2. Force function. Equipotent1al surface

Let us use the differential equations obtained in the previous section to investigate some questions of fluid equilibrium.

Multiplication of each of Eqs (A.2) by dx, dy and dz, respectively, and their addition yields

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The term in parentheses in the right-hand side of this equation is the total differential of the function p (x, y, z), i.e.,

and Eq. (A.3) can be rewritten in the form

We may conclude from this that the left-hand side of Eq. (A.3') must be the total differential of some function U (x, y, z). This function must possess the following property: the partial derivatives with respect to it along the x, у and z coordinates must be equal to X, Y and Z, respectively, i. e.,

The function U called a force function. It will be recalled from the course in theoretical mechanics that it equals the force potential taken with the sign reversed.

The conclusion is that fluid equilibrium is possible only under the action of body forces possessing potential energy. Introducing the function U into the basic Eq. (A.3'), we obtain

or

Integrating, we obtain in general form

The integration constant is found from the boundary conditions. If, at U = U₀, p =p₀, then,

A surface satisfying the condition

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is called an equipotential surface. It follows from Eqs (A.4) and (А.5^’) that for all points of such a surface the condition

dp =0

holds, or

p (x,y,z) = const.

Hence, an equipotential surface is also a surface of equal pressure.

Furthermore, it can be concluded from the foregoing that the density of a nonhomogeneous liquid is a function of pressure. This means that equipotential surfaces in a nonhomogeneous liquid are also surfaces of equal density, i. e., a nonhomogeneous liquid in equilibrium is stratified by equipotential surfaces into layers of equal density. This property is used to separate the components of nonhomogeneous liquids in centrifuges.

Let us introduce a resultant unit body force j whose projections on the coordinate axes are X, Y, Z, i. e., we take the total acceleration caused by the action of all body forces at a given point. We also take into account that the quantities dx, dy and dz are the projections of an infinitesimal line dl representing the distance between two very close points M and M' (see Fig. 178).

On the basis of the foregoing we can write the following equalities

and

The formula for the cosine of an angle between two straight lines: in space yields

Substituting this expression for dU into Eq. (A.4) and solving for dp, we have

It is seen from this equation that the greatest maximum pressure increment is in the direction of the resultant body force j, as in this case cos (f,dl) = 1.

For any line dl in an equipotential surface the increment dp = 0. But in the general case j and dl are not zero, and consequently, cos (j, dl) = 0, i. e., the resultant body force is normal to an equipotential surface.

Using this property of equipotential surfaces it is simple to determine the configuration of the free surface of a liquid when there are other body forces besides gravity acting on it.

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