Fluid Mechanics

Invariance Properties of the Entropy Production, and the Entropic Pairing of Inertial Frames of Reference by Shear-Flow Systems

“…(I) Dimensionless invariance: As established by nearly two centuries of mathematical and physical insight [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], a fundamental property of a conservation equation is its invariance to dimensionless transformations. Carvallo [10] and Vaschy [12] referred to this as similitude, later also described as similarity or self-similarity, e.g., [5,15,[46][47][48][49][50][51][52][53] (n.b., some researchers use similarity as the general term, reserving self-similarity for systems with a solution similar to itself, such as a fractal [21]). Birkhoff [20] interpreted similarity from the perspective of group theory, as invariance under the "dimensional group of positive scalar transformations of units".…”

“…The geometric scaling conditions {θ, X} impose the condition of geometric similarity; the velocities and fluxes {U, J S } impose the condition of kinematic similarity; while the remaining groups {R, Σ, ς} impose the equivalence of forces, or dynamic similarity [47][48][49][50][51][52]. Substitution of ( 11)-( 14) into ( 6) and simplification gives the nondimensional entropy production equation:…”

“…In the first example, consider the steady irrotational non-vibrational motion of a fluid relative to a solid object, such as a moving sphere or aircraft, in fluid mechanics referred to as an external flow, e.g., [47,48,[50][51][52]. This is represented by the control volume illustrated in Figure 1a.…”

“…Insofar as the subsystem consisting of the object and its surrounding fluid is concerned, there is no distinction in Newtonian mechanics between the flows described in these two inertial frames of reference, nor between these and any other inertial frames. The flow regime in Figure 1a,b is commonly described by the Reynolds number [46][47][48][49][50][51][52][53]62]:…”

“…We allow Re s < 0 when U < 0, expressed in the coordinate system of Figure 1a. For simple geometries, the frictional and pressure drag force is expressed by the drag coefficient [46][47][48][49][50][51][52][53]62]:…”