• DEEPAK KUMAR

Articles written in Proceedings – Mathematical Sciences

• Stokes drag on axially symmetric bodies: a new approach

In this paper a new approach to evaluate the drag force in a simple way on a restricted axially symmetric body placed in a uniform stream (i) parallel to its axis, (ii) transverse to its axis, is advanced when the flow is governed by the Stokes equations. The method exploits the well-known integral for evaluating the drag on a sphere. The method not only provides the value of the drag on prolate and oblate spheroids and a deformed sphere in axial flow which already exists in literature but also new results for a cycloidal body, an egg shaped body and a deformed sphere in transverse flow. The salient results are exhibited graphically. The limitations imposed on the analysis because of the lack of fore and aft symmetry in the case of an eggshaped body is also indicated. It is also seen that the analysis can be extended to calculate the couple on a body rotating about its axis of symmetry.

• Slow rotation of a sphere with source at its centre in a viscous fluid

In this note, the problem of a sphere carrying a fluid source at its centre and rotating with slow uniform angular velocity about a diameter is studied. The analysis reveals that only the azimuthal component of velocity exists and is seen that the effect of the source is to decrease it. Also, the couple on the sphere is found to decrease on account of the source.

• Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionIn+1 (F) ∩In+1 (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I3 (F)) of F are identified whenR and S are arbitrary subgroups ofF.

• Some Augmentation Quotients of Integral Group Rings

Let 𝐺 be a group and 𝐻 be a subgroup of 𝐺. A complete description of $\Delta(G)\Delta^n(H)/\Delta^{n+1}(H)$ is given, and as a consequence the structures of $\Delta(G)/\Delta(H)$ and $\Delta^2(G)/\Delta^2(H)$ are determined. Also, the structure of $\Delta^n(G)/\Delta^n(H)$ for all $n\geq 1$ is determined when 𝐺 is a free group.

• $C^{\ast}$-algebra-valued partial metric space and fixed point theorems

In this paper, we introduce the notion of $C^{\ast}$-algebra-valued partial metric space which is more general than partial metric space. Some fixed point results using C-class functions on such spaces are obtained. Moreover, some illustrated examples are also provided.

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