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(Solved): 2. Derivation of Navier-Stokes equations. In what follows, we outline the six ingredients necessary ...
2. Derivation of Navier-Stokes equations. In what follows, we outline the six ingredients necessary to develop equations of motion for a viscous fluid in terms of the fields of fluid velocity, pressure, and gravity. The actual problem statement is at the end of this list, and asks to combine these ingredients by substitution to derive the so-called Navier-Stokes equations.
To allow for concise expressions, we'll adopt the Cartesian coordinate system , in place of the traditional , and . Consequently, components of a vector will be , instead of . In a few locations, we'll refer to both notations to help familiarize. I. Conservation of mass. For an incompressible fluid, the equation for conservation of mass reduces to In coordinates, this would be written as II. Stress equations of motion. Considering the forces acting on an infinitesimal volume of fluid, we arrived to the following vector equation of motion from Newton's second law where is the stress tensor, is the component of the gravitational acceleration along the -th direction, and is the acceleration vector of the fluid along the -th direction. For example, along the 1 (or ) direction, the equation is III. Stress decomposition. Previously, in developing the Euler equations of motion for an inviscid fluid - a fluid that cannot support any shear stress - the state of stress corresponded to one of pure pressure. This corresponds to the stress where is the fluid pressure and if , and 0 if (ie., is an identity matrix whose values are 1 along the diagonal and 0 off the diagonal). To consider the stress experienced by a viscous fluid, we decompose the stress into a pure pressure plus terms that simply represent the departure from a pure pressure state (i.e., terms that imply the existence of shear stress): where the fluid pressure is the mean normal stress and the tensor is the so-called deviatoric stress tensor (for its role in capturing the deviation from pure pressure). IV. Strain-mate tensor. The gradient of the velocity field, , captures the first-order variation of velocity about a point. In class, we saw that this gradient could be decomposed into two terms, one of which captures the rate of fluid deformation at a point, and the other which captures the rate of rotation of the fluid. The former term is called the strain rate tensor and is defined as
For example, quantifies the rate of shear deformation in the 1-2 plane ( - plane) and is which is also equal to , by definition (i.e., is symmetric: ). quantifies the rate of stretch along the direction and is V. Newtonian viscous relation of stress to strain rate. A fluid's viscosity is its propensity to resist deformation. A simple model for that resistance is Newtonian viscosity. This model assumes that the rate of deformation is proportional to shear stress, expressed as where is simply the coefficient of proportionality and is referred to as dynamic viscosity. VI. Acceleration in a Lagrangian coordinate system. The acceleration of fluid particles, when the velocity field is defined for positions in space (rather than for particles in space), is given by Now, the actual problem: combining equations (1-6), arrive to the Navier-Stokes equations for a viscous fluid Aside: Recall that as done for Euler's equation, if we define to be the vertical elevation with respect to an arbitrary datum, we may rewrite the Navier-Stokes equations as