Intro to Fluid Mechanics for Engineering Students Part 3

Explore the Lagrangian and Eulerian descriptions to analyze fluid flow at a point using differential equations and differential relations, focusing on velocity, pressure, and the material derivative.

Examine the velocity field as functions of x, y, z and time, use unit vectors i j k, and derive acceleration via the material derivative with local and convective terms.

Derive the 2d velocity field accelerations a_x and a_y from u and v, evaluate at (1,2), and find velocity component in the 40-degree direction and the max-acceleration direction from (18,26).

Analyze a three-dimensional, unsteady velocity field to compute the acceleration vector via the material derivative. Identify a unit vector normal to the acceleration at a point.

Derive the mass conservation principle and the continuity equation from a control-volume, detailing inflow and outflow across faces. Extend to polar coordinates for cylindrical boundaries.

Explore mass conservation under steady, incompressible flow, deriving simplified continuity equations in Cartesian and cylindrical coordinates, and using density constancy and Mach number to justify incompressible modeling.

Solve a one-dimensional continuity problem in a piston-cylinder, derive the time-dependent gas density using the continuity equation, and calculate the rate of density change at a given instant.

Derive cylindrical continuity for steady incompressible flow with v_z = B z, solving for v_r to satisfy mass conservation, yielding v_r = - (B/2) r^2 + C/r.

Explore solving continuity for incompressible plane flow in cylindrical and Cartesian coordinates in examples 29 and 29.1, deriving v_theta and the y-velocity function.

Derive Navier-Stokes equations for Newtonian fluids, linking density, pressure, gravity, and viscous stresses to velocity fields; discuss incompressible flow and stress–strain relations.

Explore solving the Navier-Stokes equations for a frictionless, steady, 2D flow to derive the pressure gradient in the x direction, noting density may vary and gravity is neglected.

In example 31, we analyze a Newtonian, constant-viscosity oil flow with velocity components and no gravity, verify the Navier–Stokes equations, and derive a pressure field p(x,y) = p0 − 8ρ(x^2 + y^2).

Analyze a 3d incompressible flow example, verify the continuity equation with velocity components, and derive the pressure field under gravity using the Navier-Stokes framework.

Study the stream function for two-dimensional, steady, incompressible flow. Define ψ so u = ∂ψ/∂y and v = −∂ψ/∂x, with constant ψ tracing streamlines.

Plot four streamlines from a psi equation by solving for y, then compute velocity components u and v from psi, and compare with MATLAB contour and quiver visuals.

Derive the stream function from given velocity components for steady 2D flow, sketch streamlines, and verify flow direction with a MATLAB contour and quiver plot.

Explore vorticity in a three-dimensional flow by examining fluid particle rotation about x, y, and z and the curl of velocity that defines this rotation.

Explore velocity potential, where the velocity vector equals the gradient of a scalar function, enabling velocity components from partial derivatives. Apply this to incompressible flow via the continuity equation.

Evaluate the gradient squared of the velocity potential, confirm it equals zero, derive the stream function from phi, and find the streamline value at (1,1).

From continuity, derive the y-velocity as v(y) = -6y for a 2d steady flow, then verify irrotationality and use Bernoulli's equation to compute the pressure difference.

Derive the velocity potential from a given stream function in 2d flow and verify that curl is zero to confirm feasibility, as shown in examples 37 and 38.

Explore dimensional analysis and similarity for interpreting experimental data, learn dimensional homogeneity and non-dimensionalization, and apply to motion equations with constants and variables.

Explore dimensional analysis by selecting primary dimensions such as length and time, choosing scaling parameters, and deriving dimensionless variables to reveal parameter relationships and similarity conditions for modeling fluid systems.

Apply Buckingham pi theorem to achieve model-prototype similarity. Construct non dimensional pi terms from parameters and determine dependent and independent pi terms to ensure dimensional consistency.

Apply the Buckingham pi theorem to a thin rectangular plate in crossflow to derive drag relationships. Identify pi1 = D/(rho V^2 W^2), pi2 = H/W, pi3 = mu/(rho V W).

Use Buckingham pi theorem to express thrust F as a function of diameter D, angular velocity omega, speed v, density rho, and viscosity mu, yielding non-dimensional groups pi1, pi2, pi3.

Discover Reynolds number as the criterion for flow regime in pipes and ducts, linking inertial and viscous forces to distinguish laminar versus turbulent flow, with thresholds around 2300 and 4000.