• Bernoulli's Equation for a Steadily Flowing Liquid

    The Navier - Stokes equation is - a complicated equation which is difficult to solve, except in situation allowing much simplification. If we make the following assumptions, many of important features of the flow are retained. The fluid is inviscid, so...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1562-bernoulli-s-equation-for-a-steadily-flowing-liquid.html

  • Classificatin of Open Channel Flow

    Consider the steady, uniform flow of an inviscid, incompressible liquid in an open channel with a rectangular cross section of constant width. If the width is and the depth is the volume flow rate where is the speed of the liquid. is constant from the...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1564-classificatin-of-open-channel-flow.html

  • Deep Water Gravity Waves - Finite Depth

    The equations satisfied by waves are for water of depth (1) at (2) at (3) where is the velocity potential. (1) can be solved by separation of variables technique. Assume (there will also be an arbitrary factor of which we deal with later). (1) becomes...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1565-deep-water-gravity-waves-finite-depth.html

  • Deep Water Gravity Waves - Infinite Depth

    The equations satisfied by waves are for water of depth (1) at (2) at (3) (1) can be solved by separation of variables technique. Assume (there will also be an arbitrary factor of which we deal with later). (1) becomes since the left hand side is a...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1566-deep-water-gravity-waves-infinite-depth.html

  • Deriving the Equations of Water Waves

    Water waves obey simple differential equations derived using simplifying assumptions of incompressibility and irrotationality. If the flow is irrotational we can define a velocity potential satisfying If the fluid is incompressible then for waves...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1567-deriving-the-equations-of-water-waves.html

  • Equation of Motion For a Viscous Fluid

    The eqution of motion for an inviscid fluid is < > where is the density, is the pressure and is the potential as a function of position. < > To include viscosity effects we can just add a viscosity term to the right hand side. < > The equation becomes...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1568-equation-of-motion-for-a-viscous-fluid.html

  • Equation if Motion For an Inviscid, Incompressible Fluid

    For an incompressible fluid, so If the body force per unit volume is applying Newton's Second Law to a cube of unit volume to obtain We can also write where is the pressure in the fluid. Equating these gives (1) Consider the motion of a particle in the...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1569-equation-if-motion-for-an-inviscid-incompressible-fluid.html

  • Incompressible Fluids

    If a fluid is incompressible and is the velocity vector field that describes the velocity of the fluid at each point then This equation follows from the continuity equation First write the continuity equation as Since the fluid is incompressinble There...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1570-incompressible-fluids.html

  • Kelvin's Theorem

    Kelvin's circulation theorem states: In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant. where is the circulation around a material contour which may vary with...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1571-kelvin-s-theorem.html

  • Newtonian Fluids

    Viscosity may be thought of as a fluids resistance to the shearing forces acting on the fluid. We may picture a fluid between two plates, one stationary and one moving parallel to the first. The relative motion of the two plates cause shearing forces...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1572-newtonian-fluids.html

  • Pathlines

    Pathlines in fluids are the paths taken by the individual particles of the fluid as they travel from point to point. We can find the pathlines taken by the particles of a fluid if we know the velocity of the particles of the fluid. In two dimensions,...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1573-pathlines.html

  • Paths of Particles in Water Waves

    The velocity potential for water waves in water of finite depth, h, can be written From this we obtain and Integrating these equations with respect to gives and then This is the equation of an ellipse. For large and close to 0, far from the bottom...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1574-paths-of-particles-in-water-waves.html

  • Proof That Loss of Fluid Per Unit Volume Equals Divergence of Velocity Field

    The x component of the velocity at the centre of the face ABCD above is The x component of the velocity at the centre of the face EFGH above is The volume of fluid crossing ABCD per unit time is The volume of fluid crossing EFGH per unit time is The...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1575-proof-that-loss-of-fluid-per-unit-volume-equals-divergence-of-velocity-field.html

  • Streamlines

    The field lines of the velocity vector field at a particular instant of time are called streamlines. The streamlines are visualised by taking photographs of the fluid. If the velocity field changes with time, then the streamlines will change also, but...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1576-streamlines.html

  • The Navier - Stokes Equation

    The Navier – Stokes equation models the behaviour of a fluid element because of the forces acting on it, including viscous forces. When the viscous forces are ignored, the equations become Euler's equation. The equation is difficult to solve, and...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1577-the-navier-stokes-equation.html

  • The Reynolds Number

    The Reynolds number Re is a dimensionless number, the ratio of inertial forces to viscous forces i.e. where is the mean relative velocity between object and fluid (m/s). is a characteristic linear dimension (m). is the dynamic viscosity of the fluid...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1578-the-reynolds-number.html

  • Total Force Acting on a Cube of Fluid

    For a incompressible fluid at rest there are no shear or deforming forces and Any stresses are normal to any surface drawn in the fluid and the pressure inside the fluid is the same in all directions. Since though the fluid is incompressible the...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1579-total-force-acting-on-a-cube-of-fluid.html

  • Velocity Distribution of Fluid Between Rotating Pipes

    Consider a fluid of viscosity between concentric rotating pipes of radius with rotating with angular velocities respectively. For two dimensional flow in the z – direction the shear stress where is the shear strain and is the shear stress. If the flow...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1580-velocity-distribution-of-fluid-between-rotating-pipes.html

  • Velocity Potential

    Velocity potential is used in fluid dynamics, when a fluid occupies a simply-connected region – no sources or sinks - and is irrotational, so the velocity field has zero curl: As a result, can be represented as the gradient of a scalar function: in...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1581-velocity-potential.html

  • Vorticity

    Vorticity, labelled is the tendency of fluid to spin. It may vary from point to point in a fluid, and with time. It arises because different parts of the fluid are in relative motion, so any fluid element between these two parts exhibits a tendency to...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1582-vorticity.html

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