86_Newton's Second Law of Motion

Continuum mechanics assumes that a material, even when infinitely divided into smaller elements, retains the properties of the oiginal whole in each element. To consider something as a continuum means to consider it mathematically continous, implying that there are no discontinuities in derivatives and that the continuous properties of the material evident in the original element are maintained. It is from these continuous characteristics that fundamental mechanical laws, such as Newton's second law, are applied to interpret the necessary information.

 

 

Assuming there is a moving fluid parcel as shown in the figure above, the rate of change of the linear momentum of the fluid parcel, ${{\bf{M}}_{{\text{parcel}}}}$, must equal the sum of the forces acting on the parcel.

 

 

Representing the surface forces as traction ${\bf{f}}$ applied on the surface of the parcel, and body forces as the product of fluid density $\rho $ and gracitational acceleration ${\bf{g}}$, we have the following.

 

 

To relate the rate of change of parcel momentum to fluid density and velocity, first, by subdividing the parcel into volumes $d{V_{{\text{parcel}}}}$ with corresponding masses $d{m_{{\text{parcel}}}} = \rho d{V_{{\text{parcel}}}}$, we can integrate each part to obtain the expression for momentum.

 

 

SInce the integration is calculated within the volume of the parcel, which varies over time, in the right-hand side of the equation, the differentiation under the integral sign with respect to time can be replaced with a material derivative, allowing for the interchange of the order of time differentiation and volume integration.

 

 

Here, the material derivative of the elementary mass $dm$ must be zero to ensure mass conservation.

 

 

By summarizing the above equations, the parcel motion equation can be derived.

 

 

The specific $x$, $y$, and $z$ components can be represented as follows.

 

 

The above equation is valid regardless of whether the fluid is compressible or incompressible, and in two-dimensional flow, the motion equation can be represented as follows.

 

 

Where $dA$ is the differential area, and $l$ is the arc length of the parcel's boundary in the $xy$-plane.

Similarly, the components in $x$ and $y$ are follows.