Muhammad Abdullah

- Conduct research and write papers in the areas of symmetries and curvature (following the Grove Symmetry Program)
- Present research findings at seminars.

Grade homework assignments for MATH 555: Differential Equations I, Spring 2024.

In this article, we consider closed, positively curved n-manifolds that admit an isometric and effective action of the group Z_p^r with a fixed point, where p is a prime. We classify such manifolds up to homotopy equivalence when r satisfies the lower bound given in our main theorem and p is one of the primes {3, 5, 7, 11, 13, 17, 19, 23, 29}. In particular, for p = 3 and n ≥ 1908, we show that r > (9/32)n. The lower bound we obtain for this range of primes represents an improvement over the approximately (3/8)n bound found by Fang and Rong in even dimensions and by Ghazawneh in odd dimensions. To obtain this result, we derive a finite-length Elias-Bassalygo bound for q-ary codes, which is of independent interest.

A computational fluid dynamics (CFD) model is generated to study the fluid flow and heat transfer from a series of six cylinders arranged in a horizontal row. The momentum and energy equations were solved using the finite volume method, with convective terms discretized using a second-order upwind method and diffusion terms discretized using central differencing. A second-order implicit technique was used for time integration. Simulations were run with four different values of the free stream Reynolds numbers 100, 200, 300, 400 and three different values of the longitudinal pitch (SL) 1.5D, 2D, 2.5D to analyze the heat transfer behavior with fluid flow over a series of horizontal cylinders. First the grid independence test and CFD model validation are performed with the exact relation available in the literature. Next, the numerical results obtained from the developed CFD model for the row of cylinders were compared with the previously published analytical model, and correlation available in the literature, and were found to be in good agreement. The average Nusselt number calculated from the analytical and numerical models falls in the range of 8 to 10 for free stream Reynolds number equal to 100 for all values of dimensionless longitudinal pitches. Moreover, for the free stream Reynolds number equal to 400 with longitudinal pitch of 2.5D, the values of the average Nusselt number reach up to double the above-mentioned range. The maximum percentage error between the average Nusselt number obtained from the numerical and analytical solutions was less than 20% for larger free stream Reynolds number. The study found that the average Nusselt number for the array of cylinders increases with both the free stream Reynolds number and the dimensionless longitudinal pitch ratio increments. This information can be used to design more efficient heat exchangers or other fluid systems involving arrays of cylinders.