Research Article
Analytical Results of the Motion of Oscillating Dumbbell in a Viscous Fluid
Mohammed Mattar Al-Hatmi*,
Anton Purnama
Issue:
Volume 9, Issue 1, February 2024
Pages:
1-11
Received:
31 December 2023
Accepted:
24 January 2024
Published:
5 February 2024
Abstract: The aim of this paper is to investigate analytically the motion of oscillating dumbbell, two micro-spheres connected by a spring, in a viscous incompressible fluid at low Reynolds number. The oscillating dumbbell consists of one conducting sphere and assumed to be actively in motion under the action of an external oscillator field while the other is non-conducting sphere. As result, the oscillating dumbbell moves due to the induced flow oscillation of the surrounding fluid. The fluid flow past the spheres is described by the Stokes equation and the governing equation in the vector form for the oscillating dumbbell is solved asymptotically using the two-timing method. For illustrations, applying a simple oscillatory external field, a systematic description of the average velocity of the oscillating dumbbell is formulated. The trajectory of the oscillating dumbbell was found to be inversely proportional to the frequency of the external field, and the results demonstrated that the oscillating dumbbell moves in a circular path with a speed that decreases inversely with the length of the spring.
Abstract: The aim of this paper is to investigate analytically the motion of oscillating dumbbell, two micro-spheres connected by a spring, in a viscous incompressible fluid at low Reynolds number. The oscillating dumbbell consists of one conducting sphere and assumed to be actively in motion under the action of an external oscillator field while the other i...
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Research Article
Algebraic Representation of Primes by Hybrid Factorization
Issue:
Volume 9, Issue 1, February 2024
Pages:
12-25
Received:
4 February 2024
Accepted:
27 February 2024
Published:
20 March 2024
DOI:
10.11648/j.mcs.20240901.12
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Abstract: The representation of integers by prime factorization, proved by Euclid in the Fundamental Theorem of Arithmetic −also referred to as the Prime Factorization Theorem− although universal in scope, does not provide insight into the algebraic structure of primes themselves. No such insight is gained by summative prime factorization either, where a number can be represented as a sum of up to three primes, assuming Goldbach’s conjecture is true. In this paper, a third type of factorization is introduced, called hybrid prime factorization, defined as the representation of a number as sum −or difference− of two products of primes with no common factors between them. By using hybrid factorization, primes are expressed as algebraic functions of other primes, and primality is established by a single algebraic condition. Following a hybrid factorization approach, sufficient conditions for the existence of Goldbach pairs are derived, and their values are algebraically evaluated, based on the symmetry exhibited by Goldbach primes around their midpoint. Hybrid prime factorization is an effective way to represent, predict, compute, and analyze primes, expressed as algebraic functions. It is shown that the sequence of primes can be generated through an algebraic process with evolutionary properties. Since prime numbers do not follow any predetermined pattern, proving that they can be represented, computed and analyzed algebraically has important practical and theoretical ramifications.
Abstract: The representation of integers by prime factorization, proved by Euclid in the Fundamental Theorem of Arithmetic −also referred to as the Prime Factorization Theorem− although universal in scope, does not provide insight into the algebraic structure of primes themselves. No such insight is gained by summative prime factorization either, where a num...
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