Bingham fluid in a falling film

[slyman]

2) For Bingham fluid in a falling film. [tex]\tau[/tex]0.
a) Find the equation for [tex]\tau[/tex]xz as a function of x using a shell balance.
b) Solve for the minimum thickness of film that will allow flow of this Bingham fluid using the equation for a Bingham fluid and the correct boundary conditions.
c) Solve this equation for vz as a function of x for a fluid whose thickness is greater than the minimum.
d) Solve for the volumetric rate of flow Q.

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For part a, we know the shell balance is exactly the same for a Newtonian or non-Newtonian fluid so. So the equation is
τxz = ρ g x cos β

I'm having trouble with the rest of the questions. For part b we use the no slip boundary conditions but I don't know how to solve it for a bingham model. Can anyone help me out?

Thanks in advance.

 

[Valkarie]

For part b, the equation for a Bingham fluid is given by: τ = τ_0 + μ (∂vz/∂x) Where τ_0 is the yield stress and μ is the viscosity. Using the no-slip boundary condition at the wall, ∂vz/∂x = 0 at x = 0, we can solve for the minimum thickness of film that will allow flow of this Bingham fluid: τ = τ_0 => x_min = (τ_0 / ρ g cos β) For part c, we use the equation for a Bingham fluid to find the velocity profile as a function of x: v_z(x) = (τ_0/μ) x + C where C is a constant of integration. For part d, we can then use the velocity profile to solve for the volumetric rate of flow (Q): Q = ∫_0^L v_z(x) dx = (τ_0/μ) L^2/2 + CL where L is the total film thickness.

 

[piercas]

a) The equation for \tauxz as a function of x can be found by applying the shell balance to the falling film. The shell balance states that the shear stress at any point along the film is equal to the weight of the fluid above that point. In this case, we can write the equation as:

\tauxz = \rho g x cos \beta

where \rho is the density of the fluid, g is the acceleration due to gravity, x is the distance along the film, and \beta is the angle of inclination of the film.

b) The minimum thickness of film that will allow flow of a Bingham fluid can be found by setting the yield stress, \tau_0, equal to the shear stress at the wall, \tauxz. This gives us the equation:

\tauxz = \tau_0 = \rho g x cos \beta

Solving for x, we get:

x = \frac{\tau_0}{\rho g \cos \beta}

This is the minimum thickness of film that will allow flow of the Bingham fluid. Any film thickness less than this will result in no flow.

c) For a fluid whose thickness is greater than the minimum, we can solve the equation for \tauxz to find the velocity, vz, as a function of x. Using the Bingham model, the equation for \tauxz is:

\tauxz = \tau_0 + \mu \frac{dvz}{dx}

where \mu is the viscosity of the fluid. Substituting this into the equation from part a, we get:

\tau_0 + \mu \frac{dvz}{dx} = \rho g x cos \beta

Rearranging and solving for vz, we get:

vz = \frac{\rho g x^2 \cos \beta}{2 \mu} + \frac{C}{\mu}

where C is a constant of integration. This gives us the velocity as a function of x for a fluid whose thickness is greater than the minimum.

d) Finally, to solve for the volumetric flow rate, Q, we can integrate the velocity equation from part c over the entire thickness of the film. This gives us the equation:

Q = \int_0^h vz dx = \int_0^h \left( \frac{\rho g x^2 \cos \beta}{2 \