Consider a liquid in a tub and this tub is accelerated. And so the liquid will have a slanted surface. I am clear about finding pressure difference between two points in the fluid having same height with respect to horizontal. But what if the two points have a height difference with respect to horizontal? How to account for pressure difference due to this difference in vertical height?
[Physics] Pressure difference between 2 points in an accelerated fluid
fluid dynamicsnewtonian-mechanics
Related Solutions
What is the relation between pressure and velocity of a fluid in a closed pipe flow?
Bernoulli's equation:
By continuity equation velocity at all points is the same. Then shouldn't the pressure be same at both points?
No the pressure won't be the same at all points in the pipe. Considering Pascal's Law, a change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid. But it holds for static fluids essentially. And equation of continuity stems from the principle of conservation of mass but in this case the mass is not conserved per unit time. Consider 2 different cross-sections of the pipe and fluid flow through it per unit time is not the same since some component of $g$ along with some resultant force is acting on the water column at the 2 different cross-sections and the 2 different cross-sections have a distinct height difference.So equation of continuity is not VALID in this case and hence conclusions cannot be made from this.
Then how to find pressure difference between two points having height difference between them?
Use Bernoulli's principle and you get the relation $$P_2-P_1=(h_2-h_1)\rho g$$
The pipe is say tilted at an angle.
Use the components of pressure in that direction and use the above equation.
Pseudo-forces and/or accelerated frames simplify this mathematically, but here’s a way to understand it conceptually:
Consider a little bit of fluid in the middle. The pressure it exerts to the right has to be enough to accelerate all the fluid to the right. (“Has to” in the sense that fluid will flow around & seek levels so that this is true)
Now consider the bit of fluid just to the left of that. It must be exerting more pressure: it has to accelerate the bit to its right (the bit in the middle) plus all the stuff to the right of that.
So as you go from right to left across the middle, the pressure goes up.
From here you can create an exact model with $\Delta x$ etc, but this is the basic idea.
Best Answer
Look on the water from the point of view of the accelerated reference frame oriented in such way that the surface of the water is parallel to plane $x'y'$ and depth below the water surface is measured by $z'$. In this frame, the total gravity (due to Earth's gravity and due to inertial force of acceleration) is directed perpendicular to the water surface and has intensity $\sqrt{g^2+a^2}$ (hypotenuse from the Pythagorean theorem). By the same argument as in usual circumstances, the pressure is function of depth $z'$: $$ p = z' \rho \sqrt{g^2+a^2}. $$
So to find out pressure at any point, find out its coordinate $z'$ and use the above formula.