Misplaced Pages

#997002

66-597: The equations of motion for Stokes flow, called the Stokes equations, are a linearization of the Navier–Stokes equations , and thus can be solved by a number of well-known methods for linear differential equations. The primary Green's function of Stokes flow is the Stokeslet , which is associated with a singular point force embedded in a Stokes flow. From its derivatives, other fundamental solutions can be obtained. The Stokeslet

132-411: A ( x ) {\displaystyle L_{a}(x)} becomes y = f ( a ) + M ( x − a ) {\displaystyle y=f(a)+M(x-a)} . Because differentiable functions are locally linear , the best slope to substitute in would be the slope of the line tangent to f ( x ) {\displaystyle f(x)} at x =

198-441: A {\displaystyle a} is f ′ ( a ) {\displaystyle f'(a)} . To find 4.001 {\displaystyle {\sqrt {4.001}}} , we can use the fact that 4 = 2 {\displaystyle {\sqrt {4}}=2} . The linearization of f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} at x =

264-409: A {\displaystyle x=a} is y = a + 1 2 a ( x − a ) {\displaystyle y={\sqrt {a}}+{\frac {1}{2{\sqrt {a}}}}(x-a)} , because the function f ′ ( x ) = 1 2 x {\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}} defines the slope of

330-444: A {\displaystyle x=a} . While the concept of local linearity applies the most to points arbitrarily close to x = a {\displaystyle x=a} , those relatively close work relatively well for linear approximations. The slope M {\displaystyle M} should be, most accurately, the slope of the tangent line at x = a {\displaystyle x=a} . Visually,

396-475: A ) ) {\displaystyle y=(f(a)+f'(a)(x-a))} For x = a {\displaystyle x=a} , f ( a ) = f ( x ) {\displaystyle f(a)=f(x)} . The derivative of f ( x ) {\displaystyle f(x)} is f ′ ( x ) {\displaystyle f'(x)} , and the slope of f ( x ) {\displaystyle f(x)} at

462-399: A ] {\displaystyle [b,a]} ) and that a {\displaystyle a} is close to b {\displaystyle b} . In short, linearization approximates the output of a function near x = a {\displaystyle x=a} . For example, 4 = 2 {\displaystyle {\sqrt {4}}=2} . However, what would be

528-539: A global optimum . In multiphysics systems—systems involving multiple physical fields that interact with one another—linearization with respect to each of the physical fields may be performed. This linearization of the system with respect to each of the fields results in a linearized monolithic equation system that can be solved using monolithic iterative solution procedures such as the Newton–Raphson method . Examples of this include MRI scanner systems which results in

594-417: A Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin , the constant of proportionality being the coefficient of viscosity . In a non-Newtonian fluid, the relation between the shear stress and the shear rate is different. The fluid can even exhibit time-dependent viscosity . Therefore, a constant coefficient of viscosity cannot be defined. Although

660-439: A Stokes flow, are conducive to numerical solution by the boundary element method . This technique can be applied to both 2- and 3-dimensional flows. Hele-Shaw flow is an example of a geometry for which inertia forces are negligible. It is defined by two parallel plates arranged very close together with the space between the plates occupied partly by fluid and partly by obstacles in the form of cylinders with generators normal to

726-403: A dramatic demonstration of seemingly mixing a fluid and then unmixing it by reversing the direction of the mixer. In the common case of an incompressible Newtonian fluid , the Stokes equations take the (vectorized) form: where u {\displaystyle \mathbf {u} } is the velocity of the fluid, ∇ p {\displaystyle {\boldsymbol {\nabla }}p}

SECTION 10

#1733086255998

792-442: A finite yield stress before they begin to flow (the plot of shear stress against shear strain does not pass through the origin) are called Bingham plastics . Several examples are clay suspensions, drilling mud, toothpaste, mayonnaise, chocolate, and mustard. The surface of a Bingham plastic can hold peaks when it is still. By contrast Newtonian fluids have flat featureless surfaces when still. There are also fluids whose strain rate

858-399: A good approximation of 4.001 = 4 + .001 {\displaystyle {\sqrt {4.001}}={\sqrt {4+.001}}} ? For any given function y = f ( x ) {\displaystyle y=f(x)} , f ( x ) {\displaystyle f(x)} can be approximated if it is near a known differentiable point. The most basic requisite

924-450: A large tub of oobleck without sinking due to its shear thickening properties, as long as the individual moves quickly enough to provide enough force with each step to cause the thickening. Also, if oobleck is placed on a large subwoofer driven at a sufficiently high volume, it will thicken and form standing waves in response to low frequency sound waves from the speaker. If a person were to punch or hit oobleck, it would thicken and act like

990-425: A line, given a point ( H , K ) {\displaystyle (H,K)} and slope M {\displaystyle M} . The general form of this equation is: y − K = M ( x − H ) {\displaystyle y-K=M(x-H)} . Using the point ( a , f ( a ) ) {\displaystyle (a,f(a))} , L

1056-491: A moving sphere, also known as Stokes' solution is here summarised. Given a sphere of radius a {\displaystyle a} , travelling at velocity U {\displaystyle U} , in a Stokes fluid with dynamic viscosity μ {\displaystyle \mu } , the drag force F D {\displaystyle F_{D}} is given by: The Stokes solution dissipates less energy than any other solenoidal vector field with

1122-511: A non-Newtonian fluid is a suspension of starch (e.g., cornstarch/cornflour) in water, sometimes called "oobleck", "ooze", or "magic mud" (1 part of water to 1.5–2 parts of corn starch). The name "oobleck" is derived from the Dr. Seuss book Bartholomew and the Oobleck . Because of its dilatant properties, oobleck is often used in demonstrations that exhibit its unusual behavior. A person may walk on

1188-476: A point p ( a , b ) {\displaystyle p(a,b)} is: The general equation for the linearization of a multivariable function f ( x ) {\displaystyle f(\mathbf {x} )} at a point p {\displaystyle \mathbf {p} } is: where x {\displaystyle \mathbf {x} } is the vector of variables, ∇ f {\displaystyle {\nabla f}}

1254-805: A point force acting at the origin. The solution for the pressure p and velocity u with | u | and p vanishing at infinity is given by where is a second-rank tensor (or more accurately tensor field ) known as the Oseen tensor (after Carl Wilhelm Oseen ). Here, r r is a quantity such that F ⋅ ( r r ) = ( F ⋅ r ) r {\displaystyle \mathbf {F} \cdot (\mathbf {r} \mathbf {r} )=(\mathbf {F} \cdot \mathbf {r} )\mathbf {r} } . The terms Stokeslet and point-force solution are used to describe F ⋅ J ( r ) {\displaystyle \mathbf {F} \cdot \mathbb {J} (\mathbf {r} )} . Analogous to

1320-448: A solid. Quicksand is a shear thinning non-Newtonian colloid that gains viscosity at rest. Quicksand's non-Newtonian properties can be observed when it experiences a slight shock (for example, when someone walks on it or agitates it with a stick), shifting between its gel and sol phase and seemingly liquefying, causing objects on the surface of the quicksand to sink. Ketchup is a shear thinning fluid. Shear thinning means that

1386-526: A solid. After the blow, the oobleck will go back to its thin liquid-like state. Flubber, also commonly known as slime, is a non-Newtonian fluid, easily made from polyvinyl alcohol –based glues (such as white "school" glue) and borax . It flows under low stresses but breaks under higher stresses and pressures. This combination of fluid-like and solid-like properties makes it a Maxwell fluid . Its behaviour can also be described as being viscoplastic or gelatinous . Another example of non-Newtonian fluid flow

SECTION 20

#1733086255998

1452-477: A spoon, will leave it in its liquid state. Trying to jerk the spoon back out again, however, will trigger the return of the temporary solid state. Silly Putty is a silicone polymer based suspension that will flow, bounce, or break, depending on strain rate. Plant resin is a viscoelastic solid polymer . When left in a container, it will flow slowly as a liquid to conform to the contours of its container. If struck with greater force, however, it will shatter as

1518-410: A system of electromagnetic, mechanical and acoustic fields. Non-Newtonian fluid In physics and chemistry , a non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity , that is, it has variable viscosity dependent on stress. In particular, the viscosity of non-Newtonian fluids can change when subjected to force. Ketchup , for example, becomes runnier when shaken and

1584-446: Is a common example: when stirred slowly it looks milky, when stirred vigorously it feels like a very viscous liquid. A familiar example of the opposite, a shear thinning fluid , or pseudoplastic fluid, is wall paint : The paint should flow readily off the brush when it is being applied to a surface but not drip excessively. Note that all thixotropic fluids are extremely shear thinning, but they are significantly time dependent, whereas

1650-403: Is a function of time. Fluids that require a gradually increasing shear stress to maintain a constant strain rate are referred to as rheopectic . An opposite case of this is a fluid that thins out with time and requires a decreasing stress to maintain a constant strain rate ( thixotropic ). Many common substances exhibit non-Newtonian flows. These include: An inexpensive, non-toxic example of

1716-412: Is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems . This method is used in fields such as engineering , physics , economics , and ecology . Linearizations of a function are lines —usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating

1782-421: Is approximately 2 + 4.001 − 4 4 = 2.00025 {\displaystyle 2+{\frac {4.001-4}{4}}=2.00025} . The true value is close to 2.00024998, so the linearization approximation has a relative error of less than 1 millionth of a percent. The equation for the linearization of a function f ( x , y ) {\displaystyle f(x,y)} at

1848-436: Is chilled caramel ice cream topping (so long as it incorporates hydrocolloids such as carrageenan and gellan gum ). The sudden application of force —by stabbing the surface with a finger, for example, or rapidly inverting the container holding it—causes the fluid to behave like a solid rather than a liquid. This is the " shear thickening " property of this non-Newtonian fluid. More gentle treatment, such as slowly inserting

1914-499: Is found by solving the Stokes equations with the forcing term replaced by a point force acting at the origin, and boundary conditions vanishing at infinity: where δ ( r ) {\displaystyle \mathbf {\delta } (\mathbf {r} )} is the Dirac delta function , and F ⋅ δ ( r ) {\displaystyle \mathbf {F} \cdot \delta (\mathbf {r} )} represents

1980-469: Is known as the Stokes' paradox : that there can be no Stokes flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial solution for the Stokes equations around an infinitely long cylinder. A Taylor–Couette system can create laminar flows in which concentric cylinders of fluid move past each other in an apparent spiral. A fluid such as corn syrup with high viscosity fills

2046-415: Is that L a ( a ) = f ( a ) {\displaystyle L_{a}(a)=f(a)} , where L a ( x ) {\displaystyle L_{a}(x)} is the linearization of f ( x ) {\displaystyle f(x)} at x = a {\displaystyle x=a} . The point-slope form of an equation forms an equation of

Stokes flow - Misplaced Pages Continue

2112-408: Is the gradient , and p {\displaystyle \mathbf {p} } is the linearization point of interest . Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by

2178-431: Is the ambient flow, U {\displaystyle \mathbf {U} } is the speed of the particle, Ω ∞ {\displaystyle \mathbf {\Omega } ^{\infty }} is the angular velocity of the background flow, and ω {\displaystyle \mathbf {\omega } } is the angular velocity of the particle. Faxén's laws can be generalized to describe

2244-516: Is the gradient of the pressure , μ {\displaystyle \mu } is the dynamic viscosity, and f {\displaystyle \mathbf {f} } an applied body force. The resulting equations are linear in velocity and pressure, and therefore can take advantage of a variety of linear differential equation solvers. With the velocity vector expanded as u = ( u , v , w ) {\displaystyle \mathbf {u} =(u,v,w)} and similarly

2310-569: Is thus a non-Newtonian fluid. Many salt solutions and molten polymers are non-Newtonian fluids , as are many commonly found substances such as custard , toothpaste , starch suspensions, corn starch , paint , blood , melted butter and shampoo . Most commonly, the viscosity (the gradual deformation by shear or tensile stresses ) of non-Newtonian fluids is dependent on shear rate or shear rate history. Some non-Newtonian fluids with shear-independent viscosity, however, still exhibit normal stress-differences or other non-Newtonian behavior. In

2376-397: Is very nearly the value of the tangent line at the point ( x + h , L ( x + h ) ) {\displaystyle (x+h,L(x+h))} . The final equation for the linearization of a function at x = a {\displaystyle x=a} is: y = ( f ( a ) + f ′ ( a ) ( x −

2442-443: The P n m {\displaystyle P_{n}^{m}} are the associated Legendre polynomials . The Lamb's solution can be used to describe the motion of fluid either inside or outside a sphere. For example, it can be used to describe the motion of fluid around a spherical particle with prescribed surface flow, a so-called squirmer , or to describe the flow inside a spherical drop of fluid. For interior flows,

2508-441: The conservation of mass , commonly written in the form: where ρ {\displaystyle \rho } is the fluid density and u {\displaystyle \mathbf {u} } the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, ρ {\displaystyle \rho } , is a constant. Furthermore, occasionally one might consider

2574-518: The eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium. This is the content of the linearization theorem . For time-varying systems, the linearization requires additional justification. In microeconomics , decision rules may be approximated under the state-space approach to linearization. Under this approach, the Euler equations of

2640-447: The leading-order simplification of the full Navier–Stokes equations, valid in the distinguished limit R e → 0. {\displaystyle \mathrm {Re} \to 0.} While these properties are true for incompressible Newtonian Stokes flows, the non-linear and sometimes time-dependent nature of non-Newtonian fluids means that they do not hold in the more general case. An interesting property of Stokes flow

2706-467: The utility maximization problem are linearized around the stationary steady state. A unique solution to the resulting system of dynamic equations then is found. In mathematical optimization , cost functions and non-linear components within can be linearized in order to apply a linear solving method such as the Simplex algorithm . The optimized result is reached much more efficiently and is deterministic as

Stokes flow - Misplaced Pages Continue

2772-417: The Stokes equations in the domain V {\displaystyle V} , each with corresponding stress fields σ {\displaystyle \mathbf {\sigma } } and σ ′ {\displaystyle \mathbf {\sigma } '} . Then the following equality holds: Where n {\displaystyle \mathbf {n} } is the unit normal on

2838-405: The accompanying diagram shows the tangent line of f ( x ) {\displaystyle f(x)} at x {\displaystyle x} . At f ( x + h ) {\displaystyle f(x+h)} , where h {\displaystyle h} is any small positive or negative value, f ( x + h ) {\displaystyle f(x+h)}

2904-620: The body force vector f = ( f x , f y , f z ) {\displaystyle \mathbf {f} =(f_{x},f_{y},f_{z})} , we may write the vector equation explicitly, We arrive at these equations by making the assumptions that P = μ ( ∇ u + ( ∇ u ) T ) − p I {\displaystyle \mathbb {P} =\mu \left({\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{\mathsf {T}}\right)-p\mathbb {I} } and

2970-441: The body shape via cilia or flagella . The Lorentz reciprocal theorem has also been used in the context of elastohydrodynamic theory to derive the lift force exerted on a solid object moving tangent to the surface of an elastic interface at low Reynolds numbers . Faxén's laws are direct relations that express the multipole moments in terms of the ambient flow and its derivatives. First developed by Hilding Faxén to calculate

3036-445: The colloidal "shear thinning" fluids respond instantaneously to changes in shear rate. Thus, to avoid confusion, the latter classification is more clearly termed pseudoplastic. Another example of a shear thinning fluid is blood. This application is highly favoured within the body, as it allows the viscosity of blood to decrease with increased shear strain rate. Fluids that have a linear shear stress/shear strain relationship but require

3102-529: The concept of viscosity is commonly used in fluid mechanics to characterize the shear properties of a fluid, it can be inadequate to describe non-Newtonian fluids. They are best studied through several other rheological properties that relate stress and strain rate tensors under many different flow conditions—such as oscillatory shear or extensional flow—which are measured using different devices or rheometers . The properties are better studied using tensor -valued constitutive equations , which are common in

3168-475: The density ρ {\displaystyle \rho } is a constant. The equation for an incompressible Newtonian Stokes flow can be solved by the stream function method in planar or in 3-D axisymmetric cases The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function , J ( r ) {\displaystyle \mathbb {J} (\mathbf {r} )} , exists. The Green's function

3234-641: The equation the linearized system can be written as where x 0 {\displaystyle \mathbf {x_{0}} } is the point of interest and D F ( x 0 , t ) {\displaystyle D\mathbf {F} (\mathbf {x_{0}} ,t)} is the x {\displaystyle \mathbf {x} } - Jacobian of F ( x , t ) {\displaystyle \mathbf {F} (\mathbf {x} ,t)} evaluated at x 0 {\displaystyle \mathbf {x_{0}} } . In stability analysis of autonomous systems , one can use

3300-600: The fact that the pressure p {\displaystyle p} satisfies the Laplace equation , and can be expanded in a series of solid spherical harmonics in spherical coordinates. As a result, the solution to the Stokes equations can be written: where p n , Φ n , {\displaystyle p_{n},\Phi _{n},} and χ n {\displaystyle \chi _{n}} are solid spherical harmonics of order n {\displaystyle n} : and

3366-442: The field of continuum mechanics . For non-Newtonian fluid's viscosity , there are pseudoplastic , plastic , and dilatant flows that are time-independent, and there are thixotropic and rheopectic flows that are time-dependent. The viscosity of a shear thickening – i.e. dilatant  – fluid appears to increase when the shear rate increases. Corn starch suspended in water ("oobleck", see below )

SECTION 50

#1733086255998

3432-416: The fluid viscosity decreases with increasing shear stress . In other words, fluid motion is initially difficult at slow rates of deformation, but will flow more freely at high rates. Shaking an inverted bottle of ketchup can cause it to transition to a lower viscosity through shear thinning, making it easier to pour from the bottle. Under certain circumstances, flows of granular materials can be modelled as

3498-433: The force, F {\displaystyle \mathbf {F} } , and torque, T {\displaystyle \mathbf {T} } on a sphere, they take the following form: where μ {\displaystyle \mu } is the dynamic viscosity, a {\displaystyle a} is the particle radius, v ∞ {\displaystyle \mathbf {v} ^{\infty }}

3564-525: The function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} at x {\displaystyle x} . Substituting in a = 4 {\displaystyle a=4} , the linearization at 4 is y = 2 + x − 4 4 {\displaystyle y=2+{\frac {x-4}{4}}} . In this case x = 4.001 {\displaystyle x=4.001} , so 4.001 {\displaystyle {\sqrt {4.001}}}

3630-419: The gap between two cylinders, with colored regions of the fluid visible through the transparent outer cylinder. The cylinders are rotated relative to one another at a low speed, which together with the high viscosity of the fluid and thinness of the gap gives a low Reynolds number , so that the apparent mixing of colors is actually laminar and can then be reversed to approximately the initial state. This creates

3696-407: The moments of other shapes, such as ellipsoids, spheroids, and spherical drops. Linearization In mathematics , linearization ( British English : linearisation ) is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems , linearization

3762-467: The output of a function y = f ( x ) {\displaystyle y=f(x)} at any x = a {\displaystyle x=a} based on the value and slope of the function at x = b {\displaystyle x=b} , given that f ( x ) {\displaystyle f(x)} is differentiable on [ a , b ] {\displaystyle [a,b]} (or [ b ,

3828-472: The plates. Slender-body theory in Stokes flow is a simple approximate method of determining the irrotational flow field around bodies whose length is large compared with their width. The basis of the method is to choose a distribution of flow singularities along a line (since the body is slender) so that their irrotational flow in combination with a uniform stream approximately satisfies the zero normal velocity condition. Lamb 's general solution arises from

3894-511: The point charge in electrostatics , the Stokeslet is force-free everywhere except at the origin, where it contains a force of strength F {\displaystyle \mathbf {F} } . For a continuous-force distribution (density) f ( r ) {\displaystyle \mathbf {f} (\mathbf {r} )} the solution (again vanishing at infinity) can then be constructed by superposition: This integral representation of

3960-561: The same boundary velocities: this is known as the Helmholtz minimum dissipation theorem . The Lorentz reciprocal theorem states a relationship between two Stokes flows in the same region. Consider fluid filled region V {\displaystyle V} bounded by surface S {\displaystyle S} . Let the velocity fields u {\displaystyle \mathbf {u} } and u ′ {\displaystyle \mathbf {u} '} solve

4026-423: The surface S {\displaystyle S} . The Lorentz reciprocal theorem can be used to show that Stokes flow "transmits" unchanged the total force and torque from an inner closed surface to an outer enclosing surface. The Lorentz reciprocal theorem can also be used to relate the swimming speed of a microorganism, such as cyanobacterium , to the surface velocity which is prescribed by deformations of

SECTION 60

#1733086255998

4092-427: The terms with n < 0 {\displaystyle n<0} are dropped, while for exterior flows the terms with n > 0 {\displaystyle n>0} are dropped (often the convention n → − n − 1 {\displaystyle n\to -n-1} is assumed for exterior flows to avoid indexing by negative numbers). The drag resistance to

4158-430: The unsteady Stokes equations, in which the term ρ ∂ u ∂ t {\displaystyle \rho {\frac {\partial \mathbf {u} }{\partial t}}} is added to the left hand side of the momentum balance equation. The Stokes equations represent a considerable simplification of the full Navier–Stokes equations , especially in the incompressible Newtonian case. They are

4224-409: The velocity can be viewed as a reduction in dimensionality: from the three-dimensional partial differential equation to a two-dimensional integral equation for unknown densities. The Papkovich–Neuber solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two harmonic potentials. Certain problems, such as the evolution of the shape of a bubble in

4290-492: The viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations: where σ {\displaystyle \sigma } is the stress (sum of viscous and pressure stresses), and f {\displaystyle \mathbf {f} } an applied body force. The full Stokes equations also include an equation for

4356-560: Was first derived by Oseen in 1927, although it was not named as such until 1953 by Hancock. The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian and micropolar fluids. The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier–Stokes equations . The inertial forces are assumed to be negligible in comparison to

#997002