Impulse on a water bag being dropped

[abrewmaster]

I have been trying to derive an equation to represent how a bag full of fluid would react when dropped dependent on the fluids viscosity. Basically i have the general impulse equation: F∂t=∂P but the time is dependent on the fluids viscosity (higher viscosity the longer it takes for the impact to finish) so I tried using the equation of viscosity: η=F/A*t but in this case of the equation I have a changing area that is dependent on the time as well as the dimensions of the bag. I don't have the slightest idea how to model this varying area since it is an nonuniform shape. Anyone have any ideas how to model this?

 

[Chestermiller]

I have been trying to derive an equation to represent how a bag full of fluid would react when dropped dependent on the fluids viscosity. Basically i have the general impulse equation: F∂t=∂P but the time is dependent on the fluids viscosity (higher viscosity the longer it takes for the impact to finish) so I tried using the equation of viscosity: η=F/A*t but in this case of the equation I have a changing area that is dependent on the time as well as the dimensions of the bag. I don't have the slightest idea how to model this varying area since it is an nonuniform shape. Anyone have any ideas how to model this?

Hi abrewmaster. Welcome to Physics Forums.

This is a difficult fluid mechanics problem, and would probably require solution of the partial differential fluid mechanics equations (Navier Stokes equations) or turbulent flow approximations, even if the bag were assumed to be spherical and the deformation were assumed to be axisymmetric. You would also probably have to model the deformational mechanics of the bag, and its interaction with the fluid. I'm guessing that modeling this problem would require use of finite element fluid mechanics software coupled with finite element solid mechanics software. The moving boundaries would add even more complexity to the analysis.

 

[abrewmaster]

I was afraid of something like that although I already knew that was probably the answer anyway. I will probably just make a lot of assumptions for something like this to save time during the calculations to get an estimate that makes sense and just work from there.

 

[Baluncore]

There may be a model already available for this type of problem. I see two situations that might have similarities. Firstly the impulse sound of rain drops falling on a hard surface, and secondly, flood modelling.

In flood modelling a sphere of water equal to a reservoir's volume is released at one point on the landscape to model the catastrophic failure of the containment structure. For the real world, finite element analysis is used for flood modelling, but the time scale needs to be checked against a simple theoretical model. That model would include viscosity. You might google { dam burst flood modelling } and trace references back to verification of the model.

 

[Chestermiller]

I was afraid of something like that although I already knew that was probably the answer anyway. I will probably just make a lot of assumptions for something like this to save time during the calculations to get an estimate that makes sense and just work from there.

Here are some thoughts along these lines. You can bound the answer by considering the maximum and minimum possible changes in momentum that can occur. After impact, the upward velocity can't be greater than the impact velocity, and it can't be less than zero (i.e., the ball sticking to the ground). So an upper bound to the impulse of the force is 2mv, and a lower bound is mv. Taking the average (1.5 mv) might be adequate for your purposes.

Chet