Consider the Navier-Stokes equations with constant density it their dimensional form: Consider the Navier-Stokes equations with constant density it their dimensional form



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Consider the Navier-Stokes equations with constant density it their dimensional form:

  • Consider the Navier-Stokes equations with constant density it their dimensional form:







The Reynolds number Re is the only dimensionless parameter which is always important in simplifying the equations of fluid motion for various applications.

  • The Reynolds number Re is the only dimensionless parameter which is always important in simplifying the equations of fluid motion for various applications.

  • Reynoldsification helps in simplifying NS equations by ingnoring less important terms.

  • This simplification helps in obtaining analytical solutions to engineering parameters like friction factor.

  • Better understanding of the complexity of real fluid flow is achieved by fitting the situation into Reynolds frame work.















These are flows with Reynolds number lower than unity, Re<< 1.

  • These are flows with Reynolds number lower than unity, Re<< 1.

  • Since Re = UL/, the smallness of Re can be achieved by considering

  • extremely small length scales, or

  • by dealing with a highly viscous liquid, or

  • by treating flows of very small velocity, so-called creeping flows.



The choice Re << 1 is an very interesting and important assumption.

  • The choice Re << 1 is an very interesting and important assumption.

  • It is relevant to many practical problems, especially in a world where fluid devices are shrinking in size.

  • A particularly interesting application is to the swimming of micro-organisms.

  • This assumption, unveils a special dynamical regime which is usually referred to as Stokes flow.

  • To honor George Stokes, who initiated investigations into this class of fluid problems.

  • We shall also refer to this general area of fluid dynamics as the Stokesian realm.

  • This is of extreme contrast to the theories of ideal inviscid flow, which might be termed the Eulerian realm.



Re is indicative of the ratio of inertial to viscous forces.

  • Re is indicative of the ratio of inertial to viscous forces.

  • The assumption of small Re means that viscous forces dominate the dynamics.

  • That suggests that to drop entirely the term Dv/Dt from the Navier-Stokes equations.

  • This renders the linear system.

  • The linearity of the problem will be a major simplification.





Redefine the dimensionless pressure as pL/(2μU) instead of p/(U2).

  • Redefine the dimensionless pressure as pL/(2μU) instead of p/(U2).



The basic assumption of creeping flow was developed by Stokes (1851) in a seminal paper.

  • The basic assumption of creeping flow was developed by Stokes (1851) in a seminal paper.

  • This states that density (inertia) terms are negligible in the momentum equation.

  • Under non-gravitational field.





How does sedimentation vary with the size of the sediment particles?

  • How does sedimentation vary with the size of the sediment particles?

  • What electric field is required to move a charged particle in electrophoresis?

  • What g force is required to centrifuge cells in a given amount of time.

  • What is the effect of gravity on the movement of a monocyte in blood?

  • How rapidly do enzyme-coated beads move in a bioreactor?

  • The flow geometry of all above mentioned applications is flow past a sphere.

  • Define the term vorticity in spherical coordinate system.





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