Derivation of the Pressure Gradient Force

Derivation of the Pressure Gradient Force#

To understand the pressure gradient force acting on a small fluid element, let’s start by examining the forces exerted by pressure differences across this element.

Step 1: Consider a Small Fluid Parcel#

  1. Let’s take a small rectangular parcel of fluid with dimensions \(dx\), \(dy\), and \(dz\) in the \(x\), \(y\), and \(z\) directions, respectively.

  2. Assume the fluid has a pressure field \(p\) that varies in space, meaning \(p = p(x, y, z)\).

  3. The parcel will experience different pressures on opposite faces due to spatial changes in pressure (pressure gradient).

Step 2: Calculate Pressure Forces on Opposite Faces#

To derive the pressure force, we consider the pressure acting on opposite faces in the \(x\) direction.

  • Pressure on the left face (at \(x\)): The pressure on the left face is \(p(x, y, z)\), acting over an area \(dy \cdot dz\).

    • The force due to pressure on the left face is therefore:

      \[ F_{\text{left}} = p(x, y, z) \cdot dy \cdot dz \]

  • Pressure on the right face (at \(x + dx\)): The pressure on the right face is slightly different due to the pressure gradient in the \(x\) direction. Using a Taylor series expansion:

    \[ p(x + dx, y, z) \approx p(x, y, z) + \frac{\partial p}{\partial x} \cdot dx \]

    • The force on the right face is:

      \[ F_{\text{right}} = \left( p(x, y, z) + \frac{\partial p}{\partial x} \cdot dx \right) \cdot dy \cdot dz \]

Step 3: Calculate Net Pressure Force in the \(x\) Direction#

The net pressure force in the \(x\) direction, \(F_x\), is the difference between the force on the right face and the force on the left face:

\[ F_x = F_{\text{right}} - F_{\text{left}} \]

Substituting the expressions from above:

\[ F_x = \left( p(x, y, z) + \frac{\partial p}{\partial x} \cdot dx \right) \cdot dy \cdot dz - p(x, y, z) \cdot dy \cdot dz \]

Simplifying by canceling terms:

\[ F_x = \frac{\partial p}{\partial x} \cdot dx \cdot dy \cdot dz \]

Similarly, we can calculate the net pressure forces in the \(y\) and \(z\) directions:

\[ F_y = \frac{\partial p}{\partial y} \cdot dx \cdot dy \cdot dz \]
\[ F_z = \frac{\partial p}{\partial z} \cdot dx \cdot dy \cdot dz \]

Step 4: Write the Total Pressure Gradient Force#

The total pressure force acting on the fluid parcel in all three directions can now be written as a vector:

\[ \mathbf{F}_{\text{pressure}} = - \left( \frac{\partial p}{\partial x} \hat{i} + \frac{\partial p}{\partial y} \hat{j} + \frac{\partial p}{\partial z} \hat{k} \right) \cdot dx \, dy \, dz \]

Or, using the gradient operator \( \nabla \):

\[ \mathbf{F}_{\text{pressure}} = - (\nabla p) \, dx \, dy \, dz \]

Step 5: Express the Pressure Gradient Force per Unit Volume#

For a small fluid parcel, the volume \(V\) is \(dx \cdot dy \cdot dz\). The pressure gradient force per unit volume (which we use in the Navier-Stokes equations) is therefore:

\[ \mathbf{f}_{\text{pressure}} = -\nabla p \]

Summary#

The pressure gradient force per unit volume is \( -\nabla p \). This force results from spatial changes in pressure within the fluid and acts from regions of higher pressure to regions of lower pressure. It is a key component in the Navier-Stokes equations, influencing the motion of fluid by accelerating it in the direction of lower pressure.

Contents