Incompressible Newtonian fluids, creeping

Creeping Flow of an Incompressible Newtonian Fluid. It is reasonable to assume that p constant for liquids that are not subjected to large variations in temperature and pressure. This assumption of incompressibility leads to the following form of the equation of continuity (i.e., see 8-35) and Newton s law of viscosity ... 

Vorticity Equation for Creeping Flow of an Incompressible Newtonian Fluid. Since all scalar components of the velocity vector are exact differentials, it is permissible to reverse the order of mixed second partial differentiation without affecting the final result. If this procedure is performed twice, then inspection of summation representations of the following two vector-tensor operations reveals that they are equivalent ... 

Since the curl of Ihe creeping flow equation of motion for an incompressible Newtonian fluid yields... 

At first glance, three coupled linear third-order PDEs must be solved, as illustrated above. However, each term in the x and y components of the vorticity equation is identically zero because =0 and Vj and Vy are not functions of z. Hence, detailed summation representation of the vorticity equation for creeping viscous flow of an incompressible Newtonian fluid reveals that there is a class of two-dimensional flow problems for which it is only necessary to solve one nontrivial component of this vector equation. If flow occurs in two coordinate directions and there is no dependence of these velocity components on the spatial coordinate in the third direction, then one must solve the nontrivial component of the vorticity equation in the third coordinate direction. 

The final result given by equation (8-162) is generalized for creeping flow of an incompressible Newtonian fluid that impinges on a stationary sphere with constant approach velocity Vapproach from any direction ... 

Summary of Results for Creeping Viscous Flow Around a Gas Bubble. The shortcut method described above and boundary conditions at a gas-liquid interface are useful to analyze creeping flow of an incompressible Newtonian fluid... 

Creeping Viscous Flow Solutions for Gas Bubbles Which Rise Through Incompressible Newtonian Fluids That Are Stagnant Far from the Submerged Objects. A nondeformable bubble of radius R rises through an incompressible Newtonian fluid such that... 

Answer Use the postulated form of the one-dimensional velocity profile developed in part (a) and neglect the entire left side of the equation of motion for creeping flow conditions at low rotational speeds of the solid sphere. The fact that does not depend on cp, via symmetry, is consistent with the equation of continuity for an incompressible fluid. The r and 9 components of the equation of motion for incompressible Newtonian fluids reveal that dynamic pressure is independent of r and 9, respectively, when centrifugal forces are negligible. Symmetry implies that does not depend on cp, and steady state suggests no time dependence. Hence, dynamic pressure is constant, similar to a hydrostatic situation. Fluid flow is induced by rotation of the solid and the fact that viscous shear is transmitted across the solid-liquid interface. As expected, the -component of the force balance yields useful information to calculate v. The only terms that survive in the (/ -component of the equation of motion are... 

Calculate the stream function for axisymmetric fully developed creeping viscous flow of an incompressible Newtonian fluid in the annular region between two concentric tubes. This problem is analogous to axial flow on the shell side of a double-pipe heat exchanger. It is not necessary to solve algebraically for all the integration constants. However, you must include all the boundary conditions that allow one to determine a unique solution for i/f. Express your answer for the stream function in terms of ... 

Consider creeping viscous flow of an incompressible Newtonian fluid past a stationary gas bubble that is located at the origin of a spherical coordinate system. Do not derive, but write an expression for the tangential velocity component (i.e., vg) and then linearize this function with respect to the normal coordinate r within a Ihin mass transfer boundary layer in the liquid phase adjacent to the gas-liquid interface. Hint Consider the r-9 component of the rate-of-strain tensor ... 

The curvature correction factor in parentheses in (11-29) is calculated explicitly for creeping flow of an incompressible Newtonian fluid around a solid sphere, where... 

Effect of Flow Regime on the Dimensionless Mass Transfer Correlation. For creeping flow of an incompressible Newtonian fluid around a stationary solid sphere, the tangential velocity gradient at the interface [i.e., g 9) = sin6>] is independent of (he Reynolds number. This is reasonable because contributions from accumulation and convective momentum transport on the left side of the equation of motion are neglected to obtain creeping flow solutions in the limit where Re 0. Under these conditions. 

INCOMPRESSIBLE NEWTONIAN FLUIDS IN THE CREEPING FLOW REGIME... 

The Stokes-Einstein equation for binary molecular diffusion coefficients of dilute pseudo-spherical molecules subject to creeping flow through an incompressible Newtonian fluid is (see equation 25-98) ... 

The governing equations used in this case are identical to Equations (4.1) and (4.4) describing the creeping flow of an incompressible generalized Newtonian fluid. In the air-filled sections if the pressure exceeds a given threshold the equations should be switched to the following set describing a compressible flow... 

Consider a Newtonian incompressible fluid containing a component A in high dilution (<0.05M) and moving under creeping flow conditions within a relatively high porosity porous medium. The solid surface adsorbs instantaneously the eomponent A. The mass transport regime (convection and/or diffusion) is expressed by the value of the Peclet number, defined... 

The fluid is Newtonian and incompressible, and its approach velocity approach remains constant in the creeping or laminar flow regimes. 

Motion of a fiber in flow is described by Jeffery s model [3]. It is assumed that the fiber is a single rigid ellipsoidal partide suspended in a viscous fluid, the flow is a creeping flow of a Newtonian and incompressible fluid, and Brownian motion and inertia terms of the fiber are neglected. Jeffery s model was used for prediction of fiber orientation in the early period of injection molding CAE. Since it is, however, for dilute suspension, the model is replaced with the Folgar-Tucker model for concentrated suspension. 

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