Lecture - 7 (shear force required for linear velocity profile)

Shear force required for linear velocity profile 

Let us consider a thin plate moving over the fluid surface with velocity 'V'. The distance between the bottom of the fluid and surface of fluid is represented by 'y' and this distance is very small. In the below diagram the deformation of fluid particles is shown which is due to the force represented as 'F'. Now take two layers of fluid into consideration and distance between these two layers is represented by 'dy'. 'θ' represents the angle of deformation of fluid particles. In the figure notice that triangle ABC is formed between the plate and bottom of the fluid and triangle A'B'C' is formed between two layers of fluids. Distance AC is represented by 'Vt' because
Distance (AC) = velocity (V) * time (t)
Similarly distance A'C' is represented by 'dut'. 'du' is the differential speed. 
velocity profile,mechanics,fluids,shear force

In ΔABC 
Tan θ = Vt/y 
In ΔA'B'C' 
Tan θ = dut/dy
∴ Vt/y = dut/dy 
    V/y = du/dy 
From Lecture - 4 we know that τ = μ*(du/dy)
Therefore by replacing 'du/dy' with 'V/y' we get, 
τ = μ*(V/y) 
shear force 'F' = shear stress 'τ' * area 'A'
∴ F = (μAV)/y  
Here 'μ' represents dynamic viscosity
'A' represents surface area in contact with fluid
'V' represent velocity with which plate is moving
'y' represents thickness of fluid film

very important assumption is used in above derivation that we are considering a linear velocity profile instead of parabolic velocity profile because distance between plate and bottom of fluid is very small. Theoretically we assume linear velocity profile but actually it is parabolic velocity profile.
velocity profile,mechanics,fluids,shear force


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