Calculate the shear stress at the wall y 0 when u 10 x2 3

Calculate the shear stress at the wall (y = 0) when u = 10 (x^2 + 3xy + 2y^2) and v = 30 (4xy + y^2). Show the results in terms of x and appropriate properties.

Solution

Let the dynamic viscosities are ₁ N-s/m² and ₂ N-s/m² for the two velocity profiles respectively.

We know, shear stress for Newtonian fluid= *(du/dy)   -------- (i)

Where, = dynamic viscosity; u = velocity of fluid layer and y = distance of fluid layer from the wall.

For the velocity profile u =10(x² + 3xy + 2y²),

            du/dy = 10(0 + 3x + 4y) [differentiating both sides w.r.t. y]

=> du/dy_(y=0) = 10(3x + 4*0) = 30x

From the equation (i), the shear stress at the wall = ₁*(30x) = 30₁x.

For the velocity profile v =30(4xy + y²),

            dv/dy = 30(4x + 2y) [differentiating both sides w.r.t. y]

=> dv/dy_(y=0) = 30(4x + 2*0) = 120x

From the equation (i), the shear stress at the wall = ₂*(120x) = 120₂x.

Hence, the shear stresses on the wall are 30₁x N/m² and 120₂x N/m² respectively.