How Does Temperature Variation Affect Surface Tension and Flow in a Liquid Film?

[nitsan]

could not solve this exam problem completely please help:
Variations in surface tension can cause a flow in a liquid.One of the chief causes for variation of surface tension is temperature.Consider a thin film of liquid on a flat surface.The liquid film also has a free surface where the temperature varies with x.There is no variation in the z direction, the flow is 2 dimensional.Assume the flow to be incompressible and the temperature has no other effect but to change the surface tension.

a) Determine T(x) for fully developed flow?
b) What is the pressure gradient within the liquid film?
c) set up the governing equations.What are the boundary conditions?
d) Solve to get the velocity profile?
e) Obtain the non-dimensional parameters characterizing this flow?

 

[Jhero]

Hello! I am a scientist and I would be happy to help you with this exam problem.

a) To determine T(x) for fully developed flow, we need to use the energy equation for the liquid film. This equation takes into account the temperature variation and its effect on surface tension. By solving this equation, we can obtain the temperature profile T(x).

b) The pressure gradient within the liquid film can be obtained by using the Navier-Stokes equations, which describe the conservation of momentum in a fluid. The pressure gradient is related to the velocity profile and the viscosity of the liquid.

c) The governing equations for this problem would be the Navier-Stokes equations for an incompressible fluid, along with the energy equation for the temperature variation. The boundary conditions would include the no-slip condition at the surface of the flat surface, and the continuity of velocity and temperature at the free surface.

d) To solve for the velocity profile, we would need to use numerical methods such as finite difference or finite element methods. These methods involve discretizing the governing equations and solving them iteratively to obtain the velocity profile at different points in the liquid film.

e) The non-dimensional parameters characterizing this flow would depend on the specific geometry and properties of the liquid. Some possible parameters could include the Reynolds number, which describes the ratio of inertial forces to viscous forces, and the Prandtl number, which describes the ratio of momentum diffusivity to thermal diffusivity. These parameters can help us understand the behavior of the flow and make comparisons with other similar systems.