Test: Laminar Flow Between Plates - Civil Engineering (CE) MCQ

The laminar flow between two parallel fixed plate



Shear stress on the boundary


Shear Stress at Boundary:

Q = 600× 10^-4 m³ /sec , B = 3×10^-2 m
μ = 1.2 pa -sec

τ  = 480 pa

The velocity distribution across a section of two fixed parallel plates is parabolically given by

where ∂P/∂x = pressure gradient along the length of the plate
y = point of consideration from lower fixed plate
t = distance between the two fixed parallel plates

From the above velocity and shear stress distribution, the shear stress is zero at the centre where velocity is maximum.

For the flow between two parallel spaced plates (as shown in the figure below): 

The velocity distribution is given by:

Shear stress is given by:

Hence, shear stress varies directly as the distance from the midplane.
Thus, the stress distribution across a section is depicted below:

∴ The shear stress is maximum at the boundaries and zero at the centre.

Concept:
Viscous flow in Plates:

Velocity equation:

where t = distance between two plates and y = distance measured from lower plate towards the upper plate.

Shear stress equation:

Maximum velocity happens at, y = t/2, which is given by 
Ratio of Maximum velocity to the Average velocity in case of plates:

Calculation:
Given:
Parallel plates:
U_max = 6 ms^-1, Uavg = ??

Shear stress

Concept:
The tangential shear stress between two adjoining is proportional to velocity gradient in direction of perpendicular to the layer. This is known as law of viscosity.

Shear force acting on the moving plate is given by F = τ × A
Power required to maintain velocity U is given by P = F × U

Calculation:
Given μ = 0.001 Ns/m², U = 0.3 m/s, Y = 0.01 cm = 0.01 × 10^-2 m, A = 0.1 m.;
Velocity gradient or rate of shear strain is given by

τ = 0.001 × 3000 = 3 N/m²
F = 3 × 0.1 = 0.3 N,
P = 0.3 × 0.3 = 0.09 W

Reynold's number: 

• It is a dimensionless number that determines the nature of the flow of liquid through a pipe or flat plate. 
• It is defined as the ratio of the inertial force to the viscous force for a flowing fluid.
• Reynold's number is written as Re.
• 

Flat plate flow:

• If Reynold's number lies below 5 × 10⁵, then the flow of liquid is streamlined or laminar.
• If Reynold's number is greater than or equal to 5 × 10⁵, the flow of liquid is turbulent.

The velocity distribution across a section of two fixed parallel plates is parabolically given by

where ∂P/∂x = pressure gradient along the length of the plate
y = point of consideration from lower fixed plate
t = distance between the two fixed parallel plates

Note: The shear stress distribution across a section of two fixed parallel plates is Linear.

Concept:
Laminar flow between the parallel fixed plate

Velocity distribution equation is given by:

Where,
u = velocity of the fluid at any distance y from the boundary
H = Distance between two parallel fixed plates.
Calculation:
H = 6 mm, ucentre = 1.8 m/sec

Now, at y = 1 mm

u = 1 m/s.

For the flow between two parallel spaced plates (as shown in the figure below): 

The velocity distribution is given by:

Shear stress is given by:

Thus, the stress distribution across a section is depicted below:

∴ The shear stress is maximum at the boundaries and zero at the center.