Global Journal of Researches in Engineering, I: Numerical Methods, Volume 23 Issue 1

( ) 4 = 1 3 � 2 ��⃗ ⋅ ∇ 1 2 32 − 3 3 ∂ 3 ∂ 3 + 2 ⃗ ⋅ ∇( 3 )� It has been shown in Moschandreou et al [5] that this decomposition holds and that, ( ) 1 + ( ) 2 + ( ) 4 = 3 Φ( ) The function Φ( ) is the surface integral of pressure terms minus the volume integral of tensor product term. At the end of this paper, a proof that on a volume of an arbitrarily small sphere embedded in each cell of the lattice centered at ( , , ) (centers of cells) we have, ( ) 1 + ( ) 2 + ( ) 4 = 0 From this equation we then can solve for ∂ ∂ 3 algebraically and differentiating with respect to 3 and using Poisson’s equation (setting the representation of each of the two partial derivatives with respect to 3 equal to each other we obtain, = 0, which is precisely the following PDE, = � ∂ 3 ∂ � 2 ( − 1) ∂ 3 3 ∂ 3 ∂ 1 2 + � ∂ 3 ∂ � 2 ( − 1) ∂ 3 3 ∂ 3 ∂ 2 2 + � ∂ 3 ∂ � 2 ( − 1) ∂ 3 3 ∂ 3 3 + � ∂ 3 ∂ � ( 3 ) 2 � ∂ 3 3 ∂ 3 2 ∂ � − ( 3 ) 2 � ∂ 2 3 ∂ 3 ∂ � 2 − 2 �� 2 − 1 2 � � ∂ 3 ∂ � 2 − 3 � ∂ 3 ∂ � � ∂ 3 ∂ 3 � + � 3 � 1 ( 1 , 2 , 3 , ) + ∂ 1 ∂ � ∂ 3 ∂ 1 + 3 � 2 ( 1 , 2 , 3 , ) + ∂ 2 ∂ � ∂ 3 ∂ 2 + Λ( 1 , 2 , 3 , ) 2 + Φ(s) 2 � � ∂ 2 3 ∂ 3 ∂ + ��( − 1)( 1 ( 1 , 2 , 3 , ) − 1) ∂ 3 ∂ + 2 3 � 1 ( 1 , 2 , 3 , ) + ∂ 1 ∂ �� ∂ 2 3 ∂ 3 ∂ 1 + �( − 1)( 2 ( 1 , 2 , 3 , ) − 1) ∂ 3 ∂ + 2 3 � 2 ( 1 , 2 , 3 , ) + ∂ 2 ∂ �� ∂ 2 3 ∂ 3 ∂ 2 + 3 3 �− 2 3 + � + 2 3 � � � ∂ 3 ∂ � ∂ 2 3 ∂ 3 2 + 2 3 � ∂ 3 ∂ � ( − 1) ∂ 2 3 ∂ 1 2 + 2 3 � ∂ 3 ∂ � ( − 1) ∂ 2 3 ∂ 2 2 + �(−1 + (3 + 1) ) � ∂ 3 ∂ 3 � 2 + ( − 1) �� ∂ 1 ∂ 1 � 2 + 2 � ∂ 1 ∂ 2 � ∂ 2 ∂ 1 + � ∂ 2 ∂ 2 � 2 �� ∂ 3 ∂ + 2 ��� 1 ( 1 , 2 , 3 , ) + ∂ 1 ∂ � ∂ 3 ∂ 1 + � ∂ 3 ∂ 2 � � 2 ( 1 , 2 , 3 , ) + ∂ 2 ∂ �� ∂ 3 ∂ 3 + 3 � ∂ 3 ∂ 1 � ∂ 1 ∂ 3 + 3 � ∂ 3 ∂ 2 � ∂ 2 ∂ 3 + 1 2 ∂Λ( 1 , 2 , 3 , ) ∂ 3 � � ∂ 3 ∂ = 0 (1) Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 48 Year 2023 © 2023 Global Journals ( ) I Exploring Finite-Time Singularities and Onsager’s Conjecture with Endpoint Regularity in the Periodic Navier Stokes Equations 2 � ∂ 2 1 ∂ 3 ∂ � 3 � ∂ 3 ∂ 1 � + 2 � ∂ 2 2 ∂ 3 ∂ � 3 � ∂ 3 ∂ 2 � +