Momentum Conservation for Control Volume
Introduction: Momentum Governing Equation: the Reynolds Transport Theorem (RTT) was applied to mass conservation. Mass is a scalar (quantity without magnitude). The Reynolds Transport Theorem (RTT) is applicable to any quantity and the discussion here will deal with forces that acting on the control volume. Newton’s second law for single body is as the following
It can be noticed that bold notation for the velocity is U (and not U) to represent that the velocity has a direction. For several bodies (n), Newton’s law becomes
The fluid can be broken into infinitesimal elements which turn the above equation into a continuous form of small bodies which results in
Note that the notation D/Dt is used and not d/dt to signify that it referred to a derivative of the system. The Reynold’s Transport Theorem (RTT) has to be used on the right hand side.
The Momentum Governing Equation: The right hand side, according Reynolds Transport Theorem (RTT), is
The liquid velocity, U, is measured in the frame of reference and Urn is the liquid relative velocity to boundary of the control volume measured in the same frame of reference. Thus, the general form of the momentum equation without the external forces is With external forces equation is transformed to
The external forces, Fext, are the forces resulting from support of the control volume by non–fluid elements. These external forces are commonly associated with pipe, ducts, supporting solid structures, friction (non-fluid), etc
Equation is a vector equation which can be broken into its three components. In Cartesian coordinate, for example in the x coordinate, the components are
Where µx is the angle between ˆn and ˆi or (ˆn ·ˆi).