# Pressure in a fluid at rest

I hope this is the right place to post such a question. I'm studying Continuum Mechanics from Gurtin's book, and so far in my class we've seen Cauchy's the about the existence of stress, and nothing about fluids yet. I know this exercise is easy, but all I know so far is Cauchy's thm and I am supposed to use it only.

Let $$B$$ a body immersed in a fluid at rest, under the action of gravity. Assume that the stress tensor $$T$$ is a pressure, i.e. by $$T= - \pi I$$. Determine the pressure field $$\pi$$.

First of all, I can write the equation of motion: $$\operatorname{div}(T) +b-\rho \dot v =0$$

Since the body is at rest, $$v=0$$. Also, the only body force acting on the system is the gravity, so $$b= \rho g$$. I'd say that the equation becomes

$$\operatorname{div}(- \pi I) + \rho g =0$$

but now I don't know how to solve this for $$\pi$$. Any hint or help is highly appreciated.

EDIT

By expanding the divergence of $$- \pi I$$, I found:

$$- \frac{\partial \pi }{\partial x_1} = 0$$ $$- \frac{\partial \pi }{\partial x_2} = 0$$ $$- \frac{\partial \pi }{\partial x_3} + \rho g=0$$

and from the last one I have $$\pi(x_1,x_2,x_3)=+\rho g x_3$$ which indeed has the dimensions of a surface force density, as expected.

Is this correct?

• Buoyancy is not present? Or is that included in your $\nabla \cdot \mathbf{T}$? Commented Feb 26, 2021 at 23:38
• I think it's not present, because so far we've not talked about fluids and buoyancy @honeste_vivere Commented Feb 27, 2021 at 8:51