Note on the stability criteria for a new type of helical flows

In this paper, we proceed exploring the case of non-stationary helical flows of the Navier-Stokes equations for incompressible fluids with variable (spatially dependent) coefficient of proportionality between velocity and the curl field of flow. Meanwhile, the system of Navier-Stokes equations (including continuity equation) has been successfully explored previously with respect to the existence of analytical way for presentation of non-stationary helical flows of the aforementioned type. The main motivation of the current research is the exploring the stability of previously obtained helical flows. Conditions for the stability criteria of the exact solution for the aforementioned type of flows are obtained, for which non-stationary helical flow with invariant Bernoulli-function is considered. As it has been formulated before, the spatial part of the pressure field of the fluid flow should be determined via Bernoulli-function, if components of the velocity of the flow are already obtained.Comment: 10 pages, 1 figures; Keywords: Navier-Stokes equations, non-stationary helical flow, Bernoulli-function; this note corresponds to the article which was accepted for publication in "Journal of King Saud University - Science" (03 July 2018), DOI 10.1016/j.jksus.2018.07.00

Numerical Investigation of Physical Parameters in Cardiac Vessels as a New Medical Support Science for Complex Blood Flow Characteristics

تقترح هذه الدراسة نهجًا رياضيًا وتجربة عددية لحل بسيط لتدفق الدم القلبي إلى الأوعية الدموية للقلب. تمت دراسة نموذج رياضي لتدفق الدم البشري عبر الفروع الشريانية وحسابه باستخدام معادلة نافييه-ستوكس التفاضلية الجزئية مع تحليل العناصر المحدودة (FEA). علاوة على ذلك ، يتم تطبيق FEA على التدفق الثابت للسوائل اللزجة ثنائية الأبعاد من خلال أشكال هندسية مختلفة. تتحدد صلاحية الطريقة الحسابية بمقارنة التجارب العددية مع نتائج تحليل الوظائف المختلفة. أظهر التحليل العددي أن أعلى سرعة لتدفق الدم تبلغ 1.22 سم / ثانية حدثت في مركز الوعاء الذي يميل إلى أن يكون رقائقيًا ويتأثر بعامل لزوجة منخفض قدره 0.0015 باسكال. بالإضافة إلى ذلك ، تحدث الدورة الدموية في جميع الأوعية الدموية بسبب ارتفاع الضغط في القلب ويقل الضغط عندما يعود من الأوعية الدموية بنفس المعايير. أخيرًا ، عندما تكون اللزوجة عالية ، تميل المقادير القصوى لتدفق الدم نحو جدار الوعاء الدموي بنفس سرعة التدرج ونصف قطره تقريبًا.This study proposes a mathematical approach and numerical experiment for a simple solution of cardiac blood flow to the heart's blood vessels. A mathematical model of human blood flow through arterial branches was studied and calculated using the Navier-Stokes partial differential equation with finite element analysis (FEA) approach. Furthermore, FEA is applied to the steady flow of two-dimensional viscous liquids through different geometries. The validity of the computational method is determined by comparing numerical experiments with the results of the analysis of different functions. Numerical analysis showed that the highest blood flow velocity of 1.22 cm/s occurred in the center of the vessel which tends to be laminar and is influenced by a low viscosity factor of 0.0015 Pa.s. In addition, circulation throughout the blood vessels occurs due to high pressure in the heart and the pressure becomes lower when it returns from the blood vessels at the same parameters. Finally, when the viscosity is high, the extreme magnitudes of blood flow tend toward the vessel wall at approximately the same velocity and radius of the gradient

Solving the Hydrodynamical System of Equations of Inhomogeneous Fluid Flows with Thermal Diffusion: A Review

The present review analyzes classes of exact solutions for the convection and thermal diffusion equations in the Boussinesq approximation. The exact integration of the Oberbeck–Boussinesq equations for convection and thermal diffusion is more difficult than for the Navier–Stokes equations. It has been shown that the exact integration of the thermal diffusion equations is carried out in the Lin–Sidorov–Aristov class. This class of exact solutions is a generalization of the Ostroumov–Birikh family of exact solutions. The use of the class of exact solutions by Lin–Sidorov–Aristov makes it possible to take into account not only the inhomogeneity of the pressure field, the temperature field and the concentration field, but also the inhomogeneous velocity field. The present review shows that there is a class of exact solutions for describing the flows of incompressible fluids, taking into account the Soret and Dufour cross effects. Accurate solutions are important for modeling and simulating natural, technical and technological processes. They make it possible to find new physical mechanisms of momentum transfer for the design of new types of equipment. © 2023 by the authors

Arbitrary high order discontinuous Galerkin methods for the shallow water and incompressible Navier-Stokes equations on unstructured staggered meshes

In this work we present a new class of well-balanced, arbitrary high order accurate semi-implicit discontinuous Galerkin methods for the solution of the shallow water and incompressible Navier-Stokes equations on staggered unstructured curved meshes. Isoparametric finite elements are used to take into account curved domain boundaries. Regarding two-dimensional shallow water equations, the discrete free surface elevation is defined on a primal triangular grid, while the discrete total height and the discrete velocity field are defined on an edge-based staggered dual grid. Similarly, for the two-dimensional incompressible Navier-Stokes case, the discrete pressure is defined on the main triangular grid and the velocity field is defined on the edge-based staggered grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier-Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. High order (better than second order) in time can be achieved by using a space-time finite element framework, where the basis and test functions are piecewise polynomials in both space and time. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse system for the scalar pressure involving only the direct neighbor elements, so that it becomes a block four-point system in 2D and a block five-point system for 3D tetrahedral meshes. The resulting linear system is conveniently solved with a matrix-free GMRES algorithm. Note that the same space-time DG scheme on a collocated grid would lead to ten non-zero blocks per element in 2D and seventeen non-zero blocks in 3D, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier-Stokes equations. The special case of high order in space low order in time allows us to recover further regularity about the main linear system for the pressure, such as the symmetry and the positive semi-definiteness in the general case. This allows us to use a very fast linear solver such as the conjugate gradient (CG) method. The flexibility and accuracy of high order space-time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time. We will further extend the previous method to three-dimensional incompressible Navier-Stokes system using a tetrahedral main grid and a corresponding face-based hexaxedral dual grid. The resulting dual mesh consists in non-standard 5-vertex hexahedral elements that cannot be represented using tensor products of one dimensional basis functions. Indeed a modal polynomial basis will be used for the dual mesh. This new family of numerical schemes is verified by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data. Furthermore, the comparison with available experimental results will be presented for incompressible Navier-Stokes equations