The velocity distribution for laminar flow between parallel plates is given by: u u max...

Messiah Sutton

Messiah Sutton

Answered

2022-11-09

The velocity distribution for laminar flow between parallel plates is given by:
u u max = 1 ( 2 y h ) 2
where h is the distance separating the plates and the origin is placed midway between the plates. Consider a flow of water at 15 C , with u max = 0.30 m s e c and h = 0.50   m m . Calculate the shear stress on the upper plate and give its direction.

Answer & Explanation

dobradisamgn

dobradisamgn

Expert

2022-11-10Added 17 answers

Step 1
Given data:
The velocity distribution for laminar flow is,
u u max = 1 ( 2 y h ) 2
The temperature at which the flow is to be considered is,
T = 15 C
The value of maximum velocity is,
u max = 0.30   m / s
The value of the distance between the plates is,
h = 0.50   m m
Origin is midway between the plates, so the distance of upper plate from the axis is,
y = h 2 = 0.25   m m
Step 2
The shear stress in terms of velocity for a Newtonian fluid like water is given by formula,
τ = μ d u d y
Here, τ represents the shear stress and μ is the dynamic viscosity of the flow of water which is known to be 1.14 × 10 3   k g × m 1 × s 1 at given temperature.
Substitute the expression for u to find the expression for shear stress.
Therefore,
τ = μ d d y ( u max ( 1 4 y 2 h 2 ) ) = 8 μ u max y h 2
Substitute the known values,
τ = 8 × 1.14 × 10 3 k g   m 1 s 1 0.30 m / s × 0.25   m m ( 10 3 m m m ) ( 0.50   m m × ( 10 3 m m m ) ) 2 = 2.74 k g × m 1 × s 2 ( N k g × m × s 2 ) = 2.74 N × m 2 = 2.74 P a
The negative sign shows that the shear stress on the upper plate is opposite to the direction of the flow of water.
The shear stress on the upper plate is, τ = 2.74 P a acting opposite to the direction of laminar flow of water.

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