A Newtonian fluid (viscosity μ) is flowing through a narrow slit formed by two horizontal parallel plates (length: L; width: W), under steady state and laminar flow conditions. The distance between the plates is ‘2h’. The fluid is flowing in z-direction due to pressure gradient ΔP/L. Assume two-dimensional flow. Obtain the expression of (a) the velocity profile (b) the average velocity, and (c ) the z- component of the force of the fluid at the solid surface.?

Question

Answer

Velocity Profile:
The velocity profile describes how the velocity of the fluid varies across the slit formed by the parallel plates. In this case, we can assume that the flow is fully developed and laminar, meaning that the fluid velocity does not change with time and the fluid layers move parallel to each other.

In a two-dimensional flow, the velocity of the fluid is only dependent on the z-coordinate (direction of flow). We can express the velocity profile as a function of z, denoted as V(z).

Assuming Newtonian Fluid:
For a Newtonian fluid, the velocity profile follows a parabolic shape due to the no-slip condition at the solid surfaces. At the center of the slit (z = 0), the velocity is maximum, and it gradually decreases towards the solid surfaces (z = ±h).

Deriving the Expression:
To derive the expression for the velocity profile, we can apply the Navier-Stokes equation for an incompressible fluid under steady-state conditions:

dP/dz = μ * d^2V(z)/dz^2

where dP/dz is the pressure gradient and μ is the viscosity of the fluid.

Integrating this equation twice, we get:

V(z) = (dP/dz) * (h^2/2μ) - (z^2/2h^2) * (dP/dz)

This expression represents the velocity profile of the fluid in the z-direction.

Average Velocity:
The average velocity of the fluid can be obtained by integrating the velocity profile over the entire slit:

V_avg = (1/LW) * ∫[0,L] ∫[-h,h] V(z) dz dx

Simplifying the integral, we get:

V_avg = (dP/dz) * (h^3/3μ)

This expression gives the average velocity of the fluid in the z-direction.

Z-Component of the Force:
The z-component of the force of the fluid at the solid surface can be calculated by integrating the shear stress over the surface area of the plates. The shear stress is given by:

τ = μ * dV(z)/dz

Integrating the shear stress over the surface area, we get:

F_z = -2W * ∫[0,L] μ * dV(z)/dz dx

Simplifying the integral, we obtain:

F_z = -2W * μ * V_avg

This expression represents the z-component of the force of the fluid at the solid surface.

Conclusion:
In summary, the velocity profile of a Newtonian fluid flowing through a narrow slit formed by two parallel plates follows a parabolic shape. The average velocity is determined by the pressure gradient and the distance between the plates. The z-component of the force of the fluid at the solid surface is proportional to the viscosity of the fluid and the average velocity.

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