow:
The transition from laminar to turbulent flow on a flat plate occurs around a critical Reynolds number. The Reynolds number is given by the equation:

Re = (ρ * V * L) / μ

Where:
- Re is the Reynolds number
- ρ is the density of the fluid
- V is the velocity of the fluid
- L is the characteristic length of the plate
- μ is the dynamic viscosity of the fluid

5. Calculation of Fraction of Length:
To find the fraction of the length from the leading edge where the drag force on the front portion is equal to half of the total drag force, we need to calculate the critical Reynolds number and determine the corresponding fraction of length.

6. Conclusion:
In conclusion, the fraction of the length from the leading edge where the drag force on the front portion is equal to half of the total drag force can be determined by analyzing the flow around the flat rectangular plate and considering the transition from laminar to turbulent flow. The critical Reynolds number can be used to calculate the fraction of length at which the transition occurs.