Does Hydrostatic Pressure Act in Lateral Directions?

Hydrostatic pressure can produce a sideways force on a wall, but pressure itself does not have a lateral direction. In a fluid at rest, pressure at a point is the same in every direction. The direction appears when pressure acts on a particular surface: the resulting force is perpendicular to that surface.

This distinction explains why a submerged vertical wall is pushed horizontally even though the pressure is not a sideways-pointing vector.

Pressure versus pressure force

Pressure is a scalar quantity: it has magnitude but no direction. At a point in a static fluid, its value depends on the point’s depth, not on which direction you choose to measure it.

For a fluid of constant density, the gauge pressure at vertical depth $h$ below the free surface is

$$p_g = \rho g h,$$

where $\rho$ is the fluid density and $g$ is gravitational acceleration. Absolute pressure includes the pressure at the free surface:

$$p_{\text{abs}} = p_{\text{surface}} + \rho g h.$$

Pressure acting on a small surface area $A$ produces a force perpendicular to that area, with magnitude $F=pA$ when pressure is uniform over it. In vector form, the pressure traction on a surface with unit normal $\mathbf{n}$ is often written as $-p\mathbf{n}$, where the minus sign indicates compression, directed into the surface. The sign depends on which side of the surface and which normal convention you use; the physical force is normal to the surface either way.

Why a vertical wall gets a lateral force

The normal to a vertical wall is horizontal. Therefore, the pressure force on that wall is horizontal too. The pressure increases with depth, so the force is stronger near the bottom than near the top.

At any one depth, imagine a tiny flat patch on the wall. Its pressure force points perpendicular to the patch, not along the wall. For a vertical wall, that perpendicular direction is lateral. On the opposite wall of a container, the force points in the opposite horizontal direction.

This does not mean pressure has become a vector or that it has a different value in the horizontal direction. It means the same local pressure acts normally on each surface it touches.

Worked example: force on a rectangular wall

A rectangular wall is 2 m wide and holds water to a depth of 3 m. Find the resultant hydrostatic force on the submerged wall, using water density $\rho=1000\ \text{kg/m}^3$ and $g=9.81\ \text{m/s}^2$. Assume the top of the water is at atmospheric pressure, so the calculation uses gauge pressure.

At depth $h$, pressure is $p_g(h)=\rho gh$. A horizontal strip of wall at that depth, with thickness $dh$, has area $dA=b\,dh$, where $b=2\ \text{m}$. Its force magnitude is

$$dF = p_g(h)dA = \rho ghb\,dh.$$

Integrating from the surface to the bottom gives

$$\begin{aligned} F &= \int_0^3 \rho ghb\,dh \\ &= \rho gb\left[\frac{h^2}{2}\right]_0^3 \\ &= \frac{(1000)(9.81)(2)(3^2)}{2} \\ &= 88{,}290\ \text{N} \approx 88.3\ \text{kN}. \end{aligned}$$

The force is horizontal because the wall is vertical. Its line of action is at the center of pressure, not halfway down the wall: for this rectangular wall, it is $h/3=1\ \text{m}$ above the bottom. This follows because pressure is zero at the surface and increases linearly to its maximum at the bottom.

Common mistakes

• Giving pressure a direction: Pressure is scalar. The pressure force is a vector, and its direction is set by the surface normal.
• Using distance along the wall instead of vertical depth: Hydrostatic pressure depends on vertical depth below the free surface, even when the wall is sloped.
• Assuming pressure is uniform over a deep wall: It varies with depth. Use $F=pA$ only when pressure is uniform, or when $p$ is the area-average pressure; otherwise integrate.
• Mixing gauge and absolute pressure: Gauge pressure omits atmospheric pressure. For net force on a wall exposed to the same atmosphere on both sides, atmospheric contributions cancel. If pressures differ across the surfaces, account for each side consistently.

These results assume a fluid at rest with ordinary isotropic pressure. In moving fluids, viscous stresses can also act tangentially, so the total surface force need not consist of pressure alone.