# Bingham fluid in a falling film

• slyman
In summary, for a Bingham fluid in a falling film, the equation for τxz can be found using a shell balance, the minimum thickness of film that allows flow can be calculated using the Bingham fluid equation and correct boundary conditions, and the volumetric rate of flow can be solved for using the velocity profile.
slyman
2) For Bingham fluid in a falling film. $$\tau$$0.
a) Find the equation for $$\tau$$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?

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.

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 \

## 1. What is a Bingham fluid?

A Bingham fluid is a type of non-Newtonian fluid that exhibits both solid-like and liquid-like behavior. This means that it has a yield stress (a minimum stress that must be applied for the fluid to flow) and a viscosity that is independent of the shear rate.

## 2. What is a falling film?

A falling film is a type of flow in which a thin layer of fluid flows down a vertical or inclined surface due to gravity. This is commonly seen in industrial processes, such as in heat exchangers or coating applications.

## 3. How does a Bingham fluid behave in a falling film?

In a falling film, a Bingham fluid will initially behave like a solid until the yield stress is exceeded and it begins to flow like a liquid. As the film falls, it will maintain a constant thickness and velocity until it reaches the bottom surface.

## 4. What factors affect the behavior of a Bingham fluid in a falling film?

The behavior of a Bingham fluid in a falling film is influenced by factors such as the fluid's yield stress, viscosity, and density, as well as the surface geometry, gravity, and flow rate. Other factors such as temperature and surface tension may also play a role.

## 5. How is the behavior of a Bingham fluid in a falling film different from a Newtonian fluid?

A Newtonian fluid, such as water, has a constant viscosity regardless of the applied stress or shear rate. In contrast, a Bingham fluid has a yield stress and its viscosity can vary with the shear rate. This results in different flow behavior and patterns in a falling film for each type of fluid.

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