Equations of Change for Isothermal Systems

• In the previous lecture, we showed how to derive the velocity distribution for simple flows by the application of the shell momentum balance or the force balance. • It is however more reliable to start with general equations for – the conservation of mass (continuity equation) – the conservation of momentum (equation of motion, N2L) to describe any flow problem and then simplify these equations for the case at hand. • For non-isothermal fluids (heat transfer + flow problems), the same technique can be applied combined with the use of the equation for the conservation of energy Time Derivatives Any quantity c which depends on time as well as position can be written as c=f(t, x, y, z) then it differential dc is: æ cö æ cö æ cö æ cö dc = ç ÷dt + ç ÷dx + ç ÷ dy + ç ÷dz è t ø è x ø è yø è z ø æ cö • Partial Time Derivative ç ÷ è t ø The partial time derivative is the derivative of the function c with time holding x, y, z constant. (fixed observer)

ædc ö • Total Time Derivative ç ÷ è dt ø The Total Time Derivative accounts for the fact that the observer is moving (how c varies with t because of changing location). dc æ cö æ cö dx æ cö dy æ cö dz = ç ÷ +ç ÷ + ç ÷ + ç ÷ dt è t ø è xø dt è yø dt è z ø dt Dc • Substantial Time Derivative çæ ö÷ è Dt ø The substantial Time Derivative is a particular case of the Total Time Derivative for which the velocity v of the observer is the same as the velocity of the flow. It is also called the Derivative Following the Motion Dc æ cö æ cö æ cö æ cö = ç ÷ +ç ÷v x + ç ÷v y + ç ÷v z Dt è t ø è xø è y ø è z ø æv x ö ç ÷ where the local fluyid velocity is defined by: v = vy ç ÷ èv z ø

• Analogy of the river, the fisherman and the boat... Equation of Continuity

• Write the mass balance over a stationary elementary volume DxDyDz through which the fluid is flowing

z position (x+Dx, y+Dy, z+Dz)

r vx |x r vx |x+Dx Dz y Dy Dx x position (x, y, z) • Rate of Mass accumulation = (Rate of Mass In) - (Rate of Mass Out)

Rate of Mass In through face at x is r vx |x Dy Dz Rate of Mass Out through face at x+Dx is r vx |x+Dx Dy Dz Same Thing as Above for other faces æ ö ç ÷ Rate of Mass Accumulation is Dx Dy Dz è t ø æ ö DxDyDzç ÷ = Dy Dz (r vx |x - r vx |x+Dx ) + Dx Dz (r vy |y - r vy |y+Dy) è t ø + Dy Dx (r vz |z - r vz |z+Dz )

• Dividing on both sides by DxDyDz and taking the limit as DxDyDz goes to zero yields: æ æ ö ö æ ( vx )ö ( v y ) æ ( vz )ö = - çç ÷ + ç ÷ + ç ÷÷ = - Ñ · ( v ) t ç x ç y ÷ z ÷ ( ) èè ø è ø è øø which is known as the Continuity Equation (mass balance for fixed observer). The above equation can be rewritten as:

æ ö æ ö æ ö ææ vxö æ vy ö æ vz ö ö + v x ç ÷ +vy ç ÷ + vz ç ÷ =- ç ç ÷ + ç ÷ + ç ÷÷ t è x ø è yø è z ø è è x ø è y ø è z øø which is equivalent for an observer moving along the flow to: D Note that for a fluid of constant density = - Ñ · v Dt ( ) Dr /Dt = 0 and Div(v) = 0 (incompressibility) The Equation of Motion • We will again consider a small volume element DxDyDz at position x, y, z and write the momentum balnce on the fluid element (similar approach to mass balance) • Rate of Momentum Accumulation = (Rate of Momentum In) - (Rate of Momentum Out) + (Sum of Forces Acting on System) (Note that this is a generalization of what we did in the case of the Shell Momentum Balance for unsteady state conditions) for transport of x- t yx |y +Dy momentum through t xz |z +Dz z each of the 6 faces (the transport of y- t xx |x t xx |x +Dx Dz and z-momentum y Dy through each of the Dx t yx |y faces is handled x similarly) t zx |z • Note that momentum flows into and out of the volume element by two different mechanisms: 1- Convection (bulk fluid flow) and 2- Molecular Transfer (velocity gradient in Newton’s or related laws) • Molecular Transfer:

Remembering that t yx is the flux of x-momentum through a face perpendicular to the y-axis, the rate at which x-momentum enters

the face at y (perpendicular to the y-axis) is DxDz t yx |y and the rate at which x-momentum leaves the face at y+Dy is DxDz t yx |y+Dy . Similarly, the rate at which x-momentum enters the face at x by

molecular transfer is DyDz t xx |x and the rate at which x-momentum leaves the face at x+Dx is DyDz t xx |x+Dx . Similarly, there is another term for the x-momentum entering by the face at z and leaving by face at z+ Dz DyDz( xx|x - xx|x+ Dx ) + DxDz( yx|y - yx|y+Dy )+ DyDx( zx|z - zx|z+Dz )

The same should be done with y-momentum and with z-momentum. • Note that t xx is the normal stress on the x-face Note that t yx is the x-directed tangential (shear) stress on the y- face resulting from viscous forces

Note that t zx is the x-directed tangential (shear) stress on the z- face resulting from viscous forces • Convection: Dy Dz (r v v | ) Rate at which x-momentum x x x The same can enters face at x by convection be done for y- Rate at which x-momentum Dy Dz (r vxvx|x+Dx) and z- momenta leaves face at x+Dx by convection Rate at which x-momentum Dx Dz (r v v | ) enters face at y by convection y x y

Rate at which x-momentum Dx Dz (r vyvx|y+Dy) leaves face at y+Dy by convection

Rate at which x-momentum Dy Dx (r vzvx|z) enters face at z by convection

Rate at whic x-momentum Dy Dx (r vzvx|z+Dz) leaves face at z+Dz by convection • The other important contributions arise from forces acting on the fluid (fluid pressure and gravity) in the x-direction these forces contribute:

DyDz(px|x - px|x+Dx )+ DxDyDz( gx ) p is a scalar quantity, which is a function of r and T

g is a vectorial quantity with components (gx, gy, gz) • Summing all contributions and dividing by DxDyDz and taking the limit of DxDyDz to zero yields æ ö ( vx ) ( vxv x ) ( vyvx ) ( vzv x ) = - ç + + ÷ t ç x y z ÷ è ø

æ xx yx zx ö p - ç + + ÷ - + gx è x y z ø x Similarly for the contributions in the yu- and z-directions: • Note that r vx, r vy, r vz are the components of the vector r v Note that gx, gy, gz are the components of g p p p Note that , , are the components of grad(p) x y z

Note that r vxvx , r vyvx , r vzvx , r vxvy , .... are the nine components of the convective momentum flux ( a dyadic product of r v and v (not the dot product).

Note that t xx , t xy , t xz , t yx , t yy , ....are the nine components of the stress tensor t . ( v ) Gravitational = - Ñ · ( v v ) - Ñp - Ñ · + g t [ ] [ ] force per unit volume

Rate of increase Convection Pressure force Viscous Transfer of momentum contribution per unit volume Contribution per unit volume • The previous equation is Newton’s second law (equation of motion) expressed for a stationary volume element • This equation can be rewritten for a small volume element of fluid moving along the flow. Dv = - Ñp - Ñ · + g Dt [ ] • To determine the velocity distribution, one now needs to insert expressions for the various stresses in terms of velocity gradients and fluid properties. For Newtonian Fluids one has: æ vx vy ö vx 2 = = - ç + ÷ = - 2 + Ñ · v yx xy y x xx x 3 ( ) è ø v vy 2 æ vz y ö yy = - 2 + Ñ · v yz = zy = - ç + ÷ y 3 ( ) è y z ø

vz 2 æ vx vz ö zz = - 2 + (Ñ · v ) = = - ç + ÷ z 3 zx xz è z x ø • Combining Newton’s second law with the equations describing the relationships between stresses and viscosity allows to derive the velocity profile for the flow system. One may also need the fluid equation of state (P = f(r ,T)) and the density dependence of viscosity (m = f(r ))along Boundary and Initial conditions. • To solve a flow problem, write the Continuity equation and the Equation of Motion in the appropriate coordinate system and for the appropriate symmetry (cartesian, cylindrical, spherical), then discard all terms that are zero. Use your intuition, while keeping track of the terms you are ignoring (check your assumptions at the end). Use the Newtonian or Non-Newtonian relationship between velocity gradient and shear stresses. Integrate differential equation using appropriate boundary conditions General Equation of Motion in Cartesian Coordinate System

Equation of Motion In Cartesian Coordinate System For Newtonian Incompressible Fluids General Equation of Motion in the Cylindrical Coordinate System

Equation of Motion In the Cylindrical Coordinate System For Newtonian Incompressible Fluids General Equation of Motion in the Spherical Coordinate System Equation of Motion In the Cylindrical Coordinate System For Newtonian Incompressible Fluids Stress Tensor for Newtonian Fluids in the Cylindrical Coordinate System Stress Tensor For Newtonian Fluids In Spherical Coordinates The function : Ñv = F v for Newtonian Fluids Axial Flow of an Incompressible Fluid in a Circular Tube of Length L and Radius R • Use Cylindrical Coordinates

• Set vq = vr = 0 (flow along the z-axis)

• vz is not a function of q because of cylindrical symmetry • Worry only about the z-component of the equation of motion é 2 ù vz p 1 æ vz ö vz vz = - + gz + ê ç r ÷ + ú z z ë r r è r ø z2 û v 2 v Continuity equation: z = 0 Þ z = 0 z z2 p æ vz ö 0 = - + g + ç r ÷ Integrate twice w/respect to r using z z r r è r ø vz=0 at r=R and vz finite at r=0 2 æ[ P - P ]R öæ r2 ö ç 0 L ÷ vz = ç ÷ç 1 - 2 ÷ è 4L øè R ø