Pressure vs speed in an immersed object

[serbring]

Hi all,

In case of a fully immersed and standing object (i.e. a plate or sphere) in a steady moving fluid (i.e. water or oil), what is the type of the mathematical relationship between the pressure of a point of the object? Might exponential be right? I need this information because I measured the pressure on an moving plate inside a granular material (particles are in order of a hundred thousandth of the smallest size of the plate) and I found out that the relationship is close to be exponential so I want to make a comparison with fluid dynamics because the field is much more studied.
Hopefully I have well stated the question, if not please give the details you need.

Thanks

 

[boneh3ad]

Yeah I'm not following you exactly. What do you mean by "pressure of a point of the object"? Can you maybe draw a picture of what you are doing?

 

[serbring]

Look at this picture

For pressure on point, I meant the pressure over the surface of the sphere or a plate. In case of the shpere it may be described by $\theta$

 

[boneh3ad]

If you ignore the effects of viscosity, and assuming the flow is steady, then the relationship follows Bernoulli's equation:
$$p_1 + \dfrac{1}{2}\rho v_1^2 = p_2 + \dfrac{1}{2}\rho v_2^2.$$
In this case, let the 1 conditions be the free stream (call the pressure $p_{fs}$ and velocity $U$) and the 2 conditions be against the surface of the body, then the pressure is going to be
$$p = p_{fs} + \dfrac{1}{2}\rho\left( U^2 - v^2 \right).$$

What exactly those pressures will be depends on the velocity distribution. The pressure will vary with the square of the velocity, though. I am not familiar enough with granular flows to be able to tell you if there is some exponential relationship in those cases.

 

[serbring]

Thanks for your reply. I didn't know, Bernoulli's equation is valid also for external fluid flows. I found out also this formula: http://s12.postimg.org/dbr17jyt9/Immagine.png
If that it is true the pressure is linear with the speed, right? That it is different from what it is predicted by Bernoulli's equation, right?

Actually, looking more into the data, a quadratic model can fit rather well the data as it is predicted by Bernoulli's equation.

 

[Baluncore]

If that it is true the pressure is linear with the speed, right? That it is different from what it is predicted by Bernoulli's equation, right?
Actually, looking more into the data, a quadratic model can fit rather well the data as it is predicted by Bernoulli's equation.

Pressure is proportion to fluid density multiplied by the square of the speed.