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Paper on Navier-Stokes Equations

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  Mathematical analysis of Navier-Stokes Equations Sahib Chawla a , Vishvendra Singh Tomar a a Mechanical Engineering, Delhi Technological University, Delhi-110042   ABSTRACT   We present a Mathematical Analysis of the Navier–Stokes equations (which describe the motion of fluid substances), including fully implicit formulations, operator splitting methods (pressure/velocity correction, projection methods) etc dealing with internal, isothermal, unsteady flows of a class of incompressible fluids with both constant, and shear or pressure dependent viscosity that includes the Navier-Stokes fluid as a special subclass. Emphasis is put on showing the close relationship between (seemingly) different and competing solution approaches for incompressible viscous flow. We present alternative solution for these equations and show the existence of a weak solution to the Stokes equations and to the steady-state Navier-Stokes equations in the case where we have zero boundary value and the dimension of the space is less than or equal to four. Various Numerical techniques are also employed for obtaining the solutions of Navier-Stokes Equation. Using Vector Calculus & various Mathematical Operators, applications of Navier-Stokes equations are discussed. Briefly, we also discuss further results related to further generalizations of the Navier-Stokes equations. ( Keywords : Navier-Stokes Equations, Fluid, Types of Flows, Types of Forces, Incompressible, Unsteady Parallel Flows) 1.   INTRODUCTION The numerical simulation of fluid dynamics is one of the main fields in computational mathematics. It is today both an alternative and a complement to experiments in many engineering disciplines as it helps in predicting the behavior of fluids [1]. The Navier-Stokes equations are a mathematical model aimed at describing the motion of an incompressible viscous fluid, like many common ones as, for instance, water, glycerine, oil and under certain circumstances also air. They were introduced in 1822 by the French engineer Claude Louis Marie Henri Navier and successively re-obtained, by different arguments, by a number of authors including Augustin-Louis Cauchy in 1823, Simeon Denis Poisson in 1829, Adhemar Jean Claude Barre de Saint-Venant in 1837, and, finally, George Gabriel Stokes in 1845 [2]. These equations are obtained by applying Newton's second law to fluid motion, by assuming that the fluid stress is the sum of a diffusing viscous term which is proportional to the gradient of velocity, plus a pressure term. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations they can be used to model and study magneto-hydrodynamics [3]. In his immortal Principal, Newton [4] states: “The resistance arising from the want of lubricity in parts of the fluid is, other things being equal, proportional to the velocity with which the parts of the fluid are separated from one another. What is now popularly referred to as the Navier-Stokes model implies a linear relationship between the shear stress and the shear rate. The Navier-Stokes Equation, acting on an elemental fluid element for three-dimensional in vector form is given by: and the equation of continuity, also called the incompressibility constraint is given by: ∇   ・   v = 0   ..…….. (2) In these equations, v is the velocity field, p is the pressure, ρ  is the fluid density, g denotes body forces (such as gravity, centrifugal etc), µ  is the kinematic viscosity of the fluid, and t denotes time. The initial conditions consist of prescribing v, whereas the boundary conditions can be of several types: (i) prescribed velocity components, (ii) vanishing normal derivatives of velocity components, or (iii) prescribed stress vector components. The pressure is only determined up to a constant, but can be uniquely determined by ………. (1)  prescribing the value (as a time series) at one spatial point. Equations (1)–(2) are referred as the Navier–Stokes equations [5]. The Navier–Stokes equations are nonlinear partial differential equations in almost every real situation. However in cases like one-dimensional flow, the equations can be simplified to linear equations. The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity, an example of convective but laminar (non-turbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle [6]. 2.   Preliminary: Vector Calculus  Vector calculus is the branch of mathematics that is involved with differentiation and integration over vector fields. In this section we present a brief overview of this area. We begin with a very important mathematical operator called del ( ∇ ). Del is defined as the partial derivatives of a vector. Letting i, k, and j denote the unit vectors for the coordinate axes in real 3-space, the operator is defined [7]: With del ( ∇ ) defined, we may now look at four key differential operators that are based on del. First we have the gradient. The gradient is defined as the measurement of the rate and direction of change in a scalar field. The gradient maps a scalar field to a vector field. So, for a scalar field f [8]:  (   )= ∇ (   ) For example, consider the scalar field   =  2 +  .We take the partial derivatives with respect to  x, y, and  z . So, the gradient is: (   )=  󰀲󰀲󰀲󰀲  +2  +  . Next we have curl, which is defined as the measurement of the tendency to rotate about a point in a vector field. The curl maps a vector field to another vector field. For vector F, we define [8]:  (  )= ∇  ×  For example, consider vector field  =  −  +  󰀲󰀲󰀲󰀲  . We can express the (  ) symbolically as the following determinant: Letting F  1  = x, F  2  = −   , and F  3  =  󰀲  this can be expressed using the cross product form as From this we obtain: (0−0)   − (0−0)   + (−  −0), that is, C(  )=−  Third, we have divergence.  Divergence models the magnitude of a source or sink at a given point in a vector field. Divergence maps a vector field to a scalar field. For a vector filed F [8]:  (  )= ∇ ·  At any point in a vector field, divergence is positive if there is an outflow, negative if there is an inflow, and zero if there is no convergence or divergence [9]. For example, the upper left vector field, F = xi +yj , where (   )=1+1=2, there is an outflow, which makes sense as the divergence is positive. If we now look at the bottom left vector field, F = yi +xj , where (  ) =0+0, there is neither outflow or inflow, which again makes sense due to the divergence being 0. As an example, consider once again  =  −  +  󰀲  . And finally, we have the Laplacian, represented as ∆ . The Laplacian is defined as the composition of the divergence and gradient operations. This maps a scalar field onto another scalar field. The Laplacian of  f is defined as [8]: ∆   = ∇ 󰀲   = ∇ · ∇   For example, consider field   =  󰀲  +  3      3.   Navier-Stokes Equation in Cartesian coordinates Navier-Stokes Equation is derived from the Cauchy’s Equation, which acts as governing equation for it. We consider a differential fluid element, as a material element (instead of taking it as control volume). Applying Newton’s Second Law which gives, The total force on the same fluid element can be expressed as the sum of body force and the surface force. Hence, these two forces can be described as, Body Forces include Gravity force, Electromagnetic force, Centrifugal force, Coriolis force and Surface forces include Pressure forces, Viscous forces. By considering the x-component of equation (4), Now We denote the stress tensor σ  ij  (pressure forces+ viscous forces) The viscous Stress tensor will be τ  ij , And the strain (deformation) rate tensor   ij  will be, Let, Be the stress vectors on the planes perpendicular to the co-ordinate axis.   ………. (3) Since ………. (4) ………. (5)    Then the stress vector    F at any point associated with a plane of unit normal vector n  = (n 1 , n 2 , n 3 )   can be expressed as, We consider the x-component of the net surface force using the figure below. 󰁕󰁳󰁩󰁮󰁧 󰁔󰁡󰁹󰁬󰁯󰁲󲀙󰁳 󰁦󰁯󰁲󰁭󰁵󰁬󰁡 󰁷󰁥 󰁧󰁥󰁴󰀬 Thus, If we assume that the only body force is the gravity force, we have, Hence equation (5) will become, We divide by dxdydz  and get the equation for the  x-component: Hence, this can be written as: ……. (For x- Direction)
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