Chapter 10 Approximate Solutions of the Navier-Stokes Equation 10.1 - Introduction 10.2 - Nondimensionalization equation of motion 10.3 - The creeping flow approximation

Objectives Appreciate why approximations are necessary, and know when and where to use. Understand effects of lack of inertial terms in the creeping flow approximation. Understand superposition as a method for solving potential flow. Predict boundary layer thickness and other boundary layer properties.

About the whole chapter N-S equation is difficult to solve, therefore approximations are often used for practical engineering analyses. Be sure that the approximation is appropriate in the region of flow being analysed. Several approximation will be examined according to the flow situations.

About the whole chapter First, nondimensionalize the N-S eq which later yielding several nondimensional parameters: Stroudal number (St) Froude number (Fr) Euler number (Eu) Reynolds number (Re)

Introduction In Chap. 9, we derived the NSE and developed several exact solutions. In this Chapter, we will study several methods for simplifying the NSE, which permit use of mathematical analysis and solution These approximations often hold for certain regions of the flow field. Figure 10.2

Nondimensionalization of the NSE Purpose: Order-of-magnitude analysis of the terms in the NSE, which is necessary for simplification and approximate solutions. We begin with the incompressible NSE Each term is dimensional, and each variable or property ( V, t, , etc.) is also dimensional. What are the primary dimensions of each term in the NSE equation? Eq (10.2)

Nondimensionalization of the NSE To nondimensionalize, we choose scaling parameters as follows

Nondimensionalization of the NSE Next, we define nondimensional variables, using the scaling parameters in Table 10-1 To plug the nondimensional variables into the NSE, we need to first rearrange the equations in terms of the dimensional variables Eq. (10.3)

Nondimensionalization of the NSE Now we substitute into the NSE to obtain Every additive term has primary dimensions {m1L-2t-2}. To nondimensionalize, we multiply every term by L/(V2), which has primary dimensions {m-1L2t2}, so that the dimensions cancel. After rearrangement, Eq (10.5)

Nondimensionalization of the NSE Terms in [ ] are nondimensional parameters

Nondimensionalization of the NSE

Creeping Flow Also known as “Stokes Flow” or “Low Reynolds number flow” Occurs when Re << 1 , V, or L are very small, e.g., micro-organisms, MEMS, nano-tech, particles, bubbles  is very large, e.g., honey, lava

Creeping flow

Creeping flow

Creeping flow Solution of Stokes flow is beyond the scope of this course. Analytical solution for flow over a sphere gives a drag coefficient which is a linear function of velocity V and viscosity m.

Example of application of general transport equations: viscous flows

Viscously dominated flows Low Reynolds numbers. Sometimes called as CREEPING FLOWS. Assumptions are: Incompressible. viscosity constant Gravitational forces negligible or driving flow Steady flow Fully developed(velocity profile does not change with position).

Creeping flow in a circular pipe: control volume approach

How to solve a problem???

Flow development

Consequences of fully developed flow

Creeping flow between flat plates

Creeping flow in a film on a wall

Hydrodynamic lubrication

Slipper bearing

Journal bearing

Creeping flow in a circular pipe