We designate the pressure at the centre to be p, then the average pressure on the various faces can be expressed in terms of p and its derivatives as shown.

You might alarm by the expressions used for the surface forces. Don’t fret. We are using a technique called the calculus of variations here and also throughout the other lessons in fluid mechanics. Basically, without elaborating too much on the theory, it involves taking the partial derivative of a certain quantity, in this case it is p, with respect to the direction along which the quantity varies,  or  and then multiplying that with a small change in that direction,  or . Thinking of it as the gradient multiplied by a small change of the independent variable should help.

A more friendly way to describe the variation in pressure is that it is a Taylor series expansion of the pressure at the element center to approximate the pressures a short distance away and neglecting higher order terms that will vanish as w let  and  approach zero. The resultant surface force in the y direction is

or

Doing the same for the x and z directions gives us

The resultant surface force acting on the element can be expressed in vector form as

Substituting the previous results gives us

We recognize that the group of terms in parentheses represents the vector form of the pressure gradient and by employing the del operator, it can be written as

To recap on the ‘del operator’, you can check out my lesson entitled “Directional Derivative and the Del Operator” in the Vector Integral Calculus for an in-depth explanation. Briefly speaking, when we apply the operator, , to a certain scalar function, we get a vector function given by the rule.

The resultant surface force per unit volume can be expressed as

Since the z axis is vertical, the weight of the element is

where the negative sign indicates that the force due to the weight is downward and in the negative z direction. Adapting Newton’s second law to be used on a small element is

where  is the resultant force acting on the element,  is the acceleration of the element, and  is the element mass, which can also be written as . This gives us

or

simplifying to have

This is our general equation of motion for a fluid in where there are no shearing stresses. In fluid mechanics, there are a lot of fluids we can analyze – steady, accelerated, stationary, compressible, and incompressible. We will use this equation to analyze the pressure field in these fluids but applying the appropriate restrictions. The next few lessons will discuss this further.