3.An incompressible Newtonian laminar flow with the universal velocity, u, passe

Question

3.An incompressible Newtonian laminar flow with the universal velocity, u, passes through an infinite parallel plate from the leading edge (see Figure 3). The origin of coordinate is at the leading edge of the plate. We assume that the thermal and physical fluid properties are constant. A two-dimensional laminar boundary layer with the thickness, ô , can be formed, the velocity in the layer can be written as where a, b, and c are constant. Determine the coefficient, a, b, and c using the non-slip boundary conditions, velocity at free boundary edge of the boundary layer, and the first-order and second differential of velocity boundary conditions at edge of the boundary layer. Using the von Karman profile method to solve the integral momentum equation, determine the boundary layer thickness, the displacement thickness, the momentum thickness, and local skin friction coefficient. (a) (b) Derive an expression for the y component velocity v in the boundary layer (c) A very thin long rod is put in parallel to the leading edge and at the position with a distance of L from the leading edge. The rod is just at the free boundary layer as illustrated in Figure 3. Determine the local force direction to hold the rod. Please also show it with schematics 14 Figure 3

Explanation / Answer

By given velocity profile,

(u/U)=a+by+cy5/2+dy3................(1)

At y=0 Due to no slip condition velocity of fluid which is in contact with plate also zero.

by substituting the values in equation 1 we get,

'a'=0 ..........answer

Now,

du/dy=U[b+(5/2)cy3/2+3dy2]

Boundary condition,

At y=delta

du/dy=0........(that is no velocity gradient)

b=(5/2)c(delta)3/2+3d(delta)2

At y=delta u=U

b(delta)=-c(delta)5/2

From above we get values of b and c

Displacement thickness (delta)*=int 0 to infinity(1-u/U)dy

Momentum thickness (theta)=int 0 to infinity[(u/U)(1-u/U)]dy

Energy thickness (delta)**=int 0 to infinity[(u/U)(1-(u/U)2)]dy

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